Ne w-YorK 70 , S c h i rm e r THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES Music Library A PRIMER OF PDERN MUSICAL TONALITY; INTENDED FOR SELF-STUDY, AS AN INTRODUCTION TO THE STUDY OF H A E M O ^ Y. BY J. H. CORNELL. THIRD EDITION, REVISED AND IMPROVED. NEW YORK: G. SCHIRMEF-J, 35 UNION SQUARE, 1904 COPTRIGHT, 1877, BY G. SCHIRMER Note. — The author of this little work has availed himself of the oppor- tunity of correcting, in this third edition, some slight inaccuracies in the first, to which his latei studies in Harmony have called his attention. Electrotyped by Smith & McDodgal, 82 Beekman St. Music Library / INTRODUCTION MUSIC, whose noblest mission it is to awaken in the soul cer- tain sentiments and affections, uses for this purpose fleeting sounds or to7ies, whence it may be briefly defined as the Science or Art of Tone — as, indeed, it is called in German die Tonhunst, literally, " Tone-art." The explanation of the physical laws of the production of mu- sical sounds, or tones, belongs to the Science of Acoustics. Suffice it to say, in this place, that tone is the result of rapid vibrations (oscillations) of some elastic body in a state of alternate tension* and relaxation. These vibrations, agitating the air, cause sound- waves, and thus reach our ear, producing upon us the impression of a tone, which is liigh (acute), or low (grave), as to its pitch, ac- cording as the vibrations are very rapid or less so, and the sound- waves in the air correspondingly short or long. The extremes of pitch recognized in our musical system are represented by the lowest tone of the largest modem Organ and the highest tone of the modern Piano-forte. The lowest tone used in the Organ, viz.: C^ — found only in instruments of the first class — is produced by a pedal-pipe 32 ft. in length (whence the tone is * The word " tone " comes from tlie Greek " tonos," originally signifying tension. M - ■»■ ATL.. 4 INTRODUCTION. called 32 ft. C), and has 16^ double vibrations* in a second: it is, liowever, pretty generally conceded that this tone is too deep to be perceived by the ear as a tone of a certain pitch, and that it is not until we reach 41^ -snbrations per second, givuig E^, the lowest tone sounded by the double-bass, that we obtain a musical tone proper. The highest tone of the modern Piano-forte of full com- pass, viz. : C'g, is produced by 4096 vibrations per second. From the lower to the higher of these two extremes there is a contumous rise in pitch, passing through every possible gradation, thus afford- ing a multitude of tones mathematically different f (i. e., each tone having its own fixed number of vibrations), from which multitude a certain series of tones, with no smaller difference of pitch between any two than that of a half-step, has been selected to constitute the material of our modern musical system. Tones, as forming the material of music, have three principal * By a dovUe vibration is meant the complete excursion — say of a sound- ing string — backwards and forwards, just as the movement of a pendulum to the right and back to the left, or vice-versa, constitutes a complete or double oscillation. The word " vibration " is used by English-speaking writers in the complete sense ; by the French and Germans, however, generally in the sense of a single vibration. f These extremely fine distinctions of pitch may be produced to a certain extent on such stringed instruments as the violin, etc., but most perfectly by the aid of the " Sirene," an acoustic apparatus invented expressly for this purpose, by which the exact number of vibrations to every possible musical tone is unerringly indicated. It is asserted by scientists tbat irifhin a single Octave from one to two hundred distinct degrees of pitch can be rendered per- ceptible to the ordinary ear, and to that of the trained violinist a much larger number. (See Sedley Taylor's Treatise on the Physical Basis of Music.) From these considerations we may form an idea of the great number of possible different musical sounds, and consequently of the extent of the reduction of tone-material implied in our modern musical system. INTRODUCTION. 5 characteristics, viz: Filch, Duration, aud Force* — in other words, they are high, or low (as explained above) ; long, or shori ; loud, or soft. Moreover, they may be considered as separate, or in combina- tion. Hence the whole science of music may be comprised under these four general heads : 1. Melodics, the doctrine of the com- position of melody from tones differing in pitch and forming Inter- vals; 2. Rhythmics, which treat of the disposition of tones in a piece of music according to their relative duration, implying the grouping of them into measures, with regular accents, etc. ; 3. Dy- ifAirics, whose subject-matter is the various degrees of power, i. e., loudness and softness, to be given to the diflFerent tones — and in general, what is called "musical expression;" and 4. Harmonics, the theory of Harmony, i. e., of combining tones so as to sound simultaneously and form chords. Passing over the branches of Rhythmics and Dynamics, as not coming Avithin its special scope, the present little work is concerned with the necessary preliminaries to the study of Harmony and Counterpomt.