GOODRICH'S ANALYTICAL HARMONY. A THEORY OF MUSICAL COMPOSITION FROM THE COMPOSER'S STANDPOINT. INTRODUCING AN EXPLANATORY TREATISE UPON UNRELATED TONES: A NEW SYSTEM OF HARMONIC COUNTERPOINT AND DIAGRAM ILLUSTRA- TIONS OF MUSICAL FORM AND CONSTRUCTION. BY A. J. GOODRICH, AUTHOR OF "COMPLETE MUSICAL ANALYSIS," "MUSIC AS A LANGUAGE," ETC. PUBLISHED BY THE JOHN CHURCH COMPANY. CINCINNATI. NEW YORK. CHICAGO. ( OPYRIGIHED, 1893, BY THK JO1IS CllVK. II CO. All Rights Reservc.l. Muaic Library PREFACE. *"pHE advantages accruing from a knowledge of Harmony are not sufficiently understood, except by those who are ambitious to compose. Every singer, performer, teacher and critic is benefited in knowing the principles of chord succession, harmonization, etc. Pianists who possess this information have an immense advantage in the knowledge that modulatory tones, suspensions and appoggia- turas are accented ; that dissonances are to be connected with the consonances to which they resolve ; that passing tones are unac- cented; that anticipations are slightly marked, and that different kinds of cadences require different kinds of punctuation. As an aid to sight-reading (that most necessary accomplishment) a knowl- edge of Harmony is indispensable, for it enables one to anticipate a considerable portion of music by being familiar with the notation, resolution and progression of chords in general. Our present system of music has been gradually evolved during centuries of artistic and scientific progress. Some of the world's greatest geniuses laid the foundation, built up the structure and added the ornamentation. The theorist, has, therefore, but little to do beyond that of presenting the material of composition and showing how this has been employed. Certain principles and theories may be deduced from the music of a Beethoven, and these are to be systemized and explained. But while the creative impulse in music continues to manifest itself it must be unfettered by arbi- trary rules and prohibitions. Recent composers, in their use of harmony, have gone far beyond the formulas and precepts of text- books. It is no longer possible, according to existing systems of theory, or of acoustics, to explain the harmonic structure of such works as Saint-Saens' Danse Macabre, Grieg's Norwegian Dances, Tschaikowski's Francesca da Rimini ', or the later music-dramas 3 4 I'KKI-ACE. of Wagner. These creators of music followed a higher law than didactic theorem, and the theorist should act only intermediately, explaining to the student the artistic phenomena of cause and effect. During the past twenty-four j r ears since this system was com- menced the author has confined himself principally to this task : i. To present the material and technic of composition in system- atic and graded order. 2. To explain this analytically and clearly. 3. To illustrate the application of this material in the construction of music. 4. To show the esthetic effect (and, consequently, the object) of certain chords and progressions. These are the main features of the present system, which is based upon the actual results of composition rather than upon existing theoretical works ; and whatever merits this book may possess are thus ascribed to the influence of Scarlatti, Couperin, Bach, Mozart, Beethoven, Schubert, Schumann, Chopin, Mendelssohn, Wagner, Rubinstein, Dvorak, Gounod, Saint-Saens, Jensen, L,assen, Goldmark, Tschaikowski, Grieg, and Mascagni, not to Zarlino, Rameau, Kirnberger, Gott- fried Weber, Marx, Weitzmaun, Richter, uor Riemauu. TO THE TEACHER. 1YJOT only is the plan of this book different from that of other harmony books, but some of the current nomenclature has been rejected. This need not occasion confusion, for the old and the new names are mentioned synonymously. One instance is the 5th of a normal scale. This interval has heretofore been called : Pure, Major, Standard, and Perfect. Our tempered fifths are neither pure nor perfect, therefore these names are inappropriate. Major is mis- leading, because major can not consistently be applied to such intervals as the normal 4th and 5th, which are the same in both major and minor scales. The word standard is more acceptable, though the author prefers calling the 4th and the 5th of every nor- mal scale Normal intervals. A few words of explanation are also offered in reference to the designation of voice-parts. In the elemental chapters of this book chords appear in their close positions three parts in the treble and one in the base. As it is essential to correct chord progression that the student should follow the movement of each part, it has been deemed advisable to give names to these parts corresponding to the voices which would sing them if they were vocal. Therefore tli2 author has named the different parts of such chords as these : Soprano, the highest ; mezzo-soprano, the middle r^nii3q part; contralto, the lowest of the three treble parts. Y-fpr)'^ 1 *? & I This nomenclature is adopted for ail such chords, which sound exactly as written. The word " tenor" is not here applied to any part of a chord in close position, because a tenor part when sung from the treble staff sounds an octave lower than written. In the chapter on Harmonic Counterpoint (where dispersed harmony is first introduced) the original mezzo-soprano part becomes tenor whenever that part is to be inverted in order 5 6 TO THE TEACHER. to form open harmony. It is customary to name the lowest treble part tenor; but an equal distribution of the voices requires that all the parts shall be in open position, and this compels us to choose the middle upper part for inversion. The following examples in notation illustrate these different methods : b e At (a) the chord appears in close position. At (6) the lowest treble part of the initial chord is considered as tenor and inverted. At (f) the tenor part is an inversion of the & m -so 1 _ _ |Q^ ts-^ : g= -& middle treble part in Ex. (a). Any one who prefers the arrangement at (b} must have queer notions about vocal music and dispersed harmony. Particular attention is directed to the necessity of transposing the exercises into a variety of scales, for this is the surest way to a mastery of the different subjects. One of our most accomplished harmony teachers has noted with satisfaction this leature o! the work. After the student has completed a certain harmonization the act of transposition should apply, usually, to the melody only. Then the harmonization is to be completed in the new scale. This is more beneficial than to transpose both melody and harmony. By closing the books a class may write on a blackboard the various exercises without fear of unduly referring to the solutions in the text. Those who pursue the study without a tutor will understand that if they consult these solutions before the examples have been worked out they will acquire only a superficial knowl- edge of the subject. Throughout the text ellipsis * >- appear whenever an example is to be worked out before proceeding farther. The first twenty chapters are confined to concords, that the student may more readily learn to manage chords without being burdened with the additional rules of resolution which apply to discords. No discord is introduced until it is required, and in this way one subject leads to another. Inverted bases, being rather difficult of management, are not permitted before the 2jih chapter. For a similar reason the intro- duction of open harmony is postponed until the latter part of the book. The teacher's attention is therefore directed to the sequence of subjects as set forth in the Arrangement of Contents, and, ex- cepting for a particular purpose, the author would not advise any deviation from this order. The present Arrangement of Contents TO THE TEACHER. 7 has occasioned more anxiety and thought than any other feature of the work, and a comparison with any standard treatise on Harmony will show great dissimilarity in this respect. The author's plan embraced an exhaustive chapter on the Pro- gressive Development of Harmony during different Epochs and the tendency of recent chord combinations, together with a list of Refer- ences and a somewhat discursive chapter on the Supposed Physical Basis of Harmony. Owing to the already considerable size of the book the publishers advised that those chapters be omitted from the present edition and issued separately at some future time. As these parts are not absolutely essential to a text-book the suggestion lias been adopted. It is scarcely necessary to add that the chapters do not represent the number of lessons. Certain chapters, such as XIII, XIV, XXXV, XL, may require three or four lessons for their thorough comprehension. A. J. GOODRICH. CONTENTS. PAGE PREFACE 3 To THE TEACHER 5 PART I. CHAPTER i. Natural Intervals of the Major Scale .... 13 CHAPTER 2. " " " Minor " .... 17 CHAPTER 3. Formation of Major Concords 19 CHAPTER 4. " " Minor " 23 CHAPTER 5. Major and Minor Concords Re-arranged . 26 CHAPTER 6. Harmonization in Three Parts 28 PART II. CHAPTER 7. Theory of Harmonic Progression. Chord Suc- cessions Re-arranged '. 32 CHAPTER 8. Harmonic Progression in Four Parts. Addi- tion of the Fundamental Base 35 CHAPTER 9. Harmonization of a Given Theme. Application of Theoretical Principles 38 CHAPTER 10. Harmonization of Melodic Skips of a Third 42 CHAPTER n. " " " " Fourth 44 PART 111. CHAPTER 12. Forbidden Progressions 47 CHAPTER 13. Thirty Harmonic Progressions in a Major Key. With and Without Connecting Notes ... 51 CHAPTER 14. Harmonization of Themes, with and Without Connecting Notes. Chord Relations 55 CHAPTER 15. Another Method of Harmonizing Skips of a Third 60 PART IV. CHAPTER 16. The Harmonic Minor Scale and its Con- sonant Triads 64 CHAPTER 17. The Major and Minor Modes Combined. Introduction to Modulation . . 66 10 CONTENTS. CHAPTER 18. Primary Modulations to all Related Keys excepting the Subdominant 69 CHAPTER 19. Themes for Harmonization Illustrating the Preceding Modulations 75 CHAPTER 20. Modulations from the Minor Mode, with Illustrative Themes PART V. CHAPTER 21. Formation and Resolution of the Dominant 7th Chord CHAPTER 22. Omission of the Third or Fifth from the Dominant 7th Chord ... CHAPTER 23. Major and Minor Resolutions of the Domi- nant 7th Chord. Illustrative Themes .... CHAPTER 24. Four Resolutions of the Dominant 7th Chord Analyzed CHAPTER 25. The Preceding Resolutions Classified and Characterized. Direct and Avoided Cadences CHAPTER 26. Avoided Cadences Illustrated . PART VI. 84 89 92 95 98 CHAPTER 27. Inverted Bases. Their Object and Effect 104 CHAPTER 28. Unrulable Progressions and Resolutions 113 CHAPTER 29. Dissonant Triads Imperfect, Augmented and Diminished 117 PART VII. CHAPTER 30. Origin and Principal Resolution of the Diminished 7th Chord 126 CHAPTER 31. Natural Modulations to the Related Minor Keys by means of the Diminished 7th Chord. Another Mode of Transition . . . 130 CHAPTER 32. Diminished and Corresponding Dominant 7th Chords 133 CHAPTER 33. The Diminished 7th Chord Inverted. Ap- plication of the Corresponding Domi- nant 7th. Intermediate and Terminal Resolutions 135 CHAPTER 34. Themes Illustrating the Preceding. Far- ther View of Inverted Bases 139 PART VIII. CHAPTER 35. Principal and Secondary 7th Discords. Their Origin, Application and Effect . . 14 CONTENTS. II CHAPTER 36. Additional Chord Progressions 151 CHAPTER 37. Succession of Dominant 7th Chords Dia- tonicallyand Chromatically. Another Means of Transition 158 PART IX. CHAPTER 38- Fifteen Enharmonic Transitions by means of Three Primary Diminished 7th Chords. Theory of Notation 164 CHAPTER 39- The Diminished 7th Chord as a means of Enharmonic Transition. Chromatic Har- monization 168 CHAPTER 40. The Diminished 7th Chord as a Passing Harmony to the Tonic and to the Domi- nant 7th 175 CHAPTER 41. Harmonic Cadences in Major: i. Authentic. 2. Complete. 3. Perfect. 4. Extended-Perfect. 5. Avoided. 6. Deceptive. 7. Incomplete. 8. After 183 CHAPTER 42. Harmonic Cadences in Minor: i. Authentic. 2. Complete. 3. Perfect. 4. Extended-Perfect. 5. Avoided. 6. Deceptive. 7. Incomplete. 8. After. 9. Ambiguous 192 PART X. CHAPTER 43. Augmented 6th Chords. Their Derivation, Application and Effect, No. 1 199 CHAPTER 44. Augmented 6th Chords Continued, No, 2 . 205 CHAPTER 45. Augmented 6th Chords Continued, No. 3 . 209 CHAPTER 46. Application of the Various Augmented 6th Chords in Harmonization Concluded . . 212 PART XI. CHAPTER 47. Harmonic Progressions in General. Their Esthetic Effect 220 CHAPTER 48. Figured Bases. A Cursory View 228 CHAPTER 49. The Natural and Melodic Minor Scales. Their Harmonies 231 CHAPTER 50. Principal and Secondary 9th Chords . '. 237 PART XII. CHAPTER 51. Suspension. The Theory Illustrated ... 242 CHAPTER 52. Suspension Continued 249 CHAPTER 53. Pedal-Note. (Organ-Point) 255 CHAPTER 54. Seven Additional Resolutions of the Domi- nant 7th Chord 263 12 CONTENTS. PART XIII. CHAPTER 55. Duplication and Omission 268 CHAPTER 56. Related and Unrelated Tones: 'Harmonic, Passing, Appoggiatura, Suspension, Anticipation, Stationary Tone and Embellishment 276 CHAPTER 57. Related and Unrelated Tones Continued. Illustrative Themes 283 PART XIV. CHAPTER 58. Harmonic Counterpoint. Elementary Species 291 CHAPTER 59. " " Compound 296 CHAPTER 60. " " Ornate 305 CHAPTER 61. Harmonic Accompaniment Illustrated . . 315 PART XV. CHAPTER 62. Interdicted Progressions. False Relation . . . 325 CHAPTER 63. Analysis of Harmonic Sequence 333 CHAPTER 64. Influence of Rhythm and Phrasing upon Harmonic Movement 337 PART XVI. CHAPTER 65. Harmony in Five, Six, Seven, Eight, Nine and Ten Parts 342 CHAPTER 66. Abrupt, Enharmonic and Remote Transi- tion. Key and Chord Relations . 348 CHAPTER 67. Altered Discords. Double and Triple Dissonances 356 PART XVII. CHAPTER 68. Musical Form and Construction. Elementary Models and Forms ;68 CHAPTER 69. Musical Form and Construction Continued. Rhythm . . 373 CHAPTER 70. Musical Form and Construction Concluded. The Sonata Form in Major and in Minor: Tonality, Development, Form-Diagrams, Affinity of Motives. Conclusion 376 INDEX OF SUBJECTS 389 KEY TO EXAMPLES 393 GooDRicH's ANALYTICAL HARMONY, PART I. Chapter I. NATURAL INTERVALS OF THE MAJOR SCALE. A FTER countless experiments in scale construction, and centuries * of progressive development in musical theory and practice, modern composers have uniformly adopted what is called the Nor- mal major scale, as the most natural and important series of single tones proceeding from and returning to a given tonic or key-tone. As the scale is the foundation of all serious music study, both theoretical and practical, a knowledge of scale construction is here presupposed. Not every teacher, however, appreciates the necessity of a thorough and intimate acquaintance with scales in all keys and all forms. Suppose, for instance, this fragment of melody should appear: Ex. i. In order to an- alyze, transpose, :>r accompany these tones we must know that they belong essentially to the scale of A-flat, and that A-flat is the key- tone. The e-flat and d-flat presuppose b-flat and a-flat, and these, together with c, g, and/, constitute the scale of A-flat. interval is time between events, or space between things. \Yith the scale as a basis the staff degrees afford the simplest means of enumerating intervals. In musical theory, interval refers to die 14 GOODRICH'S ANALYTICAL, HARMON v. distance between tones, reckoned either up or down. Two pre- liminary explanations are necessary: 1. The foundation tone is counted one. 2. The numerical terms 2d, 3d, 4th, and so on. are not to be understood as fractions of a whole, but as expressing the number of degrees included in a certain interval. For instance, this is a 5th: Ex - 2 - F/W {"- because g is the fifth natural tone in the scale of c : Ex " 3 " Fffc ^ f- The black notes show that "i a a 4 5 live degrees of the staff are involved in ascertaining the numerical distance from c (i) to g (5). Intervals are computed both melodically and harmonically. When they occur separately they are melodic. Thus in singing i a the voice ascends a 3d, and in this : it descends a 5th. In both instances the first tone is counted one. When the tones occur simultaneously : Ex - 6 - F/fczl the interval is harmonic; but it is counted in the same manner as was Ex. 4. When three notes appear simultaneously they involve two intervals. These may be computed either fundamentally or componently : Ex- 7 Ffe_ According to the former method the interval from C to e is a major 3d, and that from C to g is a normal 5th. According to the latter there is a major 3d, c to e, and a minor 3d, e to g. The results are the same, though both methods must be understood. In order that this melodic motive : Ex> 8- rfft) -I " -J ^/ ~~0 t be transposed to any key, it should be desc'.ibed as beginning upon the tonic and ascending two whole steps. The result in would be this : Ex - 9- GOODRICH S ANALYTICAL HARMONY. To return to the scale : This consists of two equal parts called by the Greeks tetrachords. Each tetrachord contains two whole steps and one half step. Marx, following the Greek theory, called the tonic a " central tone " around which the other tones revolve, thus: I !>!>< i- hair. Ex 10 raftr-j ' '[?'- :^-q= | 1 r-p -- ' i i i H i *~ 3=3=j: Lower list ^f I ^ (5* ' ^HiiBi^^^=ls3^ ^q Chapter II. NATURAL INTERVALS OF THE MINOR SCALE. MINOR intervals are one chromatic step smaller than major intervals. C and e constitute a major third. By raising c, or lowering e, a minor third will result. Ex. 18. Maj. 3d. Mi. 3d. Mi. 3d. These are precise, theoretical distinctions, the same staff degrees being employed. In each instance the interval is a 3d ; but the first is large, the other two are small. The first seven natural intervals in A-minor should now' be writ- ten and named theoretically, according to directions : Those intervals not affected by the change in signature retain the same names as in Ex. 14. Those that are one chromatic step i8 GOODRICH'S ANALYTICAL HARMONY. smaller here than they were in A-major are to be marked minor (less).* The difference, therefore, between major and minor lies in the 3d, 6th, and yth, according to the signature of each mode. If the yth of a minor scale is raised, it will, of course, become major, but the natural minor scale is used here merely to show the difference between major and minor intervals and the circum- stances under which they occur. The 2d, 4th, and 5th are the same in both modes. The 3d, 6th, and yth are major in a major scale, and minor in a minor scale. Two comparative tables of intervals are presented, representing the tonic major and tonic minor of G. Ex. 20. These intervals are to be named by the student. The intervals thus far employed are the major and minor 2d, major and minor 3d, normal 4th and 5th, major and minor 6th, major and minor yth, and perfect octave. The others have been purposely omitted until they shall be required. n a. 1 " 1 r *~~\ *?-, I 2 1 3 .gy. . 1 4 1 5 1 6 r& \ 1 7 _^_1 rfe- -& - -> -' -& -? \ Maj. 2d. Mi. 3d. Nor. 4th. Nor. sth. Mi. 6th. Mi. yth. ji L 1 fe- . 1 - * 1 Ex. 21. Similar examples should be written in other natural minor scales. Theory seems to differ from practice in the designation of cer- tain intervals. For instance, f to a-flat is a minor 3d, whereas f to g-sharp is an augmented 2d. On all keyed instruments a-JJat and g-sharp look and sound alike ; yet there is a considerable differ- ence theoretically, and a slight difference mathematically. The vibration numbers are in the following proportions: G-sharp^ 412^; a-flat, 422! . In theory, /to g-sharp is an augmented 2d, while / to a-flat is a minor 3d. The usual difference in their resolutions may be observed by comparing (a) with (b). Ex. 22. natural serves as a flat in a sharp key, and as a sharp in a fl.it key. GOODRICH S ANALYTICAL HARMONY. ig In the first measure d and a-flat constitute an imperfect 5th ; in the second, d and g-sharp form an augmented 4th. The student should supply the theoretical names of the following ascending and descending intervals : 7th. 3d. Nor. Ex. 23. A thorough knowledge of scales will enable one to name all of these intervals correctly. For instance, take g and e-flat. In G-major e is natural, and therefore a major 6th. In G-minor e \s>flat, and is a minor 6th above g. Some of the names generally applied to intervals appear incon- sistent; the author has, therefore, changed them. However, this is a matter of individual opinion, and does not affect the intervals. Chapter III. FORMATION OF MAJOR CONCORDS. THE most euphonious intervals in music are major and minor thirds and their inversions, minor and major sixths. These may succeed each other without involving false progressions, though if they be too long continued the ear becomes satiated with their consonant effect. Inasmuch as the normal scales are composed of whole and half steps, the thirds are naturally large and small. Hence a phrase like the following is perfectly euphonious, because the ear recognizes all the sounds as belonging to the major scale of F: Ex. 24. 2o GOODRICH'S ANALYTICAL .-.KMONY. The major and minor thirds here succeed one another so natu- rally that none but a cultivated ear can recognize a difference ; yet a major $d is one chromatic step larger than a minor 3d, and the difference between the major and the minor mode is very pro- nounced. This somewhat superficial inquiry into the character of large and small thirds, and the promiscuous order in which they naturally occur in a scale, is intended to form the basis of an elementary knowledge of concords. The natural origin of what we call Harmony may be ascribed to the philosophy of sound. Acoustics has revealed many curious and interesting phenomena of a musical character, and the student should possess at least some slight knowledge of the general results attained through purely scientific investigation. As color is an inherent property of light, so is harmony an .inherent property of sound. Aside from the researches of philosophers and physicists, musical theorists have for centuries past demonstrated the fact that nearly all musical sounds are composite. In other words, the single funda- mental tone of a bell, string or tube generates other tones related to the fundamental by natural laws. Several of these overtones were known to the Greek philosopher and theorist Pythagoras, who evolved from them something of a system ; in fact the Egyptians, before the existence of the Greek nation, possessed a scale similar to our normal major scale. As the ratios of this old scale are more nearly perfect than our tempered scale, it is evident that the former was developed according to mathematical deduction. The most important series of harmonics or partial tones are those of the natural horn, produced by variable quantities of wind- pressure, without the aid of artificial valves. These tones are num- bered consecutively from i to 6, in the order of their arrangement : Ex. 25. & a 3 > 4 5 6 No. I Is the fundamental, the others are inherent elements or effects of this generator. There are other partial tones, though unrecognizable except with the aid of a resonator. The above har- monics can, however, be heard under favorable circumstances as overtones, and on the horn they all come out distinctly by means of variable wind and lip pressure. GOODRICH 'S ANALYTICAL HARMONY. 21 The vibration numbers of these harmonics are connected by a simple law of acoustics. If the lowest tone makes 100 vibrations a second, (2) will make 200, and so on, in the same space of time. " The proportion remains the same whatever the fundamental may be, and thus it is plain that the above harmonics belong to a fixed series of overtones." Taylor. This is why such tones as the following are so easily sounded upon the tubular instruments, like the horn, trumpet, trombone, and cornet : Ex. 26. -ay The first harmonic of our series (2) is naturally the most impor- tant, as its vibration number corresponds to i, according to the equal ratios y, f , f , and so on. In plainer terms, the octave makes two vibrations to every one of the fundamental. Hence the Greeks, who did not employ harmony as we understand it, sang in unison ; and as there is a difference, expressed by the ratio i : 2, between the voices of men and women, the result was octave pro- gressions. The next most important interval of the series (Ex. 25) is the fifth, No. 3. The ratio is 2 : 3, or f . This was the interval next employed for simultaneous progressions about the year 800, A. D. A brief specimen of this diaphony is presented : Ex. 27. Yt consists of a series of octaves with the fifths added above the iundamental base. The example sounds crude, almost barbarous to modern ears, and it is one of the numerous failures attending the efforts of theorists to establish a plastic art upon an abstract founda- vion, to the sacrifice of a higher law than mathematical deduction. The next progress was in the direction of counterpoint, which became highly developed before harmony was known in distinct chord formations. Major and minor thirds having appeared in the vocal part-music of the i5th and i6th centuries, composers such as Tallis and Viadana had but to include the thirds with the fifths in order to produce major and minor triads. These chords wiU firs engage the student's attention. 22 GOODRICH'S ANALYTICAL HARMONY. By referring to Ex. 25 it will be seen that the harmonics form a perfect major chord. Select i, 3, 5 from the series, omitting the duplicates: Ex. 28. 3-*>- This is the major chord of F, in open position Arrange it in close position for convenience and simplicity : 29 ' Eg&JLlAg The same numbers apply in both instances, for here i is the root, 3 is the third, and 5 is the fifth. The funda- mental (or generator) is the root, the tone upon which the chord is founded. As the chord originated in this way, and as the root gives to the combination its name, the author treats this as its first posi- tion. An analysis shows that this chord contains a major 3d (/ to a) and a normal 5th (/ to c). This is fundamental enumeration. Componently the chord contains a major and a minor 3d. The results are the same. This is known as a major concord, or as the consonant triad of F -major. It is the most important chord, espe- cially when founded upon the tonic, as in the last example. As this system is based upon the artistic results of actual compo- sition, the author believes all arguments futile that attempt to dis- credit these results, or that call in question the means employed by inspired composers. We not only accept these results, but find them excellent. There are but two other major concords to be found within the limits of a scale. The second is founded upon the fourth of the scale, and is known as the Sub-dominant harmony : It contains the same theoretical intervals as the tonic chord, a major 3d and a normal 5th. The root is B-flat, and the 5th is in unison with the key-tone F. Therefore there is an ini- portant connecting link between them, thus : This is true in whatever situation the chords may appear. The third major chord is based upon the fifth degree of the scale. It is known as the Dominant harmony, and is constructed in GOODRICH'S ANALYTICAL HARMONY. ^~:~ Counting from the same manner as the others : Ex. 32. the root C, it contains a major 3d and a normal 5th. The root of this chord is in unison with the 5th of the tonic chord, and the two are thus connected : EX. 33. iscE:! Some modern theorists present these three fundamental harmonies in this manner: Ex.34. r fcr ^_ The ulterior design of this arrangement is to illustrate the hypothe- sis of an under-scale system, but as the chords can not succeed one another in this manner, it is not adopted here. The regular mode of progression for these harmonies is here given: Ex. 35. Every tone in the scale is embraced in these three chords, and countless numbers of popular pieces contain no other harmonies. Write the tonic, sub-dominant, and dominant chords in their original positions in at least six other scales. Chapter IV. FORMATION OF MINOR CONCORDS. minor concord is generally considered as a derived har- *- mony. Certain writers argue that it has no place among fundamental harmonies, and much discussion has taken place re- specting its origin and character. But it would be unprofitable to enter into the controversy here. The minor triad has been freely used by all the great composers, and the only important problem is to ascertain how it has been treated. 24 GOODRICH'S ANALYTICAL HARMONY. The major scale presents sufficient material for the construction of minor chords. Though the intervals are all normal or perfect (counting up- ward from the fundamental), the internal arrangement yields more minor than major thirds : mi. mi. ma. ma. mi. EX. Here are four minor, and but three major thirds, for the reason that the seventh triad contains two minor thirds, b to d, and d to /.* If a fifth be added to the first six duophonic chords there will result six concords. Three of these will be major and three minor. It must be understood that a normal 5th contains a major and a minor 3d (or vice versa), not two major thirds; for this would result in a discord : Ex 37- There are, therefore, three minor chords in every major scale, founded upon the second, third, and sixth degrees, with a 3d and a 5th added above. Write these in their original positions. * * * The relative minor of any major is situated a small 3d below the latter, and the minor chord will contain a minor and a major 3d, instead of vice versa. The connections between the chords appear to better advantage in this form: Ex. 38. F- The principal points to be observed are : 1. That the relative minors are in each instance located a small 3d below the relative majors, as shown by the base. 2. That two notes of each major chord occur in the relative minor chord. These are notes of connection, as will appear here- after. By combining the concords which have been discovered, there will be six : three major and three minor. These are to be *The preponderance of minor thirds in tin's example is all the more remarkable, con- sidering that the major scale of one octave contains five major, and but two minor seconds. GOODRICH'S ANALYTICAL, HARMONY. written with the understanding that the 3d and 5th are to be added above each of the first six degrees of the major scale. * * * It is the author's purpose, during the first twenty chapters, to employ concords only, and as these must contain a normal 5th and a major or a minor 3d, it is evident that but two species of concords are recognized. The simplest manner of determining the character of these con- cords (whether major or minor) is to remember that i, 3, 5 in every major scale constitute a major concord, and that i, 3, 5 in every minor scale form a minor concord. The next example shows all the consonant triads in F-major : mi. mi. Ex. 39. ma. mi. (The student's exercise should correspond to this.) Each chord appears in its first position, with the root at the bottom. This merely shows the original formation of the various triads, for they can not follow one another in this manner. The triad founded upon the " leading-tone " does not contain a normal 5th, and therefore can not be a concord : Ex. 40. The interval from e to b-flat is what the author terms an imperfect 5th (generally called diminished), and as all theo- rists agree that it is not a concord it is excluded from this part of the work. The names of the minor harmonies are frequently called after the degrees of the scale upon which they are based. These technical terms in the major scale are : Super-tonic (2), the next above the tonic; Mediant (3), midway between Tonic and Dominant; Sub-mediant (6), the same distance below the tonic that the mediant is above it. Sometimes they are called by terms of relation, as relative minor of the tonic (No. 6 in Ex. 39), relative minor of the sub- dominant (No. 2), and relative minor of the dominant (No. 3). As relative major and relative minor scales have the same signa- ture it is often convenient to mention them in this manner. Examples similar to No. 39 should be written in all the major scales and named according to their major or minor character. GOODRICH'S ANALYTICAL HARMONY. Chapter V. MAJOR AND MINOR CONCORDS RE-ARRANGED. IN the previous chapters each concord has appeared in its first, or original position. But as every chord has as many positions as tones, or letters, the order of the intervals may now be changed. This can be illustrated more readily by re-arranging the letters which apply to the three tones of a concord. Beginning with the first position of the C chord, the letters will read from the lowest, C, e, g: Ex - 4I -Nfrr ^r - Next begin with the second letter (e) placing C last : e, g, C. This is the second posi- _n ? tion of the C chord ; in notation thus : Ex - 42 - F/jfe-^ As no new letters have been added it is still the C chord, with the root placed above instead of below. By placing the 3d uppermost the third position, g, C, e, is ob- tained: Ex - 43- pyfo Each of the six triads is to be re-ar- l~ V: I/ ^ __ _"~l ranged in the same manner, in regular order. Elementary students are advised to re-arrange the chords first by means of letters in this {D f a, i. f a D, 2. The figure shows the position and the capital a D f, 3. indicates the root. Other combinations are possible with the three letters of a triad, but these involve open positions, or " dispersed harmonies," which are not to be employed at present. These are produced by inverting the middle interval of each close position : Ex. 44. A, c, and e are close positions ; b, d, and f are open positions.* These *In addition to these the author uses what he terms half-open positions. Dispersed harmony is reserved for Harmonic Counterpoint. GOODRICH'S ANALYTICAL HARMONY. 27 are mentioned here to show the open positions and to caution against their present employment. The word Inversion is not used for these re-ananged chords, but is reserved for a future chapter, where the base L'.as another tone than the root. In re-arranging the six triads always begin with the first posi- tion, leading the letters upward as in chord formation. Add no new letters to the three which represent a certain concord, and follow these letters in their regular order, avoiding open positions. Num- ber the positions in each re-arrangement. After completing the exercise by means of letters, a corresponding example must be written in notation. Do not forget that the root of a chord remains the same during the different re-arrangements, and until some new element appears. A simple example will illustrate this fact : G tnaj. E mi. Ex. 45- o- ' Root. In the first measure the harmony of G-major prevails, and G is the natural foundation of these tones. In the second measure a new element is introduced, which results in the chord of E-minor, The upper parts progress through the different positions of this chord, and E remains in the base as root and natural foundation of the upper harmony. The six concords in each major scale should be re-arranged in three close positions, as already explained. After two or three re arrangements have been made with letters, they can be omitted and the remainder written in notation only. A complete example is given for comparison : Ex |6. E-flat. F-mi. G-mi. A-flat. B-flat. g. *^-,^ ^^.^g^ C-mi. W ' g g l%-~- i-^ 1 ^ - I 2 3 1 2 3 1 1 23 123 123 123 28 GOODRICH'S ANALYTICAL HARMONY. Chapter VI. HARMONIZATION IN THREE PARTS. HARMONIC Progression, or Chord Succession, is the act of moving from one tone-combination to another, according to certain principles. What chords may follow one another is not so important a question as how they shall follow. It has been observed that from the natural tones of the major scale three major and three minor concords can be formed, each of which has three close positions resulting from re-arrangement. As a preliminary solution of the problem of Harmonization the author has devised the following method. The first object is to inquire how many chords may be written beneath each note in the major scale. If the key of G-major is selected, begin with the key-tone, i. The student should have for reference a chart containing the six triads in G, re-arranged in three close positions, which must be con- sulted in order to answer the necessary questions, thus : __. - -* B Ex. 47- 1. Write a treble clef with the signature of G, and mark g on the second line for illustration. The number of chords that are to accompany this fixed tone depends upon how many of the six con- cords contain g. (It will be sufficient to refer to the first position of each triad, as the re-arrangement in each measure represents the same chord.) 2. How many chords among the six in this key contain ?* 3. Write g as many times as there are chords containing g, These should be tied together, representing a soprano part, which r -- __ remains uppermost throughout, thus : Ex - 4 8 - Fifo -^^ijF^^F Lt2_ The three triads containing g are to be written beneath this fixed note, g forming a part of each chord. * Whether g occurs above or below matters not ; it islsufficient that g occurs in a certain chord. GOODRICH'S ANALYTICAL HARMONY. 29 4. What are the three chords containing g ? [Mention them in regular order by their root names, specifying whether they are major or minor.] 5. What is the first chord containing gt 6. What position of this chord has g uppermost? (See first measure of chart.) 7. The root-note already appears in the soprano, therefore add the other two notes immediately beneath the stationary note. 8. What is the second chord containing g ? 9. What position according to the chart has g uppermost? 10. Write the remainder of the chord beneath the second tied note. 11. What is the third chord containing g? 12. What position shall be used in order to have g at the top? 13. Write this beneath the third tied note. After completing this illustration of the note g, draw a double bar, and add capital letters below, representing the root of each chord. * * * Supposing this much to have been accomplished, a complete example is presented for comparison : Ex. 48-5. The numbers included above the chords show the position of each. These progressions are perfectly correct. The following harmonic progressions have been made : G to C and from C to E-minor. G, appearing above as part of each chord, serves as connecting link in this chain of harmonies, and at the same time shows that the tonic of G-major may be accompanied with any or all of these chords. The first chord appears in its second position, the "second chord appears in its first position, and the last chord is in its third position. As a further result, observe that the stationary upper note is root of the first chord, 5th of the second, and 3d of the last chord. Compare these observations with Ex. 48^ until they are thor- oughly comprehended. Another circumstance to be noted is, that no part moves more tnan a major or a minor 2d up or down. The second note of the scale is now selected for illustration. * The last chord appears an octave lower than in the chart, but the two are to be consid- ered identical. 3O GOODRICH'S ANALYTICAL, HARMONY. How many chords in this key contain a ? \Vrite second space a as many times as there are concords t on- taining that note. Mention these chords in regular order, and specify whether major or minor. What is the first chord containing a ? What position is required when a is uppermost. (See chnrt.) Write this chord beneath a. What is the second chord containing # ? What position is required in order to retain a at the top? Write this, including the numbers above, and the capital letters below to indicate the roots. In harmonizing the third note of the scale the formula of questions and the means of answering remain the same. The tied note should be written as many times as there are chords in which it occurs. This fixed tone is always to be con- sidered as soprano or upper voice, and the chords written beneath it must correspond to the chart. In each instance include the cap- ital letters to show the roots, and number the position of each chord as indicated in the chart. All this tends to give the student a thorough acquaintanceship with the positions and character of concords, and the author ad- vises that no detail be neglected. The fourth, fifth, sixth, and seventh notes of the scale are to be illustrated in the same manner. * * * On account of the imperfect triad being here excluded, it will appear that the 2d, 4th, and yth degrees of the scale admit but two chords each in their illustration ; whereas all the other degrees may be accompanied with three chords. The yth tone may be harmon- ized with two chords, though no consonant triad is founded upon that tone. Every tone in the following major scales Should be treated in the same manner : A, B, F, E-flat. The last two keys might be attempted v.-ithout the aid of a chart, provided the student is thoroughly familiar with all the con- cords in their various re-arrangements. For the benefit of those who have not the aid of a teacher ^ completed example is appended, in order that the pupil's work may be compared with the printed example : S ANALYTICAL HARMONY. Ex. 49- La Si 1 ~* ~$ . U ^ Lg=g=r-^=g: ; *^ ^< ^ ^- The syllables above are included to show what notes of the scale are illustrated, and ia \vhat mannei. This will serve a special purpose hereafter. This example is not continuous., each measure being considered separately. GOODRICH S ANALYTICAL HARAIGtfV. PART II Chapter VII. THEORY OF HARMONIC PROGRESSION. CHORD SUCCESSIONS RE-ARRANGED. A SOMEWHAT superficial knowledge of chord-progressions * having been acquired, a more general and thorough system will now be introduced. 1. Any note occurring in two different chords is called a con- necting note. 2. Every connecting note i< to be tied, or sung by the same voice-part in both chords. 3. When there are two connecting notes between two chords in progression, both notes are to be tied, or rer~ain stationary : *= Ex. sort. I G is the connecting link, being common to the three chords. This note being in the soprano part, the C chord must appear in its first position in order to retain the note of connection in the upper part. Between the C and the E chords there are two connecting notes, e and g. As a rule these should be written first : Ex> 5 ' -- The note wanting in the last chord is its fifth, therefore the contralto part descends a minor 2cl, from C to bi F V C T I M ^^ 1 pfe %^jg~~l Only close positions are to be used at present, and the voice-parts must not progress up or down more than a 2d. GOODRICH'S ANALYTICAL HARMONY. 33 The first measure from Ex. 49 is selected for re-arrangement in two other positions. The next note of the G chord above is b, which will become the soprano-part, the chord reading d, g, b. In this position the same progressions should be made as in Ex. 50^, i. e., from G to C and C to E. It would be well to use three treble staffs, one above the other. The arrangement of the lowest should be the same as Ex. 49. A brief indication of the design of the first few measures is here given : 3. GCEADGBE Ex. 52. ytt g g ^ ^^_^^ fn ^ - ~s~ ~S ^ - ^* ^^^--^ ^ V-\J & & & ^^^ jr - - 2. GCEADGBE W 0* _. ^^ JF ^o ^^ S^ yl^- *j>^ .^. mj^^ fS^* *** f^>-^*^*^- j*^? & & c^T c^ &/ v- ly ~*r 5p[ *^ 1. GCEADGBE n i V it ^ ^. , ^ _ _ _., j^? j -. As the base is to move from root to root, the upper parts must be altered. Contrary movement will obviate the difficulty. Begin again, with the C chord, with the base as a foundation : Ex. 101. J ^j * The first note of the D chord below c (in the soprano) is a. This will indicate that the D chord is to appear with a uppermost. The note of the D chord next below g is /, and next below e is d. The progression will thus appear in correct form : Ex. 102. The base moves from root to root, while the e and g descend to d and /. This is not unusual : Ex. 103. The interval of a fifth in the first chord is followed by a third in the second chord and a false progression is thus avoided. But the skip from c to a in the upper part must be explained as the result of necessity. It prevents consecutive octaves and supplies the remain- ing note of the D chord. (See examples 100 and 102.) Another ameliorating circumstance is this : all the other voice- parts move alphabetically. Even the base, which has hitherto skipped a third, fourth or fifth, here moves but a second. The duplicated root-note is therefore the only one that skips. From the foregoing may be deduced the principle that when there is no connecting note between two chords, the treble parts must move in an opposite direction to that of the base. To be still more explicit, when the base ascends a second the other parts descend ; and when tne base descends a second the other parts must ascend- GOODRICH'S ANALYTICAL HARMONY. 53 When the base ascends or descends fundamentally a major or a minor second there will be no connecting note between the two chords. This is always true of concords. Xole. Some writers consider the progression in Ex. 102 incorrect on account of " hidden fifths " between the soprano and contralto parts. These may he avoided by resolving the e up to f, instead of down to d. But com- posers seldom concern themselves with these prohibitions, as the following extract shows : (From the Landing of the Pilgrims. By G. W. Chadwick.) Unaccompanied. t- Ex. 104. Aye, call it ho - ly ground. The author has, therefore, no hesitancy in recommending the progression as given in Ex. 102. Each progression is to be noted in three positions, the base being the same : 1. 3 Ex. 105. it Complete the example. This is the first progression. * * * No. 2 is from C to E. This is not new, for there are two con- necting links. These are to be retained in the same voice-parts as formerly. The intention is to supplement, not to contradict any of our previously acquired principles. Write three arrangements of this progression and number it 2. N"o. 3 is from C to f. Each progression is to be arranged in three positions and num- bered according to the index. In the fifth progression the base should descend a third rather than ascend a sixth. The upper parts present no new difficulties : Ex. 1 06. 54 GOODRICH 'S ANALYTICAL HARMONY. These are all the progressions that can be made from the initial chord ; therefore begin with D, No. 6. Here the base moves up a ad, and there is no note in common. Use the D chord in its first position and write the base, D to E. The soprano moves to that note of the E chord next below the 5th of the D chord. This fixes the position of the second chord : Ex. 107. Therefore f goes to gressions sound inharmonious is merely because we have become pocustomed to certain cadence-harmonies. It is certainly no fault of the progressions, for they have been used by all classic composers. Our present object, however, is to acquire the art of managing chords correctly and systematically. After this is accomplished the student GOODRICH 'S ANALYTICAL HARMONY. 57 may harmonize a theme with those chords that sound the most agree- able, or that represent a melodic idea to the best advantage. The two harmonizations of Ex. 109 will be found in the Key. A section of melody is presented for harmonization in two differ- ent ways : Ex. in. It might be better to harmonize one note at a time in each example, in order to produce these different arrangements readily. When completed, each copy should be examined critically for the purpose of detecting possible errors. Transpose the last example into A, C, D, and E-flat. This part of the chapter closes with a more extended theme, which is designed to include all the principles of progression and harmonization thus far explained. It should begin and end in D : Ex. 112. > .&- ~: 1 -& <9- fz 1 J "*" . & , "*~ _ CjCdZ3C2ZZZ2 1 in (? | i ! 1 Ex. izg.LJL-b-'.'* (? 1 I H F-'S^ 5 ditto. The last measure is to consist of the tonic harmony. Transpose the melody into D-flat and B and harmonize similarly. *fC 5fC Sji There are now two methods for harmonizing melodic skips of a third. It will be well to put these into practice before concluding this chapter. Which ever of these methods is employed in a certain pa.'v- sage will depend upon the nature of the sentiment, or the fancy of the composer. For instance, the notes c and e may be accompanied in three different ways : Ex. 126. The first arrangement (a) is bright ; (b) is rather sombre ; (c) is dis- connected and somewhat bold. All are correct. Transpose this example into B-flat and D. The following motive may be harmonized in different ways : Ex. 127. Examine and transpose the example. * * * The skip of a fourth can be harmonized in but one way : \vitri that chord which contains both notes of the skip, as explained. A theme will now be presented in which the skips of a third are to be harmonized according to the different methods already set forth : GOODRICH S ANALYTICAL HARMONY. x. 128. Pursue, as much as possible, the plan adopted in Ex. 127. The theory thus far developed embraces the whole modus operandi of handling chords. From these principles there will be no deviation before reaching the chapter entitled Unrulable Progressions. Even these, however, will not detract from the value of the theory of chord progression and harmonization. Transpose the last example into various keys. GOODRICH s ANALYTICAL HARMON y. PART IV. Chapter XVJ, THE HARMONIC MINOR SCALE AND ITS CONSONANT TRIADS. The minor scale was probably so called on account of the 'thirc and sixth, which intervals are smaller than in the major. There are several species of minor scale, all serving a purpose ir musical composition. Only the harmonic form will now be examined Modern tonality requires thai the seventh of every normal scale shall be a minor 2d below the t^nic, as a leading tone. Beginning upon A, the result is as follows : EX. The chief peculiarities of this scale are : three half steps and one step of an augmented 2d, 6 to 7.* This is true so long as we remaii: exclusively in the minor key. An elementary view of the harmonic possibilities of this scale will now be given. Write a triad upon each note of the scale, being careful to use only the notes of the scale already quoted. Then examine each triad in order to determine the concords. (A consonant triad must con tain a normal 5th and a major or a minor 3d.) Every triad that is not consonant should be excised, leaving the concords. * * * The second and seventh triads are imperfect and the third is aug- mented. An example of this is presented for comparison : *Any maior or normal interval becomes "augmented" when enlarged by achromatic step. UUODRICH S ANALYTICAL HARMONY. Discords excised. Ex. 130. Concords remaining. These concords are composed of the natural notes of the harmonic minor scale. Observe that g-natural does not appear. These four concords are known by the following names : i. Chord of the Tonic, founded upon the key-tone. 4. Chord of the Sub-dominant, founded upon the 4th above, or the 5th below the tonic. 5. Chord of the Dominant, so called because it dominates or con- trols the key. In major and minor this chord is founded upon the fifth natural degree. 6. Chord of the Sub-mediant. From these four concords a chart is to be made, showing how- many chords will accompany each note of the scale. Note. In founding chords, write a 3d and 5th above the foundation-note, or root, but in preparing a chart or harmonizing a theme, the chords are writ- ten below the melodic notes. * * * This diagram is presented for comparison : Do A short theme can now be harmonized, exclusively in this minor scale. The fact that three notes of the scale can be accompanied with only one chord each, leaves us no choice in the harmonization of those degrees. But as we wish to remain entirely in this scale, and have nothing but concords to work with, we must be governed by the chart : THEME IN A-MINOR. Do si do do re mi fa mi re do do si do. ^ n . Ex. 132. The letters above the melody correspond to those of the chart, and render the harmonization easy of accomplishment. The repeated notes are to be considered connecting links, and the harmony is to change at such places. Like previous themes, this is continuous, and should begin and end with the tonic chord. 66 GOODRICH'S ANALYTICAL HARMONY. The chord progressions must in every instance be according to previous directions. The interval of an augmented 2d (/to g-sharp) appears in the second measure ascending, and from the third to the fourth measures descending. The author regrets that this interval has been forbidden by theorists, for if it is incorrect, then the scale must be incorrect. Nothing but a spirit of mechanical and anti- artistic pedantry could have sought to interdict an interval so neces- sary to composition, and so characteristic of our modern minor scale. Do not, therefore, hesitate to use the augmented 2d ; it is perfectly proper, and, in fact, indispensable.* Another theme is offered, affording more scope in the harmoniza- tion : Ex. 133. & f 2 ^ This is exclusively in G-minor. As it presents no new difficulties, its completion is left to the student, with the advice that it be arranged carefully and transposed into other minor scales. Chapter XVII. THE MAJOR AND MINOR MODES COMBINED. INTRODUCTION TO MODULATION. WHILE we remain strictly in a minor key it is evident that our means are limited, and our field of operations narrow. The four concords can be combined into twelve progressions ; still we feel restrained, having but one chord with which to accom- pany the 2d and yth, and but one for the 4th of the scale. We will, therefore, combine the relative major and minor modes, as they are intimately related and connected. *This applies to instrumental music. In vocal compositions the augmented ad is fre- quently undesirable. GOODRICH'S ANALYTIC^:, HARMONY. 67 The minor scale constitutes a mode, or characteristic series of sounds having a recognizable fundamental, or key-tone, to which the melodic or harmonic cadence naturally resolves. The major scale constitutes another mode, possessing equally recognizable, though different characteristics. It is proposed to combine these two modes. Reference is here made to relative major and relative minor, having the same external signature. Each scale contains a tone foreign to the other. A-minor includes g-sharp as leading tone, while C-major embraces g-natural as normal 5th and foundation of its dominating harmony. Thus each scale has an individual tone not contained in the other. From this it is seen that g-sharp points to A-minor and g-natural to C-major. In combining the chords for this dual key, three of the chords in A-minor already occur in C-major. Name the chord in A-minor that does not appear among the six in C-major. With the addition of this, there will be seven triads to work with. Concords occurring in both scales are given : Ex. 134. The upper figures refer to the major, the lower figures to the minor scale. Write a chart for both modes, beginning upon-C. In adding the chords beneath the scale degress the chord containing g-sharp is to be included every time one of its notes occur in the upper part. This dominant chord to the relative minor is to be written last, for it is not well to follow the chord containing g-sharp with a chord containing g-natural. The indication of the chart is to be completed by the student : Ex. 135. The theory for applying this chart to harmonization has already been partially explained. When g-natural appears, either melodically or harmonically, the tonality of c-major prevails. But when g-sharp makes its appear- ance, the minor key is anticipated, for it is leading-note to A-minor, 68 GOODRICH'S ANALYTICAL HARMONY. and 3d of the dominant harmony. The minor mode will prevail until one of these chords occur : Consequently the chords E-major and E-minor should be considered entirely distinct from each other. In connection with this subject the author will promulgate but one arbitrary rule, and this must remain in force until more experience on the part of the student shall justify its violation : Whenever use is made of the dominant chord to the related minor, it must be followed by one of these chords : Tonic-minor, sub-dominant or sub-mediant. In other words, g-sharp must not be followed by g-natural. No objection can be made to the reverse of this order. (See chart.) The theme which will presently be introduced is so contrived as to be alternately in C-major and A-minor. In this respect it is similar to many compositions of the present day. The tonality is generally influenced by the harmony. For instance, this melodic phrase might be harmonized in three or four different ways : Ex. 137. i= (It would be well for the student to attempt several arrangements of this phrase.) * * * It may begin and end in either C-major or A-minor. As a prep- aration to what follows, these examples are presented : Ex. 138. EJ fr | j r r ^E| All these harmonizations are correct, though each has its peculiar effect and distinct application in actual composition. But the esthetic character of the examples can not be considered here. EX. 139. THEME UN C-MAJOR AND A-MINOR. Begin in C-major. Temporarily in A-minor. End in C-major. GOODRICH'S ANALYTICAL HARMONY. 69 This melody is principally in C-major. In the fourth measure there is a transient passage to the relative minor. Harmonize as usual in four parts, and then transpose. * * * The next theme is to be harmonized principally in A-minor. In the fifth measure there is a temporary digression to the relative major : Ex. 140. The g-sharp is the only note that admits no choice of harmony. Transpose as in the last example. * * * One more theme is offered for harmonization according to the same chart. This should begin in C and end in A-minor : Ex. 141. r- 1 The main requisites are to' have the chord progressions correct, and to know where the changes of mode occur, as this lesson is in- tended to foreshadow the principles of transition. Transpose each example until the subject it illustrates is thor- ougly comprehended. Chapter XVIII. PRIMARY MODULATIONS TO ALL THE RELATED KEYS EXCEPTING THE SUB-DOMINANT. MODULATION signifies a change of key, a passage to some other base of operations. It is generally used synonymously with Transition, though the latter is the stronger term. This dis- tinction will appear hereafter. To accomplish even a temporary modulation some chromatic alteration must be introduced that is suggestive of the key to be established. For instance, in modulating from C-major to the 70 GOODRICH'S ANALYTICAL HARMONY. relative minor it will be necessary to introduce into the modulating harmony some note that is characteristic of A-minor, and does not belong naturally to the original scale, C. What is this note ? The dominant to A is e, and as the dom- inant chord is supposed to contain a major 3d and normal 5th, the characteristic note is found in this chord : Ex. 142. This leads naturally to the chord of A-minor. The dominant chord is best adapted to perform a natural modulation, and it is the only modulatory chord at present available for our purpose. A dominant chord is founded upon the fifth tone of any major or minor scale, and contains a large 3d and a normal 5th from the root. The root is a dominating note, and when it appears in the base, that part ascends a fourth or descends a fifth to the key-tone with considerable strength and determination, thus : Ex. 143. The other parts of the chord correspond to this. The root remains as fifth of the tonic chord ; the third ascends a half step to the key- tone ; and the fifth ascends to the third of the tonic chord :* Ex. 144. The 3d, being the leading-tone, is the most important note of the dominant chord, particularly in modulation. The dominant chord is the same in tonic major or tonic minor. See example : -S A ' J w ' * G-mAjor. G-minor. EJUI45. ''When the third is said to ascend it is understood to refer to the voice or instrument that sounds this tone, for, strictly speaking, the tone itself is an independent sound and can uot move. GOODRICH 'S ANALYTICAL HARMONY. It is somewhat singular that this first chord should occur naturally in both modes, presupposing that the /-sharp, as leading-note, occurs in the minor scale. There is no difference between the dominant chord at (a) and the one at (b), though the first leads to G-major and the second to tonic minor. This is an Authentic, or regular Cadence, and in future examples will play a more important part. The first modulations are to the related keys. The related keys are "those which differ from the original by not more than one sharp or one flat." Therefore the related keys to C-major are, F-major, G- major, the related minors to these (having the same signatures) and the relative minor to the tonic. This family group consists of six keys, three major and three minor. The six concords that have been used are the tonic triads of these related keys. But when the key of any of these is mentioned, the full scale, and, consequently, the signa- ture of that key, must be comprehended. The chords of these related keys are these : Ex. 146. With their respective signatures they would appear like this : c A F D G E. EX. 147- E3E In each exe r cise the relative minor follows its major with the same signature. Therefore when we think of a modulation, for e. g., to E-minor, we must also think of this scale : Ex. 148. for in order to establish the tonality of E-minor it is necessary to comprehend this series of tones. The technical names of the related keys and the chords which represent them were explained in Chapters III and IV. Ex. 147 shows the groups of natural chords, with their corre- sponding numbers, and their connection with each other harmonic- ally. We may move the parts from one chord to any other chord in the scale without affecting the original key-tone, and without taking cognizance of any other scale. This would be Progression. 72 GOODRICH'S ANALYTICAL HARMONY. But in order to effect a transition and create a new tonality we must introduce that tone which represents the difference between the original key and the one to which we are in transit.* In this in- stance the old key disappears (even though but temporarily), and a new base of tonal operations is established. In the latter instance a chromatic alteration is employed as transition note ; in the former, no chromatic sign is used. All the related chords in progression follow : Ex. 149. This is a mere chord succession ; the tonality of C-major not being in any way affected. But when we consider any of these chords as tonic of a related key, it implies that a modulation has been, or is to be accomplished. To proceed with the modulations in regular order : C to D-minor ; C to -E-minor ; (C to F-major here omitted) C to G-major and C to A-minor. The mode of each related key is governed by the signature of the original key. In other words, the tonic chords to the related keys are to remain as we find them naturally : Ex. 150 Mi Ma Ma Mi. If we should modulate from C to D-major, it would not be an ele- mentary modulation, but a transition. The modulations should be written in this order : Begin with the C chord, c being uppermost. The first passage is to D-minor. What is the dominant to D? Write the root in the base. What is the leading tone to D ? (It must be a minor 2d below the key-tone.) What is the dominant chord to D ? Does it contain the new leading- tone? (Every dominant chord must have a major 3d and normal 5th.) Write this chord over the root note. The chord movements must be correct according to previous directions. Therefore the *By means of a dominant 7th chord with its 3d omitted we might prove an exception to this, but it would be premature here. GOODRICH'S ANALYTICAL HARMONY. 73 connecting notes must remain in the same parts as usual. Resolve the second chord (the modulatory one) to the chord of the new key, merely following the rules of progression in passing from chord to chord. * * * Supposing this much to have been written by the student, the example may be compared to the following, in order to correct pos- sible errors, or to confirm impressions rightly formed : i i ^j ^ Ex. 151. This first example is to be arranged in two other positions. The base will remain the same during the re-arrangements of any particu- lar modulation. In this example the chromatic note, c-sharp, serves to erase our impression of the original key, and, together with the dominant chord on A, establishes the key of D-minor. Considering this as an individual example, we begin again with the C chord and proceed to modulate to the next key in regular order. What is the dominant to E f Write it in the base. What is the full dominant chord ? Does it contain the leading tone to E? Does it contain any other tone not common to C? Are these chromatically altered notes common to the scale of E-minor ? Write the scale to prove this : Ex. 152. J The crosses show the notes that comprise the dominant chord to E-minor, Therefore the chord B, d-sharp,f-sharp, is perfectly nat- ural to this key. Now write the modulatory chord. In doing so, remember there is no connecting note between the first two chords, and proceed ac- cordingly. The E-minor chord naturally follows the dominant chord on B, and this progression is easily written. There is always a note of connection between dominant and tonic chords; tonic, in this in- stance, referring to the new key-tone. This modulation to^is to be numbered 2, and re-arranged in two other positions. * * * 74 GOODRICH'S ANALYTICAL HARMONY. The next modulation is to F. The dominant chord is C, e, g, but it contains no note foreign to the key of C, and therefore will not perform the transition : Ex. 153. This is a mere progression from the C to the F chords, and the key still remains C. Consequently this modulation to the sub-dominant must be omitted, as it can not be accomplished without a discord. The key of G is next to be established. The same theory will serve our purpose. What is the dominant to G ? (Always the 5th of any scale.) What is the dominant chord to G? Does -it contain the leading-note to G ? Write the base first after the chord of C, with c uppermost, add the dominant chord above the root, and end with the chord of the new tonic G. (Between the C and D-major chords there is no connection. The upper parts must accordingly move in an opposite direction to that of the base.) Two other positions of this example are to be written, as usual. In going from dominant to tonic, the fundamental base may ascend a 4th or descend a 5th to the key-tone. The fourth of the present modulations is to the relative minor. Write the base first, then the other parts, being careful to use the major 3d of the dominant chord as leading tone to the new key. Number this 4, and re-arrange in two other positions. When all these are completed, write the same modulations from D-major, E-major and B- flat-major. In no instance is the modulation to the sub-dominant to be attempted. GOODRICH'S ANALYTICAL HARMONY. Chapter XIX. THEMES FOR HARMONIZATION, ILLUSTRATING THE PRECEDING MODULATIONS. HE separate modulations which have been accomplished from an original key-tone will now be included in a continuous, transi- tional melody. The student is first to discover what modulations are intended at certain points in this theme, and then how these modulations are to be made. (The latter problem has been solved.) MODULATORY THEME. The chromatic notes, even without the dashes, would indicate where the modulations take place. By referring to the Table of Modulations in B-flat, something of a chart will be found for the harmonization of this modulatory theme. The only difficulty is that this melody is continuous. The modulations do not begin with the chord of the original key-tone, as was the case in the mechanical ex- amples. For instance, the first chord in the third measure is not B-flat, but either G-minor or D -minor. The chord movements must, however, be written correctly. The e-natural, in the third measure, might be mistaken for a modulatory indication other than that intended. But the following c-sharp removes the doubt, as it is not intended to modulate to the same key in two different places. After performing the modulation to C-minor it would be well to introduce the G-minor chord on the first of the third measure in order to temporarily restore the original tonality ; for it is more ele- mentary to modulate from G-minor to F-major than from C-minor to F-major. In the last of the fifth measure (third quarter) the impres- sion of the modulation to D-minor is to be erased by using the domi- nant chord to the original key. After the modulation to G-minor, GOODRICH'S ANALYTICAL HARMONY. include the dominant chord to B-flat in order to restore the original tonality. In case any student should experience difficulty with this example it may first be transposed into C. When the example is completed, the mezzo-soprano and con- tralto parts should each be taken as a theme, including the previous modulations. This will merely result in two other arrangements of the same modulations, but it will also show how modulations may be effected without any outward sign, thus illustrating the harmonic possibilities of a melody.* These resultant themes have been extracted from the original harmonization, and are here presented for the student's benefit. As the same harmonies and modulations are to apply to these additional arrangements the base will require no alteration ; Ex. i 2 55 Mezzo-soprano part as theme. ZBB$ gS 3 d * b ' ' [-""" 19 F | 1 H t*- CpC w 1 ^=3=E- h^rv} r I | i i-^H Sf. 1 1 Transpose these themes into C and . D-flat, and harmonize. For this purpose a chart may be necessary. The next theme contains no chromatic notes to indicate the modulations, but a reference to the chart in C will show the solu- tion. Two d's for instance will be found in the transition to G-major. In the original table the two d's appear in the second arrangement. This will serve as a clue to the others. THEME. Ex. 156. Harmonize and transpose into A, B, and D.~\ The chords should also be re-arranged if the pupil needs still more practice in this subject. * A minor key may be established by means of minor chords without employing a tran- sition chord. The same may be done with a major key. But this can not be explained at the present time. t Remember that the J3 is a sign of elevation in a flat Ley, and of depression in a sharp key. GOODRICH'S ANALYTICAL HARMONY. Chapter XX. MODULATIONS FROM THE MINOR MODE, WITH ILLUSTRATIVE THEMES. TN making modulations from a minor key- tone there are certain -A- principles to be understood that have merely been touched upon in previous chapters. These principles will be explained as they occur in the following modulations. The intention is to perform modulations from a minor key-tone, not to make mere progressions. The following triads are selected : The upper figures indicate the order in which the keys are classified, first minor and then major ; the lower figures show the degrees of the scale upon which these triads are founded. Observe that the dominant chord here appears as minor, because a natural modulation can not be made to E-major. But as the signature of E-minor is only one sharp it is included among the related keys. The leading-note to A-minor does not ap- pear, because the object is to show the keys to which modulations are to be made. The last chord ((7) is founded upon the subtonic, or imperfect leading-tone. Begin with the A-minor chord (as representing the key of A- minor} and modulate to D-minor. This is to be done in the same manner by means of the dominant major chord. (The transition chord is the same as was used in going from C to D-minor.} Then begin again in A-minor and modulate to E-minor, as before. The modulation to C-major can be effected in the same way ; but it must be understood that g-sharp is the only characteristic tone in A- uiinor that does not occur in C-major. In commencing in A-minor and introducing the dominant chord to C, which is founded upon G-natural, the latter note acts as a chromatic alteration and effects the change of key. According to the same principle a passage can 78 GOODRICH'S ANALYTICAL HARMONY. be made to modulate from A-minor to the relative major of the sub- dominant, which would have been impossible in C-major without employing a discord. For instance : Ex. 158. A-min. to F-major. The chromatic sign in the second chord complies with the principle at first set down, that every transition chord must contain a chromatic alteration. The g-natural does not belong to the scale of A. The modulation to the relative major of the dominant is effected by the dominant major chord containing f -sharp. These modulations should all be written in three positions. A theme will now be presented, in the harmonization of which the student is to begin in A-minor, modulate to five related keys, and back to the original tonic in the final cadence. Ex. 159. ^B * i i ' 1 1 i ' ' -r-rH By referring to the separate modulations from A-minor, the stu- dent will discover what keys are to be established at the places indicated by dashes. After harmonizing this, the mezzo-soprano and contralto parts may be taken as themes (uppermost) and harmonized in the same way. Also write a table of modulations from E and other minor keys to their relatives. GOODRICH'S ANALYTICAL HARMONY. PART V. Chapter XXI. FORMATION AND RESOLUTION OF THE DOMINANT SEVENTH CHORD. "YT TE have arrived at a point beyond which no material progress * * can be made with concords. A discord is therefore introduced. In modulations from a major key it will be remembered that the sub- dominant was omitted, because the tonic chord could not appear simultaneously as tonic and dominant. So the inquiry is made what note in the F scale does not appear in the C scale ? This note is situated a minor third above the 5th of the concord, with which it may well be combined. The chord on C should not be rejected because it is insufficient lo perform the modulation to F, but a transition element should be i:dded to it : Ex. 160. This destroys the impression of the key of C and creates the key F, because these notes occur naturally in no other scale : Ex. 161. 1 The crosses indicate the dominant seventh chord just formed. This is a four-toned chord, the most agreeable and one of the most important in our harmonic vocabulary. It is called a discord, >fo GOODRICH S ANALYTICAL HARMONY. merely in opposition to concord, for the root and yth form a dissc nant interval and require resolution to a consonance : Ex. 162. This is still more noticeable when inverted : To analyze it farther, it contains, theoretically, a major 3d, norma 5th and minor yth : Ex. It may also be described as consisting of. one major and two mine thirds. Or, it includes a normal and an imperfect 5th combined . Ex. 165. In certain resolutions it will be necessary to view it in this light. THE RESOLUTION. As already observed, this discord belongs to the key of F, Th 3d is leading note, and ascends a minor 2d to the tonic ; the 5th \\i. no fixed resolution, but for the present will be directed downwar a whole step; the yth has a decided natural tendency to resolv down to the 3d of the tonic chord ; the root-note, when it appear among the upper parts, remains stationary, and becomes 5th of th tonic concord ; the base moves from root to root. The illustratio is presented in score in order to show more plainly the resolution o the discord : GOODRICH S ANALYTICAL HARMONY. 8l The most important notes of the discord are the 3d and yth : -a,^~r1 "-: 1 Ex. 166. Ex - In whatever position the chord may appear, these directions will apply : These tones are known in theory as " elements of transition," and are usuallj' called by their technical terms, "leading-tone and sub- dominant." (These terms here refer to the key of F, for the chord no longer belongs to C.} Leading tone always means the minor 2d below any tonic, and sub-dominant refers to the 4th of the key to which the discord belongs. These important elements will be met with in other discords, and their thorough comprehension will facili- tate future labors. The dominant yth chord must now be written and resolved in every major key and in every position. As the base merely moves from root to root, the examples may be written without the base staff, as : In C. EX. i6g. bfer: I I As each note of the discord appears lowermost in regular succession, the re-arrangement of the discord will present no difficulty. Observe that each note of the discord is resolved in the same -manner at 2, 3 and 4, as at i. Proceed by fifths in the transpositions and include the signature of each key. The root of the discord will then appear upon the 5th of each scale, and no further chromatic sign will be re- quired. Here is a partial indication of the next example in order: In G. Ex. 170. As the 3d ascends and the 5th descends to the tonic, there are two 82 GOODRICH S ANALYTICAL HARMONY. voices singing this latter tone, and it is well to indicate this fact fo? the present by writing the tonic note in unisons or primes. This is explained by the third position ot each example, and also by the five part resolution in score, Ex. 166. In the first exercises it is advisable to resolve the most important elements first. These are the sub-dominant and leading-tone ( jth and 3d of the discord), and in whatever position the chord may be these names still apply, and the resolution is the same. B, F-sharp and C-sharp should be written in their enharmonic equivalents ; i. e., with the same sounds, but different notation. An example of this process is presented : Ex. 171. F-sharp becomes g-flat, a-sharp becomes b-flat, c-sharp becomes d-flat, and e will appear as f-flat. (A) and (b) are enharmonic equivalents. The first belongs to B-major, the second to C -flat-major. Both keys are practically identical. F-sharp and G-flat are also equivalent keys, and so are C-sharp and D-flat. Continue the transpositions (Ex. 170) by fifths, and thus return to C. Such process is called the cycle of keys. The student shou/d complete the task. * * * If the base be included with the exercise, five-part harmony will result, as in Ex. 166. This shows the resolution of all the parts of a discord ; but it is not advisable to employ more than four parts in the present exercises. The remaining modulation omitted in previous lessons should now be supplied. Begin with the C chord with c uppermost ; retain the root, 3d and 5th, and move the soprano part from c down to b-flat. In the next measure resolve the discord according to previous direc- tions. The base may ascend a 4th, or descend a 5th, from root to root. * * * Write two re-arrangements of this. An example of the three arrangements is offered for comparison : Ex. 172. This completes the modulations from C. GOODRICH'S ANALYTICAL HARMONY. 6 r " Z ' <&>- ~~ 1% W 1 ^_5! 1 r ?&} V& H^J 1 ^ 1 2 3 c* ^> C* |- 1 2 ^ The sub-dominant, b-flat, is here the most important element of transition, as it represents the difference between the scales of F and C. The modulation corresponding to this in the key of G is to C-major. It is to be accomplished in the same manner : Ex. 173. Continue these exercises in several keys, until the principle is well understood. This mode of treatment, though perfectly proper, leaves the final chord incomplete. Observe that no 5th appears in the concord. The root and 3d, however, give a fair representation of the tonic chord, for those notes occur in but one other concord, the relative minor, and we are not inclined to imagine this latter chord. The root in the base gives a very strong indication of being tonic, and the resolution of the discord, according to its most natural ten- dency, serves to confirm the impression created by the base. This, therefore, may be considered as synonymous with the cadence in which the concord is fully represented : Ex. 174, (A) has been used as frequently as (b), especially on the final ca- ieuce. Following is an instance from a standard English Glee : Callcott. :?c=: Ex. 175. From this and innumerable similar instances it is reasonable to con- clude that the 5th of the concord is not essential in a final cadence. 84 GOODRICH'S ANALYTICAL HARMONY. But in an intermediate progression the incomplete concord is some- times difficult to manage. Witness these examples : EfeSEfeEfei etc. m U-^-J-j-LJZZ^ etc. The progressions indicated by the dashes are awkward, if not positively incorrect.* To obviate these difficulties will be the principal object of the next chapter. Chapter XXII. OMISSION OF THE THIRD OR THE FIFTH FROM THE DOMINANT SEVENTH CHORD.f A NOTHER mode of treating the dominant yth chord, so as to -^"^ leave the concord complete, will now be shown. The root of the discord is the same as the 5th of the tonic triad, and is, there- fore, a connecting note : Ex. 178. But if the base be added there will be five parts. Therefore, if the complete tonic chord is required it will be necessary to omit some note from the discord. If we leave out the root or the jth, a dominant yth chord will no longer appear. But as the 3d and the 5th each resolve to the tonic we may omit either of those tones : *The resolution of the dominant 7th chord here results in what the author terms a half open position. tThis chord is also known as the Principal 7th, and as the Essential 7th. GOODRICH'S ANALYTICAL HARMONY. 85 Ex. 179. \- the second or fourth reso- lutions of the dominant 7th chord might be used, but that in the Key of F-major these resolutions would be unnatural, and i or 3 snould be selected. In either mode a direct or an avoided cadence may be chosen. In the two examples thus far classified it should be noted that the resulting triads appear naturally, without chro- matic signs. The student should now classify all the examples of the previous lesson. Write the first and third resolutions in the key of the tonic- major first : 3 These belong to C-major. Then write the second and fourth reso- lutions in the key, and with the signature of the tonic-minor : EX. 212. These belong naturally to C-minor. The four resolutions of the dominant yth chord on G appear in the two examples, but as the discord belongs equally to two differ- ent modes, the first and third resolutions occur naturally in C-major; whereas the second and fourth are associated with the tonality of C- minor. As C-major is a natural scale, and as C-minor has three flats, the signatures are too much at variance to admit of intimate relationship. The student is expected to complete the classifications in the manner illustrated. * * * Having ascertained the peculiar tonal characteristics of the various concords into which a principal discord many disappear, the next step is to know the objects of these different kinds of resolu- tion. The first and second constitute Direct (Authentic) Cadences ; the third and fourth constitute Avoided Cadences. In a direct cadence the discord is resolved in a natural and de- emed manner. In an avoided cadence the discord is resolved in an unnatural, undecided manner. GOODRICH'S ANALYTICAL HARMONY. 97 The object of a direct cadence is to decide a key or close a period. The object of an avoided cadence is to prevent a final close, and thus prolong the period. One is determinate, the other is indeter- minate. The direct resolution is most useful at the close of a period or movement. The indirect resolution serves the best purpose in the middle of a period, or before the last ending, as a means of sustain- ing the interest by leaving the ear unsatisfied until the direct ca- dence occurs. The number of cadences to be employed in any tonal genus must now be ascertained. The previous classification will help to solve this problem. The tonal genus comprehends the tonality of the six keys represented by these familiar concords : Ex. 213. The resolutions of the discord, whether direct or indirect, must, for the present, be into some of these triads. The dominant yth chord that represents the first triad is founded on the top note of the triad. Write this. * * * Of the four resolutions of this discord how many can be used naturally in this key ? Mention the chords and give their numbers in relation to the four resolutions? Write these. Which is an avoided and which a direct cadence ? * * * (This has already been illustrated in Exs. 211 and 212.) Write the dominant yth chord that represents the second triad. How many resolutions of this dis- cord can be employed naturally in this key? By referring to the original example in D, it will be seen that the four resolutions of \vhich the discord is capable are: i, D-major ; 2, D-minor ; 3, B- minor ; 4, B-flat-major. Only one of these can be used in the present instance: x 2I4 \^p MZ This is a direct resolution. Write the dominant yth chord that represents the third triad. The fifth of the E-minor triad is b : a principal discord, founded on B, will contain d-sharp,f-sharp, and a, besides the root. An exam- ination of the original example in E, which is here used as a chart, will reveal the fact that the four resolutions of the discord on B are : 98 GOODRICH'S ANALYTICAL HARMONY. i, E-major ; 2, E-minor ; 3, C-sharp-mi nor ; 4, C-major. J.t is evident that 2 and 4 answer present purposes. "Write these, and keep all examples together. The discord representing the fourth triad comes next. After writing this, inquire what are its four resolutions, and,which will be proper to use in this key. Here, likewise, is a direct and an avoided cadence. The discord on D, representing the fifth triad, is next in order. Carry out the same formula as to questions and answers. Next select the dominant yth chord on E, representing the sixth triad, A-minor, and the examples will be completed. This has a direct and avoided cadence in this key. * * * The available resolutions of a dominant yth chord have been employed on every tone of the scale excepting the fourth. This does not represent a related key in any of its resolutions, but belongs either to B-flat major or B-flat minor. Hence its rejection here. The student is to remember that though the resulting concords occur naturally and are not altered, the discords require chromatic alteration because they are transition chords in their original ap- plication. This is true of all except the discord to the central, or principal key, which belongs to its own scale and occurs naturally. Complete examples are to be written in D, E, B-flat, A-flat and G-flat. These will be required in the next chapter. Chapter XXVI. AVOIDED CADENCES ILLUSTRATED. >~rVHE avoided cadences previously classified and characterized are to be applied as in actual composition : CHART. 3 "i GOODRICH'S ANALYTICAL HARMONY. 99 By omitting the direct cadences d and 2) it will be a simple task to extract from this chart the avoided cadences (3 and 4), and this should now be done. * * * The next step is to ascertain how these are to be arranged with regard to their harmonic progression. The first position of the dis- cord is not favorable to an indirect resolution, for in order to avoid false fifths it is necessary to double the 3d of the concord, and this leaves the latter in an abbreviated form : Ex. 216. <' This is not here recommended. Invert the upper parts, as the base can not be altered: Ex. 217. The 3d of the discord descends, though in the original position it was compelled to ascend, to prevent parallel fifths with the f above. But here the 5th (b and /) appears as a 4th (/and ), and there is no prohibition against consecutive fourths when they are accom- panied by another interval. This example is, therefore, correct. The 7th must descend to the 5th of the concord ; the 5th of the discord is the sub-dominant, and must descend to the 3d of the triad as though it were the yth of the dominant : Ex. 218. The root of the discord in the base must ascend a 2d to the root of the triad. These are the most important resolutions, and from these direc- tions there will be no deviation. The resolution of the 3d of the discord is variable. When it is below the 7th it must ascend, when above the yth it may descend. This latter plan will be adopted here.* * As a discord may disappear in many ways, the author does not intend to issue instruc- tions except for particular instances. These rules merely apply to the third and fourth reso- lutions of an essential discord. joo GOODRICH'S ANALYTICAL HARMONY. Another position of the discord is selected for resolution : Ex. 219. Each note of the discord is here resolved the same as in Ex. 217, which see. No false progressions result ; the upper parts move in an opposite direction to that of the base, and the example is correct. At present only these two arrangements will be used, i. e., with the 3d or 5th uppermost. These positions and their accompanying directions will apply to every discord of the dominant 7th when resolved indirectly. Herewith a few indirect resolutions are presented, as they are to be applied in the harmonizations that follow : EX. 220. 3. 3 The first two measures are third resolutions; the next two are fourth resolutions. All are good. In the fourth resolution one of the upper parts moves down an augmented 2d. This is correct, according to the testimony furnished by the most eminent compos- ers. Here are two corroborative illustrations from the immortal Beethoven : Op. 10, No. 2. b . - Op. 53. "2- EX. 221. At (a) the augmented 2d appears ascending and descending, and at (b) the b-sharp descends plainly enough to a. *The last resolution will be utilized iu Harmonic Counterpoint. GOODRICH'S ANALYTICAL HARMONY. 101 Ex. 222. THEME FOR HARMONIZATION. . _*** * _ .0..f ~f -j -F F &- L-LX The dashes show where avoided cadences take place. The chart containing the five avoided cadences in this tonal genus, together with such aid as the theme affords, will enable the student to deter- mine upon the proper chords. All five irregular resolutions are to be employed, and that one which avoids the tonic cadence to C is to be used twice. The last cadence marked + is to be direct and final. The author will repeat the directions here, in order to free them from previous explanations and examples : 1. The indirect resolution must be to some of the related con- cords. 