\ It treats incidentally of the primary chord-forma- * Another characteristic of tones ia Quality, or Color (in German, Klang- farhe; in French. Tuabre) ; but tliis forms a specific division, in treating of music. The great German authority in acoustics. Professor Helmlioltz, baa demonstrated that the difference in the quality of one and the same tone (as to pitch) on various instruments, is due to difference in the form of ttie vibrations, i. e., in the outlines of the sound-waves, as observed by him with the aid of the microscoie. f Counterpoint may be called the art of combining melodies, in distinction from that of combining tones into chords, which is properly called Harmony. " When we look at a piece of harmonized music from the contrapuntal \)(nnt of view, we mostly direct our attention to the melodies of which each part should consist. . . .We look at these melodies as it were horizontally, along the paper," Inot perpendicularly, as in the case of chords) "and the harmonic deri- vation of the chords they may jointly produce is kept out of sight," etc.— ^> F. Ouseley, Counterpoint. 6 INTRODUCTION. tions (Triads), thus touching the fourth branch — Harmonics, but is devoted chiefly to Melodies, and to this branch, indeed, only to a certain extent, viz : so far as it comprises the analysis of the mate- rial from vs^hich melodies and harmonies are constructed, {. e., of tones, regarded as differing in jntch, as modified by chromatic alter- ation, as variously notated (from the standpoint of musical orthog- raphy), as forming the different Intervals, and as arranged in the two diatonic series or tone-families respectively known as the Major and the Minor Key. Hence this little work claims to be nothing more than its title implies — a Primer of Modern Tonality, using the word "tonality" to express a system or arrangement of tones, as best exhibited — in the case of modern tonality — in the Diatonic and the Chromatic Scale.* The usefulness of such a work, however, is deducible from the fact that it is impossible — as almost every teacher knows by experience with one or the other of his pupils — to make solid progress in the study of Harmony and Composition without a previous thorough and exhaustive knowledge of the material of music, i. e., of tones, regarded under the various aspects above indicated. Accordingly, the author of this Primer has endeavored to ex- plain, to the best of his ability, step by step, and with accuracy * Thus we say, speaking in this sense, that the " tonality " of the plain- chant or " Gregorian " melodies, for instance, which are used to some extent in the public worship of the Roman Catholic Church, is so unlike our modern tonality tliat it is difficult, if not altogether impossible, to Imrmonize most of these melodies satisfactorily. So, too, we say that in the respective musical systems of the Chinesp, the Turks, etc., there is a different " tonality " from ours, I. e., a different selection and arrangement of tones, a different scale, etc., involving, of course, melodies of a peculiar and to us strange character — just as, on the other hand, our melodies are strange to the natives of barbarous, or semi-civilized countries. IXTRODUCTIOy. 7 and conciseness, all that is necessary to be known in this matter, and to dispel the confusion of ideas — not to say the positive igno- rance — on the subject of touahty so prevalent as to be sometimes found even among those who pass for "musicians." Indeed, in most of the educational works on music which have hitherto ap- peared in English, the matter of tonaUty is by no means exhaust- ively treated, and in such explanations of it as are vouchsafed there is so much obscurity of language, so httle care to give exact dejinu tions of technical terms, with appropriate exemplitications, that tho confusion of ideas alluded to as being so prevalent is hardly to b» wondered at. The author's experience with pupils in Harmony long ago convinced him of the necessity of an elementary treatise hke the present work — for, surely, it is inconsistent with thorough- ness of instruction to enter upon the subjects of Chord-combina- tion, Suspension, Modulation, etc., with the student who cannot— for example — ^give a scientific definition of the tonal " interval," or explain the structure of the Diatonic Scale. It was thought that this little Primer might be welcome especially to teachers of Har- mony and Composition, as an inexpensive hand-book to be recom- mended to their pupils for self-study. A few words of explanation are offered, in view of some pecu- liarities of this work. Certain expressions will perhaps be found strange, even awk- ward — but only, as the author believes, because they are netv. Our English musical vocabulary is on the one hand miserably defective, and on the other hand encumbered with expressions which are not significant, or do not clearly define, frequently necessitating awk- ward circumlocutions. We need to supply the place of such ex- pressions by exact, concise terms, and to introduce new ivords. 8 INTRODUCTION. coined from other languages, or adapted or compounded from our own tongue, after the German manner. At the risk of incurring the odium which attends every innovation, the author has endeav- ored to do something towards supplying these needs, convinced that it is, generally speaking, less vital to the interests of musical science to retain established usages merely hecause estallished, than to enunciate the principles of that science in exact and concise terms. The system which acknowledges only two hinds of Intervals — major and minor — as normal in the Diatonic Scale, is that of the eminent musical theorist and teacher, 0. F. Weitzmann,* and com- mends itself by its perfect simplicity. Thus, in comparing, for in- stance, the several Fourths in the Diatonic Major Scale of C, we find one — -f — h — greater by tonal measurement than every other one : how natural, therefore, to call this one major, and the others, relatively to this one, minor, thus following the same rule as when we compare Seconds, Tliirds, Sixths, and Sevenths. The same reasoning applies to the case of what is called in this system the major Fifth, or the minor Fifth. Undoubtedly, there is here, aa in other similar cases, a question o^ name chiefly, the thing remain- ing the same: nevertheless, even in names there is sometimes a choice ; and, as between two systems of naming, that one would seem preferable which — as we claim for Weitzmann's system — con- duces to greater simplicity of doctrine by eliminating such excep- tions to rules as are not founded on actual necessity. * Carl Friedrich Weitzmann was born Aug. 10, 1808, in Berlin, where he still resides. His " Harmoniesystem " (Prize Essay) has attracted great atten- tion throughout the musical world. MODERN TONALITY. OHAPTEB I. Tones, and their Notation. 1. The expression Toxe is used in this work to signify a musical sound, produced by the agency of the human voice, or of a musical instrument, etc. As every tone is generated by a cer- tain number of vibrations of an elastic body, when we speak of a tone we mean a definite musical sound, distinguished from other sounds by its pitch, i. e., its relative acuteness or depth. It is exclusively in this sense that we shall use the word "Tone":, to express the ideas conveyed by the popular terms, " whole tone '* and "half tone" (or ''semitone"), other and more appropriate terms are employed.* 3. Out of the multitude of possible diiferent tones (differing in pitch, as above explained) twelve have been adopted as an ample material for the purposes of modern musical art, and their ar- rangement in a scientific system constitutes our modern tonality, * It is every way desirable that the word " Tone " should be used exclusively in its proi)L'r sense. According to the too common method of teaching, the pupil is told, for instance, to strike or sing c (which is undeniably a tone), then d (which is another tone), and is informed that the progression or inter- val is a tone ! or, it is taught that the tiro tones, e and/, or h and e. give a half- tone! The English translator of Dr. Marx's Allgcmeine Musiklelire is driven to the expedient of ])utting the word " tone " in italics when it is used in its proper sense, and in Roman letters when it means (as he says) " distance or interval" ( ! ). See Note, page 16. 10 MO D E R N T JSfA L IT Y, as distinguished from the tonal systems of antiquity. These twelve tones are repeated over and over again in a higher and a lower pitch, as shall presently be explained. Among them, seven are classed as primary tones, and named after the first seven let- ters of the alphabet. A, B* 6', D, E, F, G. We shall turn our attention first to these seven primary tones. 3. Tones are expressed to the eye by notes (^, f, ,• r, 5, etc.f ) written on thf degrees (lines and spaces) of the Staff, or'system of five parallel lines. The degrees are generally counted upward from the lowest line. Fig. 1. -1^ -3 4, :r=^ -7- -g- 4. Each staff-degree takes a distinctive, fixed name, represent- ing a primary tone,J as A, C, G, etc. To this end certain charac- ters called Clefs are used, each one naming a certain degree, from which the names of the other degrees are determined by following the proper order of the letters. A, B, C, D, E, F, G. The G * The Gentians use S" instead of B, and B for what we call B pit. \ The consideration of the diflferent musical notes, as indicating relative duration of tones, belongs to Rhythmks. X Any staflFdegree may, however, be the seat, not only of its primary tone, but also of a second, third, fourth, or fifth tone, regarded as a modif ration of the primary tone, consequently written on the same degree (see 20, p. 19). Again, one and the same tone, as V, G, etc., may be written (with different names) on three different staff-degrees. These things find their exjjlanation later — suffice it to say here, that there is great flexibility (so to speak) in the staff, admirably adapting it to the purposes of a very manifold notation of musical sounds. For, though a given tone is, as such, absolutely speaking, always the same (being produced by a certain number of vibrations), yet its iiaine is not an absolute, invariable thing, but changes according to the laws of musical orthography — hence its seat on the staff is variable also. MODERy TONALITY. 11 (. lef, placed on the 3d degree {2d line) of the staff, names that de- gree G : Flff. 2. IT and the F Clef (Bass Clef), placed on the 7th degree (4th line), names that degree F : Fig. 3. -jj A ^ 'E' -D -B- The F Clef is used in the notation of the deejier tones, the G Clef in that of the liiglier or more acute. 5. The key-board of the piano-forte exhibits to us the entire range or compass, from the highest to the lowest, of the tones most used in music. 6. If now we start from any luldte key, and strike the succes-* sive white keys in an uninterrupted ascending order till we reach the eighth above, we shall find tliis eighth tone to be a repetition of the first, but in a higher pitch.