2. The melody note is to be considered as 3d or 5th of the essential discord whenever an avoided cadence is made. 3. The base is to ascend a 2d from root to root. 4. The upper parts must descend in order to form contrary movement to the base. 5. The root-note of the discord is not to be duplicated in any of the upper parts. 6. The jth of the discord must not (at present) appear in the melody. Students are not required to commit these directions to memory, but to understand the principles involved. To do so it may be necessary to refer back to the illustrative examples, for every direc- tion has been duly exemplified. The time has come for us to dis- card the machine methods of the school-room, where teachers are still groping in the dark, oblivious of the fact that cramming the memory does not cultivate the mind, or that one may " memorize lessons " without comprehending them. A few of the progressions in the last example may cause mis- givings on the part of the student, as where no connection appears in the upper parts, and where the base moves more than one degree : Ex. 223. c _ _ 102 GOODRICH'S ANALYTICAL HARMONY. While the progression at (a) is not positively wrong, it has some- what the appearance of evil on account of the similar movement of all the parts. The arrangement at (b) is preferable, and should be employed when it is possible. The harmonization of the theme should now be completed. * * * Almost every good composition verifies this application of the four resolutions of a dominant yth chord. In fact, this system is based upon actual composition, not upon mathematical theories or vague hypotheses. The harmonization of the last theme will be presented, as it con- tains some features comparatively new to the student : Ex. 224. The E-minor triad is perhaps better in the first and fifth meas- ures, as it furnishes a connecting note with the following discord. Observe the contrary movement in such places as from the third to the fourth chords, first measure. The last avoided cadence might have been arranged without the yth, in thi? way, Ex. 225. etc. especially as the avoided cadence to A-miw WPS included in the first of the example. The last complete cadence might also have been written Jik? this: Ex. 226. 2 <: m ^\ & 1 1 1. ..._. ~*~\ In such instances the yth is not absolutely essential, though the GOODRICH'S ANALYTICAL HARMONY. connecting note gives more consistency to the harmony, and is gen- erally preferable. The theme should be transposed to B-flat, D, E-flat, and F, Then write the avoided cadences associated with each key and pro- ceed with the harmonizations. In one of these examples the tonic triad may be substituted for the minor triad on the mediant. The last example does not admit re-arrangement. The third resolution of a dominant yth chord is inclined to be plaintive, to express disappointment and regiet. Schubert has em- ployed it in this sense. The fourth resolution is bolder and brighter, though generally unexpected. Both cadences have this in common : They avoid the regular cadence, and thus serve to postpone the ending ; or preserve the in- terest, instead of allowing it to subside. For most interesting illustrations of avoided cadences, the reader is referred to the Romance from Tannkduser, " O, evening star." The first five cadences are avoided by means of third and fourth resolutions of the dominant 7th chord. Note. Numerous instances occur in which a fourth resolution is used differently than in the author's classification. These occur in the nature of abrupt transitions, and usually after the prevailing tonality has been more or less exhausted. Under these circumstances the ear more readily follows any unusual progression. Rossini was very partial to this expediency so much so that he employed ' it in nearly all his operatic finales and Overtures ! See Overtures to Othello (after the return of the first subject); Tancredi (the finale). Semaramis (end of first subject). An illustration from a popular overture is quoted. The fourth resolution occurs after several periods in A-major : Herald. Ex. 227. The strain beginning in F is the same as that of the preceding in A, but after the avoided cadence it appears in a new light. -04 GOODRICH S ANALYTICAL HARMONY. PART VI. Chapter XXVII. INVERTED BASES. THEIR OBJECT AND EFFECT. THE various close positions of major and minor concords and the re-arrangements of the dominant yth chord have been pre- sented and explained. It has also been shown that a concord or a discord has as many positions as notes. Heretofore the movement of the base has invariably been fun- damental, from root to root. In modern music, however, the base is considered equally important with the other parts, and it may therefore assume any position in a chord that its melodic progression requires. The word Inversion has frequently been applied to intervals in the previous lessons, but in an Inverted chord the base has some other tone than the root. " Real-base, " or actual base, serves to designate the lowest part of the harmony, and also to indicate that the base note is not a root-note. The simplest example occurs when the base part executes different intervals of a chord, thus : Mozart. Ex. 228. The solo in the base part consists of a chord motive. The othei parts (violins, etc.) merely accompany the solo with the harmony of GOODRICH'S ANALYTICAL HARMONY. 105 D, indicated by the chord figure below. This idea is continued during the first sixteen measures of the Symphony (No. 23, B. and H.) and it would be well for students to examine and perform the entire passage. A similar instance occurs at the end of periods in popular music, where the base passes through the different tones of the tonic chord while the upper parts sustain the same harmony : Jos. Strauss. Ex. 229. ^ =tr. X X (2) (1) The figures (2) (i) indicate the inversions. This is so evident that no farther explanations seem necessary. The management of inverted bases requires considerable practical experience and theoretical information, and for this reason their in- troduction has for so long a time been deferred. When the base has the 3d of the triad it is customary to omit that note from the upper parts, thus : Mozart. Ex. 230. The soprano skips up a 3d, thus forming a counter melody to the base. The other parts remain stationary, as may be seen by arranging the example in this manner: Ex. 231. (D (D Aside from the two melodic parts, which here result from changing positions, the principal reason for omitting the 3d above when it io6 GOODRICH'S ANALYTICAL HARMONY. appears below is, that this interval determines the character of the chord (whether major or minor), and on account of its strength it becomes too prominent if doubled above. If this interval Ex - *3 2 - P/U J appears to create a void in the harmony on account of its ambiguity, the ear will experience an agreeable sensation in discovering the characteristic tone below, which completes the tonal effect : Ex. 233. Perform the examples separately and listen to the effect. This characteristic quality .of the 3d is stronger in major than in minor chords. Another reason for omitting the 3d above when it occurs in the base is, that if the 3d be doubled this duplication is liable to result in false progressions, on account of the tendency of the two thirds to move in parallel movement. This is especially true in chord progression where the parts move alphabetically : Ex. 234. This results in consecutive octaves between the extreme parts, and should be avoided. There are several ways in which the same chords and the same base may be arranged correctly : Ex. 235. 4j- C J ^K ^F =s i */ s* h""\* T i i ~Jt! =q ^ J -* I (i) (i) u I I \- rr , i \ , ff 4 1(- -*- H -- 2 j 5 ^ * * * * * 1 , - 1 ! j J 1 ,~*~ *~ -/- 4- * ' 1 (i) GOODRICH S ANALYTICAL HARMONY. IO7 The inversion is accompanied by a half-open position of the G chord, as a convenient method for avoiding the duplicated 3d. In the second measure the soprano and contralto parts move together at the distance of an octave, producing a counter-theme to the base. These octaves are not objectionable ; in fact, both arrangements are decidedly preferable to Ex. 234. The only caution necessary is this : if these half-open positions be continued beyond the influence of the connecting tone (g, in last example) they will generally result in evil. The fifth of a concord as real-base now claims attention. Nothing more need be said of bases that merely pass through the different tones of an unchanging harmony, except that such instances are numerous and effective. In Progression, when the base occupies the fifth of a concord, the conditions are altered, and some care is required in its manage- ment. There is an old thorough-base law to this effect, that " a f chord must be followed by the dominant or dominant 7th harmony." This signifies that when the 5th of a concord is in the base, the latter remains and becomes the root. Ex. 236. The 5th of the C chord in the base is somewhat out of balance, and the following dominant chord serves to restore the equilibrium ; the base remaining as root and connecting-tone. This forms a part of . the perfect cadence, as will be seen later. The formula just quoted has been much used by composers, and though not now followed so literally as it once was, the student can not do better than adopt -it until some other method is offered. The same directions apply to both modes. The inversion of the chord of the dominant 7th is next in order. Any of its tones may occur in the base. Therefore the base is said to be inverted when the 3d, 5th, or 7th of the discord appears below instead of above. In each of these instances the base note is to be *This might have been the dominant ?th. io8 GOODRICH'S ANALYTICAL HARMONY. omitted from the upper parts. The fundamental position may ap- pear ia any of these forms : Ex. 237. The first inversion is to be written with the 3d omitted above ; so with the second inversion, and especially with the third, in which the 7th is below : Ex. 238. ^EE (1) (2) (3) Observe that in each position the discord appears complete. The notes omitted above are supplied by the real-base below. Each of these measures is capable of being re-arranged in three positions : Ex. 239. _^ + + *__ -* *-*! i* 5 J J 1 J -9- ' -9- ' 1 ^2 * t (& i 1 (1) (2) (3) Compare each upper position with the base. Throughout all these inversions and re-arrangements the root, of fundamental remains C, the theoretical generator of the discord. The resolutions of these inversions must now be undertaken. The directions remain in force so long as the resolutions are to tonic major or minor. The 3d must be resolved up a 2d, and the yth down a 2d without regard to the position of the discord. The 5th usually resolves down to the tonic. The only difference between base and treble parts is this, that the root in the base ascends a 4th or descends a 5th, where- as the duplicated root-note in any of the upper parts remains station- The figures i, 2, 3 refer to the number of the inversion. GOODRICH S ANALYTICAL HARMONY. 109 ary as a connecting link. The student many now proceed with the resolutions, of which a sample is given : Ex. 240. Minor. (1) (2) (3) (I) (I) (2) (3) (1) (3) (!) With exception of the inverted bases there is nothing new in these resolutions. The third measure of the minor example shows a half- open position of the concord. But this merely results from the regular resolution of the 3d and the 5th (e-natural and g} to the tonic, as in the third measure of the major example. The contrary movement in the last measure is also good, provided the following progressions are in keeping with it. Several examples similar to the last ought to be completed, some of which should contain resolutions of the re-arrangements. See Ex. 240. The third inversion is the most troublesome to man- age proper!}-, because its resolution leaves the concord either in an abbreviated form as here, Ex. 241. or in a half-open position like this : Ex. 242. m (3) (1) (3) (I) In either case the inexperienced harmonist would be liable to en- counter some difficulty in progressing beyond the second chord, which is likewise inverted. For the purpose of anticipating these difficulties a number of examples are offered in which the yth ap- pears as real-base : Ex. 243. no GOODRICH S ANALYTICAL HARMONY. The first two exercises embrace a modulation to the sub-dominant and back to the original tonic. They are alike, excepting the treat- ment of the C chord with its 5th as real-base. This scheme, with various modifications, has been much used. At (b) there is no ob- jection to the duplicated 5th in the C chord. Example (c) embraces a transition to the dominant. This contains two peculiarities. The C chord is followed by the D chord, each in the same position. Hut as no fifths appear, and as the base acts as connecting note, no objec- tion can be raised against this. The second, c in the base becomes 7th of the chord on D, and this is resolved correctly, although different in one respect from the other resolutions of a thiid inver- sion. The 5th of the discord ascends to the 3d of the concord, thus doubling that tone which the base is obliged to sound. But a good reason appears for the duplicated 3d. The mezzo-soprano part has a regular melodic progression, g, a, b, c, which accords well with the base. These two parts are presented : , i J 1 I In separating from each other they sound b simultaneously, and as this was necessary to the design, the temporary prominence of the major 3d in the middle of the progression is not objectionable. Be- sides, the other parts contribute to the good effect. An important esthetic principle is here enunciated : the regular melodic progression of any voice-part may justify the most severe dissonances or the most unusual harmonic progressions which would otherwise be intolerable. Sequences also justify many transgres- sions of grammatical rules and harmonic precepts. The following extract may be explained in the same manner : A. Scarlatti. Ex. 245. The base has a regular melodic progression upward, so have the other parts downward. That a duplicated 3d and duplicated 5th result, is not to be objected to, for the design is more important than the preservation of an arbitrary formula. OOODRICH'S ANALYTICAL HARMONY. in Another method of employing inverted bases, and one that is more easily reduced to practical theory, is the following: Whenever a modulation is made to the key a third below, the tone between the two tonics may be given the base. This tone is to be the 5th of the transition chord, and must of course be omitted from the upper harmony. Suppose the pupil is writing in F-major and wishes to modulate to the 3d below. In elementary modulation the base moved from root to root : Ex. 246. m. But the tone between F and D being a part of the dominant yth chord to D, may appear in the base, thus : Ex. 247. This gives to the base a distinct melodic progression, and is a con- siderable improvement upon Ex. 246. With all these transitions to the 3d below, this second inversion may be used, and in every in- stance the real-base will be the 5th of the discord. This tone is to be omitted above. The upper parts of the last example should be re-arranged in two other positions, without altering the base part. A modulatory theme, with the treble parts added, is here pre- sented to illustrate this theory. The student should write the base part. Every discord except the last appears in its second inversion : Ex. 248. ' I^BZFf^r*; feZ ': Only such places as are marked -f are to be real-bases ; otherwise the base is to have the root of each chord. The solution of this will be included in the Key. The example should be re-arranged and transposed into several keys. * * * 112 GOODRICH'S ANALYTICAL HARMON \. When the modulations are to the 3d above, the same principles may be applied ; but in these instances the real-base will be the 3d not the 5th of the discord. No connecting note appears between the discord and its antecedent ; but when the modulation is from niiuo] to relative major, the chord progression is easily managed : Ex. 249. (i) The parallel fifths, c g, d a-flat, are allowable, because the firs fifth is normal and the second is imperfect. The reverse of thi: order is not good. What adds most to the effectiveness of the firs progression is the contrary movement of the octave, c to d, below and c to b-flat above. The resolution of the discord is perfect!] regular. The figure i refers to the first inversion, the 3d of the dis cord being below, as real-base. This example is shown in three positions, that the student ma} observe its different phases. The 3d and 5th of the triad ascend will the base, while the duplicated root-note above descends a whole step The real-base must be a minor 2d below the resulting tonic, and ii some places a chromatic alteration must be supplied by the student A theme and upper parts are presented as illustrations. Tin student is to add the base part according to the same principles tha governed the previous example. The real-bases are indicated. In these places first ascertain thi root of the discord, then write the 3d in the base. The note omitted from the treble part is thus supplied by th< base: Ex. 250. Re-arrange the treble parts in two other positions. Transpos( to C-minor, D-minor, and F-sharp minor. One or two examples should also be worked out from the base Such ground-work is here transcribed for the student to build upon GOODRICH S ANALYTICAL HARMONY. 4- +7 Skip. 4- "3 II) (1) (1) The notes marked 7 are tc oe roots of dominant yth chords. (1} signifies the first inversion of a dominant jth chord, the 3d being in the base. The following exercise, if worked out in various keys and positions, will be found useful : Ex. 252, ;fe? * t ^T=^=l (1) -* 0- Chapter XXVIII. UNRULABLE PROGRESSIONS AND RESOLUTIONS. / HpHERE are so many seeming contradictions to the rules of com- * position that the author of this system has set forth as few as possible. Musical rule can exist only as a deduction from musical usage. The creative artists are the highest authority respecting the material of composition and its application. If Beethoven caused a yth to ascend, it is the duty of the theorist to show why the com- poser did so ; not to stand aloof and shake his head with the remark, "this is a violation of our rules ! " This has always been the custom. Yet who made these rules? Were they engrossed and put forth by some one greater than Beethoven ? Directions concerning the resolution of a discord can not be given until it is known what application is to be made of this dis- cord. When the chord that is to follow the discord is determined upon, as well as the situation in which both occur, then may certain precepts be followed to advantage. But there is an underlying principle that affords the solution of every musical problem, and that is : the object in view, or the esthetic effect desired. GOODRICH'S ANALYTICAL HARMONS Music students should endeavor to grasp these principles and apply them ; not to imagine that the memorizing of rules and for- mulas will be sufficient. The example illustrates a connecting note passing into another voice-part instead of remaining stationary : Raff. Ex. 253. .^ ; j-j-^Tteq B^ This does not agree with the connecting-note theory, but it is un- avoidable here. All that can be said against the progressions in- dicated thus ^ is, that they are not smooth and connected. It is unprofitable to look upon them as .contradictions of a rule, especially since no error results. Though such progressions have been fre- quently used to advantage, the student must not conclude that the connecting-note principle is to be dispensed with. In the great majority of progressions the previqus directions will be applicable, and should be followed. But in harmonizing a theme ascending from the tonic, the most available method is that employed in the last example, in which the base moves contrarily to the other parts. If, however, smoothness and connection were desirable, the melody could be harmonized in this manner : Ex. 254. There is a connecting note throughout, in the mezzo-soprano part, and the base moves alphabetically. Each example serves a particu- lar purpose, and the purpose must justify the method employed. Here are other progressions of a character similar to those in Ex. 253: &=*=l=tA Ex. 255. * *=* -: & GOODRICH 'S ANALYTICAL HARMONY. At (a) three parts skip, while the mezzo-soprano part moves alpha- betically. The same is true of (b). In neither case does the con- necting note remain in the same part. It would be childish to con- demn these progressions merely because they do not comply with the directions as to chord succession in general. At (c) all the parts leap a considerable distance. But the only change in harmony con- sists in the introduction of the yth. Besides, the parts move in- opposite directions. All these progressions are correct, though some- what irregular. The contrary resolution (progression) of the 3d and yth of a dom- inant yth chord is now in order. Take the 3d first. The rule is that it must ascend a minor zd in going to the tonic chord. It may also descend a $d. In the middle of a strain, or wherever it is not desirable to make an authentic cadence, the 3d may descend to the 5th of the tonic chord : Ex. 256. The author's explanation of these seeming contradictions of musical rule is, that they are Progressions not Resolutions ; that they occur in passages where the decisive, compulsory character of a direct resolution is not desirable, and that in such places they serve a dis- tinct purpose. This must be understood as an intermediate, not a final application. The next example illustrates this : /_ ! , 2-K * A*_ @fr I - r^ i -p~" 1 L -* ^ i* ^~ 1 =i I H * "^i p ^ > 3 fi *^ i -* j Chapter XXIX. DISSONANT TRIADS IMPERFECT, AUGMENTED AND DIMINISHED. THE IMPERFECT TRIAD. OpHOUGH the imperfect triad was excluded from the preliminary -*- examples, its intervals are included in the essential yth chord. Consequently but little remains to be explained. ti8 GOODRICH'S ANALYTICAL HARMONY. This triad is founded upon the leading-tone of every major scale, and upon the super-tonic and leading-tone of every minor scale. It consists of two minor thirds, component!? ; or of a minor 3d and imperfect 5th : ' ^.^~^^^ ] Not being a concord, it must be classified among discords, and some general directions will be given for its resolutions. If it be considered as belonging to tonic major, the root should ascend and the 5th descend : Ex. 263. z=^= These elements of transition correspond to the 3d and jth of the essential discord, which disappear in the same manner : Ex. 264. Fij^^iib: The 3d usually descends a 2d. These are the mobt natural tendencies of the different tones toward resolution, though the triad has been used in various ways. A few illustrations in three-part harmony are presented as the imperfect triad usually appears in this manner : j I I I I I | I I In whatever position the dissonant triad may be placed it can be resolved in this manner without fear of impropriety. This is per- haps the best reason for following the above plan. The imperfect triad is indicated by a cross. By assuming a melodic license the 3d may ascend a 4th or de- scend a 5th, provided the two elements of transition (root and 5th) resolve regularly : I Ex. 266. *=f^^= i3z^^-d This supplies the remaining note of the tonic chord and is correct. Transpose each of these examples. In four-part harmony the root or 3d may be used as base, and these notes may also be doubled always provided that no improper progressions result. The 5th is seldom used as a real-base. An exception may be observed in Ex. 245. GOODRICH'S ANALYTICAL HARMONY. 119 In associating this triad with the minor of the tonic, the same observations will apply : _,J^ J *k I Ex. 267. ^^^^f There is nothing new here, except the difference in mode. But when this imperfect triad is used in connection with A-minor the treatment should be different, as the tonic will be below the root. If this root-note in the base is moved down to the tonic, parallel 5ths will invariably result. Ex. 268. If the root be resolved up to C, the final chord will be recognized as that of C-major, not A-minor. The only remaining choice would be to lead the 5th (/) up a major 3d to the minor key-tone ; for the 5th of a concord may be dispensed with, but a root can not be omitted without establishing in its place some other root. The example would appear like this : Ex. 269. These examples are sufficiently correct, but they partake of the nature of progression, rather than of resolution. The natural ten- dency of the f is to descend a minor 2d to e, and the skip up to a is therefore an expediency. But the imperfect triad founded on the leading-tone to A-minor corresponds exactly to that one used in Ex. 267. With this, no difficulty will be experienced, for there is here a note that resolves naturally to the tonic, and the other intervals disappear according to their melodic tendency without endangering the smoothness and correctness of the \vhole resolution : Ex. 270. I2O GOODRICH'S ANAIA TICAL HARMONY. Before dismissing this somewhat ambiguous triad, a few instances will be recorded in which it is treated in chord progression the same as a concord. That is, it may supply an accompaniment to that tone of the scale upon which it is founded : Ex. 271. In each of the first three measures the imperfect triad takes its place among the perfect triads in order to carry out the sequence* indicat- ed by the slurs. The fact that the dissonant triad ascends and de* scends without any fixed resolution shows that it assumes here the functions of a concord, w r hich, of course, has no resolution. But at the close, after the sequence has been carried out, the dissonant triad is resolved as though its tones belonged to the essential jth on C. The following extract from a favorite English song illustrates the same theory in a different manner : fiishoft. In the second and fourth measures of the accompaniment the im- perfect triad is treated like the concords, these being progressions, not resolutions. Attention is also directed to the manner in which parallel fifths are avoided. The 5th between the base and lower treble part becomes a 6th before the base ascends from c. Then the c above, which now produces a 5th with the f-sharp, ascends to d. By moving these two parts alternately, false progressions are avoided. * Sequence is the repetition upon different degrees of the scale of anv figure or design considered as a model. The upper part during the first three measures constitutes a rielodic sequence; the chords all being in the same position, constitute what is nere caUed Har- monic Sequence. GOODRICH'S ANALYTICAL HARMONY. 121 The impel feet triad is frequently used in a minor cadence in place of the sub-dominant harmony : Haendel. A Ur-t- Ex. 273 (2) (2) The 3d is doubled to avoid similar movement in the upper parts. Observe that the root (b) may descend when it appears above the 5th. The usual method is to use the 3d as a real-base : J'.E. Bach. Ex. 274. (1) (2) The imperfect triad becomes a sub-domimant harmony in this in- stance, b being substituted for a on account of the melody. The real-base is doubled above, but one d ascends to e while the other descends to c. The last example may be written in this form : Ex. 275. P= m Care must be bestowed upon the re-arrangement of these dissonant triads. Transpose the last three examples into several keys. THE AUGMENTED TRIAD. The word augmented, as applied to intervals, refers to the en- largement of a major or normal interval by one chromatic tone. This may be accomplished by sharpening the upper tone of the in- terval or flattening the lower tone. The former process is more common. The 3d or 5th of a maj or chord may be augmented, but in this chapter the augmented 5th only will be considered. Select any major chord, and by sharpening the 5th an augmented triad will 122 GOODRICH'S ANALYTICAL HARMONY. result : Ex - 2 ? 6 - The nature of this chord, containing as it does two major thirds, is harsh, and it has a strong tendency towards immediate resolution. The object of d-sharp is to ascend to e. The other two notes may remain, or the upper major 3d may ascend to c and e, while the root remains as connecting note. Tlu- two examples follow : Ex. 277. The second of these resolutions is of more frequent occurrence,, though both are useful. The base to these exercises is easily man- aged. In either instance it may proceed from root to root, as though the augmented interval did not appear : Ex. 278. -*1 5hK t =g Arrange this in two other positions and transpose. (There is another mode of treating the base, but it can not properly be introduced here.) The augmented triad, on account of its dissonant character, re- quires preparation. But as it is already prepared in the examples. any farther explanation of this subject may be left to a future chapter. Frequently the 3d of the augmented triad is used as a real-base, that tone being omitted above : J* Ex. 279. In four-part harmony the root, 5th or base may be doubled ; making two treble and two base parts. Rubinstein has even given the augmented 5th to the base and with charming effect, as this quota- tion will prove : GOODRICH'S ANALYTICAL HARMONY. 123 BALLET MUSIC FROM " FERAMORS." Ex. 280. The augmented 5th is included in the middle part in order to pre- serve this design in the accompaniment : Ex. The full effect of the augmented triad on the second beat of meas- ures i and 3 is also more satisfactory and complete than if the upper d-sharp had been omitted. THE DIMINISHED TRIAD. Theorists are agreed that a diminished interval is one chromatic step smaller than a minor interval. Therefore diminished presup- poses minor, but with regard to this interval : Ex ' 282 - which is generally called, a diminished 5th, there is a contradiction to be noted. By enlarging this so-called diminished 5th one chro- matic step the result should be a minor 5th in order to make the theory consistent. But these fifths : Ex - 28 3- Ffen have never been called minor. The old theorists called them "perfect," though in strict designation they are not absolutely pure. For this reason Weitzmann terms them " major fifths." Riemann says they are " standard fifths." But since these intervals are the same in both modes, as well as by inversion, the author has applied the term Nor- mal to the 4th and 5th of every normal scale. For this interval : Ex. 284. the most appropriate name seems to be Imperfect. It might be called minor, but it can not consistently be called dimin- ished. By flattening the / or sharpening the b, a diminished 5th 124 GOODRICH 'S ANALYTICAL HARMONY. will result. This interval has been used and will be included here- after among altered intervals. The same process is carried out in forming a diminished yth or a diminished 3d, thus : Mnj. Mill. Dim. Ex. 285. w Min. LX> 1 Dim. & P-frs* \?s> W The diminished triad is now resumed. This will be formed from a minor triad by raising the root : Ex - a86 ' \3gEEE$sr:E:{ From -j//a;^ to c is a diminished 3d, and from this the triad is called diminished. In this position the parts are brought so near together that it is not favorable to practical application. The root, or the middle note, may be inverted, thus securing a better position. With regard to the resolution, two of the notes have a fixed progression to the unison b : The 5th of the triad may descend a whole or a half step . Ex. 288. The former is more unusual, and for this reason seems less satisfac- tory. However, as it is sufficiently correct it may serve a purpose. Transpose the last example, using both resolutions. A diminished triad may also be produced by raising or lowering the 3d of an imperfect triad. In the first instance the upper 3d ap- pears diminished; in the second instance the lower 3d is diminished." Ex. 289. GGODRiCII'3 ANALYTICAL HARMONY. 12.5 An open position is more lavorabie tor tnese discords, on account of the ascending and descending tendency of the extreme parts : A A. f Ex. 290. At (a) the 3d is placed below, as a real-base ; at (b) the root remains below. The latter is more satisfactory, and accordingly more use- ful. The resolution at (a-) is so ambiguous as to suggest some con- tinuation beyond this point, as : Ex. 291. The discord is, therefore, an intermediate one, and not suited to a final close. The same may be said of the discord at (b) and its resolution, though this is more satisfactory, at least in a mere theo- retical exercise, where the ulterior design does not appear. Transpose the last two examples. The latter should be contin- ued to a satisfactory close by adding two or three measures. 126 GOODRICH'S ANALYTICAL HARMONY. PART VII. Chapter XXX. ORIGIN AND PRINCIPAL RESOLUTION OF THE DIMINISHED SEVENTH CHORD. BY sharpening the root of any dominant 7th chord there results a diminished yth chord; as any minor interval lessened by one M x^j^"^*"x^ chromatic step becomes diminished : E>x - 2 9 2 - NK^EjiizjJazE From the root to the highest note of the first discord (a to g) is a minor 7th; a-sharp lessens the interval and makes it diminished. The 3d, 5th, and 7th of the dominant chord remain stationary as parts of the next discord, while the base is raised one chromatic step. The chromatic alteration of the first discord changes its name, nature, and resolution, and results in a principal diminished jth chord upon A-sharp. The latter consists of a minor 3d, imperfect 5th and diminished 7th; or, a compound of three minor thirds. The first discord belongs to D-major, the second to B-minor: E, 29 ,Ffe%=iz3ER|^ I>-maj. B-min. The roots of the two discords represent the difference between the scales of the two modes, D-major and R-minor. The natural resolution of the diminished jth chord is to the minor concord founded a minor 2d above the root of the discord. The root of the diminished jth chord is the leading-tone of the key to which it naturally resolves. This is one of the strongest argu- ments in favor of the harmonic minor scale, as a characteristic series GOODRICH'S ANALYTICAL HARMONY. o; tones ; for the diminished ~th chord and its principal resolution comprise every tone in the scale, tnus : , i 3 i IT Ex. 294 ^ ^K=r5&=zgs -"- - z= bg : g = g : ? As the diminished chord occurs in this scale only, it may be con- ( eluded that it belongs naturally to A-minor. Observe that the dimin- ished yth chord contains both elements of transition (leading-note and sub-dominant) to A-minor. The student should now write a diminished yth chord in each of the fifteen keys. In every instance the diminished yth is to be de- rived from the dominant yth chord. Write the signature of each key in regular order, and locate the dominant yth discord upon the 5th degree of the major scale: Ex> 295 ' tfezz ^ Each example is to begin in major and terminate in the relative minor. The minor concords are to appear in their first position. The next example in regular order follows : Ex - 2Q6 - E-=i jjjjj == =| L [ -H[ -L-^=3 Proceed with the remainder as directed. When the root of the first discord is sharpened by the signature it will be necessary to use a double sharp for the root of the second discord, example : . ..J -. Ex. 297. The example in C-flat will correspond to the one beginning in Pi- rn a j or : Ex. 298. <%^ These are enharmonic equivalents. * This is not the only derivation of the chord. It is also a product of the harmonic minor scale, and may be entirely independent of an essential- yth chord. This will appear hereafter. The principal resolution of the diminished yth chord will now be considered. The root, being the leading-tone to the key in which it belongs, ascends a minor 2d ; the jth must descend a minor 2d < be- cause there is no other interval of the tonic triad to which it will 128 GOODRICH'S ANALYTICAL HARMONY. resolve) ; the 5th of the diminished chord is the sub-dominant, &n< must descend a major 2u. Here are the three most important notes : Ex. 299. EJE ^: The 5th of the diminished yth chord must, accordingly, be treate< the same as the dominant 7th to the same key, because this note (c here) has the same relationship in both chords: Ex. 300. Hr g""^ The white notes show the two elements of transition, and their reso lution, first in the diminished, and then in the corresponding dom inant yth chords. The black notes show that while the 7th of tin diminished chord descends to the 5th of the concord, the root of tin dominant jth chord remains stationary. (Compare (a) with (b), Ex 300.) The leading-note at (a) is the root of the discord ; at (b) it i; the 3d of the discord ; but as both are transition chords to A, botl elements of transition must be treated alike. The 3d of the dimin ished 7th chord, like the 5th of the dominant 7th chord, has no de cided, fixed resolution, but may ascend or descend, according to th< position of the chord : Ex. 301. *& f\ rfe*=g= The intervals which have a fixed resolution are indicated in Ex 299. By analyzing the discord it will be found to contain two in tervals of a 5th, from i to 5 and 3 to 7 : Ex. 302. Consequently if either of these fifths be resolved in similar move ment parallel fifths will result, as in the second measure. There GOODRICH'S ANALYTICAL HARMONY. 129 fore, whenever the fifths appear, they must be resolved in contrary movement, as here : Ex. 303. This is correct. The parts that move in similar directions are sit- uated a loth from each other, thus : i and 3, 5 and 7. By resolving the principal intervals of the discord (root, 5th and 7th) as directed, no raise progressions can result in any position. But when the 7th is uppermost the 3d must ascend. When the 3d or 5th is upper- most the 3d may (and generally does) descend to the tonic. The two positions most available for present purposes are these : Ex. 304. ~ The upper 5th (as f-sharp and c] appears in both measures as a 4th. Therefore the 3d or 5th of the diminished chord is to be in the melody whenever the chord is used in its principal resolution. It is not advisable at present to invert the base. As a prepara- tion for the following chapter, transpose the last example into several keys. GOODKICII S ANALYTICAL HARMONY. Chapter XXXI. NATURAL MODULATIONS TO RELATED MINOR KEYS BY MEANS OF THE DIMINISHED SEVENTH CHORD. ANOTHER MODE OF TRANSITION. THE chord of the diminished yth affords another means o) tran- sition to the related minor keys. The classification of these here follows : Ex. 305. I a -i The upper staff here contains the three dominant yth chords to the three major triads ; the lower staff contains the three diminished y th chords to the three minor triads. Each minor triad below is the relative of the major triad above. Compare the exercises vertically : i (a) with i (b), 2 with 2, and 3 with 3. Each diminished yth chord is the natural consequent of the antecedent dominant yth chord above it. The three diminished yth chords representing the three related minor keys to the normal scale of C are, strictly speaking, the only diminished yth chords in music, all others being mere enharmonic alterations of these primary chords. Consequently it is important that the pupil should acquire thorough control of these discords. As already ascertained, the positions of a diminished chord best adapted to present requirements are those in which the 3d or 5th is uppermost. In employing the diminished yth chords in the follow- ing harmonization introduce the antecedent dominant yth chord before each diminished chord. The difference between the two discords is in the roots ; the 3d, 5th and yth remaining the same. GOOI'RICH'S ANALYTICAL HARMONY. This melody contains transitions by means of diminished 7th chords to the three related minor keys : Ex. 306. The first chord may be either C-major or E-minor, the second must be a dominant yth, the third must be the consequent diminished jth, and the fourth chord must be a minor triad founded a minor 2d above the root of the diminished 7th chord. The dashes show where the resolutions of the diminished 7th chords take place. If the student requires a chart for the harmonization of this theme, Ex. 305 may be consulted. It would be well, however, to write without a chart, if possible. The repeated notes are to be considered as connecting notes. %. % ^c The only unusual progression in the harmonization occurs in the third measure. There is but one concord which can properly accompany the first g of the third measure. From this triad to the following dominant 7th chord involves a progression not heretofore employed, though there are two connecting notes. As the base ascends a second in all the resolutions of the dimin- ished 7th chords, the treble parts must all descend. No connecting note appears in the principal resolution of a diminished 7th chord, therefore contrary movement is in keeping with previous directions. (This is not to be re-arranged.) Transpose into G, B-flat and A-flat. An illustration of the facts to be comprehended appears in the following chart in A-flat : Ex. 307. Dom. 7th. Dim. 7th. Minor triads. Major triads. The dominant 7th chord marked i represents the key of the major triad with a corresponding number. So with 2 and 3. The dimin- ished 7th chords numbered i, 2, 3 represent the same keys as the minor triads whose numbers ar2 the same. Also, the antecedent of each diminished chord is its corresponding number among the dom- inant 7th chords, i among the diminished chords is derived from r among the dominant 7th chords. The numbers and the chords correspond in four different ways, the last of which will be men- 132 GOODRICH'S ANALYTICAL HARMONY. tioned : The minor triad marked i, is the relative of the major triad marked i ; and the same may be said of the other triads : their num- bers correspond to each other. Examine Ex. 307 attentively, and re-read the explanations follow- ing the example. Also transpose the chart into several keys. One other simple theme is presented in order to illustrate another position of the diminished chords : Ex. 308. The dashes indicate the resolution of the diminished yth chords. Begin and end in C-major. Transpose to D and E. * The next exercise (mostly in minor) introduces the diminished chord without its antecedent dominant yth. The diminished chords are here introduced independently, but their resolution remains the same. The student should complete the harmony by supplying the base : Ex. 309. The dashes indicate the three diminished chords resolving to the three minor chords in this key. Real-bases are to be included at (2) and (i). The first chord in the fifth measure is to be an imperfect triad with the 3d in the base. During the last of this measure an avoided cadence is outlined. This postpones the final cadence until the very last of the example. Transpose the theme alone into C-minor and B -flat-minor , and "'armonize it in similar manner. GOODRICH'S ANALYTICAL HARMONY. 133 Chapter XXXII. DIMINISHED AND CORRESPONDING DOMINANT SEVENTH CHORDS. THE two principal discords which represent a given key are already familiar. One is founded upon the leading-tone of a minor scale ; the other upon the 5th of a major or minor scale. Three notes of one chord occur in the other. The root, 3d and 5th of the diminished yth chord are the same as the 3d, 5th, and yth of the dominant yth chord to the same key. They are presented together: Ex- 3IO- rBtP*i|f ^%t?r Each chord appears in its fundamental posi- tion in order to show its formation ; but it would not be proper to employ them in this manner. The rules of progression will aid in this matter. As the diminished chord is to come first, write this and tie the three connecting notes: Ex - 3"- F/rv-^^:^ After this it is apparent that the yth (/) descends a minor 2d to -- 1 -^S> 1 \-^9 E^, 1 S P \ \ 1 \-\ -1 F=l (2) (2) (3) (3) (3) The first measure shows the transition chord in its original position. As this resolves to the tonic minor triad uninverted, it is not here necessary to work this out by means of the corresponding dominant 7th. It is included to show the chord selected for illustration. At (a) the first inversion is to be resolved to tonic minor according to directions. This will be an intermediate resolution, and, per conse- quence, the corresponding dominant jth chord is to be introduced in the following measure and resolved as a terminal cadence. The next example (b) is the same inversion re-arranged above. In each instance the resolutions are similar, with exception of the 3d of the first discord, which may descend a whole step when it does not form a 5th with the base or any upper part. The root, 5th and yth of the diminished chord have fixed resolu- tions, and always disappear in the same manner, without regard to position or inversion. This fact simplifies the task considerably. Each example, as (a), (b), or (c), is to be complete in itself, and the last chord of every example is to be that of A-winor, with its root in the base. The entire modus opcrandi has already been explained and illustrated with the diminished chord on D-sharp. Students will therefore have no trouble in completing the outlines furnished by Ex. 323. * * The third diminished jth chord should now be taken for similar illustration : Ex - 3 2 4- The inversions will consist of e, "The present intention is to work out merely the three primary diminished 7th chords, . B, and C. GOODRICH'S ANALYTICAL HARMONY. 