* Or, starting from the same first tone and descending, we shall find the tone given by the eighth white key below to be a repetition of the first in a lovoer pitch. Con- sequently, if that first tone is C, for instance, its repetitions above and below will also have the same name, C. Each repetition * We say, a higher pitch ; for if the pitch of this eighth tone were not merely higher, but also absolutely different, as is, in respect of the starting-tone, that of each of tlie six other tones, the (Mi^-lith tone could not be a reiwtition of the first. But, in fact, when both the latter are sound or \^\^\ the latter, when neutralizing the ^ or X. 20. These five characters, the ^, x, [?, [?[?, and ^, are called Accidentals, or better. Chromatic Signs, and they effect what ia called in general chromatic alteration, which, again, gives rise to the chromatic nomenclature, or naming of the altered tones. Thus, when the \, x, I;, or [;[; is placed on any staff-degree, the degree retains its primary name, but takes an addition, or qual- ■ification, being called, for instance, C sharp, C double-sharp, O flat, or C douhle-flat, as the case may be. In this way eacli staff- degree may be the seat of four additional tones, as in the following examples : Fig. 8. —1 ^^ tm S9 \m ^^» i n — ~~ 4,^ Sf — ^ " :-!= —■ «,-_::^»__^ ^ bb* 20 MODERN TO NA LI TY QUESTIONS. 16. How many diflbrent tones, counting also the intermediate tones, are contained within an Octave?. .. .17. In what sense are the intermediate tones derUatiees of the primary tones?. ... 18. How is a tone modified as to pitch? What does the Sharp indicate? The Double-sharp? The Flat? The Double-flat ? When is the Double-sharp used ? When the Double- flat ?....! 9. What is the effect of the Cancel? Does the Cancel raise, or lower ?. . . .20. What is the general expression for the effect of the chromatic signs ? How does the chromatic alteration of tones affect their names * CHAPTER lY. The Chromatic Scale. 21. A series of twelve different tones, ascending or descending by consecutive half-steps (the Octave of the starting-tone being generally added, to complete the series), is called a Chromatic — more strictly, CHROMATic-DiATO]!fic*— Scale. This Scale is in fact a Diatonic Scale with the intermediate or derivative tones added to it, the latter being an effective means for embellishing and enriching both melody and harmony, thus obviating the com- parative stiffness and monotony necessarily attendant upon ex- clusively diatonic progression. 23. The Chromatic Scale is usually Avritten in two ways ; viz. : as an ascending series with the chromatic signs of raising, and as a descending series with those of depressio7i, as in the following ex- amples, in which the primary tones are represented by white notes, to distinguish them from the derivative tones, written in black notes : * See Remark to 24. M D E R X TO XA L I T Y. 21 Fuj. .'>.— Chromatic Scale op C. Ascending. H^- Z E5 ^ ^ ^ ^ ^*"^ -«^ =^= 1^ -^ ps^ ■f::^ t* 2 3^ :: c c$ d dt e f ft g gt Descending (to be read backward from the right). a at hm J) c' db d eb e f g^ ff ab a Vo Fig. 10. — Chromatic Scale of A. Ascending. 1"^' ^ T- '^ i» --^ -^s- zzi-n =2= -Jt«= —& T, A A% B c ct d dt e Descending (to be read backward from the right). ti ;=: bm ^ f ft 9 Qt 1^ a _^ ^ ^ c- P* ^ A B*) B c db d So e f ^ g (A> a 23. The theory of the chromatic nomenclature, as above il- lustrated, may he thus summed up : each iutermediate tone is regarded as a raising, in the ascending series, and as a lowering, in the descending, of the primary tone sounded immediately before, consequently takes tlie name, with the suitable addition to it, of the tone of which it is a modification.* Two-fold Distinction of Half-Steps, and of Steps. 24. It is now proper to state that there are t7i'0 kinds of Ilalf- steps, viz. : Diatonic and Chromatic — a distinction which is im- portant, yet involves no difTiculty. Illustrations are found in the above figures 9 and 10. In the diatonic half-ste]) the second tone takes another name, and is written on the stajf-degree imme- diately uhove or below that occupied by the first tone {e. g., c^d, * See Appendix I, p. 83. 22 MODERN TO NA LIT Y. d^e, e-f, etc.) ; in the cliromatic half-step the second tone stands on the same staff-degree as the first, and has consequently the same name, with a qualification added to it {e. g., c-c^, «-f^^, etc.). One and the same tonal progression (half-step) is capable of a two- fold musical orthography; thus, the progression from any (', for instance, to the tone immediately above may be written C-6 |f (chromatic half-step), or C-D^ (diatonic half-step), according to circumstances.* It should be well understood that it is not the mere use of the cliromatic signs which makes the cliromatic half- step, since from F^ to G, or from B^ to A, is as much a diatonic half-step as from E to F, or from c to B, because in each case the name and staff'-degree are clianged — a peculiarity of every diatonic progression, to which we have already called attention (see Re- marh to 13). Remark. — From the two-fold distinction of half-steps as dia- tonic or chromatic, it will be seen why the scale generally called " chromatic " is more properly called " chromatic-diatonic." The diatonic scale progresses by diatonic degrees exclusively; the chro- matic scale, however, does not progress by chromatic half-steps exclusively, but, as the examples show (Figs. 9 and 10), by diatonic and cliromatic half-steps intermingled. Hence the greater fitness of the name "chromatic-diatonic," or "diatonic- chromatic." 25. The Step may also he written diatonically or chromati- cally. Thus, e. g., the progression g-a (diatonic step) may also be written g-gx (chromatic step). However, the step mostly used is the diatonic, and it is in this sense only that we shall speak of it henceforth : it is composed of a diatonic plus a chromatic half- step, or vice-versa, as in the following examples, in which the sign I — distinguishes the chromatic half-step from the diatonic ^^. * This is explained by the law of correct musical notation (musical orthog- lapliy), a subject of which we have briefly treated in Appendix II, p. 8& M D E R X TO NA LITY. 33 Fi(j. 11. i i=fea -^ ^ Iz =}&=: =:&2= Exercises on Fig. 9. 1. Go through all the half-steps, first ascending, then descend- ing, and distinguish the kind of each, i. e., whether diatonic or chromatic. 2. Point out all the diatonic steps, first ascending, then de. scending {e. g., not only the steps c-d, d-e, etc., hut also fj|-^j^, and all the others involving the use of the chromatic signs). QUESTIONS. 21. What is tlie Chromatic Scale? Does it include the Diatonic Scale? ... .22. How is the Chromatic Scale usually written ?. . . .23. Give a sum, mary of the theory of the Chromatic nomenclature illustrated in Figs. 9 and 10. . . .24. How many kinds of Half-step are there? What is the dis- tinction between them ? Does the use of chromatic signs in a half-step neces« sarily make it a chromatic half-step ? Give examples of diatonic half-steps written with chromatic signs. The Diatonic Scale progresses by diatonic degrees exclusively ; the Ghromutic, not by chromatic half-steps exclusively, but by diatonic and chromatic, intermingled ; what name, therefore, is most appropriate for the latter scale? (Remark.). .. .25. Give an example of a chromatic Step. Which kind of Step is mostly used ? How is it formed ? 24 MODERN TONALITY. CHAPTER Y. Intervals in the Diatonic Scale. 26. We have hitherto considered but two different tonal pro- gressions — the Half-step and the Step; we must now turn our atten- tion to greater ones. Every progression from one tone to another, involving a change of name and of staff-degree, forms what is called an Interval; in other words the Interval is the result of such a progression from one tone to another as involves a change of name and of staflf-degree.* The term " Interval " tlierefore sup- poses two different tones,\ and has reference to.their degree-rela- tionsJdp, or relative positions on the staff. In writing, e. g., the two tones c, d, I place d on the degree next above c, and the c-degree being counted as 1, the next above will be 2, and I say that d is a "Second" to c {L e., above c, as we generally count upward from the lower tone. The word "above" is generally understood, but in counting doimiward from the upper tone the word "below" or "lower" must be expressed). Again, I write c, g, aud since the ^-degree is the fifth above c, I say that g is the * Hence the Octiwe cannot be said, strictly speakinor, to form with its lower tone an Interval. " The Octave," says the late Dr. Moritz Hauptmann, in his Harmonik, " irajn-esses us as a repetition of the same tone in a higher pitch. It has no special significance for harmony, and is, in this sense, not a liar monic interval, as is the Fifth or the Third," etc. Similarly, what is some- times classed as an altered Interval, viz., the " augmented Prime," as c-cH, etc., does not appear in this work among the Intervals. For as c-c, is merely a Unison, r-c$ is simply a chromatic half-step, or an altered Unison. + It is as confusing and incorrect as it is common, to apply tho word " Interval" to one of two tones, as, e. g., in the case of the Second, c-d, to call d the " upper interval." Why not upper tone (of the interval) ? MODERN TOyALlTY. 25 "Fifth " to c, or, counting downward, tliat c is the '"' Fifth below" g ('• lower Fifth " to g). Fig. 12. ^—^w--^'— , 4 — 5« ^"-^ ^ * H 27. We distinguish in the Interval two elements — Denomina- Tiox i;nd KiXD. The scale affords, within a compass of thirteen staff-degrees* (involving, of course, some added lines), six different Dexomixatiojts of Intervals, viz. : the Secoj^d, the Third (or Tierce), the Fourth (or Quart;, the Fifth lor Quint), the Sixth (or Sext), and the Seventh (or Sepfiina).\ Each of these terms expresses the relative positions of two different tones on the Staff, fls explained above. 2S, Each Dexomination of intervals in the Diatonic Scale is tiL'ofohl in Kind, viz. : either minor (lesser), or major (greater). These are the normal intervals, as found in the Diatonic Scale j or, to put it differently, this Scale contains only two Kinds of intervals, viz., minor and major, of the several denominations. A major interval is greater by a cliromatic half-step (^24) than the minor interval of the same denomination. For instance, given * If we did not go beyond tlie Octave, say in the Diatonic Scale of C, we could not fonn the Seventh to any tone higher than D, nor the Sixth to any tone above E, nor, in short, the TJdrd, Fourth, and other intervals abova b' f It would seem desirable to have distinct names (as in German) for the Intervals, rather than the ordinals, Third, Fourth, Fifth, etc., and for this pur l>ose the names joriven above as o|)tional, %iz., Tierce, Quart, Quint, Sext, and