139 g, and b-flat in regular order as real-bases. Each inversion is to include three close positions in the upper parts as explained. The immediate resolution of the diminished chord, in whatever inversion, \vill result in an inversion of the concord. These intermediate reso- lutions must be followed by terminal resolutions. The means em- ployed are the same : a corresponding dominant 7th ending with the tonic in the base. There is a difference of but one note between the two principal discords herein employed : Ex. 325. Compare the two discords first, and then their respective resolutions. In the second measure e may descend to d, but in the first it must ascend to /I The example, when completed, ought to correspond exactly to the other two illustrations, A and B. The student should finish mis task. Chapter XXXIV. THEME FOR HARMONIZATION, ILLUSTRATING IN- TERMEDIATE AND TERMINAL RESOLUTIONS. FARTHER VIEW OF INVERTED BASES. A CCORDIXG to the previous lesson, all resolutions of a principal -*~^" discord are intermediate when the resulting concord appears inverted. Therefore, when the 3d or 5th of a concord occurs as real- base we must proceed until a terminal resolution is effected. The reason for this is, that the base is the foundation of harmony, and in the final cadence the root of the tonic chord must appear in the 140 GOODRICH'S ANALYTICAL HARMONY. base in order to produce a sense of repose or completeness. An; note of the tonic triad may appear uppermost and leave a satisfac tory impression, provided the chord reposes upon its root. Th diminished chord was selected as illustrating this principle, for al its inversions resolve to an inverted concord. But the dominant 71! chord is less subject to this influence ; and, as a matter of fact, it root, 3d and 5th may resolve direct to the tonic, and thus constitut a terminal cadence. An example of this is appended : Ex. 326. -- (1) v (2) ^ Observe particularly that the tonic appears in the base in each in stance, and yet the discord is resolved strictly according to formula The only inversion that results in an intermediate resolution is th third. The tendency here to resolve down to the 3d of the concon is too strong to be ignored : Ex. 327. (3) (1) (3) (1) This must continue until a more complete ending is reached, as here Ex. 328. (3) (1) (2 ^ Both phrases are correct. The skip of a 4th in the upper part at (a is desirable, as no other part skips. At (b) the major 3d is doubled but this results from the melodic progression of the theme niovini GOODRICH'S ANALYTICAL HARMONY. 141 ^^-^H contrarily to the base. Such duplications are theoretically and esthetically correct. There is another resolution of this third inversion, in which the 5th of the discord ascends a 4th. As the base does not skip, the soprano may assume this privilege and skip from 5th to 5th. Beethoven. J I ^ Jix. 329. (3) (1) (2) In addition to the connecting note the 3d and yth of the discord are resolved most naturally, and no fault occurs as a result of this unusual progression. Besides, the half-open position of the B-flat chord is very agreeable. A similar instance may be mentioned in connection with the following : Ex. 330. (2) (1) (2) Observe the final resolution : a in the base and c in the mezzo-so- prano part descend to g and b-flat. These, being tenths, are always euphonious; the leading note ascends a minor 2d, thus avoiding fifths with the c above. The soprano part skips from dominant to tonic, root to root. In this instance the base moves alphabetically. Usually this is more effective than to retain the 5th above in the last chord. (The figures indicate the inversions, and the fermata ^ shows where the cadence is complete.) The last four examples should be transposed until the student is familiar with these applications. * * * Before presenting the illustrative theme, attention will be called to the fact that the corresponding dominant jth may be introduced before the diminished chord is resolved to its tonic. The previous method will be presented first : GOODRICH S ANALYTICAL HARMONY. A* S 5 A _& a & 1 2-1 VK ! */ F I II 1 ^~v _j ^ 1 B ^^ /p ^g' c ] 1 (2) Vi/ Ex. 331. The diminished chord is resolved to the tonic triad in the seco measure, the 5th being in the base. This necessitates another a more final resolution. But by introducing the corresponding ; chord on the last of the first measure it may resolve directly to t tonic chord, and so end : Ex. 332. ii i (3) (3) This is more brief, but is not materially different from the exerci: in Chapter XXXII. The yth of the diminished chord resolving down a minor 2d the root of the corresponding dominant yth chord may occur in a of the parts, though the resolution will not always be terminal, as the last example. With the 5th of the diminished chord in the b; the result will not be altered by introducing the corresponding c cord, because that interval becomes the jth of the second chord, either case its resolution is to the 3d of the concord : Ex.333. The result is an intermediate resolution as though the second discc had been omitted. With the 3d of the diminished chord the res is different : *The Roman numerals indicate the kind of discord. GOODRICH'S ANALYTICAL HARMONY. Ex. 334. (1) (1) At (a) the concord appears inverted, and is, accordingly, intermediate. But by introducing the corresponding discord at (b) the resolution is directly to tonic. The explanation is, that the 5th between the base and soprano in the first discord is changed to a 4th in the second discord. These results will be the same in the various positions of this inversion. These should be written by the student. * * * A theme will now be presented in which the newly acquired information is to be applied : Ex. 335. (1-) (2J (3) (3) In s'-ipplying the middle parts of the harmonization one should be governed principally by the base. The figures apply only to inverted chords, b being numbered (2) signifies that the root is a 5th below. Therefore the yth is in the melody, and as this is to be a dominant 7th chord the remainder is easily supplied. The Roman numerals indicate the antecedent, diminished, or corresponding dominant jth chords in this order : I refers to an antecedent, followed by a consequent discord when- ever the numerals I, II follow each other successively. II always refers to the diminished chord. I following II indicates a corresponding dominant 7th. GOODRICH'S ANALYTICAL HARMONY. In moving the base to or from an inversion it is advisabL preserve as much as possible a melodic progression in that p This is equivalent to the remark that the base must not skip exc from root to root. In the present exercises this is unexception true, and as these independent melodic progressions in the base ] are generally attractive and afford variety to the fundamental ba it would be well to cultivate them, especially since their proper n agement is much more difficult than the original base movement fi root to root. But this is intended neither as a rule nor a prohibit It is simply a direction for the student's present guidance. The ol arrangements should be made from the last example, without altei the base part. Transpose into various keys. The design, though occupying only a short period, is com] hensive, and the author considers it worthy of thorough treIONY. PART VIII. Chapter XXXV. PRINCIPAL AND SECONDARY SEVENTH DIS- CORDS. THEIR ORIGIN, APPLICATION, AND EFFECT. THE dominant and diminished yth chords are to be classed as Principal discords. That is, they contain the elements of tran- sition, and when resolved naturally are capable of performing a de- cided terminal cadence. Nearly all yth chords in a given key are secondary. These lack the modulatory elements and are incapable of effecting a modulatiot or of performing a cadence. A yth chord is built upon every degree of the normal scale : Ex. 336. Here are five species of discord. I and II we know to be transition chords. The remainder will be classified in this manner : The dis- cord founded upon the tonic contains a major 3d, normal 5th and major yth. This is the harshest of all the yth chords. Number it Y. Is there another discord in this scale of the same species? \Yhat is it? This should also be numbered V. The discord founded upon the second of the scale contains a minor 3d, normal 5th and minor yth. This is less harsh, and is numbered IV. Are there other clis- t46 GOODRICH'S ANALYTICAL HARMONS cords in the example containing the same intervals? If so, they belong to the same species, and are to be numbered identically. The chord upon the 7th of the major scale is still less discordant. It is numbered III. It consists of a minor 3d, imperfect 5th and minor 7th, and may be used as a secondary or as a principal discord. The diminished chord comes next. This is an agreeable discord, and is numbered II, as in a former chapter. It can be used in a major key, but it occurs naturally in the harmonic minor scale. Hence it is included among the unaltered discords. The most agreeable of all discords (so-called) is the one founded upon the 5th of the scale, the familiar dominant yth chord. This was the first discord explained, and being the most important it is numbered I. Corresponding Roman numerals should be written above each chord in Ex. 336. The discords marked V are so harsh, if sounded independently, that they must be submitted to some process of preparation. As this is explained elsewhere, the author will merely give directions concerning the present application of these dissonances. The dis- sonant interval is the major yth : Ex< 337- fry ^ ~ and, as with- out the yth there results a consonant major triad, it may be concluded that the b is the disturbing element. In order to prepare the ear for this dissonance, begin with a consonant chord containing b, and gradually introduce the c either above or below : IL Ex. 338. The base begins with a consonant root-note and moves down by- natural degrees. The first step results in a moderate discord, duly prepared. The second step (d to c), though perfectly natural, results in a discord of the fifth species. But the suspended notes above, together with the objective progression of the base part, have a tendency to ameliorate the dissonating effect of this harsh combina- tion. And what is more important, the base, progressing by scale- ^like degrees, while the other parts remain passive, presents a distinct object in what transpires, and is its own raison d'etre. GOODRICH S ANALYTICAL HARMONY. The dissonant interval may be introduced above by causing the upper voice-part to descend a half-step, while the lower parts, as consonances, remain stationary : Ex. 339. The first chord is purely consonant, and while the three lower parts are retained, the soprano merely descends a minor 2d. Any disso- nance, however harsh, would be justifiable if produced in this manner, on account of the seemingly slight difference between the consonant and dissonant harmonies. This difference, though theoretically very great, is merely what is heard or perceived in this example, Ex. 340. j^ p? ^__ while the same fundamental harmony pre- *) vails. These principles will apply to all harsh combinations, though the author doubts the propriety of preparing discords in general. Before proceeding, attention is directed to the fact that in both the previous examples three notes remain passive while one moves alphabetically. This plan will be followed in the practical applica- tion of these discords. Each of the secondary yth chords (including the secondary tran- sition chord on the yth of the scale) should be inverted, as students must be familiar with all positions of these discords. Omit the base note from the upper parts. One of these is presented as a sample : Ex. 341. :Eg= (1) (2) (3) The V sho A-S the species and the figures indicate the inversion. Follow the same plan with the other inverted discords. 143 GOODRICH'S ANALYTICAL HARMONY. A brief illustration will be given of the manner in which these secondary discords are to be utilized : *~-,Z- i 55- -55 ^ xj 1 fa " (2 h- p . w^ j i H 1 V IV _L_ , , . ^n i i ^Mtyt ,y f i (* + 3 & te= ff f ' ^ r- EE 1 H (3) (1) W Ex. 342. At V a harsh discord results in the manner already explained. From here the degree of dissonance is gradually reduced by moving only one part at a time until the dominant yth is reached. This is an euphonious discord. Number II is omitted, being unnecessary here. In progressing from one concord to another, or in resolving the essen- tial yth chord, one connecting note is sufficient. Re-arrange this in the upper parts and transpose, as it is prelimi- nary to what follows. In harmonizing the following theme three notes are to be tied in each succeeding chord of the dissonant progressions. The moving notes may occur in any voice-part, above or below. Real-bases are not to be duplicated here, and in the dissonant chords even the root- tone must not be doubled. (See inversions, Ex. 341.) Ex. 343. IV IV V IV III II (3) o (1) (2) (1) Previous directions as to the movement of voice-parts, together with the accompanying numerals, will enable the student to complete this in correct manner, especially since the number of each inversion is indicated by figures below. Roots in the base are marked with ciphers. The entire series of yth chords in the chart, Ex. 336, is to be employed in this harmonization, and the theme is so designed that the dissonances are gradually reduced in harshness after the second discord marked V. Observe the numerals. * * * GOODRICH'S ANALYTICAL HARMONY. 149 The next step consists in writing the contralto part uppermost as theme. Use the same chords and the same base. Another resulting theme may be obtained by copying the original mezzo-soprano part, and writing the harmony beneath it. These two themes are given : Ex. 344. ffi ^=5: This exercise is susceptible of still another arrangement, and as it illustrates a useful principle it will be included here. Reference is made to the original base part (from third measure) being transferred to the soprano as theme : Ex. 345. V IV III II The first two measures of the original theme are not adapted to the base, but from the third measure, where the parts are bound so closely together, any of the upper parts might appear below. The only difference in treatment would be required in the final cadence. In case the last note was a real-base (inversion of the tonic chord) it would be necessary to make some alteration. For instance, in the theme marked (c) the last chord could not have its 5th below, but would proceed from root to root, thus : Ex. 346. (3) (8 This theory, as well as the resolution of the diminished chord into 150 GOODRICH'S ANALYTICAL HARMONY. the corresponding dominant 7th, has been set forth in a previous chapter. Transpose these exercises into various other major scales. * * # These secondary and principal yth chords may also be treated fundamentally, especially if they form a harmonic sequence. In sucl situations the secondary discords assume the functions of a principal yth chord, and are apparently resolved in the same man- ner ; but it must be understood that the former merely occupy cer- tain positions in the scale according to the nature of the sequence, and that their resemblance (on paper) to the principal discords is apparent, not real. In rococo music this treatment was of common occurrence on account of the infrequency of transition passages. Since the advent of Boccherini and Mozart the tendency has been to change the secondary into principal yth chords. An example of the former method will be quoted as of greater present consequence: It is from a Rondo by Paganini. The discord marked V is resolved (disappears) as though it appeared as a principal discord on c-flat : Ex. 348. But in the original the a and d are natural, and the chords are all confined to the scale of B-flat until the tempo- rary modulation to the dominant in the last measure. The chord marked V is very harsh, but here it is duly prepared by the previous b-flat and d, and it also results naturally as part of the sequence : f, e-flat, d, c, in the base. The discord marked IV may be explained in like manner, though the base here ascends from d to g, the same as does the root of a dominant yth chord. In the resolution of the discord on e-flat the root ascends an augmented 4th, because the normal 4th (a-flat) is GOODRICH S ANALYTICAL 1IAKMONV. 151 not a part of the B-flat scale. Observe that the 5th of the triads are omitted. If these be added the progressions would appear like this: Ex. 349 These more closely resemble the original treatment of these discords as connecting links and suspensions. The student should particularly note the design of the sequence in Ex. 347 ; how the discords take their places in the sequence upon successive degrees of the natural scale ; how the secondary chords disappear as they follow the model, and the effect of the entire pas- sage. This will prove much more useful than all the rules of thor- ough-baser books, for it discloses the motive, and forms the basis for all available musical knowledge.* Transpose Ex. 347 into several major scales. If this can be done at the piano prima vista, it will render unnecessary the act of writing the notes. Chapter XXXVI. ADDITIONAL CHORD PROGRESSIONS. previous lessons in chord progression have demonstrated -A- the fact that the movement of a concord can not be prescribed. Certain principles can be deduced from standard compositions and according to logical theory ; but it can not be said that a certain tone of a certain concord must be followed by a certain other tone of another chord. I'.c-in with the concords containing a given tone. These are six in number. Ex. 350. bfc= *See accompaniment to Schumann's " Ich grollc nicht. 152 GOODRICH'S ANALYTICAL HARMONY. The first three occur in A-major, the last three in A-minor. But they are all related in a secondary manner to the key of A. Trans- pose into G and B-flat, and add the fundamental bases. The direc- tions for harmonic progression would apply to any of these chord successions. Observe the following miscellaneous triad progressions : Ex 351. vTZ ^ =zg^=|= l^S -2,^=L :_^"-:^J ~-%3\ :^T2?- \-4^-*~ *' ' (5* a b c d e f g \) ~-~^ * \ \ f^V* ** i \ es i , ~ ^ ^~~ 1 (1) GOODRICH'S ANALYTICAL HARMONY. 185 But this is not materially different from the others. The dominant chord makes the actual close in this as in the authentic cadence, though the former is more comprehensive. 3. PERFECT CADENCE. This includes the same harmonies as the preceding, with the addi- tion of the second inversion of the tonic chord between the subdorni- nant and dominant, thus : Ex. 440. -fl-i ^^ * ^ s 5- - 5 E52 (M) 2* __ '. ff Us P* 1- -^ ^ 1 x5~ 1 ^ 1 i L. 1 , 1 1 . 1 1 Here the E-minor chord takes the place of the tonic chord (G) at the natural termination of a melodic idea. It preserves the interest and prolongs the period, for this is an indirect resolution. An example is given showing the situation in which an avoided cadence produces the best effect : Ex. 448. The dash shows where the cadence is avoided, this being the 3d reso- lution of the essential discord. (The fourth resolution, corresponding in minor to this, will appear in the next chapter.) No satisfactory close occurs until the appearance of the tonic chord marked /TV The last example does not admit much re-arrangement, but should be transposed.* 6. DECEPTIVE CADENCE. The principal difference between avoided and deceptive cadences is that in the latter discord follows discord. Reference is here made to the last of a period where the final dominant yth chord would naturally be expected to end on the tonic chord. But if another dis- cord be substituted for the tonic triad, a deceptive cadence occurs, and the music must continue until a satisfactory close is reached. Following is an example : (No. 11, Peters' Ed.) Allegro. Sonata, Haydn. M Ex. 449. * The Andante to Schubert's B-flat Symphony contains excellent illustrations of avoided cadences ; also the Alia Marcia in Schumann's Op. 44. 190 GOODRICH'S ANALYTICAL HARMONY. This is the last of an eight-measure period ending naturally at 8. Here the deceptive cadence has the effect of considerably extending the actual period, until the essential discord resolves io the tonic triad. A similar instance is presented, to which the same remarks will apply : Ex. 450. ^ 0. #"" The last measure is a deceptive cadence.* An interesting instance may be found in the Largo of Beethoven's Op. 7, where the regular close is prolonged from the 2oth to the 24th measure. 7. INCOMPLETE, OR HALF CADENCE. The incomplete, or half cadence, consists of the tonic followed by the dominant harmony, with the 5th of the former in the base. This naturally leads to the dominant : Ex. 451. 2) The tonic chord is needed to form a perfect cadence. It is therefore incomplete, and presupposes that something else follows. Illustra- tion (b) has the same effect of keeping in abeyance the actual cadence and is lacking in that sense of repose and completeness which only the tonic can impart. After the pause, G-major, G-minor, D-major, D-minor, or even other keys may be introduced. The Sonatas of Haydn, Mozart, dementi, Dussek, and the earlier ones of Beethoven, contain many instances of the incomplete cadence, which usually occurs at the end of the principal theme. "To distinguish these intermediate cadences the reader must understand the analytical divisions of a period and where the melodic cadences occur. Musical Analysis explains this. GOODRICH'S ANALYTICAL HARMONY. 8. AFTER CADENCE (PLAGAL). 291 The subdominant harmony followed by that of the tonic consti- tutes what is known as a plagal cadence. The author calls it an after cadence as this term is more significant ; this cadence coming after the final ending of a composition, to which it serves as a short coda or extension.* In church music the after cadence is often used at the end of an Anthem or Te Deum as accompaniment to the word Amen. Hence it is frequently called the Amen Cadence. In instrumental music the application is the same : -4-^- Ex. 452. A - men. The phrase in brackets shows the application and effect of the after cadence. The period terminates before this, on the G chord. This cadence may be written in various ways, thus : Ex.453. m At (b) the passing note between e and d is included. At (c v the sub- dominant minor does not appear till the last of the measure, v ut the chromatic progressions in the middle parts naturally lead to th^ sub- dominant and tonic. Of all the cadences herein enumerated the after cadence is the most mild and undecided. The diminished /th chord as a passing harmony may be included among the Amen cadences : Ex. 454. *The use of this cadence as a substitute for the dominant in o! ! ecclesiastic music is inw obsolete, and the Greek term has no real significance in modern music. IC)2 GOODRICH S ANALYTICAL HARMONY. The tonality of G is not affected by the chromatic passing chord especially as the tonic remains above and below. The harmony oi a major 3d below may also be included, though its relationship i: apparently remote : Ex. 455. The effect is somewhat transitory and unexpected, but, like the others it serves a particular purpose. This is beautifully illustrated in tin song by Kiicken, Good night, farewell. Chapter XLII. HARMONIC CADENCES IN MINOR. i. Authentic. 2. Complete. 3. Perfect. 4. Extended- Perfect. 5. Avoided. 6. Deceptive. 7. In- complete. 8. After. 9. Ambiguous. i. AUTHENTIC CADENCE. THE principal cadences in the minor mode are based upon th< same fundamentals that were employed in major. And as th< dominant harmony is identical in both modes the auvhentic cadenc* will consist of these chords, in any position : Ex. 456. GOODRICH'S ANALYTICAL HARMONY. 193 Compare this with Ex. 435. Every note of the minor scale is here employed except the 6th. The diminished yth chord plays a more important part here than in the major cadences, for the discord and its principal resolution embrace every note in the harmonic minor scale : Ex. 457. This principal resolution of a diminished 7th chord decides the minor key as satisfactorily as does the dominant yth harmony. For termi- nal resolutions of this discord the student will dp well to use only these positions : Ex. 458. As the tonic appears in the base after each resolution, they are all final. Of the illustrations (a) and (b) are best. At (c) the 7th ap- pears uppermost and this necessitates the duplicating of the 3d to prevent fifths. If it is desirable to have the last chord complete in the upper parts, double the root and omit the 3d or 5th. When the 3d appears as real-base it resolves intermediately (a), and the corresponding dominant 7th will be necessary for the final cadence (b) : Ex. 459. (1) (2) So with the other inversions. This is the only objection to the di- minished 7th chord in a final cadence. Remember that the root, 5th and 7th have fixed resolutions, while the 3d may resolve according to circumstances. 194 GOODRICH'S ANALYTICAL HARMONY. 2. COMPLETE CADENCE. Here the tonic and subdominant are minor, while the domman remains major. Complete cadences in major and minor follow : Ex. 460. Observe that the fundamentals are identical. The complete cadence embraces all tones of the harmonic mino scale. The cadence given at (b) admits of re-arrangement according t< this model, and must also be transposed. * * * The second form of this cadence given in major does not admi of exact reproduction, because the subdominant is itself minor, an< therefore has no relative minor. But a form may be used corres ponding to that given in Ex. 438, by substituting an imperfect tria< on the supertonic for the subdominant harmony : Ex. 461. (1) The second chord contains two notes common to the subdominan harmony, and by placing the 3d in the base a very good substitut( for the harmony of the 4th is obtained. On account of the inverted triad it is better to move the parts contrarily, as at (a), (b) and (c) Contrary movement is also better in going from the first to the sec ond chord, because there is no connecting note. Transpose these *This affords another argument in favor of the harmonic minor scale. GOODRICH'S ANALYTICAL HARMONY. 3. PERFECT CADENCE. 195 ia io reptodd<_ed by using the same harmonies according to the tonality : Ex. 462. The dominant chord at the close ii sufficient, but the 7th is included in order to follow the downward tendency of the theme. Should the melody suggest it the imperfect triad may be used in place of the subdominant. Otherwise it would be the same as Ex. 462 : Ex. 463. Sf h* e^E (i) In a final close this might be preferable to the essential discord as a harmonization of the melody note at + . The subdominant harmony can be combined with the imperfect triad, making a secondary yth somewhat similar to the one employed in major: r^ t ^ Ex. 464. -j (1) (2) := B This corresponds to Ex. 442 (a). The 3d is usually treated^s a real- base in order to produce the effect of a subdominant harmony. There is a foreign harmony frequently employed in a minor ca- dence that deserves a place here. Counting from the fourth as real- base it contains a small 3d and normal 4th. Theoretically it is a major chord located a minor 2d above the key-tone ; in actual prac- tice it is treated as a derived harmony, like the augmented 6th chords. It is known as the " Neapolitan Sixth," but this appellation does not 196 GOODRICH'S ANALYTICAL HARMONY. seem to be appropriate. Every modern composer has used it with- out regard to local origin or association. In his Italian symphony Mendelssohn does not employ it, but in the Scotch symphony it occurs several times. This would seem to prove that no local color- ing is to be derived from this cadence, for Mendelssohn was quick to- avail himself of any extraneous aid to legitimate tone painting. Ros- sini uses the cadence in his Swiss overture, as well as in his Neapol- itan tarantella, La Danza : BESS^^BE 1>W 4 * 1 5 2_ X9 U^5 V 1 1 * x -J. J. * a ) ) . ?.' lr? \ *S jO * /^ (* S rt i't i [ZSZIj 9jJ 1 f 4: U . 1 f i Ex. 465. This so-called Neapolitan 6th furnishes a major harmony with the subdominant as real-base, in place of the imperfect triad. It is much brighter than the latter, and tends to relieve the sombre hues of the harmonic minor scale. Transpose the example into several other scales. The passing diminished yth employed in Ex. 444 has no equivalent in the minor mode, because there is no recognized tone between 2 and 3 of the scale. But 4 in the base might be sharpened as a passing-note to 5. See Chapter XL. 4. EXTENDED-PERFECT CADENCE. This may progress by thirds as in major. It is only necessary to follow the signature : Ex. 466. Either form may be utilized. 5. AVOIDED CADENCE. The same general principles apply to both modes. The example is therefore presented : GOODRICH'S ANALYTICAL HARMONY. 197 Ex. 467. The fourth resolution corresponds to the third in major. These are employed whenever it is desirable to prolong the cadence. The pro- gression from f-sharp down to e-flat in the middle of the example is perfectly correct, notwithstanding the attempted prohibition of this augmented 2d. 6. DECEPTIVE CADENCE. This may occur in the minor, as well as in the major mode. The principle and the general effect remain the same. Examples in major are, however, more common. 7. INCOMPLETE CADENCE. By comparing the next example with No. 451, the student will perceive the general similarity : Ex. 468. In either mode it is an incomplete cadence. 8. AFTER CADENCE (AMEN). This consists of the subdominant (or some harmony correspond- ing to that of the fourth) followed by the tonic, as in major. With exception of the difference in mode the effects are identical. There is, however, this important distinction to be made: In major use either a major or minor chord on the subdominant ; but in a minor key the subdominant harmony is naturally minor:* Ex. 469. Good. bad. e) * In the Sicilienne from " Cavalleria Rusticana " this natural order Is reversed and with excellent effect. 1 98 GOODRICH'S ANALYTICAL HARMONY. The major chord in the second example is usually ineffective. The situation in which the amen cadence occurs was illustrated in the previous chapter. The harmony of a major 3d below the tonic might be used as a modification of the after cadence, and in a minor scale the effect would be less abrupt than in major. The diminished chord is not available as it was in major. g. AMBIGUOUS CADENCE. In Chapter XLJX this cadence occurs among the harmonies ot the natural minor scale. It consists of the dominant minor, in place of dominant major, with the subtonic as melody note. This is fol- lowed by the tonic harmony : Ex. 470. It is rather weak and melancholy, and should be used in accordance with these sentiments. Its effect is more satisfactory if not brought into immediate comparison with the more positive and familiar dom- inant major harmony. An instance is here quoted from Grieg, whom the author considers the greatest harmonist now living : Ex.47i. n p tt ' ' ^ ^ ^-^^ Op. 22. V I *T 1* 1 ^ F " ^ ^ <* 1 y2_ j : ~^J -1 Rit. n "^" i*F "a * * & ") ft "< ^ q n - \ PP ~ c^* i ^' g f -^-" The composer evidently anticipated that some academic musician would " correct " this cadence, for the natural is found before d in both parts of the duet. This would be less effective in the majoi mode. ''All these cadences should be performed iu various minor scales. GOODRICH'S ANALYTICAL HARMONY. 199 PART X. Chapter XLIII. AUGMENTED SIXTH CHORDS THEIR DERIVA- TION, APPLICATION AND EFFECT. No. i. A MAJOR 6th enlarged, becomes what is known as an " extreme sharp," or augmented 6th. It is two chromatic tones larger than a minor 6th, and one chromatic tone larger than a major 6th. In notation, thus: Min. Maj. Aug. Aug. l-i -T Ex. 472. The augmented 6th may be produced by raising the upper, or low- ering the lower tone of a major 6th, as illustrated. In both instances the direct resolutio-n is to the octave situated a minor 2d above and below the interval of the augmented 6th : Ex. 473. This interval may appear as a i3th or a 2oth from the real-base (c or c-Jlat in last example). It is still called by the same name and treated hi the same manner as though the interval of an augmented 6th appeared uninverted ; for it is this interval that gives the chord its name, independently of the intermediate tones omitted from the last example. 2OO GOODRICH'S ANALYTICAL HARMONY. There are various chords containing the interval of an augmentet 6th. The author has systemized these, and endeavored to explaii them as they are used by standard composers. They are numberec from i to 3. No. i, with its most natural derivation, is given first Begin with a yth chord, species IV, as at (a), invert it once (b) am raise the original root, now the 6th, (c) : Ex. 474. No. 1. The theoretical derivation of the augmented 6th chord, No. i (c), i here shown as a formula. In actual practice the original positioi (a) seldom appears, and it is unnecessary to go through this process From the real-base (/) to the d-sharp is an augmented 6th, and a this chord, to be most characteristic, must have its 3d in the base the propriety of calling it a 6th chord is seen, though the 6th (/t< two have already been wrkten (besides the one founded on the sec- ondary yth), these will include all that are practicable in this kev. 208 GOODRICH a AAALVTICAt, HARMONY. The most feasible derivations of the augmented 6th chord. No, are therefore as follows : From dominant yth chords founded up 2, 3, 5, and 6 of any major scale, and from the secondary discord the yth of the scale. Each of these is to be inverted twice, and 1 real-base is flattened in those derived from essential discords. T 6th chord derived from the secondary yth on the leading tone requi: 110 alteration of the real-baae, because the original 5th is imperfe and this produces the augmented 4th. See Ex. 493 (a). All these instances result in the same kind of an augmented ( chord, and their treatment is therefore identical, excepting the fi that two end in minor and three in major, according to the signatu With exception of the 6th chord produced from the inverted doi nant yth chord the other derivations may come from secondary i well as from principal) yth chords on the ad, 3d, and 6th of any ma_ scale. If a secondary discord be chosen it will be necessary to ra the 6th in addition to the lowering of the real-base. By using inverted dominant yth the major 6th will be ready to hand, and th by lowering the real-base the augmented 6th, No. 2, will result. Th< are the only differences between the two methods ; the general suits are identical. Another derivation is here presented : Gleason. Ex. 496. The 6th chord, No. 2, is preceded by an altered triad, having the effc of an augmented 6th chord. The derivation is perfectly natural. T final chords to be written and re-arranged will (in the key of D) le to D-major, E-minor, G-major, A-major and B-minor. Reduce the to numbers and they will apply to any major key in transposin They ought to be worked out in at least four other major scales ; ai the practical performance is not to be neglected, for one is a nec< ^ary complement to the other. /p. RTl BZ i i " K y & n* " etc. J J. j^! j 5 nzg 1* t-^ 2J r j ^ GGODRICH'S ANALYTICAL HARMONY. 209 Chapter XLV. AUGMENTED SIXTH CHORDS CONTINUED. No. 3. BY selecting an augmented 6th chord, No. 2, and raising the aug- mented 4th still farther, there will result an augmented 6th chord of the third species : Ex. 497. This contains a major 3d ; a doubly-augmented 4th,* and an aug- mented 6th. Its resolution is to the second inversion of a major chord. The real-base and the augmented 6th resolve as usual ; the 3d remains as connecting link ; and the doubly-augmented 4th as- cends a minor 2d. The dominant, or dominant 7th, follows the in- verted concord, and then the final cadence naturally takes place. Complete the exercise, and make two other arrangements, preserving the same base. * * * As the resolution of No. 3 is necessarily to a major chord, the first task is to select the three major chords that occur naturally in every scale. These represent the three related major keys, tonic, subdomi- nant, and dominant. The augmented 6th chords are located a large 3d below each of these key -tones. Or, to be more explicit, the real- base of any augmented 6th chord is a major 3d below the final tonic. The three discords from which these augmented 6th chords are de- rived are therefore founded upon the 2d, 5th and 6th of a major scale. These are inverted twice, and altered into augmented 6th chords. In Ex. 497 the chord numbered 2 was derived from the 7th chord on the dominant to C. Those discords founded upon the 2d and 6th of the scale may at first be principal or secondary. The student can complete the task of writing out in different posi- tions the two remaining altered 6th chords. The} 7 end in C and in G, and are transitions. Together with the one ending in F as final * The doubly-augmented 4th is two chromatic steps larger than the normal 4th. 210 GOODRICH S ANALYTICAL HARMON V tonic, these will include all the related major keys in the scale of C. It is of no present consequence whether the student flattens the real- base before sharpening the augmented 4th, or vice versa. * * * In case it were desirable to extend the progressions, begin with a secondary jth chord on the 2d or 6th of the scale and gradually altei the intervals until an augmented 6th, No. 3, appears : Ex. 498. II 3 =,2=tez i==t Such arrangements are well adapted to the organ. The second chord is the essential yth to F, and yet the section begins and ends in B-JJat, But the student should understand that the nature of this discord is almost completely changed at 3, and that these changes direct it intc a different channel.* The following illustrates in a more musical manner the particulai application of this chord : Jl'i/sitn G. Smith. Ex. 499. B&(\, . . and yearnings sad my lone heart doth en-dure. It is sometimes desirable to raise the 4th and 6th simultaneously aftei having lowered the real-base : Ex. 500. " The discord marked I is a fundamental harmony; the one marked 3 is a derived bar- tnony. GOODRICH'S ANALYTICAL HARMONY. 211 The student should be familiar with the various methods of intro- ducing and arranging the altered 6th chords. Any of the three diminished yth chords that represent the related minor keys may be utilized in producing the discord under notice. This is partially illustrated by the chord marked II, in Ex. 498.) In the scale of D these diminished chords are the following : Ex. 501. : In their present appearance they represent B-minor, E-minor, and / ' '-minor. Invert each one twice, lower the real-base a chromatic step, and the result will be three augmented 6th chords of the 3d species, leading to G-major, C-major, and D-major. These are to be worked out in different positions as directed. * * * The intervals from the real-base must be : a major 3d ; doubly- augmented 4th, and augmented 6th. The immediate resolution is to a major chord with its 5th in the base. These points are particularly important, since there are three discords that sound exactly the same, the differences being in their notation and resolution. These are here presented, with their resolutions indicated by quarter notes : Ex. 502. * The first belongs naturally to A-flat ; the second to G -minor ; the third to G-major. The discord at (a) is an essential yth. The other two are augmented 6th chords of the first and third species. The different signatures show the different situations in which these dis- cords would naturally occur. The minor yth at (a) becomes an aug- mented 6th at (b) ; and the normal 5th at (b) appears as a doubly- augmented 4th at (c). Examples (b) and (c) should be worked out to their final tonics. The examples explained and illustrated in this chapter are to be transposed (both theoretically and practically) into several other scales, and every exercise should be written in three positions. The different modes of treatment with regard to the endings are also to be used in different examples. 212 GOODKICH'S ANAT_YTICAT V HARMONY. The derivation, notation, and resolution of the various augmented 6th chords i, 2, and 3 must be thoroughly understood, for in the next chapter attempts will be made to use them in the harmonization of themes, as they are employed in actual composition. Chapter XLVI. APPLICATION OF THE VARIOUS AUGMENTED SIXTH CHORDS IN HARMONIZATION CONCLUDED. THE three principal augmented 6th chords are to be considered as transition chords, even when their resolution is indirect. No. i contains a major 3d, normal 5th, and augmented 6th, and resolves to a minor chord in its second inversion. No. 2 contains a major 3d, augmented 4th, and augmented 6th. This resolves direct to a major chord in its first position. No. 3 consists of a major 3d, doubly-augmented 4th, and aug- mented 6th. Its immediate resolution is to the second inversion of a major chord. The final resolution of No. i is to minor ; No. 2 ends in major or minor, according to the prevailing tonality ; No. 3 belongs expressly to the major mode. The immediate resolution of i and 3 determines the final resolution. The resolutions are all intermediate, and must proceed beyond the point at which the augmented 6th chord dis- appears. In actual practice the principal difficulty will consist in properly introducing these altered 6th chords. First of all decide the follow- ing points: What chord do you wish to use in a certain place, i, 2, or 3? What is the real-base? What other chord will contain one or more connecting tones ? Suppose this chord was required : Ex. 503. RgF x ;;^ It is. GOODRICH'S ANALYTICAL HARMONY. 213 derived primarily from the second inversion of a secondary yth chord. The root will be A, and the real-base e. So much being ascertained, the next step is to find a connecting chord with this. As the base is to be ? * * ; i 1 1 ... 1 L_l_^ \ r ^ {m^r i rr r f i M ' 1 q 1 ^ 1 1. 2. 3. c&4-f-p J^^ 1 -& \ f ^H ^ % \ & .i (2) The figures i, 2, 3 refer to the different species of augmented 6th chords. The last two ^'s in the first measure signify that the discord marked i, and the following minor triad to which it resolves, both contain c, This minor triad must appear with its 5th in the base, followed by the dominant and tonic. The chord at 2 resolves direct to a major chord (dominant), after which the minor yth is introduced c being retained in the middle part. 3 indicates an augmented 6th chord of the third species, resolving to the second inversion of a major chord. This is a transition to G-major. The first two half notes represent two quarter notes each in the other parts. Illustrations and explanations in the previous chapters may be referred to. 214 GOODRICH'S ANALYTICAL HARMONY. After completing the harmonization in four parts it should be written in two other positions with the same base. The mezzo-so- prano part may be copied as a theme, but the original contralto part should be inverted an octave higher when it appears as melody. Afterwards transpose into A-flat, B-flat, D, and E-flat. (Those who have not the benefit of an instructor will find the solution in the Key, but the author again urges students not to con- sult this part of the work except when necessary.) Another theme is given w r ith the same general object in view : Ex. 506. JLJa. r J a i ' i " i r r?Tr & 9 & \ i *V 1 1 i i i lill Tr 1 1 1 J 1. 2. 3. C""\* I _ i J. f ! \ 3 28 .- f # -^ rt ..3 * _' * J 2 ty 22 1 1 1 BJI i- ' j (2) Modulations are here made to D-minor, G-minor, and E-flat-major, by means of the augmented 6th chords, in addition to the passing modulations through the essential jth chords. The resolutions here are the same as in the theoretical exercises. The only difficulty con- sists in ascertaining what particular key is intended at the different points indicated by the figures. These latter tell the species of 6th chord to be used, and as only the related keys are to be reached, students who have learned how to apply mental force will have no trouble in completing the harmonizations as intended. During the fifth and sixth measures the regular formula as to intermediate and final resolutions is slightly varied. An avoided cadence is indicated by the dash, in place of the tonic resolution to E-flat-major. Avoided cadences in such situations are always proper and usually effective. * * * The altered 6th chords introduced into the last themes have be- come known as " Italian," " German," and " French Sixths," though for \vhat reason none can tell. These chords can not be nationalized, and the names have, therefore, no significance whatever. The augmented 6th chord, No. 2, is the most masculine and inci- sive of all the transition chords. The resolution of these intervals, Ex. 507. GOODRICH'S ANALYTICAL HARMONY. 