Septima, are suggested. The ad()i)ti()n of them may be urged at least in ths formation of compound words. For instance, the expressions, " Tierce- r^si- tion," " Qnint-pnsition " (of the Triad), are surely less confusing than " Third- position," "Fifth-Position"; again. " tierce-rdatcHl," "quint-related," '•tierce- relationship," "quint-relationship," are less cumbersome expressions than " re lated by the Third," etc., " relationship by the Fifth," etc. 26 MODERN TONALITY. two diflFerent progressions, say B-f and c-g, the number of slaif- degrees to be counted will in each case be found the same (5), ao that in each case we have a Fifth. So much for the denomina- tion of the two intervals. We determine the kind of each Fifth by counting the half-stejjs involved in each. Thus, B-f contains one half-step less than c-g ; therefore B-f is a minor (lesser ;, c-g a major (greater) Fifth. To make it clear that m this, as in every sim- ilar case, the difference between minor and major is that of a chr'o- matic half-step, the followiug illustrations are given (Figs. 13 and 14), from which it is plainly seen that while the Fifth, B-f, con- tains the same number (4) of diatonic half-steps (marked ^^ ) as the Fifth, c-g, the number of its chromatic half-steps (marked i — i ) is styialler hy one. Fig. 13. — MixoR Fifth. z$m •— S *^ Fig. 14. — Major Fifth. ■ ^-—T ,=^ ^ ~g »= -m—^—^ This can be made still clearer, if necessary, in the following way : we take this same minor Fifth, B-f, and make it major by adding a half-step to it. But what kind of half-step ? If we add a dia- tonic half-step, we obtain the following results: Fig. 15. Piatoiiic Half-step added abnve. Diatonic Half-step added lielow. ^E =J»=:*=i?^ *: =35=: but in either case we have no longer a Fifth ; whereas, by adding a chromatic half-step, as in Fig. 16, the interval remains a Fifth, being merely changed in Mud, i. e., from minor to major, the proof of which is that it now contains the same number of half- steps (7) as the major Fifth, c-g (Fig. 14). MODERN TONALITY. 37 Fig. 16. Chromatic Half-step adiloil above. Chromatic Half-step added below. ^=^i^^*3^=^^^=te^^i^$i^^: 29. We proceed to examine the various Intervals, drawing our illustrations from the Diatonic Scale of C, and beginning with Seconds. 30. Any two different tones, written on adjacent degrees, form a Second. Eemakk.— It is said, "any two different tones," for two tones might appear on adjacent degrees without forming a Second, or any progression whatever, as in Fig. 17, which is an example of " Enharmonic change," to be explained later. Fiy. 17. it ■±MZ Jt '-=S»- 31. The Diatonic Scale of C contains the following Seconds : C-D, D-E, E-F, F-G, G-A, A-B, B^c. The progressions E-F and B-c are diatonic half-steps, all the other progressions are steiJS. "We have seen (35) that the step is composed of a diatonic plus a chromatic half-step ; hence the Seconds E-F and B-c are lesser or minor, all the others being major Seconds. Remark. — Every diatonic step is a major Second, and con- versely, every major Second is a diatonic step. 33. The minor Second is the smallest of all the intervals, being simply the diatonic half-step. Remark. — Though every minor Second is a. half-step (diatonic), not every half-step is a minor Second. In the chromatic half-step the two tones stand on the same degree, therefore cannot form a Second, as appears fnjm iJO. 33. A progression by Seconds, minor or major, is called grad' 28 MODERN TONALITY. ual [i. e., from one degree to the next above or below), as distin guisbed from one in Avhich one or more degrees are passed ovei which is called a shipping progression, or simply a skip or Ica;^ To the latter class belong of course the remaining Intervals to b( considered. Thirds. 34. We find in the Diatonic Scale of C the following Thirds : C-E (two steps), D-F (step and half-step), E-G (half-step and step), F-A, G-B (in each two steps), A-c (step and half-step), B-d (half-step and step). The progressions D-F, E-G, A-c, and B-d, being severally a half-step smaller than the others, are minor, all the others are major Thirds. Fourths. 35. The Diatonic Scale of C gives the following Fourths: C-F, D-G, E-A, F-B, G-c, A-d, B-e. It will be found that one of these progressions, viz. : F-B, involves three steps, and every other only two steps and a half-step. Therefore the Fourth F-B is major, all the others are minor. Eemark. — The minor Fourth is sometimes called the " per- fect," and the major the "augmented" Fourth. As to the former appellation, see Note, p. 53 ; as to the latter, Remark 2, p. 46. Fifths. ,*>6. The Diatonic Scale of C gives the following Fifths : C-0, D-A, E-B, F-c, G-d, A-e, B-f. The last progression, B-f con- tains two steps and two half-steps, each of the others three steps and a half-step. B-f is therefore the 7ninor Fifth, all the others are major. Remark 1. — The minor Fifth is sometimes called the "dimin- ished," and the major, the "perfect" Fifth. As to the former ex- pression, see Remark 2, p. 45 ; as to the latter. Note, p. 53. Eemark 2. — The Fifth is a very important interval, especially J/ ODE li ^' T O XA LITY. 29 in harmony. Two tones forming the 7najor Fifth are in the closest relationship to each other after that of the -Octave. Sixths. 37. In the Diatonic Scale of C are the following Sixths : C-A, D-B, E-c, F-d, G-e, A-f, B-g. The progressions E-c, A-f, and B-g are minor Sixths, being severally formed by tlivee steps and two half-steps ; while each of the others — major Sixths — involves four stejjs and one half-step. Sevenths. 