215 ascending and descending by half steps, produces a very decided effect, while the two major thirds add considerable boldness to the transition. The augmented 4th (d) supplies the dissonant quality (aside from the sharpened 6th), for d produces a discord with a-Jlat, as well as with c. Composers frequently omit the augmented 4th, even in full har- mony. In such instances the- 3d (from the real-base) is usually doubled, and then resolved differently in each part, thus; ,/, Ex. 508. The upper c descends to b, while the lower c ascends to d, leaving the G chord complete. Besides, it is usually better to resolve a du- plicated tone either in contrary or oblique movement. This may be done whenever the augmented 4th can not be introduced conven- iently. There is another form of augmented 6th chord, consisting of an essential discord in its fourth position with the original 5th sharp- ened : Ex. 509. But this is generally accompanied with the fundamental in the base, as here : Ex. 510. The resolution of the essential discord is not materially changed on account of the altered interval, and as the chord here stands it is simply a dominant ~th with augmented 5th. Hence it belongs more to the fundamental harmonies than to the altered 6th chords. Augmented 6th chord, Xo. 2, is sometimes treated in the same 2l6 GOODRICH'S ANALYTICAL HARMONY. way, the original root being given the base. When this occurs the 6th chord proper resolves as usual, while the independent base skips from dominant to tonic : Ex. 511. -P-ir m This is a somewhat rugged, forcible harmonization, and its use i rare. It also brings the base into greater prominence on account of its independent, fundamental character. The augmented 6th chords, 1,2, and 3, are, however, the most important and characteristic. With the derivation and resolution of these the student should be familiar in every practicable key. This subject will be concluded with a few quotations from stand- ard works, intended to show the deviations from, rather than the conformations to, our theoretical formulas. The first extract is from the allegro to Mozart's 2d G- minor Symphony : Ex. 512. * :# 7* S Mrj3ti -^. The 6th chord, No. 2, does not here resolve directly to the dominant harmony, but to the inverted tonic chord first. This is on account of the melody above, g, a, b-flat. Observe the chromatic progression in the tenor and base parts, as the augmented 6th chord is produced principally by this contrary movement. The next extract is from the "Jupiter" Symphony : v Ex. 513. No. 3. i^%4 4 | 1 3P GOODRICH'S ANALYTICAL HARMONY. 217 The quotation commences at the end of a transitional section in D- flat, and this shows the manner of returning to the original tonic, C. The second chord in the first measure is a passing diminished har- mony on the dominant, a-flat. By changing the essential yth chord enharmonically there results an augmented 6th chord, No. 3, resolv- ing naturally to C. In the andante to the same symphony the composer has resolved the base first, and then the other parts alternately, thus : I Mozart. Ex. 514. Observe that the treble parts are suspended after the lower resolu- tion, and the c does not move to b until after the second beat. This may be done with any four-toned discord. The next illustration is still more exceptional. It is from the slow movement to the 2d G-minor symphony : Mozart. 5*5- Observe firstly the two lower staffs. On the third beat an augmented 6th chord appears, and in the resolution o -sharp descends to g-natu- ral, in place of ascending to a. But the sequence embraces a chro- matic progression similar to the succession of essential yth chords. B-flat appears in the base on account of the prevailing tonality, and this is what produces the augmented 6th chord in connection with the dand g-shar.p above. Observe the two ^'s moving contrarily, 218 GOODRICH'S ANALYTICAL HARMONY. and especially the chromatic progression in the lower treble part The violin figure above is a variation of the middle parts. The student is to be cautioned against writing a succession of such progressions as this from the last quotation : Ex. 516. Their effect is the same as two minor sevenths, which can not, as a rule, follow each other. Several such dissonant successions would be liable to offend good taste, and even a single progression of these intervals must be justified by some such design as the one quoted from Mozart. Another exceptional resolution of an augmented 6th harmony is here cited from The Dream of Jubal, by A. C. Mackenzie Ex. 517. There is nothing unnatural about this, and, doubtless, similar in- stances exist. By including c in the tenor a connecting tone would result, and this might be utilized. By lowering the 5th of No. i a minor 2d it will result in a similar chord of the 2d species, and ma} r therefore go direct to the dominant chord. Such an instance is here extracted from the F-minor con- certo by Chopin : Ex. 518. The change from i to 2 at + may be considered an expediency, an in such situations a most useful one. Beethoven frequently change^ an augmented 6th, No. i into No. 2, when he desired to pass direcii> GOODRICH 'S ANALYTICAL HARMONY. to the uninverted major chord, and to avoid parallel fifths. In these instances the minor chord in its second inversion does not appear, but the final resolution is the same. In a descending sequence of dominant yth chords an augmented 6th chord of the first or third species is sometimes substituted at the close, as a means of establishing more directly a particular key. This is partially explained by E::3. .^02 and 513. The student should write such exercises, using augmented 6th chord, i for the minor, and 3 tor the major, cadence. Examples of this will be included in the Key. 22O GOODRICH'S ANALYTICAL HARMONY. PART XI. Chapter XLVII. HARMONIC PROGRESSIONS IN GENERAL. THEIR ESTHETIC EFFECT. HERETOFORE chord succession has been considered principal! in its fundamental order and relation. Excepting in the t\v chapters relating to harmonic cadences the author has refrained fror presenting chord progression in its purely euphonious aspect. Th harmonic cadences will serve as a basis for this study, which, in it general features, is a rather superficial one. The relations of tonic and dominant are so intimate that the admit of unlimited alternate repetition, after the manner of an at thentic cadence. The following stretto illustrates this; Allegro. Ex. 519. -* r .* f <>. =g s*-g ** ^'53= -Jr -f- No other two harmonies have been used so frequently as these, not withstanding their affirmative character. The subdominant is next in order. On account of its connectioi with the tonic harmony the subdominant maintains the tonic impres sion more nearly than does any other chord : GOODRICH S ANALYTICAL HARMONY. Ex. 520. 221 Schiilz. The tonic remains in the base throughout this section and the changes to subdominant in measures 3 and 7 represent the least disturbance of tonic impression ; for the theme might have been accompanied exclusively with the tonic chord without materially altering the effect. In the cadence-forms these two chords were considered in their well- known capacity as constituting an after cadence. They are not to be so regarded here, for the last example is an initial section, not a final close. The subdominant harmony, preceding that of the tonic, is extremely mild and does not possess that positive quality that is char- acteristic of dominant and tonic. They are in fact almost opposite in their effects, for one is progressive, the other is non-progressive. Compare Exs. 520 and 521 for this purpose: Ex. 521. (TO * 1 . .._ 1 i i I SF *.*-*. +.+.*. +.+.+. R:> 5^5 S S S Z Z Z 0r *r ~ 8^1 r 1 I -r- ' T This is very simple pabulum from Czerny, but it serves the present purpose. Observe how much more decided are the changes of har- mony in the last example. By combining these three fundamental harmonies we have the characteristic effects noted, and at the same time the most pleasing and easily comprehended means of harmonization. And it must be said that they have been used innumerable times for mere ear-tingling purposes. Haydn employed them almost constantly, Mozart less so, 222 GOODRICH'S ANALYTICAL HARMONY. and the popular opera composers have worn them threadbare. I his mature works Beethoven made very little use of the perfe< cadence-form. Schumann considered it a symptom of philistinism Chopin was rich in harmonic invention ; Berlioz and Wagner d< spised all such devices, and at the present time these cadence-ha monies have fortunately passed into a state of inusitation. The chief value to the young composer is to furnish a simple and euplu nious basis upon which to erect a broader and more diversified ha: monization. Even in an elementary chapter acquaintance was mad with thirty chord progressions in a single major scale, and all thes are of practical utility. Since then hundreds of chord succession have been herein illustrated, and it only remains to note the rels tionship of certain harmonizations and observe the effect of differer orders of chord movement. A quotation from the allegro to Bee hoven's Adelaide is presented : Ex. 522. ft - (t -1 I + s -x - -f- f I*- fcFS=M: W ^F~ J=t The tonic chord occurs but once ; the subdominant twice (the laj time inverted) ; and the following chords once each : C-minor, F-u jor, D-minor, G-minor^ and finally the dominant yth harmony. Thi scheme introduces all the concords in B-flat-major (besides the esser tial yth chord), and these occur principally in a dominant relatior thus: B-flat to E-flat ; Ctof; and D to G. These form sequence; There is also an alternate third relation : E-flat to C ; F to D ; an G to E-flat. In the former instance there is one connecting tone ; i the latter there are two. This proves that a perfectly natural an euphonious series of harmonies may consist of something more tha tonic, subdominant and dominant, and without even a temporar transition. GOODRICH'S ANALYTICAL HARMONY. 223 The next quotation is of a different nature. It is the second theme in Schubert's Unfinished Symphony : Ex. 523. -*- Ef =: --r* 1 I N N I rfrJ* ->- -- S Onl}' a limited number of harmonies are here employed, for the grace and simplicity of the melody require very little elaboration. The syncopated middle parts, the chord figure in the base, and the modu- lations to and from A-minor, are to be noticed. Melodic passages that remain very nearly upon a monotone re- quire more harmonic elaboration than do such themes as the one last quoted. The very nature of the case makes this necessary, for if there is no melodic progression in the upper part, there must be in the harmony. In vocal music this is of frequent occurrence, espe- cially where the solo part is of a declamatory nature. Such an in- stance is the following from Mililotti : THE POOR MARINER. Ex. 524. Aub p__C_3 1 ^ P C W iT - ^ ffTr N i V * * S 41 * * * / ('old-ly the dav is end-ing, Niujht fmm the licav'ns descciid-ing; 1 y -' " ^ jjf_ t, | | F?V>^ i <^~ --* /^L " V + **: w3?~ ^ & i ' % "9" w "f Z '3Z. " "ft* * ~2^ .j , i j 9 m i^ s~ i^ -5. "*! rtM fj K -^ y&i 24 GOODRICH'S ANALYTICAL HARMONY. JV JS K^P^-^-""^ -V*--V+^0 - L - Hoars - !/ the bil-lows moaning, Sounds in the fate-ful gloaming; Here are seven different harmonies accompanying the g and b-flat oC the voice-part, which is almost non-melodious. The design is well conceived and worthy of careful study. Somewhat similar instances occur in the Farewell Song, by Schubert, and in the Palms, by Faure. These are so nearly alike that an epitome of the latter only is given : E.X. 525. 53 = ^^==^z^=3^= : ? * -^ f ~^~ b* - -*-=-- 3 etc. This scheme is easily explained. The 3d of a dominant yth chord descends by half steps to the root-tone, and the root ascends chro- matically to the 3d. Meanwhile the 5th and yth remain stationary, as parts of the intervening harmonies. Transpose and re-arrange this example, omitting the upper part, A somewhat similar instance is extracted from the opera Puritan i a EX. 526. R. S. Kclley. -**'r A * - ' ".'* -* f: tfrrw^ri ~f^!^rr-r P U U ' ' V V \ \ jt - I I S n 1 - : ; - -:* GOODRICH'S ANALYTICAL, HARMONY. 225 Owing to the monotonous character of the melody the harmonic parts are considerably varied. Observe the base progression from a down to b, and how the cadences are avoided until the close. In the first part of Lohengrin, after the king's solo, there are some progressions that have been classed as " inharmonious " by certain theorists. For instance, F to E-flat ; F to C-mi?ior ; G to F-minor, and so on. This proves what has frequently been stated in this vol- ume, that any chord progression is correct if employed effectively. Besides, have we not heard too much of subdominant and dominant ? The re-arrangement or inversion of certain harmonic and melodic designs may be included here, especially where such an arrangement serves to carry out the same idea : Ex. 5270. The motive at (a) is repeated at (b) in the base part, while the melody above is continued. This is not to be confused with the inversion of separate chords, as here : Ex. 527^. JECven these may prove useful in harmonization, though no harmo* ic progression takes place in the different measures considered st; -a- rately. Beethoven produced many novel effects by means of inver- sion, as with the 26. theme in the finale to his Op. 27, No. 2. 226 GOODRICH'S ANALYTICAL HARMONY. In the following quotation from Wagner a derived harmony as- sumes a fundamental character on account of the added base below . Ex. 528. The passing diminished chord above is made much more incisive by the fundamental progression in the base dominant to tonic. Since the most natural order of successive harmonies has fre- quently been reversed it would be useless to prescribe certain for- mulas. The present object is to call attention to various methods and means of harmonization, and point out their peculiar effects and applications. Fundamental progressions by seconds are, on account of their want of connection, inclined to be abrupt and rugged. Progressions by thirds (fundamentally) are most intimately con- nected and related, and produce an opposite effect to that of the disconnected progressions.* The ascending fourths are sufficiently connected, and have a dom- inant relation that represents something of incitement and onward progress. The progressions by fifths ascending (or fourths descending) are also connected by a tone in common ; but they have a retrogressive tendency. Chromatic movements have been described in a general way, but their changing colors are so kaleidoscopic that no one could hope to explain them in detail. Since the advent of Chopin and Schumann the tendency has been to indulge very freely in chromatic transition and elaboration. The postlude to one of Schumann's songs is quoted as a simple illustration : Ex.529. ~ "^ -This important relation by the sd is partially explained by the natural series of har- mcnic tones. But it is not necessary to discuss this here. GOODRICH S ANALYTICAL HARMONY. 227 *-.*- Almost every chromatic passing tone is here employed. The effect is appropriate -and charming. We have become so accustomed during recent years to almost incessant transition that a passage like the -following scarce disturbs the key-impression : J. Low, r .Op. 485. This begins and ends in C, but it includes a phrase in A-flat and one in F-minor. In connection with this subject students are recommended to study as many as possible of the following scores : Schumann, Ops. 21 and 26 ; Ball-Scenes, Op. 26, Nicode ; Moszkowski, Spanish Dances, Op. 12, especially the No. 3, from the rogth measure to the end of the movement ; Mackenzie, The Song of the Sickle, from Dream of Jubal, first part in A-minor ; Wagner, Evening Star, Romance; I. Low, Paul and Virginia, Op. 485 ; any of Chopin's piano works ; 228 GOODRICH'S ANALYTICAL HARMONY. Grieg's Overture, In Autumn ; Franz, The Dark Eye, Op. 9, No. 2. This exquisite song, also known as Request, is one of the best illus- trations of rich harmonization to be found even in modern music. It should be transposed a minor 2d lower, if that would more plainly show its peculiar and masterly harmonic structure. Either the original song, or Mr. Clarence Eddy's organ arrangement, should be consulted. The latter may be found in Vol. II of The Church and Concert Organist. Chapter XLVIII. FIGURED BASES. A CURSORY VIEW. WHEN the author of this volume first began to formulate a system of Harmony [in the year 1866] he discarded the then prevailing " Thorough Base " methods, because he was convinced that figured bases had served their purpose, and could not be made sufficiently precise or comprehensive to support a practical theory of composition. From that date unto the present he has totally ignored the " thorough base " formulas ; but so many books have been published on this plan, and so many old scores contain figured bases, that he has concluded to give a brief explanation of the sub- ject, independently of his own system. Thorough base represents a method of musical stenography, wherein a single base part, accompanied by figures, indicates the full harmony. To ascertain the corresponding note of any figure it is only necessary to count up from the base. The computation is al- ways made from the actual base, this being counted i. The intervals of a triad from the root are a third and fifth. Therefore the base note C, marked signifies that the chord of C is to be played. The inver- sions are numbered according to the actual distance of the intervals from any real-base. If e in the base is to represent the C chord, it is figured \ . 3 represents g and 6 represents c. Together these pro- duce the full chord, e, g, c, in the first inversion. In the second inversion g is the real-base. The remaining tones of the chord of C are situated a fourth and a sixth above. Therefore the g would be GOODRICH'S ANALYTICAL HARMONY. 229 figured I to indicate the second inversion of the C chord. The first inversion is usually figured 6, the 3 being omitted. When no figures are included the base is a root. Here are a few simple illustrations : Ex. 531. f -& (a) and (b) show the manner in which the harmony would be added to the base part in three and in four parts. At (c) the shows that the C chord is sustained above, while the base passes through the various notes of the same chord. The figuring of the dominant yth chord and its inversions is as follows : Ex. 532. 7 signifies that a chord of the yth is required, the base being the root. | indicates the first inversion 3 being presupposed; | shows the second inversion 6 being omitted, or presupposed. The last inver- sion calls for * (g, b, d\ but this has been abbreviated as shown in the last example. Any position may be written above, provided the chord is generally correct. For instance, the chord of the third and fourth may be taken in any of the following arrangements : Ex. 533 The figures \ simply indicate the second inversion of a yth chord, and so all these arrangements are correct. This is one of the prin- cipal objections to the system ; for a performer can not tell with any degree of certitude what particular design the composer had in mind when he wrote the base part. 230 GOODRICH'S ANALYTICAL HARMONY. All 7th chords, whether principal or secondary, are figured in the same manner. Transitions are indicated by placing before a certain figure whatever chromatic sign may be necessary for the correspond- ing note above. Thus the inversions of the principal diminished yth chord in a minor key would be figured : Ex. 534. m Aside from the sharp the figuring is the same as that employed for a dominant or secondary yth chord. As an epitome of what has been thus far explained a short exercise in figured bases is presented to be Avorked out by the student : Ex. 535. - IS J36 4 266 6 7- 53 44- The first chord is to have its fifth uppermost. After that it is only necessary to follow the general principles of chord progression and the rules of resolution. The second base note, without figures is a root ; therefore the E-minor triad is intended. The formula, 4-7, is frequently employed in final cadences. No further explanation is necessary. * * * According to the principle of figuring just explained various har- monic combinations are indicated from the base part. In order to carry out the system so as to meet all requirements the figures become almost innumerable and frequently confusing. The author admits that considerable practice of this kind is beneficial to the student. But the fact remains that information thus acquired is both mechan- ical and superficial. One never knows why he makes a certain pro- gression according to the " thorough base " formulas, and all the work becomes artificial and uninspiring. It is, in fact, beginning at the wrong end, for all music is founded upon melody, and the harmony is its accompaniment.* Composers no longer employ this lazy-man's expediency, and it certainly ought to be classed among things that are obsolete. Herewith is presented a solution in piano score of the " thorough base " Ex. 535 for comparison : "Even in purely harmonic combinations there must be melodic progression in some ot the voice-parts. The author has demonstrated this in a previous work. GOODRICH'S ANALYTICAL HARMONY. 231 Ex. 536. Considerable of this is, of course, optional. For instance, the third measure might have been arranged in this manner : Ex. 537. As the object was to give merely the principal features of this old method, further examples need not be presented. Viadana and Cata- lano first employed the figured base method about 1598. It was called basso continue. Chapter XLIX. THE NATURAL AND MELODIC MINOR SCALES. THEIR HARMONIES. THE NATURAL MINOR. HISTORICAL research has discovered a great variety of scales that have been used at different times during the past fifteen centuries. Nearly all of these (including the ecclesiastical modes) have been discarded, with exception of the Ionian, beginning on C. This is now called the Normal Major Scale. Modern composers have undoubtedly considered the minor as a derived scale, for it has since the time of Bach been written in five different ways. Most of these are variations of the old ^olian mode. This primitive scale is de- 232 GOODRICH'S ANALYTICAL HARMONY. serving of more attention than it has received, and the author believes it will be employed more freely in the future. It consists of a natural series of seven notes, and is the same ascending and descending : EX. 538- t*===z: 1 a 3 4 5 6 7 6 5 4 3 The two small steps are here more equally distributed than they are in the melodic minor. The group i, 2, 3 has its counterpart in 4, 5, 6 ; the major element being represented by 3, 4, 5, with its corre- sponding sequence, 6, 7, 8. Likewise, a counterpart of 2, 3, 4 is found in 5,6,7: Ex. 5 39.fcjfc= Two corresponding tetrachords may also be formed by considering 4 as the ending of one and the beginning of another : Ex. 540. I These characteristic scale features are usually overlooked, but they were considered of vital importance by our musical forefathers. The harmonization of this scale forms a prominent part in its consideration. Those who hear an old melody clothed in the garb of modern harmony have no better notion of the original effect than can be obtained from listening to one of Scarlatti's harpsichord sona- tas performed in the maudlin, tempo rubato style of the present day ! But we are so accustomed to certain harmonic progressions that what- ever is at variance with the prescribed order is by many thought to be incorrect. In fact, the author has been gravely informed by a con- vention of music-teachers that such progressions as these are wrong J Ex. 541. It is no fault of these progressions that they sound bad to certain persons. GOODRICH V S ANALYTICAL HARMONY. 233 Though the natural minor scale contains a minor yth from the lower tonic, there is no good reason why the dominant major chord may not be used in the lower half of the scale, thus : Ex. 542. In the upper tetrachord the natural subtonic is used ; in the lower the modern leading-note is included. However, instances are not wanting in which the subtonic is substituted for the leading-note at the close. This is according to the natural character of the scale, whereas the regular leading-note is a foreign element. Reference is made to the final, as well as to the intermediate cadence, in which the dominant minor chord is employed. Such an instance is given : Ex. 543. - s- -7V -* :g r~ id H 15: ^ i This ambiguous cadence is mild and plaintive, and ought, therefore, to serve a very distinctive purpose, especially where decided char- acter is not required. A similar instance from Bizet's La Bohemienne is quoted : Ex. 544. This is the initial period, and occurs twice with the subtonic accom- panied by the E-minor chord. Rubinstein, Rimsky-Korsakow, Tschai- kowski, Saint-Saens, Dvorak and Grieg have produced some of their most characteristic effects by means of these quaint harmonizations. The Danse Macabre is a notable instance, for no leading-note appears during the greater part of the work. 234 GOODRICH'S ANALYTIC AT. HARMONY. It should be added to what has been said about Exs. 470, 543 and 544, that the fundamentals remain the same, thus preserving the dominant relation in the cadence. On this account they are more satisfactory than the following close : Ex. 545- This is a mere progression ; as a cadence, the relative major sounds incongruous. (See Ambiguous Cadence, Chapter XLJI.) THE MELODIC MINOR. This is a modern scale, and one that is little understood. It is derived from the harmonic form, which contains a minor 6th and a major yth. In order to obviate the effect of an augmented 2d from 6 to 7, composers frequently raise the 6th, thus making the last of the ascending scale exactly like that of the major : 546. m As a characteristic series of sounds this is inferior to the other forms, for it is too much like the scale of A-major, and too little like that of A-minor. The melodic form is chiefly valuable in a rapid ascending ca- dence. The following extract from Mozart shows its principal ad- vantage ; Ex. 547. The fact that Mozart used this form is a sufficient reason for its exist' ence, but the manner in which he employed it must be borne in GOODRICH'S ANALYTICAL HARMONY. 235 The sharpened 6th appears as a mere passing-note, and does not receive separate harmonic treatment. This has led to the erro- neous opinion (asserted as a fact in certain thorough-base books) that this scale can not be harmonized ! Considering the fact that even the chromatic scale can be harmonized in various ways it is useless to occupy much space in controverting such queer notions. It would, indeed, be unfortunate if no suitable harmonies could be found for a scale so frequently employed as is the melodic minor. A few examples are here given : Ex. 548. h I y^r^i u g-|>-r g T'fe/g&g i ^ iHu. *> gi^iKo &g T ,- t ifT LU -_*. i'| rl I Observe that the melodic cadence is the same in each phrase. These are correct and serviceable. In the descending form of the melodic minor scale the flattened 7th (subtonic) is sometimes used as a passing-note to the minor 6th : Ex. 549- This is done for a melodic purpose. The minor jth may also appear as an appoggiatura to the note below : -^ J-vJ -e?-*-^\ =T r I Or the minor yth may form part of a transition chord in a transient modulation to the subdominant : * See also Kuhlau Sonatina, Op. 55, No. 3 ; last movement, measures 21, 22, 23, 24, of the second theme. GOODRICH'S ANALYTICAL HARMONY. Ex.551. In the second measure the minor 3d is restored, and in the cadence the leading-note appears ; so the example, as a whole, gives a very fair representation of the descending scale harmonized. The concluding example illustrates the ascending and descending forms of this scale, following each other immediately : Ex. 552. The sharpened 6th is used, because it passes more naturally to the leading-note. In descending, the e-flat is restored, because it is part of the subdominant harmony, and f-natural leads to e-flat more me- lodically than would f-sharp. The scale is, therefore, an expediency, both ascending and descending, and aside from these conditions (as they appear in the last example) it is inferior to the harmonic form. If a composer chooses to employ the melodic form in certain situa- tions, that is a matter of esthetics suggested by the nature of his melody. But this does not justify the assertion that the harmonic form is non-melodious.* * If any further testimony be required, let the spirit of Mozart answer through his Requi- em, or the ad G-minor Symphony. GOODRICH S ANALYTICAL HARMONY. 237 Chapter L. PRINCIPAL AND SECONDARY NINTH CHORDS. PRINCIPAL NINTH CHORDS most important five-toned chords are the dominant gth in major and the dominant 9th in minor. They consist of a major or a minor 3d added to a dominant yth chord, thus Major 9th. Minor 9th. The root, 3d, 5th and yth are identical. The large gth is perfectly natural to the scale of G-major, and the small gth is equally natural to that of G-minor. All these combinations are double dissonances, though they do not require two resolutions when they are dominant 9th chords. On account of its dissonant nature the gth chord should be prepared. Herewith are examples showing the preparation and resolution of a major gth chord : Ex. 554. y -&- p^g fg 1 h^ ; r iE e> i^ At (a) the 5th is omitted to prevent parallel fifths. Besides, the 3d is more essential than the 5th. At (b) the 5th and gth are contained in the antecedent yth chord, and, therefore, retained. Otherwise it might not be well to omit the yth. The gth chord at (c) is the result of a melodic progression in. the soprano and contralto parts. As a general rule the 9th sounds best at the top. The reason for this is that the combination under notice contaias a double dissonance, 238 GOODRICH'S ANALYTICAL HARMONY. Ex. 555. and these tones sound incongruous and con- fused it they come too near together. The tendency of the gth is to resolve down a 2d to the 5th of the concord (or to the root of the dominant 7th chord), and the student is not advised to seek an exception to this. The remaining notes are treated as though the 9th did not appear, for this combination is merely an essential yth chord,with a major 3d superimposed upon it. (See Ex. 874.) In five-part harmony the full Qth chord may be used, but this is not essential. In four-part harmony omit the 5th, yth, or 3d. In a minor scale the gth added to the dominant 7th chord will be minor. It is slightly harsher than the major gth, but the preparation and resolution are governed by the same principles. The three illus- trations in Ex. 554 may therefore be transferred to the opposite mode: Ex. 556. m^ 3: These are correct and effective. As they do not naturally admit of re-arrangement, it will be suffi- cient to transpose them into several other scales. * * * Ninth chords are rather difficult of management when inverted, because the ear does not readily recognize their tonal structure when the root is displaced.* The following arrangements have, howevei, been employed : Ex. 557- "Schumann, in his B flat Symphony, used a major 9th chord in its third inversion. See measures 72 and 73, first allegro. GOODRICH'S ANALYTICAL HARMONY. 239 The 5th of the 9th chord might have been omitted from examples (a) and (c), but at (b) the five parts are indispensable. SECONDARY NINTH CHORDS. These do not perform any direct act of modulation or resolution, but merely form parts of a chord progression, thus: Ex. 558. The first two are secondary, the last two are principal gib. chords, though they are here all treated as preparatory discords. In the Wedding March (2d period) Mendelssohn used a secondary 9th chord on the subdominant : Ex. 559. etc. g^=1=f^: -i *-^.-3: ~at~ This is a very harsh combination, but it is here duly prepared, and considering that it occurs at the end of a period the effect is highly satisfactory. A few quotations are included as additional illustrations. The first two are from Beethoven : Allegretto. Op. 31, No. 1. Ex. 560. r=v? * * * { * * * f f ^ The gth disappears in the essential harmony, as usual. In the next example the gth chord is more fully represented and of longer dura- tion. Its disappearance into the dominant yth harmony is effected in the last measure by the simple omission of the gth : GOODRICH'S ANALYTICAL HARMONY. i 5 3 The dissonant combination here appears uninverted, though the 3d^ 5th, 7th and gth exchange places above. In the excerpt from Nicode the major gth appears as a funda- mental harmony at (b) and at (c) : Nicodf. Op. 26. Ex. 562. - * X -Jf- An unusual example is here quoted from an American song. A principal gth chord is resolved indirectly, constituting a species of avoided cadence: FCERSTEK. Op. 30, NO. 1. Ex. 563. Without the gth this would be equivalent to the 3d resolution of a dominant 7th on ^^a/. A' fuller effect is obtained by including the major gth, and this remains as a connecting note to the F-minor chord. The last example illustrates the employment of an altered gth chord of the secondary species : GOODRICH 'S ANALYTICAL HARMONY. BEETHOVEN: Adagio. Op. 31, No. 2. Ex. 564. It should be remarked that this gth chord is not an independent har- mony, but results from the suspension of b and f. These two notes resolve as they would in the diminished chord. By performing this in several other scales it will be sufficiently understood. GOODKICH S ANALYTICAL HARMON^, PART XII. Chapter LI. SUSPENSION. THE THEORY ILLUSTRATED. " This arises through th delaying of a progression of a voice, which is expected at a defi- nite time, or even necessary, and in such a manner that the voice, which has to progress one degree downwards, in order to occupy its position in the following chord, lingers still upon the tone of the first chord, while the others progress to the second, and this voice does not pass over into the harmony until later." E. F. Richter. THIS is quoted to show the ordinary explanation of an important and interesting subject. Suspension refers more particularly to some part of a chord that is held back while the other parts move to another harmony. The suspended tone thus forms a dissonance, as it does not ordinarily belong to the second harmony. The resolution of a suspended tone is the same as it would have been had no suspension taken place. SUSPENSION IN TWO-PART HARMONY. The following two-part cadence will illustrate this in the simples* manner : Ex. 565. These parts suggest the tonic and dominant harmonies. Suppose when the contralto moves to the dominant chord at (b), that the soprano note, g. is held back until after the change in harmony takes place : GOODRICH'S ANALYTICAL HARMONY. 243 Ex. 566. The principles of resolution teach that a yth naturally resolves to a 6th, and that the inversion of this is treated in the same manner; i. ;: Write the suspensions in the upper part, beginning and ending in G^ Then add a middle part between the two parts already given. \Yhen the three upper parts are completed the base should be added. Owing to the descending movement of the treble parts this will require some care, if not ingenuity. A fundamental progression in the base will be necessary, because the form of the other parts does not invite, or even admit the employment of real-bases. After the essential jth has been introduced it will be necessary to avoid the cadence. The suspension does not'palliate such progressions as these, which are never allowable : Ex. 576. But the f-sharp, a, d, above may be treated as parts of a secondary 7th chord, and thus produce an indirect resolution corresponding in general design to the previous avoided cadence. Or the B-minor chord may be used. No modulation or decided cadence is required in this intermediate passage. The complete cadence naturally follows. The last of this exercise presents a situation so different from previous examples that a few words of explanation seem necessary. If the last discord includes the 3d in the middle part, and the rule of resolution be followed, the result will be this inharmonious syn- chysis : *Tbe complete harmonization may be made first, and the suspensions added afterwards. 24*3 GOODRICH'S ANALYTICAL HARMON\. Ex. 577. The second resolved to the unison is very rarely permissible, never in such an instance as this. Better keep the /#, as well as the a, in abeyance, though this is not always effective : Ex. 578. The simplest, and, all things considered, the best method is, to move the leading-note down to the dominant, thus : Ex. 579. This leaves the last concord in a complete form, and does not destroy the effect of the suspension above by anticipating its resolution. The movement of the leading-note down a 3d is a progression, not a reso- lution ; but as the a above is kept in abeyance after this, (a) to (b) is regarded as a partial resolution, completed at (c). Though this is a perfectly justifiable expediency, the fact remains that the descending movement of a leading-note in a final cadence lacks decision and completeness. Compare (a) with (b) : Ex. 580. Therefore some good reason should appear as a justification of the progression at (a), at least in terminal passages. The complete example is given : GOODRICH'S ANALYTICAL HARMONY. fc=d=p 247 The measures lettered (d) and (e) admit of different arrangement ; but if a secondary yth chord be substituted for the B-minor triad, the ,y#, a, coming after the E-minor chord, will strongly suggest d$ in the melody. At (e) C-major may be substituted for the inverted A-minor chord. Both methods should be used as a practice, and the different effects particularly noted. It is to be observed of all these suspensions that the upper part becomes a dissonance soon as the -harmony changes to a chord of which the delayed tone forms no part, or, to which it does not natu- rally belong. This concludes the synthetical exercises. The next example consists of a few simple chord progressions. These are to be arranged with suspensions in the. upper part : Ex. 582. All the suspended notes here are to resolve down a whole or a half step according to the model, and to the signature. In every instance the delayed note is to be a dissonance. * The note that follows a suspension, when it is 3d or 5th of a chord, is to be omitted from the other parts. An instance of this was given in the 2d resolving to the unison. Even such arrangements as these are not good, and should be avoided : Ex. 583. 248 GOODRICH'S ANALYTICAL HARMONY. The effect is unnecessarily discordant and confusing. The next is much better : Ex. 584. These suspensions have their due effect, because the resolutions are not interfered with. This does not apply to root-tones, which are nearly always doubled. And even such instances as occur in Ex. 581 are allowable, because the base is removed v'from the reso- lution a distance of two octaves. The 3d is used as a real-base to prevent a false progression, and on account of C being the sub- dominant. Supposing the last exercise to have been completed, the student may compare his work with this : Ex. 585. Inverted bases are not here employed after the suspensions begin, and this plan should be followed as often as possible until the sub- ject is mastered. Transpose the theme of Ex. 582 into D-flat and E-flat t and arrange in the same manner. A^ ^ ^ } r>}' ' J ' f ' 1 ' b^ I ' 1 \M7 ~&P x y ^ ^^. (5* -(S* 5) -d ^ 1 tr ST * f\* i > Jj GOODRICH'S ANALYTICAL HARMONY. 249 Chapter LII. SUSPENSION CONTINUED. SUSPENSION IN THE MIDDLE PARTS. THK principles explained in the last chapter may be applied to any part. AN EXAMPLE OF SUSPENSION IN THE TWO MIDDLE PARTS. Ex. 586. The same directions are observed. Thus, while g is resolving to f-sharp the latter note is omitted from the D chord. When the note above the root of a chord is held back, the root may appear in the base as fundamental. SUSPENSION IN THE BASE. This is not generally effective unless the base is a solo or obligate part. A few illustrations are appended for examination : Ex. 587. 3^ The tonic is first suspended and then resolved down to the 3d of the essential discord, that note being omitted from the upper parts. The 3d of the tonic is next delayed in its progression, and resolves to e after the remaining tones of the essential discord have been sounded above. The theme being in the base, that part assumes more freedom and independence. Re-arrange Ex. 587 in this position, GOODRICH 'S ANALYTICAL HARMONY. Ex. 588. with the same base, and transpose both arrangements into C and B-jlat. All the chord progressions must be as correct as though the sus- pensions did not appear. To test this, simply omit the suspended note : Ex. 589. a b C~\' - i 1 (5 & a 1 I? 5 gy I S r r 1 i (a) is the same as (b), except that e in the base is held back until after the harmony has changed. The e then descends to d, as in the second example. UPWARD RESOLUTION. In the exercises thus far the suspension has resolved down a 2d. This is the usual tendency, with exception of the leading-tone. In all direct cadences this tone is resolved to the tonic, and the fact that the leading-tone is suspended does not alter its natural progression. The 5th of a dominant chord may also resolve to the tone above, and so may any part of a concord. These points are illustrated in the next example : Ex. 590. J-J- j" The ascending resolutions are indicated by crosses. The f-s/iarp ascends because that is its natural tendency ; but in the other in- stances the delayed note ascends in order to complete the harmony. Observe the places where d ascends to e, and where c descends to b* Transpose the last example to F. Ordinary notes of connection are not here considered suspensions. GOODRICH'S ANALYTICAL HARMUXY. DOUBLE SUSPENSIONS. 251 These should usually be either a 3d or a 6th (major or minor) apart, resolving in parallel movement ; or an augmented 4th (and its inversion, an imperfect 5th) resolving in contrary movement. A suspended 3d, and a suspended 6th, each resolving down, are given : Ex. 591. The intervals mentioned possess a certain harmonizing relationship, and naturally go in pairs. Therefore, when one tone of the interval of a 3d or 6th is suspended, the other may also be delayed in its pro- gression so as to accompany its reciprocal tone. The double suspen- sion is sometimes more, and at other times less dissonant than the single suspension. In the latter the melody is more isolated and independent than where the delayed tone is accompanied with another delayed tone in some other part. When the leading-note is suspended it is proper to delay the resolution of the subdominant also, as the simultaneous disappearance of these two elements of transition is very natural and agreeable : Ex. 592. Example (a) is preferable to (b), though the latter is correct. Unless the base is a solo part it is better to give it a regular fundamental position as in Ex. 591. Re-arrange and transpose examples 591 and 592 until they are thoroughly understood. SUSPENSION RESOLVING TO A CHANGING HARMONY. The harmony may change at the same time the suspension re- solves. In such instances the note to which the suspension resolves must belong to both chords, and the progression is, of course, to be *The tied notes may be repeated in order to s^ow the full effect. 252 GOODRICH'S ANALYTICAL HARMONY. correctly managed. Suppose b is suspended over a. When the reso- lution takes place the harmony may change, provided both chords contain a : Ex. 593. The resolution (b to a) occurs in the full measure, and the harmony changes from a dominant to a diminished yth chord. The a belongs to both harmonies. Between these two chords there are three con- necting notes. In the next example there are two notes in common, e and g : B* Ex. 594. Such instances are always proper, provided care be exercised in maintaining the connecting note in the same voice-part. The resolution of the suspended a is the same as though the E-minor triad had remained and not been succeeded by the C chord. A short theme here follows as an exercise in suspension : Ex. 595. ^r i '-^ f ^ *? j^ . ..^ . 1 rm I 1 j P' {\\j 1 1 t r ! 1 + fc 1 f? KS ^-W- ~\ This begins and ends in C, with a temporary modulation to the rela- tive minor. Change the harmony during the measures marked + . A double suspension may be included in the cadence. Transpose to A and B-flat. * * * Another short theme is given for harmonization. The student is merely to supply the middle parts. The bases are all fundamentals except in the sixth measure, where the 5th is below : GOODKICll'S ANALYTICAL HARMONY. 253 Ex. 596. The only difficulty anticipated is in the proper treatment of the ca- dence. It should be written in one of the following ways Ex. 597. -- g I * "* *" V i ^~ ^~ V *" * -- ^ -* -*- ^ -9- -9- f- Y*r ir * *W P* u * ^ ? 'f * ._? rTu <2? 1 1 ^ Ex. 619. The concord with its 5th in the base may be obtained by giving the root to the base, and causing that part to ascend a 4th or descend a 5th, as at (a) or (f ) in the previous example. The essential discord to the tonic naturally follows, as in the last illustration. The 6th resolution is similar to this, but in the minor mode thus: Ex. 620. The 5th of the discord is omitted in the second measure, but this is of no particular consequence. The same directions apply to this, as 5 and 6 correspond. Beethoven used this much more freely in the development of O>> 7, just before the reprise : Ex. 621. The effect here is that of a harmonic digression. The chord marked peonies so unexpectedly that it forces the attention away from the tonality of A-minor in a most emphatic manner. This leads to D-mi* nor, as might be presumed from the base. No. 7 resolves to a minor triad whose root is the 5th of the dis- cord. This note must not, therefore, be omitted. There are two connecting tones, 5th and 7th of the discord, while the root and 3d both resolve to the 5th of the concord. The following positions are available : GOODRICH'S ANALYTICAL HARMONY. i**?- 265 Ex. 622. These are all resolved in the same manner, because no other alter- native presents itself. But in the progression following the yth reso- lution considerable liberty may be allowed! Advantage can be taken of the I chord in order to pass to D-minor by introducing the domi- nant ; or consider the D-minor triad as a mere passing chord in pro- gression. Examples of these two methods : Ex. 623. Others are possible, but the author gives the most feasible, and leaves the ingenious young composer to the pleasures of discovery. The 8th resolution is likewise to a minor triad. The root and 3d remain ; the 5th and yth resolve up and down to the root of the triad : Ex. 624. m This is a mere progression. Its most peculiar feature is that it admits of inversion more readily than do the others. In truth, the uninverted form is not very serviceable. No. 9 is more transitional. So far as the author has been able to discover, it was first employed by Schubert in his great C-major Sym- phony. The root of the discord remains as 3d of the major concord ; the 3d descends a chromatic step ; the 5th ascends a minor 2d, and the yth descends a major 2d, to the root of the concord : Ex. 625. 266 GOODRICH'S ANALYTICAL HARMONY. It is the most abrupt of all, excepting, perhaps, No. 10. In the ex- ample from Schubert the 5th of the discord is real-base, and resolves directly to the root of the triad : Ex. 626. TTI This is much more effective than the uninverted form. Observe that the root, 3d and yth of the discord are duplicated above by the differ- ent instrumental parts, but that the 5th is given only to the bases. The immediate effect is bright, and rather bold, the concord on E-JJat being altogether unexpected. This extract is taken from the 2d sub- ject of the first allegro. No. 10 is also novel, but perfectly feasible : Ex. 627. 10. It is better to omit the 3d, as it has no natural resolution here. following positions are perhaps best adapted to practical uses : The Ex. 628. The immediate resolution is remote, and might, for this reason, be utilized in making a distant transition, as in the last example (c). This, however, is but one of the many ways in which the harmony might be directed after the loth resolution has taken place. The nth and last of the consonant resolutions of a dominant yth chord is to the major triad located a whole step above the root of the former : ;OODRICH'S ANALYTICAL HARMONY. 207 This is not a very natunl termination of the discord, but in certain progressions it might be utilizedj as here : z=z * * - Ex. 630. The end justifies the means. It is well to know, however, that this resolution is abrupt, even in the midst of the chromatic progression, and should not be used ordinarily. Nearly all these examples can be reversed, as were numbers 623 and 624. (Another resolution is possible.) Some of the preceding resolutions the author has given on his own responsibility. Less than one half have appeared in the text-books, but most of the others have been employed by living composers. In adding to those already known the main object has been to introduce a greater variety of harmonic progressions, for we are naturally disposed to employ only those sanctioned by frequent usage. But the cadence forms have been used as the harmonic basis of so many thousands of songs and instrumental pieces that they no longer interest a cultured listener, and almost any reason- able progression is a relief from their satiating effect. The substance of this chapter is to be worked out by the student in at least four other scales. It would also be well to make a diagram of the entire eleven resolutions in some key different from that of the printed examples. 268 GOODRICH S ANALYTICAL HARMONY. PART XIII. Chapter LV. DUPLICATION AND OMISSION. DUPLICATION. A LIMITED knowledge of these subjects has been acquired from A Chapters XXII, XXVII, XXIX and L,. It is well known that the root of a chord admits the greatest amount of duplication, and as that tone is the foundation of the chord this requires no farther ex- planation than the following : Brahms. Dvov&k. Ex. 631. ^B -ig ' & J ->*- )*- This principle applies to all fundamental harmonies. The major 3d is of a decided character, and its appearance in the last example shows that it will not admit the same amount of dupli- cation as does the root. In harmonic masses the 3d may appear in each group : GOODRICH S ANALYTICAL HARMONY. 269 In an orchestral score this would be perfectly satisfactory, for there are five r's against three ez: ^a ** ' U ^i n* ^, * In the last measure the 5th is doubled, even to the exclusion of the 3d ; but the progression of ascending thirds against the descending base justifies this procedure. Of the same nature is the following more artistic design quoted from Beethoven : Ex. 643. FINALE. Op. 2, No. 3. 272 GOODRICH S ANALYTICAL II AKMON Y. The fifths and thirds are here duplicated with the greatest of free- dom, though this is the result of expediency. The inversions of a dominant yth chord should not, however, be doubled, unless some particular melodic design justifies the dupli- cation. The next example, in any of its re-arrangements, may be taken as a safe model . Ex. 644. (1) The jth may be doubled in the base, but not in the upper parts. Such unison passages as these are perfectly proper : Ex. 645. Z= P = 1 (3) (1) Particular attention is directed to the third inversion of the essential discords in the first and third measures. These directions do not apply to diminished yth chords except where they are treated as principal discords. When they appear in a secondary capacity as passing chords, or in chromatic progressions, they are not subject to the same restrictions, for any note of a dimi- nished yth chord may be considered root, 3d, 5th or 7th, according to its notation. Correct progression of the parts should determine whether a certain note may or may not be duplicated ; for neither the root, 3d, 5th nor 7th possess that distinctive tonal character which is recognized in the dominant 7th chord. The same principle applies in a modified sense to the secondary 7th chords. Such instances as the following are of frequent occur- rence, though care must be exercised to avoid false progressions : GOODRICH'S ANALYTICAL HARMONY. 273 JL Ex. 646. Transpose and re-arrange all the examples. * * * OMISSION. The root of a concord can not be omitted without destroying its identity EX. 647. F j^ 3 The latter is simply a binary mj _~^i~ ^ + chord on -- E. Xfrin. Op. 18, No. 5. - . ^ fc i The same remarks apply to this. Sometimes the embellishment includes three or four notes foreign to the accompanying harmony, and at other times a passing note may be included. Both styles are here presented : GOODRICH S ANALYTICAL HARMONY. 6 Ex. 689. The group of unrelated notes at (a) is in the nature of a parenthesis. The embellishment at (b) includes a suspension, harmonic, and pass- ing note. Groups (a) and (c) may be considered as the dominant harmony suspended over that of the tonic. These adventitious notes may appear in a variety of forms, but the four examples will give the student a general knowledge of all others. In transposing the exercises write the melody part first with the necessary ciphers and figures above, and then attempt the harmoni- zation. There is but little to be gained from the mere act of trans- posing an entire example. The general treatment of the embellishment is the same, whether it begins with a suspension or an appoggiatura. The latter is neces- sarily more dissonant ; or, rather, the dissonance is unprepared, and consequently more noticeable. FARTHER APPLICATION. The artistic employment of these unrelated tones has, to a con- siderable extent, been illustrated. In addition to what has been shown of the general character of passing tones it may be remarked that an>- number of them is possible, provided they can be executed between the regular accents of a measure. Thus, the former design, Ex. 690. may be written in i6th notes without interfering with the melodic outline or the harmonic substance : In both instances the notes that fall upon the accented parts of the measure are c, e, g ; therefore the harmony of C will accompany GOODRICIl'S ANALYTICAL HARMONY. the melody in both examples. According to a strict designation all the c's and ^'s were marked with ciphers, as they belong to the har- mony ; but in actual practice all such notes occurring upon unac- cented parts of a measure are passing notes, and will be considered in this light hereafter. Chromatics may also occur as notes of passage, two or three being included between the harmonic notes. A brief exercise is given for harmonization : o j _i o 1 i ft_iit 011 o Observe that the second and third notes of each triplet are passing notes. The ciphers sufficiently indicate the harmony. * * * Chromatics may continue through a greater compass and be har- monized collectively, thus : Ex. 693. Or, they can be harmonized with the tonic chord throughout. In this instance all the chromatics would be considered passing notes with exception of the first in each measure. The harmony below is a sufficient support for the rapidly passing chromatics above. A theme containing appoggiaturas, harmonic and passing notes is here presented to be harmonized according to the indications : Ex. 694. n i 0111 kl t> J>1?S- f-^f- bi* Li U / 7 3 u. ^ Jjj * ' 1 ra*/ T i e^^L H~Z _- ==- : t t -. ii~ ' t A , } 7 PH* )',.? r i -f : , t ^- The notes unmarked are harmonic. No modulations are suggested by this theme. * * * GOODRICH S ANALYTICAL HARMONY. 289 The following exercise contains the suspension, anticipation, embellishment, and stationary note. It is to be harmonized accord- ingly : 3 Ex. 695. The first of this presents no difficulties. During the last three meas- ures the harmony should make a temporary digression and then return to the key-note on the last measure. In other words, at the beginning and ending of the sustained note the chord is to be that of G-major. The same exercise should be worked out in Fand in E-flat. * * A considerable amount of material is now available in harmoni- zation, and the first point to determine is the character of the com- position. If a melodic figure like this Ex. 696. should present itself, there would be a choice of several arrangements : i . The notes that occur on the accented parts of the measure may be considered as harmonic, and the others as passing notes. 2. The accented notes can be treated as appoggiaturas, leaving g, c, e as harmonic notes. 3. Every melodic note may be considered as har- monic, and harmonized separately. 4. A series of passing harmo- nies may be employed on a pedal-note. The four illustrations are presented : 697 A V 1 U A P^ V I i pi 2000 ^ -^"""SB fi f TLT E3 = g = *-| :i i f Z t~ -^ '-' 1 L^ 1 i F -"-^ 1 F-F 5^ i^J ^f^^ 3=^ 290 GOODRICH'S AXALYTICAI. HARMONY. All these are proper, but each has its own peculiarity. The first is milder and more smooth than the others ; (b) is the strongest and most dissonant ; (c) is rather detached and fragmentary, and gives a more prominent character to each melodic tone ; the last arrange- ment represents the least possible disturbance of tonic impression. The nature of the melody, or the situation in which it occurs must, therefore, determine which will be most suitable. Under this heading may be mentioned the following exception.*! instance from the allegro of a Piano Quartet : Ex. 698. A prepared suspension is here followed by an anticipation, effect is somewhat similar to that of the embellishment Tne GOODkiCH is ANALYTICAL HARMONY. 29] PART XIV. Chapter LVIII. HARMONIC COUNTERPOINT. ELEMENTARY SPECIES. SO much is said and so little is understood of Counterpoint that the author has for several years past endeavored to formulate a system for the simplification of this difficult subject. The results are herein submitted. The definition of Counterpoint is note against note. In its strict sense this signifies two or more voice-parts moving independently of one another, each part having its individual melody. A chorus writ- ten in canonic or fugal style affords the best illustration of counter- point. It presupposes a thorough knowledge of all possible chord formations, of inversion, of the theory of harmonic progression, mod- ulation and transition, resolution, harmonization in all its phases, sus- pension, pedal-point, chord-relation, metre, rhythm, thematic devel- opment, etc. Hence the term counterpoint is frequently defined as the art of musical composition in general. But the word is here used in its specific sense, implying the union of several independent voice-parts as in a quartet, and as opposed to mere chord progression. The author introduces what he calls Harmonic Counterpoint as the natural solution of strict, melodic counterpoint. Begin with a simple design in which notes of the same value are employed. The usual form of writing the chords above, with base on a separate staff, will serve as a nucleus. The student : complete the following exercise : 29-: GOODRICH'S ANALYTICAL HARMON' Ex. 699. Piano form. *><***- y i i ^5 -p E^ r r 3 a _^0\ ^^E ^0 r H 1 i -f- i ' f_^i g ^ "^ ^J 1 j g ^ -^ I ~ ' -^ i 1 (2) (1) It will be a simple task to arrange this in vocal form. In order to distribute the parts more equally with regard to the intervals between them, select the middle part from each chord above (called in former lessons mezzo-soprano], and write this one octave lower in the base staff. The melody (soprano), lowest treble part (contralto), and base are to be copied identically. The result is as follows : Ex. 700. This appears in open position, and the manner in which- it was accom- plished is the very simplest method of producing dispersed harmony. NOTE. When the tenor sings from the treble staff, the notes sound one octave lower than written. Therefore that part is here represented exactly as it is sung. Some books mention the lowest treble part as tenor, but if that part is inverted the result will be a very unequal distribution of the parts : S J 1 _ JJ /5 _" -^*^ \J 1 ^^ ^^ '-^ "-* r z JF t j _ 1 M^^ f^ ^ ^ ** s> \ I/ 1 Quartet form. T -^ ^;J ^ _^_ ^" ^.' -gj j_ ~ P\ ^ ^ **> * ^ ~. ^-^ ^ \J t & tf 5 /T3 E -, _^-_, 2: 5. Jf L & r ff> g ^ 2 c ff^ i 1 * , * t ! 1 I 1 J i 1 1 1 1 I IV 1 ii k ). 9 f ^i .. /e ^ <, & y ^ * ** ^ I (2) (3) (1) Complete the simple harmonization in close position, and then ar- range it in quartet form. Copy the first, third and fourth parts (counting from the soprano downward), and write the tenor part in the base staff, one octave below its original position. No unrulable progressions will result from the inversion in this example. Not all designs admit this process of inversion, for parallel fourths in the piano arrangement become parallel fifths in the open position. In such instances contrary or oblique movement must be employed. This has been explained. * * * The last quartet arrangement is here presented for comparison : This is perfectly correct and singable. There is considerable me- lodic character ( to the various voice-parts, and a very agreeable interval is maintained between soprano and contralto, contralto and tenor, and tenor and base. Elementary species of harmonic counterpoint may contain any natural chord progressions, modulations, different kind of cadences, principal and secondary jth chords, skips, etc. Unrelated notes, organ-point, and chromatic progressions are to be reserved for the other species. A more transitional exercise is now presented for arrangement : GOODRICH'S .\XALYTICAL HARMONY. 295 Ex. 706. The inverted bases are an important feature here, as they give to the lowest part a melodic movement very desirable in the quartet form. In changing this from close to dispersed harmony the stu- dent is to understand that the tenor is the only part to be inverted. The tenor and the base may occasionally come together upon the unison, but the base must not ascend above the tenor.* In such instances write the base an octave lower, for it is not well in these exercises to alter the tenor part. * * * * The theme next introduced is to be arranged in close position, and then as a quartet in the manner explained. Every melodic note is to be treated as a harmonic note. The skips of a 3d and a 4th, and the temporary modulations are to be managed according to the usual methods : Ex. 707. -t i r m When there is no accompaniment it is usually better to so arrange the essential discord that the last tonic chord will appear in its en- tirety. The 3d or 5th of the discord must necessarily be omitted, unless the base is inverted. See Exs. 700 and 706. * * * The modulation indicated by f-sharp can be written in various ways : * Some charming effects have been produced by the crossing of voices, above and below. Witness the quartet for men's voices in Mendelssohn's 42d Psalm. But this belongs to strict counterpoint, andean not be considered here. 296 GOODRICH S ANALYTICAL HARMONY. In the first example the cadence is direct ; but this renders the A- minor chord necessary on account of the skip. At (b) an avoided cadence is effected. This makes the F chord necessary, because of the second skip, e down to c. The F chord is then changed to that of D-minor to avoid the awkward progression from sub-dominant to dominant, when all the parts ascend, thus : 111 Ex. 709. In the last arrangement (c) the 5th resolution of the dominant yth is used because all the notes of the fifth measure comprise the chord of C-major. All these arrangements are available, and should be used in the different transpositions. The same process is to be worked out in B-flat and in D-flat. Chapter LIX. HARMONIC COUNTERPOINT CONTINUED. COMPOUND SPECIES. TD Y including in the quartet various kinds of unrelated notes, thus -L' making the parts unequal in rhythmical value and different in .harmonic character, we produce compound, or mixed harmonic coun- terpoint. An avoided cadence is selected for preliminary illustration : 1*1:1110. Ex. 710. GOODRICH'S ANALYTICAL HARMONY. 297 Measures (a) and (b) are sufficiently correct as mere chord progres- sions ; but in a vocal quartet the effects are awkward and ragged. The fourths at (a) become fifths at (c) between contralto and tenor. At (dj the parallel fourths between soprano and tenor are almost as inharmonious as parallel fifths. Several methods for avoiding these inaccuracies in vocal part-music are represented. The first relates to the upward resolution of the 3d of the discord : Ex. 711. The fourths and fifths are here resolved contrarily, and the example thereby assumes more of a contrapuntal character ; for this is one of the governing principles in polyphonic writing. Not that the coun- terpoint shall consist of thirds and sixths, like a Glover duet, but that these intervals represent the maximum of euphonious harmony between the parts, and when so written they can not be wrong. Observe particularly the parts that ascend and those that descend. The awkward progressions in Ex. 710, (c) and (d), may be avoided also by suspending the resolution of the 3d and 5th, or of the 7th, thus : Ex. 712. Sc3 5- B^ 3 ? 1 gp g j 1 1 1-v^ a 1 a ~""~ ry-'ft ^ ri | 1 ZZffl xa 1 The augmented, followed by the normal 4th, is obviated at (a) ; par- allel fifths between contralto and tenor are avoided by the suspension of the 7th at (b). An important feature of counterpoint, and one that did not enter very prominently into previous lessons, is the separate, inde- pendent movement of the voice-parts. This is partially illustrated in Exs. 711 and 712. A farther view is presented by the follow- ing from Haendel: 298 GOODRICH S \N.\LYTICAL HARMONY. Ex. 713. The treble part is composed exclusively of harmonic chord figures in sequence order : A, D, G, C. Against these the base parts alter- nately ascend alphabetically, which necessarily results in dissonating intervals. These are marked 2, for in strict designation they are appoggiaturas. This is Counterpoint; therefore observe particularly the dissonances, and how these result. Perform and transpose the last example. * * * Another important secret of counterpoint is to be found in the theory of Suspension. That is, any dissonance is allowable that results from the detention of one part while another part moves away from (or toward) the stationary note : Two \ o i EX. 714- The second voice enters upon the unison at (a) and descends a minor 2d at (b) while the first voice sustains g. This results in an extremely dissonant interval. At (c) the second voice descends another half step, and the dissonance is reduced to a minimum. At (d) the lower part descends to e, and this results in a most agreeable consonance. The stationary note is its own justification, while the lower part moving away from the suspended note presents a sufficient object for the progressions described. A base may be added to this duet by writing a part in contrary movement to the under treble part : Ex. 715. f he upper progression points to the subdominant ; therefore the base melody leading up to that note is desirable, especially as its notes GOODRICH S ANALYTICAL HARMONY. 299 harmonize perfectly with those of the contralto. The student may form a quartet out of this design by adding another part above the stationary note. * * * The example, with its analysis, is presented : Ex. 716. 4^. j*. \t-ex- .&- The base and soprano parts move up at the interval of a loth; the tenor descends chromatically in harmony with the base ; the contralto remains stationary.* The student is advised (if this design is well understood) to write a stationary note on the tonic of the F scale, and, without consulting the printed examples, add the other parts synthetically, so that they will correspond to the last exercise. Like all contrapuntal designs, this admits of re-arrangement and inversion, which should now be done. The -suspended note may appear in the soprano, contralto, or tenor parts ; or the base may exchange parts with the soprano. # * * Before proceeding with the exercises the author will attempt to give the intervals, considered separately, that produce the most satis- factory results in two-part counterpoint. No distinction is here made between dissonances and consonances, as the object is to show what intervals may be employed simultaneously, one being a counterpoint to the other. The unison and octave come first. They perform important parts, as show r n by Ex. 716. Then the minor, major, and augmented 2d ; minor and major 3d ; imperfect and augmented 4th ; imperfect and augmented 5th ; minor, major and augmented 6th ; diminished, mi- nor, and major yth. (The 9th, loth, nth and i2th are here considered as inversions of the 2d, 3d, 4th and 5th.) Following are these inter- vals in notation : * A phrase of similar construction may be found on the first page of Beethoven's Op. 27, No. i. It is preceded by the reiterated chord of C-major. 3 oo GOODRICH'S ANALYTICAL HARMONY. Ex. 717- Some surprise may be occasioned by the fact that the normal 4th and normal 5th are excluded from this scheme, especially since the 5th was employed so prominently by the old contrapuntists. This inter- val used to be so highly esteemed that it was frequently included in the final chord to the exclusion of the minor 3d ! Even during the 1 8th century certain composers preferred resolving the leading note down a major 3d rather than omit the 5th from the final chord. How- ever, the 4th and the 5th are generally unsatisfactory when employed alone. Musicians of different epochs have entertained different notions regarding the character of intervals. Until the advent of Mozart it was customary to terminate minor compositions with a major chord, because the small 3d was not believed to be sufficiently euphonious for a final ending. But Mozart, whose delicacy of tone-perception was phenomenal, did not accept all the old canons and theories. Among the important improvements that he effected was the less prominent treatment of the 5th, and a freer use of the small 3d. The normal 4th is even less satisfactory than the 5th. The situations in which the 5th may be used were explained in Chapter LV ; and when those conditions exist the normal 4th may also be employed, one being an inversion of the other : Reinecke. Ex. 718. In the first measure the 5th is founded on the dominant, and th& key is sufficiently established to determine the fact that the major 3d (c-sharp) is here omitted. In the second measure all the intervals appear inverted. Where these peculiar conditions do not exist, the 4th and the 5th GOODRICH S ANALYTICAL HARMONY. 301 will generally sound unsatisfactory. The following instances are cited in proof of this : Ex. 719. -! 4- The counterpoint is good, with exception of the 4th at (a) and the v 5th at (b). No reputable composer would write such examples as these in a serious work, for they are ambiguous and unsatisfactory. To preserve the original theme the counterpoint should be like this : r - fl-i ,T - 1 - 1 i -- i - 1 -- 1 An imperfect 5th takes the place of the normal 4th, and every inter- val is satisfactory. To preserve the theme at (b) the counterpoint is arranged in this manner : Ex. 721. The substitute here for the normal 5th is an augmented 4th. These are decided improvements upon Ex. 719. In full harmony the 4th and 5th are perfectly proper, because other intervals are combined with them. Thus, g in the C chord is not alone a 5th from c, but a 3d from e : Ex. 722. ft And in the next example the blank in the upper parts is filled by the inverted base : Ex. 723. No ambiguity results in such instances. The principle of agreement between tw r o prominent voice-parts 302 GOODRICH'S ANALYTICAL HARMONY. (as illustrated in Exs. 720 and 721) is frequently employed in full har- mony. The following extract from a Barcarolle is a good example : Ex. 724. Tschaiko^^>sky. The real-bases \vere especially designed to harmonize with the mel- ody, those two parts being heard simultaneously on the first and last of each measure. Note particularly the euphonious effect of the intervals marked + . This is, of course, more noticeable when the melody and base are heard without other harmon}'. Hundreds of similar instances might be quoted. Observe, for instance, the relation between melody and base in Schubert's little ballad, Hedge Roses, a section of which is quoted : Ex. 725. The designs for harmonic counterpoint are now resumed. Sus- pensions and appoggiaturas contribute materially to the independence of the parts, and are, therefore, valuable adjuncts. F-^r instance : Ex. 726. GOODRICH'S ANALYTICAL HARMONY 303 By arranging this in open position a fair specimen of the quartet style is obtained : Ex. 727. Every musician perceives at a glance that this is correct. The first interval of a 7th becomes a 6th ; the second 7th becomes a 5th, and a in the tenor part connects the two chords. From this point the intervals between the middle parts are a 6th, 7th, and 6th : Ex. 728. m The main point to observe is the intervals that should not follow each other in similar movement, such as seconds, fifths, sevenths, and octaves. In the last example the 7th resolves to a 6th ; the 5th between contralto and soprano becomes an augmented 4th (/and 6), and this resolves to the final 6th. \Ve have here the principal movements : Oblique and contrary.* Attention is also directed to the different rhythmical denomina- tions of the notes in Ex. 727. The next exercise is to be harmonized in close position, and after- wards arranged in quartet form : 2020202030203 729. V p-fe The first chord is designed to be that of tonic major. C% in the sec- ond measure can be treated as an appoggiatura, or as a harmonic note. On the last half of the third measure the harmony should be changed. The manner of producing the quartet remains the same : invert the mezzo-soprano part an octave lower and it becomes tenor. * * * The next theme contains passing notes, inversions, modulation, suspension, and an anticipation at the close : "Students should analyze aii the voice-progrressioms carefully until such analyses unnecessary. GOODRICH'S ANALYTICAL HARMONY. 0000 0101 '*-,- - i , " .. -J- -, f * , Ex. 730. M i_ --r *frl ' * '> r T Cj T y l '1 li| -g * \ < =JE (i 1 H L) (2) (3) (2) ^^ TlA ^^ B* ^ * S3 _^_g_.g_ i Y i--V-^ ^_ ' V f ' ^ ' J ' 1 * * 1 ~ 4 1 1 -i E On the last of the second measure an 8th resolution of the dominant 7th chord is outlined ; or c in the base might, if the movement were quick, be considered a passing note. Complete the arrangement in piano form, and then change it into dispersed harmony. * * * As the final cadence is peculiar, the student may find these illus- trations useful ; or he may employ one of his own invention : Ex. 731. One more theme is included. In the cadence a changing harmony is to be introduced on the resolution of an appoggiatura : Ex. 732. . r r * f T '&-r-r+b- r ?-+-m - * . p ? ^ n Jf J J J ' *^ J J J ' " N 1 i 1 4 N m * I'.* _ 1 * It * 1 A* ? *P I-P iz^ - " jg- 1 ^i9 P I 4-p P ^ & 1 7 1 ! b 1 C|^ 1 -1 ^/ ~ ^ -*> * 1 I* Notes not marked are harmonic. Transpose to E-flat and G, and arrange in quartet form. * These notes may also be treated as appojjjfiaturas. t'i'he notes numbered 3 are treated as appoggiaturas. ANALYTICAL, HARMON v. Chapter LX, HARMONIC COUNTERPOINT CONCLUDED. ORNATE SPECIES. A STILL more independent and flowery species of harmonic coun- ** terpoint is here introduced. This is accomplished in various vays. The simplest method is to include natural or chromatic pass- ing notes in any of the voice-parts that move a major 2d or a dia- tonic 3d. PRELIMINARY EXAMPLE: Ex. 733. No intermediate passing note is possible in the soprano part, nor in the tenor. But between d and e of the contralto part a note of pass- age can be introduced in ascending, and likewise in descending. As these do not interfere with the other parts (the notes of the contralto not being duplicated) they may be freely introduced : Ex. 734. The chromatic alterations possible are too numerous to mention, but a sufficient n imber will be given. It is not well to sharpen the root above, unless the root below remains as a pedal note : -Buck. Ex. 735. 306 GOODRICH'S ANALYTICAL HARMONY. Diatonic passing notes may be included in the base whenever that part proceeds by thirds. Suppose this to be the original : Ex. 736. m Between the lower root-notes we may write c and a, because they occur naturally, and give to the base a regular melodic progression : Ex. 737. Compare this with the previous example. Of course the passing tones must form correct progressions with the other parts and not produce unmusical relations, as in the fol- lowing : zt I !- Ex. 738. In such instances similar movement should be avoided. A simple harmonic design is presented for elaboration. First add the other three parts to this base : Ex. 739. (1) ' After completing the simple harmonization as indicated, the passing notes are to be written in the base between the root-notes. (See Ex. GOODRICH S ANALYTICAL HARMONY. 307 737.) When any of the upper parts move in similar direction with the base, care must be exercised that no false progressions result. I* *l* *! The chromatic passing notes should now be supplied. Those most available here have been illustrated in Ex. 734. The passing note between the 3d and root of the essential discord may be accompanied with a corresponding note of passage in the soprano part. An inversion of the design will illustrate this : Ex- 740. The root and 7th remain stationary while the other parts ascend and descend in thirds. This may be freely inverted or re-arranged. Such designs are always effective, and tend to relieve the monotony of com- mon chord progressions. (See Ex. 706.) When the example is sufficiently elaborated it is to be arranged in quartet form as usual. * * * The original base part of Ex. 739 should be transposed into G, and B-Jlat. Then work it out in the same manner. If the base runs too low it is better to skip up an octave than a 7th : etc. Ex. 741. The first measure is not good vocally ; (b) is better ; (c) is best of all. The second method of producing this kind of harmonic counter- point is to include more or less ornamentation in the different parts. Unrelated notes and organ-point will serve as a nucleus for this species. Wherever the unrelated notes may appear they are to be treated in the same manner as though they occurred in the soprano part. Here, for instance, is a design in which a melodic figure passes in sequence-form through all the voice-parts. This would be effect- ive as a vocal or an instrumental quartet : 3 o8 GOODRICH'S ANALYTICAL HARMONY. Ex. 742. The melodic figure appears alternately in the soprano, contralto, ten- or and base parts. This relieves the effect of chord movement and imparts to the design a contrapuntal character. The suspensions serve as connecting links, and to prevent such awkward progressions as this, EX. 743. : which would otherwise have re- sulted between the soprano and tenor. Transpose Ex. 742 into E- flat and F. Attention has been directed to the fact that consecutive seconds should not follow each other. The author would also include among the unmusical parallel movements two or more normal fourths, or an augmented, followed by a normal 4th. Parallel fifths are generally condemned, especially in counterpoint. Successive sevenths in simi- lar movement are as inharmonious as consecutive seconds. (It has been remarked in a previous chapter that augmented sixths should not follow'each other, for they sound the same as do minor sevenths.) The only intervals that remain for practical uses in similar movement are : The unison, major and minor thirds ; major and minor sixths^ and diminished sevenths. Unison passages (employed for the pur- pose of strengthening a certain melodic part) may occur in an}' twa parts, excepting the base and soprano. These must not be confused with what are called " parallel octaves." Normal fourths, when they form part of a triad progression, may follow one another ; but as they form fifths when inverted they must be used with discretion. In good counterpoint they are seldom em- ployed. Imperfect fifths and augmented fourths can be used in parallel movements for a chromatic harmonization, but these intervals must be accompanied with some other tone of a principal discord, as shown in Chapter XXXIX. See the coda to Au Matin, by Godard. So much for the parallel progression of any two voice-parts. GOODRICH S ANALYTICAL HARMONY. 309 Other parts, opposed to these, result in all kinds of intervals. But these occur singly, not consecutively, thus : Ex. 744. Here are seen a gth, a 7th, and an imperfect 5th, from the lowest to the highest part. But none of these intervals are consecutive, since the two parts above move in contrary direction to the lower parts. The gth becomes a yth, the jth becomes a 5th, and this resolves to a 3d. This is more contrapuntal than harmonic, and where the upper and lower parts move in opposite directions they frequently result in dissonant combinations so harsh and incongruous as to be other- wise intolerable. For instance, here is a passage frequently used : Ex. 745- ? * *. , ? +^ It would be absurd to call this harmonic progression. Of the four dissonant combinations marked + only the last one resolves accord- ing to any rule or principle. The others forcibly pursue their way to the final tonic chord. The example consists merely of the major scale in contrary movement, with an additional counterpoint added in thirds and sixths. It is sufficient that the last two chords har- monize. The student should analyze minutely all these illustrative frag- ments in order to discover the various conditions that create the dis- sonant intervals, thus : Ex. 746. 310 GOODRICH S ANALYTICAL HARMONY. 1 1 1 J-^ 1 f * - j- S fcP 4 fl 0" 532 p I S3 5th, 1 1 6th, 9th, 10th, llth, 10th. f~\* i, m f ^Hj '- g ^1 V This begins with a consonant chord. By merely moving the solo base to the notes below and above the tonic, while the upper parts remain passive, a series of dissonating intervals result. The melodic groups in the base revolve around the harmonic tone in a perfectly natural manner, and the resulting dissonances are not only justifi- able, but desirable. The next illustration is somewhat similar, though there is a counter-theme above : Ex. 747. o i o i a o The fractional figures in the middle show the intervals from the base to the soprano ; the other figures indicate the character of the unre- lated notes. As the two middle parts remain passive, merely consider the extreme parts. The yt.h is a passing note, and the Qth results from the contrary movement of base and soprano. This discord on the last beat is the dominant yth, the d below being a passing appog- giatura that occurs naturally in the descending melodic figure. The dissonant gth becomes a consonant loth immediately. At the close the solo base ends upon the tonic, while the upper parts do not resolve until later. Thus all the disagreeing intervals are the result of favor- ing circumstances, and present an unmistakable object for their ap- pearance. The pedal-note may also form a nucleus for this ornate species ; though the suspended base presents some difficulties to the singer, especially if it be long continued, or if the upper harmony is of a chromatic character. The well known quando corpus from Rossini's Stabat Mater is an instance. This was written as an unaccom- panied quartet, but the author has known professional singers to fail miserably in their attempts upon this composition. The fol- lowing extract from an original Hymn is somewhat similar, but the chorus bases are here re-inforced by the 'cellos, double bases, horns, and kettle-drum: CrOODRICHS ANALYTICAL HARMONY. ChoniB and Orchestra. 3" Ex. 748. f O - ra pro no - bis, no - bis pec - ca to - ri - bus i ; m i Tim*. -IfP 2 J. + V ' * II ^^ ~ * I | | g 20 * rrjr j. ^ -* ^3 - ^t= nunc et in ho - ra mor - tis, in ho - ra mor - tis nos - trae P^ This begins with the tonic chord, and while the bases remain as pedal-note the other parts ascend chromatically through a series of major chords. In the 6th measure the base becomes again consonant to the other parts. Such designs are impossible to the average chorus singer, unless assistance is afforded by the instrumental accompani- ment. Pedal passages like the following present no difficulties and are always effective : Ex. 749. The dissonances are well prepared and of brief duration. The upper parts are sufficiently independent to form good counterpoint with one another against the pedal-note. The embellishment and the stationary tone can be made to serve good purposes in florid harmonic counterpoint. These may occur in any of the upper parts, though the former is better suited to the soprano part if it contains several foreign tones. A simple illustra- tion of this follows : 312 GOODRICH'S ANALYTICAL HARMONY. Mnrttti t-rm I ||||||=|| Ex. 750. This is especially adapted to the quartet form, for if the tenor part were written an octave higher the embellishments would sound some- what confused. As it is, the distance between the voice-parts is both convenient and agreeable. Finally a series of suspensions may be introduced with the same general result. The principle of suspension as explained in a previ- ous chapter is not difficult of comprehension, but some of its phases are so complex as to require both skill and ingenuity in their man- agement. A familiar design is selected as a nucleus : Ex. 751. Suspend the progression of the lower part from the last half of each measure, thus : Ex. 752 i i F=f^ The elementary theory is illustrated here. This can be utilized in various ways : A third part may be added a 6th below the soprano and the sequence thus continued. An additional advantage to this plan is that it can be freely inverted, and the retardations may occur in the soprano or tenor parts. The manner in which the chords follow one another does not admit a series of fundamental bases ; but the design naturally rests upon a tonic pedal-note. An example of this, with the suspen- 'sions in the tenor part, is represented : Ex. 753. w* ^ 52 C' j^ Ej j I y A' ^ J- at"" 1 19 ' ^ H r r ^ ^ tt [^ , __ _^ , r ^ _J GOODRICH'S ANALYTICAL HARMONY. 313 The tenor part becomes more prominent on account of the suspen- sions, and a very ordinary progression, is thus made interesting. The student should re-arrange Ex. 753 with the suspensions in the soprano part. The descending tendency of the progressions must be considered, that the tenor may not interfere with the pedal- note. Therefore it will be necessary to begin the tenor part at a considerable interval from the base, as at (a), unless the former be made more florid, as at (b) : Ex. 755. In an instrumental quartet the second arrangement would be pref- erable. Or the rhythm could be enlivened by brief imitative passages between the middle parts, thus : Ex. 756. It would be a useful practice to transpose the base and soprano of this into A-flat and G, and then attempt to supply the middle parts without consulting the printed exercise. This chapter will be concluded with a quotation from a choral melody harmonized by that Past Master of counterpoint, Sebastian Bach: GOODRICH'S ANALYTICAL HARMONY. The passing notes and suspensions are here indicated that the reader may more readily appreciate the construction. Aside from these un- related notes appearing in the various voice-parts, this is based upon a simple harmonic design not materially different from some of the previous illustrations. In fact, the aim has been to deduce principles from the works of standard composers and to lead gradually to this point in quartet writing, for the practice of all masters is similar in these respects. The underlying principles are fundamental, and must be the same in all countries. We know that certain passing notes may be filled in between certain intervals of any harmonic design ; and that appoggiaturas, suspensions, anticipations, and stationary tones can be included in any part, either for the sake of grace and ornamentation, or to make a particular voice-part more independent. Added to this information we know what single intervals produce the most satisfying effects when heard simultaneously, and what intervals may safely follow each other consecutively. Thus equipped, the intelligent student will experience no great difficulty in comprehending the intricacies of canon and fugue. The examples of the different species of harmonic counterpoint may serve for organ arrangements in dispersed harmony, for a simple string quartet, or for a vocal quartet or chorus. GOODRICH'S ANALYTICAL HARMONY. Chapter LXI. HARMONIC ACCOMPANIMENT ILLUSTRATED. THE accompaniment is now to be considered separately. In a general sense it is secondary to the melody, though the best accompaniments are those that are complete in themselves. Primarily the accompaniment is to consist of certain chords sug- gested by the melody, and these chords are to be arranged according to the principles of harmonic progression. Observe the following section of a theme from Rubinstein's Op. 13 : Ex. 758. The harmonies are sufficiently indicated here. In the second meas- ure c% is, of course, an appoggiatura. A simple arrangement of this may be attempted. * * * The composer's solution is given, not for comparison, but as a study : Ex. 759. f -.. v i ^ *=r =^1^=*=: ^~~~ r^ , . ^ Any position of a chord may be used in the accompaniment, this being a matter of taste, rather than of theory. A certain position might, however, bring the accompaniment' into such proximity to the melody that the effect would be indistinct or confused. In the next fragment the melody traverses a considerable space, and in such instances the accompaniment, being very nearly station- ary, appears first below and then above the theme * 316 GOODRICH 'S ANALYTICAL HARMONY Ex. 760. Where the melody does not skip, the chords may be re-arranged in this manner : Ex. 761. This form makes the accessory parts more prominent, and is bettet suited to a sustained melody in measured notes. The broken chord, or arpeggio form, is much used in accompani- ments. It is usually advisable to conduct the chords as though they occurred in regular progression. For instance, the design at (a), if performed in sixteenths, would appear as at (b) in this example : -tf-T 1 ? rj- \kr-*-t E U* :-^-l / .5. f\* li <5" * ^ * Ex. 762. 