38. The Diatonic Scale of C gives the following Sevenths : C-B, D-c, E-d, F-e, G-f, A-g, B-a. Of these C-B and F-c are major Sevenths, as involving each a half-step more than any one of the others, which are therefore minor Sevenths.* The "Half-step Formula" of Intervals In the Diatonic Scale. 31). The Half-step Formula is a simple and sure method of determining an interval as to kind. Given two different tones written on the Stafi", the denomination of the interval is, as we have seen, determined by the number of staff-degrees counted, from the lower to the upper tone (or vice-versa) ; but the kind of interval, i. e., whether it is minor or major, is determined by the number of tones, counting all the intermediate tones from the lower to tlie upper tone inclusive. 40. The latter process gives a fixed formula for each hind of interval, e. g., one for the miiior as distinguislied from the major Third, another for the minor as distinguished from the major Fourth, and so on. For instance, looking at the Chromatic Scale * Tlie author of a new musical system published in Geniiany a few j-ears ago, advocates a doctrine of Intervals which differs from that usually taught (as represented in our Primer) in the distinction of intervals as eitber "abso- lute" or " casual." See the first part of Appendix III, p. 91. 30 MODERN TONALITY. exemplified in Fig. 9, p. 21, and comparing the major Tliird c-e with the minor •Third d-f, we find, if we count all the intermediate tones, that e (the Third to c) is the fifth tone above c, while/ (the Tliird to d) is the fourth tone above d. Every other major Third will be found to differ from every other minor Third in the same way, i. e., is a chromatic half-step greater. Hence we say thab the formula of every minor Third is 4, and that of every major Third, 5. 41. This is called the " Half-step Formula," because we count from the lower to the upper tone of an interval by consecutive half-ste^DS (diatonic and chromatic, intermingled) : of course, how- ever, the actual numher of half-steps will be less hy one than the formula-number, as the following examples clearly show : Fig. IS. — Major Third. Formitla: 5. 1st Tone, 2d Tone. 3d Tone. 4tli Tone. 5th Tone. m Half-stepTi* Half-step. *" Half-step. ** Half-step. 12 3 4 Fig. 19. — Minor Third. Formula: 4. 1st Tone. 2d Tone. 3d Tone. 4tli Tone. ^ Half-step. *•■ Half-step. " Half-step. z^^ 4:2. The following are the formulas for the different normal intervals in the Diatonic Scale, with one example of each, in illus- tration. The intermediate tones are supposed to be counted only, not sounded, hence they are i^rinted in small black notes. A Eoman numeral is added every time a staff-degree is changed, the degree on which the first tone stands being always numbered I in determining an interval, whether we count upward or downward from it. M D E E X TO XA LIT!' 31 Fig. ;20.— Majok Second. Formula: 3 13 3 PS 1" II I II (Diatonic Step.) Fiff. ;?i.— Minor Third. Formula: 4. 12 3 4 I — ' II III I m Fig. 22.— Major Third. Formula: 5, 12 3 4 5 I ^ ^-, tt* =^=?g^ II III I III Fig. 23. — Minor Fourth. Formula: 6. 12 3 4 5 6 E iS*; :3«^ I II III IV I IV Fig. ^4.— Major Fourth. Formula: 7. 12 3 4 5 6 7 Fig. 2.5.— Minor Fifth. Formula : 7. 13 3 4 5 6 7 i w E=jJ»=r^3*z II III IV V Fig. 20. — Major Fifth. Formula : 8. 12345678 i ■^ — tf* — * — s*— I II III iSir== ■ oi in; IV V 32 MODERN TONALITY. Fig, ;2 7.— Minor Sixth. Formula : 9, 123456789 =«»= ^^- =i::^2=T I II III IV V VI VI Fi(j. ^<9.— Major Sixth. Formula: 10. 123456789 10 f=7 II Ill IV V VI I VI Fig. ^.9.— Minor Seventh. Formila: 11. 1 28456789 10 11 --P^ zj^ -^^m-=tmz -JM=timZ II III IV V VI VII I VII Fig. 30. — Major Seventh. Formula : 13. 2 3 4 5 6 7 8 9 10 11 12 :«*- II =«»= =^=j»i :c2z ~g?~ III IV V VI VII I VII QUESTIONS. 26. What is an Interval ? .How many tones are involved in an interval, and what relationship between them does the expression indicate ? Give an example. .. .27. What two elements do we distinguish in the interval? Name the various Denominations of intervals in the Scale. . . .28. How many Kinds of intervals do we find in the Diatonic Scale? How does a Jtuijoi^ interval differ from the ?Hitt(9r interval of the same denomination ? Give an example. . . .30. How is the Second formed ? Can we have two tones on adjacent decrees without ha\inff a Second ? Give an example. [Remark.) 31. Which are the OT?!«or Seconds in the Diatonic Scale of C? What interval does the Diatonic ;S;^;5 constitute ? (Remark.). ...32. Which is the smallest interval, and with what kind of half-step is it identical ? Why is the chromatic Half-step not identical with the minor Second? (Remark.).... 33. What is a gradual progression? What is a skipping progression or leap ? Name the different intervals in the Diatonic Scale of U, beginning with MOB £ R X TO iV.l L IT T. 33 34. the minor Thirds; the major Thirds; 35. the minor Fourths; the major Fourth. What other name has tlie minor Fourth ? What other the major Fourth? {Remark.) 30. ISame the minor Fifth in the Diatonic Scale of C; the major Fifths. What other name has the minor Fifth ? What other the major Fifth V {Remark.). .. .37. Name the minor Sixths in the Diatonic Scale of C; the major Sixths ; 38. the minor Sevenths ; the major Sevenths . . . .31). What is the " Half-step Formula "' ? -40. Give an example of its use -41. In the use of this Formula, will the number of half-steps correspond with the formula-number ? CHAPTEE VI. The Equal Temperament. 