3 : l=C=Fg (2) (2) (2) (2) This is not always essential, but it is always correct. Until the stu- dent has acquired some experience in this matter it would be well to perform such designs as the last (b) in simultaneous form, as at (a), in order to test the correctness of the chord progressions, for they are in a harmonic sense identical. From this last we may derive the harp-like form, embracing two or more octaves : GOODRICH S ANALYTICAL HARMONY. 3-7 Ex. 763- ff i i i r f fc * f ?= - i ! * ^^ ^*^^ - ! i * ^r-^u This arpeggio style is conducted in the same manner, only the latter is more ornamental. A design like this (or like that of the previous example) once begun should be continued, at least during the length of a period. The accompaniment to the soprano solo Inflammatus represents a more independent type. It consists of passing appoggiaturas above and below the reiterated chords. One phrase of this will suffice to show the design : Ex. 764. This style of accompaniment is continued throughout the solo part. It is not the purpose here to give a great variety of styles for these adventitious parts, but merely to indicate their character. The broken chord or arpeggio forms represent the same harmonic sub- stance, and are generally conducted in the same manner as chord pro- gressions. But there are some additional features to be mentioned. i. In relation to melodic skips that could not very well be har- monized individually on account of their length. Such is the fol- lowing : GOODRICH'S ANALYTICAL HARMONY. Ex. 765. ]*** *- |-Tfel =^qg ,^> : ^-u^^g^ =p=q \-^^^^ =f= =^ =^ To attempt to follow these melodic skips with the chord accompani- ment would be impracticable, even if it were desirable. As here written the piano part is simple, correct and effective. 2. Where the theme is so rapid as to make it impossible to move the harmony at the same rate of speed. In such instances merely mark the rhythm and indicate the harmonic substance, thus : Ex. 766. The harmonic impression created by the theme is fully represented by the accompaniment, and .on. account of the florid nature of the violin part the accompaniment is made as simple as possible. This principle may be generally applied. 3. Still another situation presents itself when the melody pro- gresses in such manner as to violate the rules of re-solution if the harmony should follow the theme: Ex. 767. GOODRICH S ANALYTICAL HARMONY. The skip from the leading-note down a major jth to the tonic would certainly be incorrect as a harmonic progression, but as the accom- paniment is conducted regularly the melody may skip about as the fancy of the composer suggests. A somewhat similar instance occurs in that excellent song " O wretched slave," from Paul and Virginia: a f Mass's. L<2. Ex. 768. / 1 jf ff f ^ 5 vT) ^ * * 4 our trust be-trayed n f ^ Itf M^ * J Ku 5 ^ ^ - e-*- ^ * ' t-wj & 1 S MM 1 The le?p of a gth in the vocal part is characteristic of the senti- ment, and was a clever stroke, but this does not apply to the har- mony. Observe that the chords in the accompaniment as written by the composer are in strict conformity to our principles of har- monic progression. In reference to appoggiaturas and suspensions the author has had occasion to remark that the resolution of the dissonant tone is omitted from the remaining parts of the harmony whenever the melody and accompaniment are comprehended in one design, as here : Ex. 769. u 'i o a o s^ r^ IT =F^i=i=^-*-^\ 9 ^^0 t^~~ ^ "*"" * Either the harmonic quartet, or such piano arrangements as this, are to be treated in the manner illustrated. Each of the harmonic notes indicated by ciphers (when preceded by an appoggiatura) is to be omitted from the accompanying harmony, the root in the base always excepted. But if the melody issued from a different instru- ment, and especially in a higher or lower register so as to separate it from the accompaniment, then it would not be necessary to omit nnv part of the harmony on account of the temporary dissonances. Com- pare this with the previous example : GOODRICH'S ANALYTICAL HARMONY. a o 2-^0 T - Ex. 770. The fact that the solo is here considerably removed from the piano- part renders this plan more necessary ; and it is also better to make the accompaniment complete in itself especially when the solo issues from a different kind of an instrument, as in the last example. Arrangers frequently make the mistake of writing blank and naked intervals in the piano part, trusting that the other instrument will complete the effect. This is generally a false hope, especially if the timbre of the two instruments is different, as in this instance from a Rigaudon by Rameau : Ex. 771. The great French composer and theorist is not responsible for the unsatisfactory thinness of the accompaniment, as this is an arrange- ment for violin and piano by W. L,enz. If both parts were played by the pianist the fault would disappear, but where a violin or flute plays the melody the piano part sounds unsatisfactory. In the following cadence to a piano duo by Bach this point is farther illustrated. A blank 5th occurs in the first piano part, but this vacuum is supplied by the second piano, which sounds the major 3d: GOODRICH'S ANALYTICAL HARMONY. 32* Ex. 772. _ f - =*= . l-iuno II. The effect is the same as though the last chord issued from one instrument, excepting the unison D. Both D-minor fugues are writ- ten in this manner, and no fault appears. Another form of accompaniment consists in duplicating the un- related notes of the melody either in the unison or octave. In vocal music this affords some assistance to the singer (though this is seldom necessary), and adds to the interest of the associate part, thus: Ex. 773. Voioe. Lassen. The principal occasion for this style of accompaniment presents itself here, where the proximity of the vocal and instrumental parts renders it necessary. The effect of the unrelated tones is more noticeable because of their duplication in the piano part. The additional disso- nances that would otherwise result are here absent. Even when the melody is somewhat removed from the associate parts this plan may t>e adopted with good effect. Observe this illustration : OOODRICH'S ANALYTICAL HARMONY. //. W Nicholl. Ex. 774. ^ ^j^b ^g ~3.: I : -^-^rr^g-. .^ . l ^. : ^ J. "i . J. 333 ^^ ?S r-^-r Where the accompaniment proceeds with the regular fundamental harmonies, irrespective of the disagreeing tones in the solo, it pre- supposes that the melody is more animated and ornamental, and that it is considerably removed from the neighborhood of the accompani- ment: Oboe. Ex. 775. Under the circumstances this is sufficiently proper ; but the fact re- mains that this plan is generally inferior to that of Exs. 773 and 774. The most common form of accompaniment to a quartet or chorus is to transcribe the vocal parts almost exactly. When these are inde- pendent they neither require nor admit the same amount of extrane- ous embellishment as does a solo. Even to some of the examples of harmonic counterpoint a figufated accompaniment would sound incongruous and confused. (Such, for instance, as numbers 742 and 749.) If the vocal parts are in plain harmony we may repeat the chords in notes of quicker succession, or employ the arpeggio or broken chord forms. Or a design more or less ornamental may be given the instru- mental parts, as in the tournament of Song from Tannhauser ; the chorus " Down with the Moslem ! " from Buck's Don Munio ; the last chorus in Dream Pictures by G. E. Whiting ; or the Kermesse in the second act of Faust. In the latter the principal themes of the well-known waltz are heard in the orchestra as a musical coloring to the scene, while the chorus parts merely consist of harmonic outlines. See also the last of No. 3 in Mendelssohn's 95th Psalm, after the trip- let figures appear in the accompaniment. GOODRICH'S ANALYTICAL HARMONY. 323 A brief quotation from Haydn's Imperial Mass will show the usual mode of procedure when the vocal parts are merely harmonic : Ex. 776. Glo - ri - a, glo - ri - a, Glo - ri - a, glo - ri - a. B -ft- *- ^ & - ZT * *- The instrumental base is slightly more animated than the chorus base, and the former is also altered in a few unimportant particulars. When the choral parts are more measured, buoyancy and anima- tion may be imparted to the movement by adopting some such plan as this from the finale to " The Heavens are Telling" : The Creation.' 1 324 GOODRICH'S ANALYTICAL, HARMONY. Some designs do not, on account of their simplicity or their com- pleteness, require any elaboration in the accompaniment. Among numerous instances of this kind the following from a four-part song by Sir G. A. Macfarren is selected : Ex. 778. l#y *~~ -r rj 1 r ^=^ Sr- i* Sigh no more, la - dies, c PT^ -H ^-*- 1 1 \ \/ ^T T = ^ * i ?- V ^^ | L _ -1 9 "if f_ ~~P * E5s G' * \\ )y X ^ 1 1 Sigh no more, la - dies, sigh no more, la - dies, f\* , a * j * * ^> ; * b _l -6 , r-*-= fS -i i i i i / L - ^ - , j -^ 9 r ^ BE ^.. *y ^ JL - * ^ r- r 4 f -&- +- rv j 1 eS Ji ' 9 \ 2 > ^j i The accompaniment here is a literal copy of the vocal score, the tenor part being represented in the base staff according to the real pitch when sung by a man. The accompaniment to part songs of this character may very well be dispensed with, except for purposes of rehearsal. As a continuation of this subject the student will receive much benefit from selecting simple songs, copying the melody, and then writing an accompaniment according to the nature of the theme and the plans herein suggested. Afterwards more elaborate songs may be chosen for harmonization. The accompaniments to a volume of choice songs are in them- selves an excellent study, and in many of the modern classical songs, the instrumental part is frequently the most important. GOODRICH S ANALYTICAL HARMONY, PART XV. Chapter LXII. INTERDICTED PROGRESSIONS AND FALSE RELATION. THE author has thought best to treat of these topics secondarily, as they might occur in connection with the various lessons; for Tie believes that much of what is commonly " forbidden " is mere bugbear. Mozart and Beethoven were hampered by the countless rules and restrictions of theorists, and the young composer who is ambitious will be obliged to defy many of these injunctions, or suppress his originality. CONSECUTIVE PARALLEL FIFTHS. These have occasioned more discussion than any other progres- sion, and under ordinary circumstances they are certainly incongru- ous and unsatisfactory. The prohibition of parallel fifths should apply more particularly to the quartet style of writing, and to places wherein the fifths occur prominently. But if a composer desires to produce a rough, rigid, or blank effect, he may purposely choose these interdicted intervals, as Mr. E. S. Kelley has done in his Mac- beth music. Where the fifths occur in the lower or middle parts, and are somewhat concealed by the melody above, there is little or no reason in condemning them. So many instances like the following have been written that it is scarcely necessary to explain them : 326 GOODRICH S ANALYTICAL HARMONY. Beethcrven. Ex. 779. 5 5 The e-flat being retained throughout the measure, and the /of the melody, both contribute to the good effect, which is really this : Ex. 780. Grieg, in his Op. 35, repeats a phrase a minor 3d above, each chord being in its first position. All the parts ascend in similar movement : Ex. 781. Had such a passage come under the notice of Fiix, Kirnberger, G. Weber, or Marx, they would have been horrified ; for not only are the parallel fifths unconcealed, but they are emphasized with strong accents. And a fault equally grievous, in the opinion of past theo- rists, lies in the " cross relation " of both phrases. Yet how sug- gestive of the quaintness and incitement of northern life are these very transgressions of musical rule ! The next example presents parallel fifths moving by regular steps : Grieg. Op. 22. Ex. 782. ''* V: K GOODRICH'S ANALYTICAL HARMONY. 327 The melody is above, and the fifths occur in the accompaniment. These are paliating circumstances, if any be needed. But in truth the principal charm lies in the unusual mode of harmonization, which infuses into the music a distinctive character and coloration. A remarkable passage, containing a series of corresponding fifths, is here quoted from the ballet music in Rubinstein's Per amors : Ex. 783. "Bride of Cashmere" m These result from the resolution of an augmented 6th chord, No. r, direct to a major chord. But this is necessary on account of the chromatic progressions. Theorists agree in allowing the imperfect to follow the normal 5th ; the reverse of this is not so good : Ex. 784. 15 The effect at (b) is inclined to be rough and generally unsatisfactory. By retaining the tonic a more satisfactory result is obtained : Paine. Ex. 785. CV-U ! Such progressions as the following from Meyerbeer (a) are of frequent occurrence, though this fact does not excuse them. At (b) the faults are avoided, and without changing the melody or the har- mony : 32* GOODRICH'S ANALYTICAL HARMONY. Ex. 186. "HIDDEN" FIFTHS AND OCTAVES. Another restriction of harmony book-makers, and one that has very little practical justification, is in relation to hidden, or covered, fifths. They occur in such progressions as these, Ex. 787. which have already been explained. But these speculative gentle- men write a passing note between e and g to show that a 5th can be imagined when the contralto skips up a 3d. The author has no hesitancy in asserting that this so-called fault lies entirely in the imagination of those who consider it incorrect; for the supposed tone (/) has no relationship to the C chord, and unless the inter- mediate tones were produced .by means of portamento no fault could occur. It is a common progression to be met with in almost every composition even in the fugues of Bach. " Hidden Octaves " constitute another theoretical bugbear. One can scarcely conceive of a composition that does not contain hidden octaves, and yet they have been cataloged among the guilty things to be avoided. Here is an example : 1 \ Ex. "88. I i 9 : Observe that b in the soprano part ascends a half step to c, while the base ascends a normal 4th, from c to c, thus producing an octave. The other octave (s below) exists in the imagination, so we are told, and therefore it is classed with the " hidden," the " covered," and the " secret " fifths as something about which a suspicion is attached. GOODRICH'S ANALYTICAL HARMONY. 329 But these progressions are so necessary in modern harmony, and they have been written so persistently by standard composers, that it seems a mere waste of words to discuss the matter. Of course a distinction should be made between chord movements in free instrumental style and regular harmonic progression in strict style. Such movements as the following are frequently written in piano and organ music : Liszt. Ex. 7^9. But in a vocal composition this would be unmelodious and contrary to the principles of harmonic succession. Our illustrations show that any of these interdicted progressions can be used. It is for the composer to decide whether he desires the peculiar effect which they produce. CROSS RELATION. The quotation from Grieg illustrates this : Ex. 790. The upper part proceeds from e to g, while the middle part moves from c to e-flat. This is called cross relation. It is more noticeable at (a) and (b) in the next example : Neither of these can be recommended, for such anomalies usually leave an unpleasant impression on the mind. The student must not confuse this with a mere change in mode, where the chromatic alteration occurs in the same voice-part : . Ex. 792. 330 GOODRICH'S AMAI.VTICAL HARMONY. Such relations are strictly correct. E. F. Richter has very properly pointed out that the interdiction ought to he removed from all such progressions as these : Ex: 793. Indeed, composers have long since settled this matter. The chorale quoted from Bach (Ex. 757) presents an instance of false re 1 ?!' m where the B-major chord is succeeded by a dominant 7th ot? R Ex. 794. But the 7th chord containing d-natural is not connected with the & major chord, which latter occurs at the end of a musical and poetical phrase. Numerous examples similar to the following might be quoted : Beethoven. Ex. 795. The semi-phrases here begin upon the second quarter of each meas- ure. THE AUGMENTED SECOND. The productive musicians have used this so variously and so effect- ively that all attempts to suppress it have proved futile. The author finds nothing wrong in the progression of an augmented 2d, either melo'dically or harmonically, and he can not, therefore, condemn it Two illustrations will suffice : Ex. 796. GOODRICH S ANALYTICAL HARMONY. Mendelssohn. Glinka. fc^ESt =g-HHSg; *^L * I ^ H "=Fr=%=F =^F *H It /s simply a natural interval of an important modern scale. The reverse of this has been considered still more "faulty"; yet both progressions are rendered necessary by the nature of the harmonic minor scale. (See chorus of the Shemites in Rubinstein's Tower of Babel.} TRITONE. Another peculiar prohibition applies to the " tritone," 4 to 7 of the major scale. It is so-called because of the three whole steps included in this interval: Ex. 797- Gfc= tzac The tritone has incurred the displeasure of theoretical writers to such an extent that they have actually objected to this common progres- sion : Ex. 798. In order to be " allowable " the tritone must, we are told, belong to the same chord : "Ttt. 1=Z=- Ex. 799. ^=^ But this is no better than the preceding example. The difficulty of singing the augmented 4th led, originally, to its exclusion from poly- phonic music. The prohibition was then generally applied to all styles, and even the relation, or the suggestion, of the tritone was interdicted. In truth, the majority of these interdictions belong to 33* GOODRICH'S ANALYTICAL HARMONY. the primitive stages of musical development. But they have been repeated in all seriousness by modern writers, as though we had made no progress since Peri composed his Euridice. They ought to have been discarded during the last century, for very few of these prohibitions have any application to modern musical composition. A certain interval, or chord progression, may sound unmelodic or inharmonious, because the situation in which it occurs is not favor- able, or because no object appears to make it necessary. This does not justify us in condemning the procedure, for the composer with a definite object in view may produce excellent results with material that seemed useless to the mere speculator. Thus Mr. E. Prout writes an example of these minor triads, Ex. 800. 3^ and because they sound strangely in his ears he declares them to be "simply detestable ! " But Mr. Prout ignores the fact, as do nearly all speculative musicians, that genius is a law unto itself, and that a Saint-Saens, Dvorak, Grieg, or Tschaikowski may discover in the strangeness of a certain progression the very expression they desire to convey. Indeed, all these " detestable " harmonizations have been utilized by the greatest composers, and it is merest folly that attempts to proscribe them. It may be well for the arranger to be restricted by abstract formulas, but the composer who has something new to say through the mystic soul-language can not be bound and fettered by arbitrary, didactic theorems that must be violated on every page of original music. While musical effects remain inexhaustible, theory must play a secondary GOODRICH S ANALYTICAL, HARMONY. 333 Chapter LXIli. ANALYSIS OF HARMONIC SEQUENCE. MELODIC sequence is the repetition of a group or ngure upon different degrees of the scale ; the consequent succession of similar melodic intervals. The author applies this term to harmonic progression and transi- tion, wherever a certain arrangement of chords is repeated higher or lower. There must be some characteristic feature to the original progres- sion, which is considered as the design. The position, or the kind of chords employed, or the manner in which they follow one another, must be sufficiently characteristic to constitute a model. Here is a simple illustration of harmonic sequence : Ex. 801. The model (a) is repeated exactly at (b), a whole step lower. In each measure a dominant chord with the 3d uppermost resolves to its major tonic, and in both instances the base descends from the root to the minor 7th, which latter resolves to the 3d of the major concord. This is strict sequence, as is the following : Ex. 802. $t- ; I s *= **. . -S- -* ^P The design (a), consisting of the peculiar resolution of an essential discord, is repeated in sequence at (b) and at (c). 334 GOODRICH'S ANALYTICAL HARMONY. A free sequence mry be described as a repetition in which the same positions, but not the same species, of chords are employed. There is this important difference between the two: Strict sequence is transitional ; free sequence is not. The latter is here illustrated : Ex. 8or ) =?; The arrangement of the chords with reference to their positions is the same in every measure ; hence they appear identical to the eye. But at (a) the chords are minor, at (b) they are major. There is also a passing 7th on each second beat, but some of these are major, and some minor sevenths. At (c) the last chord is an imperfect triad, though this appears in its first inversion, as do all the others. No transitions here occur, since the natural tones of but one scale are employed. Compare this with Exs. 801 and 802. Another kind of sequence takes place when several chords follow one another in the same position. A familiar example may be quoted from Beethoven's Op. 2, No. 3, where the first eleven chords appear in their second position : Ex. 804. The sequence is indicated by the slur. Two other examples are quoted from the same opus : Ex. 805. fiijxi=r~r~?"< J -1 r^-f-f-ff^ ff ^, I ^" 4 ^*^ The first is an irregular sequence. Both are in the free style. The melodic part of a sequence from Haendel's F-major Chaconne is now presented for the student to complete according to the model : Ex. 806. GOODRICH'S ANALYTICAL HARMONY. \ N-r-F 33: .*-^-^g^pTJ . p The design to be carried out begins at (a). At (b), (c) and (d) it :'s to be sequenced, all three parts being similar to those given at (a). * * * The Chaconne from which this extract was taken is constructed principally by means of sequence, and would be a useful study in connection with this subject. The next illustration represents diminished and dominant yth chords resolving to the note above the root of each discord : /. Ldvt. Ex. 807. The 4th resolution of the dominant yth chord in the second measure corresponds to the first resolution of the diminished chord in the other measures. That is, the base in every instance ascends a minor 2d from root to root. This is free sequence. A charming chromatic sequence is contained in the following excerpt : Chopin. Op. 6, No. 1. #* ' ^t_ **m L^_|^^^fc | >v| The harmonic design consists of diminished, changed to correspond- ing dominant yth, chords. It is all strict with exception of the last 336 GOODRICH'S ANALYTICAL HARMONY. melodic note. Here ihe/% is used in place of/ 5, because the return. of the first theme requires the former. Chopin was partial to these sequence designs, and used them most adroitly. It would therefore be well to transpose the last example, and to seek other illustrations in his works. Exs. 809 (a) and (b) are to be completed by the student. The first consists in resolving a series of essential discords indirectly. The figures indicate 3d and 4th resolutions. The design of Ex. 809 (b) should be analyzed, and then con- tinued to a natural cadence on D. No chromatic signs are here necessary. Ex. 8oga. HE ~t- 5 'f* 9~" J J I P v ' \ t -f u 1 -* * - 1 m/ * r 3- 4- 3 *")'? ' ^$ 1 | Ex. 8og. I V V a Design. b etc. ^Z~r ZE 2*$ ' I : Transpose, but do not invert, these exercises. K3ODRICH S ANALYTICAL HARMONY. 337 Chapter LXIV. INFLUENCE OF RHYTHM AND PHRASING UPON HARMONIC MOVEMENT. WE enter here into a new field, and one that will require some knowledge of musical analysis. Chord movements that are contrary to all principles of harmonic progression frequently occur in seeming connection with one another. To analyze the design is particularly necessary in such instances. We must know the con- structional divisions of the work in order to understand where the harmonic connection ceases. Periods, sections, and sometimes even phrases, are to be isolated from what follows, and the principles of progression and resolution do not necessarily apply beyond these divisional or subdivisional points. Antiphonal groups, sequences, echoes have a like effect upon the harmonic connection, as well as upon the phrasing. An illustrative instance is quoted from a descriptive song. Notice particularly the progression from the second to the third measure : Ex. 810. Mililotti. & - 1 r The first two measures comprise a phrase. At (c) a different senti- ment is represented. This is indicated by the composer : " Like a boat-song heard in the distance." There is therefore no connection between the phrase ending at (b) and the one commencing at (c). If there were, the resolution of the essential discord at ^D) would certainly be unrulable, and even incorrect. The next quotation is of similar import : 338 GOODRICH'S ANALYTICAL HAK.MON Schumann. Ex. 811. ^ipEEjEj ^ .-sS^-s- The phrases marked p and / were intended by the composer to be isolated. Everything indicates this fact. Consequently the pro- gression from the A to the G chord does not come within the rule or meaning of a continuous harmonic movement ; they are disunited, both in phrase and sentiment. The next illustration is taken from a mazourka by Chopin. A part of the prelude and one phrase of the principal theme are given : Ex. 812. M=MfH-i The mazourka begins at (b), after the preliminary motive, with which the piece closes. No rest or pause appears between the two phrases (a) and (b), but the design is so apparent that the composer merely wrote two perpendicular bars to indicate the beginning of the ma- zourka. Otherwise the progression from the second to the third measures would be unaccountably and inexcusably strange. A motive or phrase repeated in form of an echo is to be included among these seeming contradictions to the principles of harmonic movement. Such an instance is cited from Beethoven's Op. 27, No. i: Ex. 813. GOODRICH'S ANALYTICAL HARMONY. 339 The short figure at (a) is repeated an octave higher at (b). The groups are not only separated by the slurs, but the tone-quality is considerably changed ; for these echoes are given to different kinds of instruments in orchestral music. As Beethoven generally had an eye to the orchestra while composing, the design may be represented in this manner : Ex. 814. String*. -3r- tf-b- J f- a f- m All such examples are to be understood in this sense. As the period is a point of repose we may very properly conclude that chord connection does not extend beyond this point, unless the directions of the composer indicate the contrary. United periods are usually bound together and treated as one period. A rest, or pause, or the addition of other parts above or below the prevailing harmony, are sufficient to suspend the rules of progression at that particular point. All these circumstances are to be duly considered by the student. Mention must now be made of an important consideration that enters here, and one that has hitherto been overlooked. It is the license frequently made necessary by the natural tendency of a har- monic sequence : Ex. 815. 34 GOODRICH'S ANALYTICAL HARMONY. The model (a) is sequenced at (b) and (c). The progressions are perfectly correct in each measure considered separately. But from one measure to another the chord movements are not so good, as may be seen by changing the order of the sequence : Ex. 816. These parallel movements, especially to an inverted base, lack poise, and are unmusical. But according to the original, only the half- cadence-figure is connected, EX. 817. 3|== the next measure being a sequence of this. Consequently the progressions at (a), (b) and (c) are all disconnected. Sequence thus renders possible progressions of this kind that would otherwise be undesirable, and even incorrect. A better illustration is quoted from the finale to Beethoven's Op, 27, No. i : Ex. 818. The second semi-phrase, marked f, does not succeed the first according to any known principle of chord progression ; for all the parts skip down a considerable distance. And if the sequence had followed in the same octave the result, as a continuous progression, would have been still more irregular : aThe slur is to be understood in its usual sense, as indicating the notes that are to be connected. fThe author includes the slurs merely to show the design. The style is staccato. GOODRICH'S ANALYTICAL HARMONY. 34! Ex. 819. Kven a less particular composer than Beethoven would avoid such inaccuracies as these, and yet the original arrangement, with the downward skip, would be equally objectionable but for the sequence- like character of the antiphonal groups, which are disconnected from each other. In the performance of such passages every intelligent pianist understands that the different groups are antiphonal, and therefore he not only separates one from the other, but changes the tone-quality also. There are many forms of harmonic sequence, but as they all possess the same feature in common it will be sufficient to present one more illustration : Ex. 820. In each measure here the chord passes into its first inversion, and the positions are the same at (a), (b) and (c). This constitutes the sequence. The example likewise embraces the license previously mentioned, and but for the sequence these progressions would be unsatisfactory. X t 1 \J 5 ^ r 1 J 2C t 5P -I] *^\\ e 4 a . "] 232 ^X ,_i 1 3 4 -'Sf i| r a 5*- - 0- t b C ~- _^- C\ 1< ^ ^-j ' , 2 r r 34* GOODRICH S ANALYTICAL HARMC'NV. PART XVI. Chapter LXV. HARMONY IN FIVE, SIX, SEVEN, EIGHT AND TEN PARTS THE simplest method of producing harmony in five parts is to duplicate the base an octave below. One quotation will suffice : 0- .*- > . Chcfiti Op. n. L *:> .V ** X 5- JL^| i* *^ SJp 4 J ^ ^ *- J ^ S. T . j ,] *j'* ! = ^^ P -*- Ex. 821. This requires no farther explanation than that of the first sentence. The next design is very simple : Sckumattx. =*=r Ex. 822. y \ s \ i :fc The two upper parts are duplicated below, with the addition of a in the middle. GOODRICH'S ANALYTICAL, HARMONY. Another method is represented in the following : Ex. 823. 343 j a H KB 5 -_ * 9, i ^ i ^~ "^ 1 J 1 1 75* ^** 1 ; * m _ m 55- ~ 1 r The main object of the baritone part is to fill in the void between base and treble parts. In piano music this is a comparatively unim- portant office, but in orchestral scores these vacuums usually sound bare and unsatisfactory. This form is more compact than where the added part is a mere duplication of the base. The parallel octaves that may result between the baritone and lower treble parts are not objectionable, but the general principles of harmonic progression should be observed. The best tones for duplication are roots and fifths. Designs similar to the last require the most care in their management, because the parts are independent of each other. SIX AND SEVEN PARTS. By writing four treble parts and adding an octave base, six-part harmony is produced. A comparatively new feature enters here : Ex. 824. At > x* * * C_ |C_S fr^ ^ i ~ ; 1 ^ B J i 1 1 7 f { ' i ^f 1 * i J * ^ 1 , -M i Octaves between the extreme treble parts are unobjectionable, be- cause the lowest treble part merely reinforces the melody. The unisons in the base have been explained. The base should move contrarily as much as possible, though when there is a connecting note above, the base may move in similar direction with the treble parts to prevent the former from descending too low. See second measure of Ex. 824. When there is no con- necting link, contrary movement is especial!} 7 necessary. 344 GOODRICH'S ANALYTICAL HARMONY. The following quotation from Schumann is in this style : Ex. 825. * &CJ 3= J*^ | I j ^** 5 5 55^-* * 5 In order to give the bases a melodic progression contrary to the upper parts the composer doubles the minor 3d in three parts at (a), and the 5th in four parts at (b), which is rather weak. But at (c) these inversions and duplications are justified. At (e) the com- poser leaves to implication the root of the A-major chord in order to avoid the open fifths : -jJFJrnr- sr-q^F Ex. 826. The perfect cadence-form here enables us to readily seize upon the object in view. Otherwise the a could easily have been supplied by including it in the previous chord : Ex. 827. Another simple design consists of two chord-groups o\ three parts each. The following form has been much used : Ex. 828. -& * i* - f - m r* - 1 - On account of the connecting note in each chord-group this is per- GOODRICH S ANALYTICAL HARMONY. 345 fectly simple in its construction. Observe that the 3d and jth of the # minor intervenes. The most important of the secondary, or transitional, relations have been indicated. These are all that can be directly connected through the tones of an original major chord. The keys located a minor or a major 2d above or below any tonic have no natural connection, and in actual composition they have generally been ignored. A transition to any of these four keys may be effected, but we can not pass to them directly with any hope of connection or relationship. Neither is the key located an augmented 352 GOODRICH'S ANALYTICAL HARMONY. 4th above to be considered as related, though some writers so regard it. Even if the mode be made minor it seems strangely incongruous. Some preparation is necessary ; in which case we may pass to A-flat- major as naturally as to G% minor : a ix- r .^ r L . : *. * iSp^dz^z^b^nl^r-g-fr i J^F-*= H^-?^=fp=* Ex. 842. This might serve a purpose ; but if we present the two tonics con- secutively there is not the slightest connection : Ex. 843. Such progressions are esthetically impossible, unless G% minor be associated with its parallel major, B. A summary of the keys related to a given major tonic, both nat- ural and transitional, is here represented : NATURAL CONNECTIONS (DIATONIC.) zP^ Ex. 844. b|t=Zi 1. 2. 3. 4. TRANSITIONAL CONNECTIONS (CHROMATIC.) 10. 11. These should be written in A, B-flat and E-flat, with a separate staff for the base, which is to be fundamental. * * * The main conclusions to be drawn from the preceding apply more particularly to the different movements of a connected work, and to GOODRICH'S ANALYTICAL HARMONY. 353 the important divisions of each movement. Chord relations and key- relations are, therefore, intimately associated with Form, especially that part which relates to the main outlines and their tonalities. (This is set forth in Chapters L,XYIII and L,XX.) ENHARMONIC TRANSITION. In the FO-called " Emperor Concerto" Beethoven introduced the slow movement in the parallel key a major 3d below. As this would require for its signature seven flats, the composer substituted five sharps, with the same result, practically : Ex. 845. v 7 I *^T^rr u The enharmonic representation at (b) is simpler, and therefore pref- erable. The greatest enharmonic possibilities are contained in the dimin- ished yth chord. These have been explained. When the appearance of a diminished yth chord is altered for the purpose of establishing a certain tonality the notation becomes a matter of necessity, not of convenience and simplicity. But in the next example" we may use nine flats, or three sharps : Ex. 846. If the harmony of the second measure appeared transiently, and returned immediately to G-flat-major, this notation would be proper. But should an entire period follow in this new scale it would be better to use the enharmonic equivalent : Ex. 847. GOODRICH'S ANALYTICAL HARMONY. Compare (a) with (b). Also, see No. i of Grieg's Waltzer Caprices, Op. 37- Another means of arriving at any key, however remote, is fur- nished by the chromatic scale. A number of these half steps in succession have a tendency to disturb the tonality, or at least to leave us in doubt as to the actual key-tone. (In a previous chapter this was more plainly set forth.) The chromatic passage may lead to the tonic, or to any part of the new scale. An instance is found in the rondo to Chopin's E-mi- nor Concerto, in which the unusual change from E to E-flat is accom- plished in the manner described : Ex. 848. At (a) the tonality of E is not .disturbed ; but when the diatonic is succeeded by the chromatic scale at (b) we are prepared for any new key the fancy of the composer may suggest. The rhythm, and the natural tendency of the cadenza toward b-flat, aid us somewhat in anticipating the actual result at (c). After a short period of this principal rondo theme in E-flat the composer lowers" the major 3d and returns almost imperceptibly to the original key-tone. The effects are as charming as the means are simple. REMOTE TRANSITIONS. These take place when no preparatory chords are used to intro- duce the new key. Such an instance is the following f;om Beet- hoven's Op. 27, No. 2 : Ex. 849. GOODRICH S ANALYTICAL HARMONY. 353 From G% minor to A-major is not a very remote modulation, but the unexpected manner in which it appears strikes one with almost tragic force. The quotation previously made from Grieg's Op. 35 may be in- cluded under, this heading. The scheme in a condensed form appears like this : Ex. 850. These represent four tonalities. They are all parallel keys, related through the minor 3d. The King's first solo in Lohengrin, and the ensemble that fol- lows, present some excellent illustrations of this subject. A few are quoted : Ex. 851. By means of the 4th resolution of a dominant yth the music passes to D-flat, thence to G-flat, and back to the key-tone through the dominant. The next are similar : Ex. 852. Wagner. Observe the enharmonic change in the last measure but one. For the remainder, students who are ambitious must examine the works of high-class modern composers, where abundant illustrations may be discovered. 35* GOODRICH'S ANALYTICAL HARMONY. Chapter LXVII. ALTERED CHORDS. DOUBLE AND TRIPLE DISSONANCES. ALTERED CHORDS. A PRELIMINARY knowledge of these subjects has been acquired --L in former lessons. But there are two different results to be ob- tained from altering chords, and these must be considered separately. The various augmented 6th chords may be mentioned here as transition harmonies ; they are all altered chords, and not treated as fundamental harmonies. But the majority of altered chords are the result of a passing note. Both objects will appear in the following examples. A yth chord V is selected. If the root be sharpened the result will be a discord equivalent to III ; but as it leads in a different direc- tion it may be included here : Ex. 853. This is a passing harmony with a slight transitional tendency toward the dominant. There would be no object in augmenting the major 3d ; but it may be lowered : Ex. 854. This can be re-arranged with the same general result. GOODRICH'S ANALYTICAL HARMONY. 357 The 5th may also be lowered in connection with the 3d : Ex. 855. These are mere passing notes and do not affect the tonality. Now raise the 5th and resolve the discord in any of the following Avavs : Ex. 856. 1 u ' i -J I r I n! . I ,,1 . I .. I i -J 1 i m 1 Here again the chromatic alteration is a passing note. The base assumes the character of a pedal-note in these instances. The corresponding discord upon the fourth of the scale is sus- ceptible to the same alterations. Both are harsh and require some preparation. The discord on the second of the scale is next in order. By raising the root there will appear a transition chord with which we are familiar in an inverted form. But it is sometimes used in its original position : Ex. 857. (See overture to Oberon ; also the Allegro Vivacissimo in Mendels- sohn's Scotch Symphony.) The effect is transitional, like that of the augmented 6th chord, No. i. In the next example the root is lowered as a passing note : etc. Ex. 858. This presents an unusual combination : major 3d, augmented 5th, and major yth. 35* GOODRICH'S ANALYTICAL HARMONY. By lowering the 5th we produce a discord identical in appearance to No. Ill, founded on the leading note to A-flat, but very different in its natural progression : Ex. 859. At (a) the d-flat is a mere passing note between 6 and 5 of the major scale ; at (b) the tonality of A-flat is supposed to have been estab- lished. If we desired to pass into F-minor the altered interval, as minor 6th, would serve a different purpose in preparing the ear for a change of mode : ' (2. Ex. 860. etc. At (a) the d-flat does not appear as a passing note. At (b) the key of F-minor is sufficiently established. The yth chord III presents some features for alteration. Lower the 3d first : Ex. 861. This is a passing harmony. The 3d and yth may be lowered, and treated as in the following example : ii Ex. 862. ? \y(? & r & \yi *y 1 - L, - L_) - 1 ' | in GOODRICH 'S ANALYTICAL HARMONY. 359 The succession of normal fourths is not very euphonious, but with the 3d or 5th uppermost the examples might be utilized. The dominant jth chord is susceptible to considerable alteration. The 5th is frequently augmented, and in this form it has the strength and decision that characterizes all augmented 6th combinations. (See Exs. 510 and 511.) The 5th might be lowered as a passing note, thus : Ex. 863. But this does not seem to be of much utility. The flattened 3d offers greater advantages, especially if the object is a transitional one : Ex. 864. The e-flat leads naturally to d, and is characteristic of G -minor. The jth is sometimes raised as a passing note without destroying the effect of the essential harmony : Ex. 865. Sp==5=i=^i iEtFjtz=iB=i: The bb is a melodic passing note and might be included with b-ftat as harmonic note below. In the following melodic sequence all the altered chords arc pass- ing harmonies : 360 GOODRICH'S ANALYTICAL HARMONY. :fc=i^=zi Ex. 866. The augmented triad at (d) corresponds, in the harmonic sequence, to the passing diminished chords at (a), (b) and (c). None of the chromatics are to be considered as transitional. The following example illustrates more plainly the difference between a passing chord and a transition chord: Fink. Mendelssohn. Ex. 867. _Mbi f^t .MC , -y-yr^ U=^=:^r g 1,1? L.'^= ^ I? At (a) the/"# is a passing note and does not affect the key of B-flat. At (b) the /# forms part of the essential harmony on D, and this modulates decidedly to G-minor. DOUBLE DISSONANCES. With exception of the principal Qth chords, nearly all double dis- sonances are a product of suspension, or of the harmonic appoggia- tura. The double dissonance, as its name implies, is a combination embracing two dissonating intervals, generally requiring separate resolution. Suppose this to be a model : Ex. 868. Suppose, farthermore, that the root of the first chord be suspended after the discord is introduced : GOODRICH S ANALYTICAL HARMONY. Ex. 869. At (b) we hear a double dissonance of suspension d and c, and - and/. Observe that each dissonance is resolved to a consonance, as though they had appeared in this form : Ex. 870. & & H^ \\ F The upper discord at (b) resolves to a consonant interval at (c) ; the lower discord at (c) is resolved at (d). The preparation may be seen at (a), Ex. 869. The 3d of the concord may be suspended with the same general results : ^ <. " Ex. 871. This is slightly more dissonant than Ex. 869, but should be analyzed in the same manner. While it is true that a 2d or a yth may resolve up as at (a) in the next example (because the e is necessary to complete the C chord), the usual tendency of the suspension is to descend, as at (b) : & I <& f & - ri ^ ^r ~ K& 1 sB ^ TT i \ \ 1 > ^2\ ^ 1 S *J Ex. 872. In the latter instance c descends, because b is wanting in the domi- nant jth chord. This is also its natural resolution here. 362 GOODRICH'S ANALYTICAL HARMONY. In the next example e descends to d, because d is the tone antici- pated, having been delayed by the prolongation of e : Ex. 873- 12) And since the upper parts contain the root, 3d and 7th of the essen- tial discord, this resolution to the 5th becomes all the more imper- ative. These double dissonances are very useful, and may be introduced into the strictest styles of composition, for they freely admit of re-ar- rangement and inversion. Principal 9th chords are an exception to this theory. They fre- quently resolve direct to a concord, thus, from Schumann : Ex. 874. The root and 7th, and the 3d and Qth, form dissonating intervals ; or they may be considered in this way : Ex. 875. In Ex. 874 the dissonances are so far removed from the fundamental that they do not sound harsh ; and the fact is to be considered that this is a principal discord. In the following extract an altered Qth chord is treated as a prin- cipal discord : Mac Powell. Op. Ex. 876. GOODRICH'S ANALYTICAL HARMONY. 363 This contains a major 3d, augmented 5th, minor yth and minor gth, an unusual combination. All the "upper parts ascend and descend chromatically while the base moves fundamentally. The secondary gth chords, though a product of preparation rather than of suspension, are to be treated as double dissonances and re- solved to some single discord : Ex. 877. One dissonance disappears at (b) 9-8; the other disappears at (c). The following secondary discords are conducted according to the same principles : Ex. 878. 9 resolves to 10, 7 to 6, and so on. (See a to b, b to c, c to d, and d to e.) TRIPLE DISSONANCES. Some combinations require more than two resolutions, though these are of rare occurrence. Such an instance is this : Ex. 879. There are four discords here, but the f-sharp in the first measure may be considered a passing note. The combination at (a) is theoretically a triple dissonance, since everything points toward the dominant jth harmony, and the JO. tt =P-r ~ -2. ' H2 I c|; 4 * & & -* fES * ' *> & 1= Op. 37, No. 2. Ex. 883. This is prepared by and resolved to the tonic harmony. With excep- tion of the base B, the upper harmony has somewhat the effect of an after-cadence. This is counteracted and made more decided by the dominant below. The dual character of this combination produces a peculiar effect here. GOODRICH'S ANALYTICAL HARMONY. THE DIAPAPON. There is a still more dissonant and extended harmonic mass, con- sisting of every tone in the scale. We may term this a Diapason, though it is known theoretically as the " chord of the i3th." It results from suspension, and rests upon the tonic. In major this combination would be prepared and resolved in some such manner as this : 3$JF^ Ex. 884. At (b) the major Qth is added to the dominant 7th harmony on the tonic pedal. At (c) the base parts are resolved first to the full tonic chord, while the upper harmony is suspended and resolved afterward at (d). The diapason at (c) includes every note in the D-major scale. This is an elaboration of the dominant yth chord suspended over that of the tonic. The complete representation, as here given, is of rare occurrence in the major scale. In the next example this mass ap- pears with the 3d of the tonic omitted : Gavotte from "Otho Visconti.' 1 ' Glettson. ?& i ^^jjiliy* * y~p *_* # n ! I ^r~ I Ex. 885. The Diapason at (c) resembles an eleventh chord on the dominant ; but as the pedal-note is tonic we recognize the upper combinations as dissonances resolving to a concord of which the root and 5th are foundation notes. A remarkable instance in minor is here quoted from Beetiiovec's last symphony : SCKTDIHCH'S ANALYTICAL HARMONY. Ex. 886. This serves to introduce the baritone solo. The only preparation consists of the dominant in the kettle-drum part, and the fact that the key was previously decided as that of A-major. The combination is enumerated from the tonic, D, thus : I, 3, 5, 7, 9, n, 13. It is the diminished yth harmony suspended over that of the tonic. One peculiar feature is the inverted base. But this almost immediately moves to other notes of the chord, whereas the root above remains stationary. That this is treated as a suspension (though there is very little preparation) appears more plainly in the second measure, where the dissonances are resolved to the tonic har- mony. This harmonic mass is really a quintuple dissonance. GOODRICH S ANALYTICAL HAKMuJN ,. PART XVII Chapter LXVIII. MUSICAL FORM AND CONSTRUCTION. DIAGRAMS OF ELEMENTARY MODELS. *~T"VHESE subjects, with their various divisions and -1- would require a volume for their adequate explanation. This system, however, would not be complete without a synopsis of form and analysis such as the author has found to be of the greatest beiie- lit to the average student. Supposing the subject-matter of the preceding pages to have been mastered, the next question is, How can this material be utilized? Form and Analysis furnish the answer : The first embraces outline ; the second includes all the details of composition. Form is the shape and structure of anything, as distinguished from the material of which it is composed. All the previously acquired information is to be con- sidered as the material from which music is constructed. MOTIVE AND SEMI-PHRASE. The musical motive is to be considered as a subject, or text, and tne composition should be an outgrowth of this. Short motives include but one measure, and as these are the smallest analytical divisions the author terms them Semi- phrases. Such are the fol- lowing : Ex. 887. Haendel. Mozart. Schubert. Chopin. GOODRICH'S ANALYTICAL HARMONY. 369 The small notes in the first extract are included merely to complete the measure. The motive consists ol the three repeated notes, .sup- posed to represent fire! fire! fire! PHRASE. This contains two semi-phrases, as may be seen in the continua- tion of the requiem motive : ^ Adagio. Ex. 888. " The phrase has three features to be analyzed: proportion, rhythm^ and melody. The student must be familiar with the analytical divisions (phrase, section, period,* etc.) and the different methods of constructing these. Considerable practice of this kind is necessary, for melodic invention and thematic development are among the first artistic requisites of a composer. To this end the following course is suggested : Select the first phrase of some natural melody and endeavor to supply the remainder of the period without consulting the original theme. (A volume of popular songs, or the vocal etudes of Concone, would answer this purpose.) Then copy the first phrase of the second period and work this out in the same synthetical manner. NOTE. The various methods of building up sections and periods are fully set forth in the author's Musical Analysis. This work is acknowledged to be complete and explicit in these respects, and as it was intended as a compendium to the harmony treatise, no apology seems necessary for recommending the former, especially since its peculiar field is not occupied by any other text-book. The outlines of some of the smaller forms are here represented by means of diagrams : -The definitions of musical phrase, section and period are here retained on account of their general acceptance, and because they seem to the author sufficiently appropriate 370 GOODRICH'S ANALYTICAL HARMONY. Diagram A. 1st Period. Ill 3d Period. I Coda. This would begin and end with the principal key, only a transient modulation being necessary. Diagram B. 1st Period in G. I : 2d Period Ditto. I Trio in some related key. I 1 D. C. al /TV. The tonality here would not change materially during the first two periods, especially since the 2d period, as final ending, must terminate upon the tonic of the principal key. In the trio a different tonality prevails as a contrast to the first two periods. Some melodic and rhythmic variety are also desirable here. The da capo is necessary in order that the tonality of G may leave its final impress. An im- portant element of form is proportion, and this requires that the repe- tition signs be disregarded after the D. C. The last is a dance form. Diagram C. PART I IN D. T\ 1st Period. 2d Period. PART II IN G OR Bi>. 3d Period. lilt Period D. C. al ^. Part II is generally misnamed " trio." See trio in Diagram B. Many of the common dance species are built on the plan of Dia- gram C. Temporary modulations may be freely indulged. Among the related keys preference should be given the dominant, and the GOODRICH'S ANALYTICAL HARMONY. 371 relative minors of the dominant and the tonic. The subdominant is usually reserved for the trio or coda on account of its retrogressive tendency. The main conditions to be carried out are that the begin- ning and ending shall be on the principal key, and that the scale of this key shall be heard more frequently than that of any other. Hence the D. C. is necessary whenever the trio or part II are written in a different scale. (Observe the difference between the diagrams marked D. C. and those which do not return to the beginning.) Diagram D. 1st Period, tonic. 2d Period, related key. 1st Period, tonic. Diagram E. PART I. Imt Period. Ill 2d Period. PART II. SAME SIGNATURE. 3d Period. 1th Period, tonic. These are rococo forms, but still useful. Diagram F. 1st Period in C. 2d Period, related key. 3d Period in C. Diagram Q. PART I. C MINOR. 372 GOODRICH'S ANALYTICAL HARMONY. PART II. C MAJOR. D. C. a! rrv. Diagram F requires a closer affinity between the three succeeding periods. The alternation of minor and major in Diagram G is revers- ible. In either instance the tonic remains the same. When these diagrams are sufficiently understood, the student should attempt the composing of a few common dances, such as mazourka, waltz or galop. This is a very useful preliminary practice, and even a galop can be made interesting. In Musical Analysis nearly thirty species of the dance form are described, together with their intermediate details. Extended and united periods, intermezzo, coda, etc., are also fully illustrated. The intermezzo may be used as a relief to a frequently recurring principal theme, or as a means of connecting two dissimilar parts written in different scales. See No. 2 of Mendelssohn's "Songs Without Words." An intermezzo begins at the 2Qth measure and continues to the 4Oth, where the main theme recurs. (The last 15 measures comprise the coda.) Another instance occurs in the Spin- ning Song, No. 34. The song ends at 25, and is resumed on the last of 29 the intervening measures being devoted to an intermezza founded upon the figure of the introduction. Another intermezzo occurs from 56 to 64. The former is a mere diversion ; the latter serves to connect the strain ending in E with the one beginning in C. The intermezzo is an important feature, though it is seldom employed in the dance form. In ball-room waltzes and other disconnected works, composers use what are called eingange (entrances) as transitions from one number to another. When the keys are unrelated the eingang serves a pur- pose, especially if the modulatory section be cleverly managed ; but it were better to begin abruptly in the new scale than preface it with an unnatural or awkward transition. (See the eingange in Grieg's Waltzer Capricen, Op. 37.) The author applies this term to all tran- sitional passages that aim at the establishing of a particular tonality. In this sense it is frequently in form of an episode. After the com- mon dances a few rococo or modern classical species should be writ- ten.* The saraband, menuetto, gavotte and musette, tarantella, bo- lero, avanera and czardas are interesting species. *See partitas and suites by Couperin, Scarlatti, Corelli, Paradisi, Rameau, Bach, Haendel, Purcell, and Haessler. GOODRICH'S ANALYTICAL HARMONY. 373 Even if these synthetical effusions prove to be artistically worth- less they should be continued until the student has acquired that ready command over the material and technic of composition which very composer must possess. The author does not regret the hundreds of pages of MSS. written at an early period of his musical career, even though they were afterwards purposely destroyed, and though he has for a number of years past discontinued all efforts at composition It teaches that which no book and no professor can impart. Chapter LXIX. MUSICAL FORM AND CONSTRUCTION CONTINUED. RHYTHM. ~T~) HYTHM is an important element of construction, and yet it -^- must be sufficiently varied to prevent monotony. Two, or even three, periods may be built up from a single rhythmic design, but in the second part a different arrangement will be necessary. Great care must, however, be exercised in the choice of rhythmical devices, for rhythm is the principal element of dance music. It represents action and motion, and nearly all characteristic rhythms are suggest- ive of mechanical effort or physical exertion. Observe the rhythm and melody in such airs as He shall feed His flock, I know that my Redeemer liveth, or the Tuba Mirum from Mozart's Requiem : Ex. 889. f f Voice. - / /> . o "T~ m Tu - ba minim spargeus so num. The trombone motive sounded in advance is especially appropriate to the text; the melody is consonant to the sentiment, and the rhythm and movement correspond to the accent and metre of the 574 GOODRICH'S ANALYTICAL HARMONY. words. This may also be said of the songs from Haendel. These are slow movements, but fast movements might be cited as well. Rejoice, rejoice, from " The Messiah"; the second part to Beethoven's Adelaide ; and Di quella pira from " II Trovatore," are instances in which rhythm and movement are sufficiently animating to express the sentiments of the words, and yet without any suggestion of danc- ing, marching, or mere physical exertion. Notice also the Slumber Song by Schumann, Op. 124. How chaste and gentle and hopeful the melody ; how artistic the harmonization, and how suggestive the agitation of the trio in G-minor! There is a certain rhythmic move- ment that accompanies the song, but this suggests the motion of a cradle, and is, therefore, a necessary feature of the song. J. S. Bach was a great master of rhythm, and displayed rare judg- ment in its application ; for though he composed considerable music in the dance form, we find in his serious works very little of terpsi- chorean suggestion. The motive of one of his clavier fugues is quoted as an example : ** * Ex. 890. EBtfefauT^J^P Intelligent discrimination must be exercised in this matter, for in a nocturne, or any composition of a contemplative nature, it would certainly be incongruous to introduce the bustle and swing of a dance rhythm. Beethoven's object in substituting the scherzo for the minuet was undoubtedly the eliminating of dance elements from pure instrumental music. Moszkowski, in his Spanish Dances, Op. 12, has succeeded in reproducing not alone the rhythm but the national characteristics with remarkable cleverness. The student must understand that national dances in some of the older countries are so characteristic as to form something of a psycho- logical index to the habits and aspirations of the people. Therefore we who have no national dances must judge from a more remote standpoint. We must know their origin and the historic associations connected with these old dances in order to utilize them in artistic composition ; for the author can conceive no greater musical vulgar- ity than the introducing of a common dance-rhythm into a serenade, overture, sonata, or string-quartet. The young composer is there- fore admonished to exclude all terpsichorean rhythms from his sen- GOODRICH'S ANALYTICAL HARMONY. 375 ous compositions until he knows their significance and is sure of their appropriate application. Beethoven introduced in the scherzo of his Pastoral Symphony a- rustic dance, and with charming drollery of effect. So in the Italian Symphony, by Mendelssohn ; the Country Wedding, by Goldmark ; Jm Walde, by Raff; Frithjof, by H. Hofmann ; Saint-Saens' Danse Macabre ; the ideal dances of Chopin, and such works as Liszt's Hungarian rhapsodies, and the Op. 38 of Ph. Scharwenka. In the first movement to Beethoven's 5th symphony the rhythm of the fate-motive is imitated almost continually ; even during the lyrical second theme it is heard from the bases Every concert-goer will remember how persistently the i r- > 044 in the allegretto of the yth symphony is maintained. Another famil- iar instance is the Erl King, by Schubert, wherein the triplets of the accompaniment are continued throughout.* In regard to mensural proportion, extended and united periods are important features, especially in works of considerable length. Introduction, eingang, anticipation, intermezzo, passage, development and termination admit considerable irregularity in their rhythmic proportions. Lyric movements must be more equal and symmet- rical in their periodic construction. Here also may be mentioned the numerous avoided and decep- tive cadences which have a tendency to prolong the periods and thus prevent the interest from prematurely subsiding. Uneven rhythmic phrases, irregular periodic construction and continued thesis all have the advantage of relieving an otherwise monotonous movement by the variety of effect which they produce. Provided the student is endowed w r ith sufficient intuitive capac- ity, there yet remain the more important secrets of thematic elabora- tion, choice of means, and significance of effect. Development and form will be discussed in the following chapter. * The author's principal definition of rhythm applies to the value and arrangement of notes in a measure. In a more general sense rhythm refers to mensural proportion, and includes accent and movement. 376 ^GOODRICH'S ANALYTICAL HARMONY. Chapter LXX. MUSICAL FORM AND CONSTRUCTION CONCLUDED. The Sonata Form in Major and in Minor: Outline, Tonality, Development, Affinity of Motives, Diagrams, etc. OUTLINE. THE sonata is a cyclical form consisting of three or four move- ments. The first of these, usually an allegro, is most important, and will be briefly outlined. This is founded upon a formal plan as to symmetrical proportions, tonal arrangement and logical develop- ment. There are three main divisions to the first allegro (sonata movement) : 1 . From the beginning to the double bar. 2. From the double bar to the end of the " development." 3. From the reprise, or return of the principal theme, to the end of the movement. The first division is a citation of the leading motives. The sec- ond division consists of an elaboration, or discussion, of the principal motives. The third division is similar to the first, excepting in tonality. The first and third divisions have three subdivisions: " Principal theme," " Second theme," and " Conclusion." The main theme contains from 16 to 60 measures, depending upon the dimensions of the work. The second subject is about the same in mensural proportion. The conclusion is shortest of the three subjects. The development was originally brief, but it has been enlarged since the advent of Beethoven. In his Op. 2, No. 2, the development contains 103 measures. GOODRICH'S ANALYTICAL, HARMONY. 377 TONALITY. The first theme is principally in tonic major. The second theme and conclusion must, according to the old formula, be in the domi- nant. The dominant modulation has already been explained har- monically. The author has also pointed out, in another work, that a new tonality presents a different view, on account of its different location above or below the original tonic. The mere difference in signature between the related keys is not what produces the effect of a new tonality. It is to be found in the metaphysical relation of keys and the different views presented by the changing tonalities. The development may begin in any scale that naturally suggests itself, excepting that of the original tonic. The free use of transition becomes a necessity here in order to present the chief motives in different lights and colors. The reprise usually recurs in the tonic. The second subject is transposed from dominant to tonic, as is the conclusion, so that the movement may end in the original scale, thus maintaining the su- premacy of the principal key. The location of the second theme can no longer be prescribed. Any of the parallel keys may be selected. DEVELOPMENT. This has an important influence upon musical construction in general. The difference between variation and development must first be understood. In the former the melodic outline and harmonic structure usually remain the same. In the latter only a part of the theme is selected, and this is led in a different direction from that of the original, being sequenced, modulated, or otherwise metamor- phosed. Variation shows the same picture in different phases; develop- ment exhibits only a part of the picture, and then presents other views of a kindred nature. (No reference is here made to those so-called variations in which the theme is repeated identically amidst the idle flurry of arpeggio and scale passages. This is merely varia- tion of the accompaniment, not of the melody.) The first section of a theme upon which Beethoven wrote three sets of variations is quoted : 378 GOODRICH'S ANALYTICAL HARMONY. Ex. 8gia. Andante. Op. 14, No. 2. After observing the melody and harmony of these two phrases they should be compared with the following variation, corresponding to this section : Ex. N P - > The harmonic substance is identical, but the rhythm and style are varied. Observe that the upper notes, indicated by double stems, represent the original theme. Variation may thus be used as a means of construction to prevent a repeated passage from sounding monotonous, or to exhibit a recur- ring theme in different colors. (The reader should here refer to Beethoven's theme and variations in the Sonata, Op. 26 ; the last half of the adagio in his first F-minor Sonata ; and to the Sonata in A, by Mozart, No. 6, Edition Litolff. Schumann's Op. 46 represents a still more artistic unfolding of a musical germ, and belongs to development rather than to variation.) Examples of development are now presented : GOODRICH S ANALYTICAL HARMONY. 379 "Scotch Symphony." Mendelssohn. Allegro. Ex. 892. Among numerous transformations of this motive, in the develop- ment, notice the following : Ex. 893. The natural melodic tendency of the theme is not followed here, but the second measure is a contrary inversion of the first. The rhythm is maintained throughout. The next quotation is more elaborate : Ex. 894. HE te^ Ii3=i=i= Eclfe Observe the isolated phrases of the highest part in connection with the original theme. Each voice-part in Ex. 894 represents a devel- opment of the motive. 380 GOODRICH'S ANALYTICAL HARMONY. Such designs excite the keenest interest because of the various voices all talking about the same subject in a different manner. Another form of metamorphosis is represented in the next quota- tion : Ex. 895. 7..n The two middle parts carry on a free canon, with ad libitum parts above and below. For other instances see the full score, or the four- hand piano arrangement. The subject of development will conclude with a few excerpts from Schubert's Tragic Symphony : Ex. 896. Principal Them* of the lt Allejrro. --*- It is scarcely necessary to remark that this is perfectly natural and melodious. Now observe the first of the elaboration : Ex. 897. fe I IIIHOIIO Tulli. . , *u A __-- -- * '*-- ff GOODRICH'S ANALYTICAL HARMONY. 381 This illustrates the general principles still more plainly, especially with regard to sequence. Only the first of the motive is here devel- oped. In the last quotation two different phases of the principal theme are elaborated : Ex. 898. ^* ^ ' ^ * - *-*- - A -4 ^ *- W. , 1. =1 . . - w- i . ^ 1 At (a) the rhythm is slightly altered, and no attempt is made to pursue the natural trend of the melody. At (b) a smaller fragment of the original motive is taken as a model, and this is continued in sequence beyond the quotation. In addition to sequence and passage, the various kinds of canonic imitation play important parts in elaboration. Augmentation, dimi- nution, repetition, rhythmic imitation and transition are also means to this end. But the lessons should not terminate here. The student must consult standard compositions in this and all other matters that relate to musical construction. Enough has been explained to enable the observing reader to examine profitably the thematic work of emi- nent composers, and they are the greatest teachers of the secrets of composition. In Mozart's last three symphonies the fourth as well as the first movements are in sonata form. The finale to the "Jupiter Sym- phony" contains the most ingenious and complicated development. The unraveling of these musical threads will sufficiently tax the mind of the reader, though it was all perfectly easy to Mozart ! AFFINITY OF MOTIVES. A feature of great importance now demands attention, and as it is a more or less latent principle the young composer will do well to give to it his most serious endeavor. Reference is made to Unity of design, or the innate affinity and relationship of the different move- ments to the original motive. A symphony, overture, concerto, string-quartet, sonata, is not a hotch-potch medley, but a congruous 382 GOODKICH'S ANALYTICAL HARMONY. and connected work ; a logical illustration of some musical impres- sion. The motive is to be considered as a subject to be discoursed upon and illustrated in various lights and colors. Such a work as Tschaikowski's E-minor Symphony is the psychological expression of a series of kindred emotional images. The leading motives from Schubert's B~flat Symphony are quoted. These represent the four movements : The principal theme (a) is a chord-motive, and the various ramifi- cations of this may easily be traced through the entire symphony. Another phase of the subject appears at the sixth measure, which is employed as a counter-subject during the repetition. A section of the 2d theme appears at (b). The outline of this is also a chord- motive. No analytical knowledge is required in tracing the affinity between the allegro and the andante. The minuet contains the same motive in different measure, and changed from major to minor. In the trio the original motive is reversed. The same coherency is observable in the finale. See (f), (g), and (h;. Attention is now directed to the Sonata, Op. 13, by Beethoven. GOODRICH'S ANALYTICAL HARMONY. 333 This should be examined in detail, for the unity of design is dis- tinctly traceable. Observe first these three notes, the germ of the sonata: They occur in the very first of the allegro, and are indicated bv accent marks : Ex. 901. In the second subject they appear in this form : Ex. 902. and in the rondo thus : Ex. 903. ^ ^~f Observe, not only the c, d, e-JJat, but the e-flat, d, c, descending, whia i is the original motive reversed. These tones also occur in the epi- sode, and in the second theme of the adagio, though the latter is founded upon the second half of the original motive : Ex. 904. Another melodic figure, of a subsidiary character, occurs in all the movements in different guises. Two of these are presented : -""im Adagrio. Even this motive is a natural outgrowth of the principal theme. The suites and partitas of Corelli, Couperin, Scarlatti, Paradi;;i, Bach and Haendel contain many interesting illustrations of coherent {.hematic development and affinity of motives. The melodists of the {Stli and igih centuries frequently lost sight of congruity and iioia<> GOODRICH'S ANALYTICAL HARMONY. geneity, though the best composers of the present century have aimed at greater unity and connection. Many of Grieg's works contain but a single motive worked out and elaborated in the most concise and masterly manner. Observe the funeral march upon the death of Ase, and the last movement in the first Peer Gynt suite. SONATA FORM IN MINOR. The principal differences here are in relation to mode and tonal- ity. The classical formula is as follows First theme in tonic minor ; second theme in the relative major; conclusion, the same. This last subdivision is sometimes modulated to the dominant on account of the repeat. Development in various tonalities ; reprise, tonic minor ; second subject, tonic major, or tonic minor ; conclusion, the same. The relative major of the subdominant is also used for the second theme in the last division. The student should now attempt the composition of a Sonatina in major, and one in minor. To facilitate these lessons a few dia- grams are given, showing the forms in outline : Allegro I. A. SONATINA IN F-MAJOR. < 1 Principal theme in F. Thematic. 16 to 20 measures. Extended period or modulation. B. C. 2d Theme (lyric) in C-major or A-minor. 16 to 20 measures.* Conclusion. Same tonality. D. Development in various tonalities, using frag- D. ments of at least two themes from A, B, or C. Modulation or cadenza leading naturally to the c The spaces represent jthe relative mensural proportion of the different divisions and subdivisions. GOODRICH'S ANALYTICAL HARMONY. 335 A. Reprise. Same as first subdivision in F. B. \ 2d theme in F-major or D-minor. Conclusion in F. At the end of the first subject in the reprise the modulation .must of course be altered. A coda is sometimes added to the conclusion as final ending. Or the conclusion may be extended. See Beethoven's Sonata in F-mi- nor, Op. 2, No. i. Andante If (a). D-MINOR, B>, OR Db-MAJOR. : ist Period. : ; 2d Period. : Coda. II (b) or this form may be substituted for (a). Principal theme extended. Intermezzo, (irregular.) Main theme repeated and somewhat varied. Coda. At least one change in measure should be included, for it is not well to have all the movements alike in this respect. Rondo III. In F. Principal theme. About 16 measures. Intermezzo, irregular and transitional. 386 GOODRICH'S ANALYTICAL HARMONY. Principal theme as at first, in F. 2d theme in some f A contrast to the parallel key. ( main theme. Eingang or cadenza, leading to the Principal theme as at first. 1 Termination, or coda in F. The difference between a regular period with coda and an ex- tended period may be illustrated in this manner : Regular period of 16 measures. Coda of 4 measures. I Extended period of 20 measures. The first is isolated, the second is continuous and uninterrupted. Irregular intermezzo is intended to bignify that the mensural pro- portion is uneven, and that the construction, being less melodious than the preceding period, does not divide itself into regular phrases and sections. This is also a peculiar feature of eingang, anticipation, coda, and termination. Tne student must not conclude from this that rhythmical balance and symmetrical proportion may be lightly set aside. Irregulai periodic construction is effective only as a relief to the even rhythmical balance of regular periods ; and the former must be so conceived that the irregularity is not especially noticeable.* It will be a pleasant task for the student to fill in these outlines, though some difficulty may be encountered in devising motives with sufficient affinity for the various movements. No diagrams are necessary for the sonatina in minor, because the outlines remain very nearly the same. * These features are illustrated in Musical Analysis. GOODRICH S ANALYTICAL HARMONY. 387 With regard to the slow movements, the first form (a) may be used for the sonatina in major, and the second (b) for the one in minor. In the rondo of the latter a recollection and stretto* may be sub- stituted for the termination indicated in the diagram. (If required, motives may be found in the Key to this work.) A few concluding sentences with regard to the manner of com- posing : Creative artists do not strum out their music from the key- board of a piano or an organ. That is manufacturing, not composing music. If one can conceive a theme, or a harmonic progression, it can be committed to paper without the aid of an instrument. And the very conception of an idea presupposes that its author knows how it will sound. f Certain students may, however, need the very practice w r hich this strumming process affords, and it might be advisable for them to test every phrase, section and period through the agency of a piano or organ ; at least until they can judge a passage independently of its actual performance. -These features are illustrated in Musical Analysis. fHans Richter asserts that while composing The Mastersin^ers, Wagner never sounded the piano in his music-room. GOODRICH'S ANALYTICAL HARMONY. 389 INDEX OF SUBJECTS. Most of the references are to chapters, indicated by Roman numerals, or 4o examples in notation. The pages are not given excepting where a subject is merely mentioned ; then they are indicated by cardinal numbers. A . .Accent. LI, LIT, LVI, LVII. Accompaniment. LXI. Affinity of Motives. LXX. Ex. 899. After Cadence. XLI, XLII. Altered Chords. XXIX, XLIU, XLIV, XLV, XLVI, LXVH. Ambiguous Cadence. Exs. 470, 471. Also Ch. XLIX. Amen Cadence Its Application. Ex. 452. Anticipation. LVII. Appoggiatura (Harmonic). LVI. Augmented 2d. XVI, XLIX, LXII. Ex. 796. Augmented 6th. XLIII, XLIV, XLV, XLVI. Augmented Triad. XXIX. Auricular Exercises. (See Practical Exercises.) Page. 185. B . .Base, Inverted. XXVII, XXIX, XXXIV, XXXV. Base, Real. XXVII, XXIX, XXXIV. Exs. 698, 809 (b.) Base, Suspended. Exs. 587, 589, 742, 871. C . .Character of Certain Harmonic Progressions Analyzed. XXXV, XXXVI, XLVI, XLVII, LXVI. Chord Connections. VII, VIII, XXII. Chord Movements. VII, VIII, IX, X, XI, XIII, XX, XXI, XXIII, XXVIII, XXXVI, LXVI. Chord Relations. XIII, XVIII, XXXVI, LXVI. Chord Representation. XXI, XXII, LV. Chromatic Harmonization. XXXVII, XXXIX. Also Exs. 748, 802. Chromatic Scale. LVII. Ex. 848. Close Position. V, VI. Comparison of Sub-dominant and Dominant in Relation to Tonic. Pages 220 and 221. Connecting Links. (See Notes of Connection.) VII, XXXVI. Consecutive Fifths. (See Parallel Movements.) Consecutive Octaves. (See Parallel Movements.) Counterpoint : Harmonic. LVIII, LIX, LX. Cross Relation. LXII. 3QO GOODRICH'S ANALYTICAL, HARMONY. D . . Deceptive Cadence. Exs. 449, 450. Derived Harmonies. XXIX, XLIII, XLIV, XLV, L, LXVII. Design. (Motive, Object.) LV, LVII, LX, LXVI. Development. LXX. Diapason. Exs. 884, 885, 886, and remarks following. Diatonic Progressions.' XIII, XIV, XVI. Diatonic and Chromatic Triad Progressions. Exs. 351, 844. Diminished Seventh Chord. XXX, XXXI, XXXIII, XXXIV, XXXVIIL XXXIX, XL. Direct Cadence : Authentic. XLJ, XLII. Directions for the Base. VIII, XXVII, XXXIV. Disconnected Progressions. XIII, XXXVI, XLYVI. Dispersed Harmony. LVIII, LIX, LX. Distinction between Variation and Development. LXX. Dominant Chord. XVIII, XX. Dominant Seventh Chord. XXI, XXII, XXIII, XXIV, XXV, XXXII. XXXVII, LIV. Double Pedal-notes. LIU. Double Appoggiatura. Ex. 678. Double Suspension. LII. Drone Base. (See Double-pedal.) Duophonic Chord. Exs. 24, 176, 177, 356. Duplication. XXII, XL.VII, LV. E.. Effect of Simultaneous Intervals in two-part Counterpoint. LIX. Ex. 717. Eingang. LXVIII. Eleventh Chords. LXVII. Embellishment. LVII. Enharmonic Representation. XXXVIII, LXVI. Also Ex. 171 and re- marks preceding ; and Exs. 846, 847. Enharmonic Transition. XXXVIII, LXVI. Esthetic Character of Rhythm. LXIX. F . .False Relation. LXII. Figurated Accompaniment. LXI. Form. LXVIII, LXIX, LXX. Fundamental Progressions. VIII to XXVI. Fundamental Harmonies. I to XXVIII. Q . .General Base. XLVIII. Ground Base. XLI. H . .Half-open Position. XXII, XXVII. Harmonic Minor Scale. XVI, XXXVIII. Harmonic Progressions. XIII, XVI, XXXVI, XLVII, LIV, LXIII, LXV, LXVI. Harmonic Tone. LVI. Hidden Fifths. LXII. Hidden Octaves. LXII. Holding Tone. (See Stationary Tone.) How to find a Root. VIII. GOODRICH 'S ANALYTICAL HARMONY. 391 I ..Imperfect Triad. XXIX, XLII. Imperfect Leading-tone. XLII, XLIX. Interval Defined. I, XXIX, XLV. Intervals in Parallel Movement. LX. Intermezzo ; objects of. LXVIII. Inversion; XXVII, XXXIII, XXXIX, XLIII, XLIV, XLV, LI, LII, LXIII. Inverted Harmonies. Exs. 5270 and b. J . Justifiable Fifths. Exs. 783, 831. Justifiable Octaves. Exs. 352, 830. K . . Key Impressions. Exs. 394, 530, 653. Key vs. Mode. pp. 169, 170. L . .Leading-tone. XVIII, XX, XXXI. Lower parts : Base, Baritone, Tenor. LVIII, LXV. Exs. 823, 828. Al. .Major Concords. III. Major Resolutions of the Diminished 7th. Exs. 436, 445. Major Resolutions of Dominant 7th. XXI, XXII. Melodic Minor Scale Harmonized. XLIX. Minor Concords. IV. Minor Resolutions of Dominant 7th. Exs. 547 to 552. Modulation. XVIII, XX, XXIII, XXX, XLIII, XLIV, XLV. Modulation and Progression Distinguished. XVIII. Monotone. XLVII. Motive : Semi-phrase. LXVIII. Ex. 887. N . .Natural Harmonics. Ex. 25. Natural Minor Scale Analyzed. XLIX. Natural Modulations. XVIII, XIX, XX, XXXI. Neapolitan 6th (so-called). Ex. 465. Ninth Chords. L, LXVII. Normal sth. I, II. Normal 4th. I, II. Normal Major Scale. I. Notation : Theory of, XXXVII, XXXVIII, XXXIX. XL. Notes of Connection. VII, XIV. O . .Omission. XXII, LV. Also Exs. 365, 366 and 654. Open Position. LVIII, LIX, LX. Organ-Point. LIU, LX. Outline. LXX. P . .Parallel Movements. XII, XXVIII, LX, LXII. Passing Chords. XXXV, XL, LIII, LXVII. Passing Modulations. XVIII, XX, XXXI, XXXVII. Ex. 530. Passing Tone. LVI, LVII, LXVII Pedal-note. LIII, LX. Pedal-note : Necessity for. Exs. 607, 610, 748, 756. Period. LXVIII. Phrase. LXVIII. Preparation of Discords. XXXV, L, LXVII. Q . .Quartet, Vocal. LVIII, LIX, LX. Quint Succession. (See Parallel Fifths.) 392 GOODRICH'S ANALYTICAL HARMONY. R . .Real-Base. XXVII, XXIX, XXXIV, LV. Related Keys. XVIII, XX, XLVII, LXVI. Remote Transition. LXVI. Resolution vs. Progression. XXV, XXVIII. Retardation. LI, LII, LX. Rhythm. LXIV, LXIX. S . .Secondary 7th Chords. XXXV. Secondary gth Chords. L, LXVII. Secondary Resolutions. XXV, XL, XLVI, LIV. Section. LXVIII. Semi-phrase. LXVIII. Sequence. LXIII. Also Exs. 347, 349, 380, 522. Single Tones as Chord Representatives. LV. Skips of a 3d in the Melody. X. Skips of a 4th in the Melody. XI. Sonata, Sonatina. LXX. Stationary Tone : Object and effect. LVII. Sub-dominant Harmony. XLI, XLII, XLVII. Sub-tonic as a Scale Degree. XLII, XLIX. Supposed Inharmonious Progressions. Exs. 541, 542. Suspension. LI, LII, LX. Also Exs. 869 to 873. T . .Tetrachord. I, XLIX. Thorough-Base. XLVIII. Tonality. XVII, XVIII, XX, XXXIII, XXXVII, XLIV, LXVI, LXX. Tone-quality considered. LXI, LXIV. Exs. 771, 772, 814. Timbre. (See Tone-quality.) Transition. XXXII, XXXVII, LIV, LXVI. Tritone. LXII. U . .Unity of Design. LXX. Unrulable Progressions. XII, XXVIII, XLVII, LX, LXII. V . .Vocal Quartet. LVIII, LIX, LX. Voice-parts. LVIII and Preface. This Index is designed more particularly for the use of advanced students in reviewing their theoretical work. The subjects treated are so numerous that a process of summarization be- comes necessary. All the information upon a given topic must, in reviewing, be gleaned from different parts of the book and focused upon that particular point. This will also enable one to make the necessary distinctions between elementary restrictions and final applications. THE AUTHOR. GOODRICH'S ANALYTICAL HARMONY. KEY TO EXAMPLES. (FOR SELF-INSTRUCTION.) NOTE. T HE principal seventh chords are indicated thus : I. Dominant yth ; II. Diminished yth ; III. Leading-note yth. Secondary seventh chords are marked IV, V, and sometimes III. Augmented-sixth chords are indicated by cardinal numbers, i, *. 3- Inversions are marked (i), (2), (3), in place of the old thorough- ba^e figures. The cipher, o, indicates a harmonic note above, or a root-note below. Solution to Ex. 72. Chapter IX. (V/ith connecting links throughout.) The chord of E-flat would be equally correct at these places + , but the G-minor chord affords more variety. 394 Ex. 77- GOODRICH S ANALYTICAL HARMONY. J _u i g &> &> .- *=* a- J J r-^^-^ *=e Skips of a 3d. Chapter X. Ex.81. g p ;^- Skips of a 4th. Chapter XI. Ex. 87. S^ -T2^&- 1=1: i =*=*3 - * -^r GOODRICH S ANALYTICAL HARMONY. 395 The chord of F^ minor may be used at (a) and (c), or the D-major chord could be substituted at (b). Thirty Harmonic Progressions in B=flat. Chapter XIII. g- r- r The first five of the thirty progressions are given, as an indication of the manner in which the others are to be written. These include every possible progression by means of concords. Preparatory Theme Harmonized in Two Ways. fcfe 3, 1 Ex. 109. Jt & ,_ SBfc J J | | These should be written in vSeveral scales. The order of progression is reversible. 396 GOODRICH'S ANALYTICAL HARMONY. Exercise on a Fundamental Base. Ex. 117. :g 3=:? *^f-9- -& 2T 3^%=\ 1 I g J (Two re-arrangements of this.) Another Method for Harmonizing Skips of a 3d. Ex. 125. 5858 -O. ,- a $: IMsconuected l'roKr-s-.ig'- 5 -r-g-ik-a L r*'-^i \ \ ^ i J i -* ^ J * L r GOODRICH S ANALYTICAL HARMONY. Chapter XVIII. Table of Modulations. 397 r=^ '&=^%- w^s- RH to C min. BJ2 to D min. BJ2 to F." BJ2 to G min. Two more arrangements of this, with the same base. Modulatory Theme Harmonized. Ex. 154. T2==4=q= r^ =1 -S^=g *-+!$ kg , jj M \-*<& 0- l ^ 1 ^ 1& * Concords only. Efc a^j--i=^=jj^E^=^ EEE 3d or sth Omitted from Dominant yth Chords. Ex. 185. m Major and Minor Resolutions. Ex. 193. tdb if, * - > H-iT i" :f r : - 1 ^ =t=ib=Jzd?Q 3EM^^=^=3 ^ O I 398 GOODKICH'S ANALYTICAL HARMONY. =fe: -1 0- Inverted Bases. Chapter XXVII. Ex. 248. HH 2? ^J (2) (2) (2) (2) (2) o i The inversions (2) are the result of melodic progressions in the base. Primary Resolutions of the Diminished jth. Ex. 306. AH P" *-^^* * ^ P * f 4 ' P t^ssstf. p V-y r o j *.__ i n i ii cy 1 * Jp f . -4 _/ - 9 -^ , ^* r i ii fE Ex. 309. y&-^2- AH fundamental progressions, excepting + , (2). Corresponding Dominant 7th Chords. Chapter XXXII. rffetF i Ti" i * > -x ' #J# ' I > , ^2~ - ff^SB3rr?Mi*S Enharmonic equivalents, (a) and (b). GOODRICH'S ANALYTICAL HARMONY. 399 Intermediate and Terminal Resolutions. Chapter XXXIII. %=3==&&^-U&^=f=*=\ -0 -m 9^- m 1 9 m ^^ m 1 t" ^ " " 1 ._ _ 1_1 _i 1 , ! 3 f^-f-rf-M* -+ \ (1) (1) (2) o (1) (1) (2) o +4*1 j J-J 1 (2) (1) (2) Two more arrangements of the last. 3200 335- Farther View of Inverted Bases. ^-4 -' - ' -J J-jtLJ iT-^ -4 1-+^. i- 3 **- ^ ;4r l==: > ^^ 3E (2) (2) (1) (2) (2) - * i- =^=3? 1 (2) (3) i -* * The yth ascends here to avoid the half-open position, and because this is not a final cadence. It is, rather, an intermediate progression. Chapter XXXVI. Exs. 365 and 366. In the first chord g and b represent the G-major triad. Therelj/e d is the remaining note. In the next two chords e is wanting, thus : 400 GOODRICH'S ANALYTICAL HARMONY. The fourth and fifth chords are complete. Small notes indicate th. remaining intervals of the sixth, seventh and eighth chords : X The harmony is sufficiently represented in the original example. But in supplying adventitious parts it is often necessary to under- stand the theory of chord representation, especially where but two notes of a four-fold discord are present. It is evident that d is want- ing in the last two chords of Ex. 366. Sequence of Essential Discords. Ex. 380. I I I I I II I I I O (2) Chapter XXXVIII. Enharmonic equivalents. GOODRICH'S ANALYTICAL, HARMONY. 401 These represent merely the changes in notation of the three primr.ry diminished yth chords, A, B and C, resolving naturally to fifteen minor key-notes. The inversions are also to be worked out. Sequence of Diminished yth Chords. Ex. 408. Ex. 429. Passing Chords. Chapter XL. - & fia * rr Ex, 481 and 482. Augmented 6th Chords. No. 1. | I Ex. 505. \ 3. rp3ZZC " f _ I* _ 1 -S >i 1 ^ Y ' ^J Re-arrange, but do not invert, this exercise. 402 Ei.. 506. GOODRICH S ANALYTICAL HARMONY. g * -fa * ** >s in 2. |^i^__j j. J- V-& *_ 1; ^ f * . *, . 2S3 T ^?~ Pu* 1 *k~ p 1Q0 - -2 * * -* --* H KF QF 7 DM 832 III ' r ^r 1 - i ' * f "r (S' 1 IV ^ 1 i i i r 1 g^ T r J I/ [?(- 1 1 p-^-f ^ ^=\ (2) Sequence of Dominant jth Chords Terminating with an Augmented 6th Chord for the Cadence. Chapter XLVI. ~**m iTu l,r ?" <2 ijJ e> b In minor. In example (a) the chord numbered 3 is an enharmonic expediency ; c% and of the essential discord on G-flat. In example (b) only one enharmonic change is made : C ? is substituted for d t> in the descending sequence in order to make the direct cadence in G-minor. In all such instances the augmented 6th chords are resolved according to previous directions. Suspensions Resolving to a Changing Harmony. Ex. 595. fr-} & (S ,5. S -fi>: & 1 t^-r "^ r ^^^f" I^-W- -= GOODRICH'S ANALYTICAL HARMONY. Appoggiaturas Harmonized. 403 Ex. 669. 20 p*. foff PTJ I *i- i=tztzz== =|= Harmonic Counterpoint. Chapter LIX. 2 3 Ex. 729. * In place of this passing diminished yth chord the essential discord may be retained, considering the <:# as an appoggiatura. Passing Tones and Appoggiaturas. Ex. 73irt. M In the first measure the tenor might sing the tone between a and c, as it would accord very well with the base and soprano. But this would compel the contralto to descend to e. A slightly different arrangement is added Ex. 732^. 404 GOODRICK'S ANALYTICAL HARMONY. This is an improvement, though the first measure is really melodic counterpoint. Elaboration of Ex. 739. 3 i t=f=? I LJ Harmonic Sequence. Chapter I. XIII. Ex. 8oga. 5 j-* __ \ | 1 fm ff * i P w ~1 " K i ^ ^35- S S 4 J J / - 1 F 3d. m / *~~ m * ^ 4th. 3d. ^ i i .^. ^ t~\* ^ J- 1 *' tf i 1 ** ^ _ ^ ^^ ' ^ i 2: IT >J r 1 J kj Lj -* fix. 8ogi>. Secondary and Principal Sevenths. is* ^ ' i^-^-h- 11 1 ; g~r^ -f-g '- zz =tg ^Z=*n^:} ' b V V III IV (3) (1) (1) ^^^^ (3) (1J Motives for Sonatinas in F-major and in D-minor- Allpgrro. a niod*rato. b Each motive is divided into semi-phrases, which may be used sepa- rately in the development.