43. We have thus far treated of the intervals in the Scale on the supposition that they are all tuned, in keyed instruments, with mathematical correctness. This assumption is, however, false, with one exception, and the present chapter is devoted to the im- portant subject of the so-called " Equal Temperament," or modern system of tuning. 44. We have seen that we have 12 tones, chosen from a mul- titude of possible ditferent tones. Now, the selection of these particular 12 tones is based on the principle of their inter-rela-' tionship. The nearest tone-relationship is that of th.Q perfect Oc- tave, the next, that of the perfect Fifth. 45. Two tones form a perfect Octave when their viljrations give the mathematical proportion 2 : 1, i. e., when the nppcr tone vibrates ttoice while the lower or fundamental tone vibrates once. 46. Two tones form a perfect Fifth when their vibrations give the proportion 3 : 2, the upper tone vibrating three times while the fundamental is vibrating tivice. Kemabk. — The perfect Fifth may be heard in the so-called 34 MOBEEX TONALITY. "harmonics" or "over-tones" of a vibrating string, e. g., of the piano-forte. If, first raising the dampers, we strike a key — say C (of the great Octave), and Hsten attentively, we shall hear (not to mention other over-tones) ascending from the fundamental C, first the perfect Octave, c, then g, its perfect Fifth. A string per- fectly tuned to this c would give 2 vibrations to 1 of C ; a string perfectly tuned to this g would give 3 vibrations to 2 of c. 47. Now, by multiplying a tone by the perfect Octave we ob- tain a series of repetitions, i. e., of tones of the same sound cvnd name, in a higher and lower pitch, for instance, C, C, c, c, c", c", etc. Then Ave adopt the next relationship, and multiply a tone — say C— by the perfect Fifth, thereby obtaining the following sys- tem of different tones, quint-related* to each other : 16 15 14 13 13 11 10 9 8 7 . . . fW c\^\^ 9'?^ ^9 6 5 4 3 2 12 4 a\, c\, h\> F C G 13 13 14 15 16 4 l^ fx cx gx ExPLAXATiois".— Begin at C: reading to the. left we have the perfect Fifths below ; to the right, the perfect Fifths above C. 48. On examining the above series of tones we find that we have lost the Ocfave-s we had gained (see 47)— neither C, nor any other tone occurs more than once ; and we might continue the series of perfect Fifths ad infinitum without ever obtaining repe- titions! Here then we have a superabundance of 35 different tones (7 primary tones, each admitting 5 modifications; thus, 7x5=35— which is as far as we can go without using the triple J or [7), yet a series of perfect Octaves cannot be had, for hj^othetically no one of these 35 tones is a repetition of any other, but every one is absolutely different from every otLei "V? <^v? ^t^b /■[? 4 9"? 3 4 5 6 7 8 9 10 11 DAE B A 4 9^ 4 «# 17 18 19 20 f/x «X ey. hx .... * See the second Note, p. 25. MODERN TOSALITY. 35 one, hence has its own separate name. Yet, perfect Octaves we must have. 49. Xow, modern music takes this superabundant tone-mate- rial and reduces it to a system of 12 different tones, so arranged that every 13th tone from any starting-point sounds the perfect Octave of that starting-point. Thus, our modern tonal system consists of a series of perfect Octaves, each Octave embracing within its limits 12 absolutely different tones. This result is ef- fected by means of a compromise (so to speak) on the part of the intervals, whereby they give up some of their normal mathemati- cal perfection. The Fifths, for instance (which are principally regarded in tuning), are somewhat flatted,* i. c., a fundamental being assumed, the vibrations of the upper tone (Fifth) are slightlv reduced in rapidity. We have seen that by tuning Fifth after Fifth perfectly we render a perfect Octave impossible. For instance, starting from Cj and tuning by perfect Fifths, we shall not meet a tone sounding like that C iln a higher pitch, of course) till we reach the 12th perfect Fifth above, which is Zf^lj: (thus : C^, Gi D A e hf,^ c4 g„^ d.J^ a^^ ej ^4^ )• ^^ow, every piano-key Avhich sounds b'^ has to serve as a c-key also: but b^^ is too sharp (has too many vibrations) to give a perfect Octave of C ; hence, if we slightly flat every Fifth from C upward, the excessive sharp- ness of b^ is sufficiently corrected, or tempered, to enable this tone to sound the perfect Octave required {i. e., the tone c), while re- maining M sufficiently exact for all practical purposes. Moreover, as the same c-key has to serve for a third tone, viz.: d\^\f, an ap- proximation to this tone, as afforded by the tone c, having with it almost coincident vibrations, is accepted instead of d\f}f mathomat- icallv exact. Thus we begin to see how 35 tones may be played * This flattinjr of the Fifths is easily tested by the(»xperimpnt suggested in the Jif'iiiar/c to 4(i ; the g, for instance, which is soun led by the correspond- ing? key of a piano properly tuned, will be perceived by .a sensitive ear to be somewhat lower than the over-tone g which is generated by the string c. 36 MODERN TONALITY. by 12 keys. For in the same way every other triplet of tones which vibrate almost alike is reduced, by merging, for example, d \^ and h x into (^, e^ and gl}l> into /, /x and «^^ into g, etc., etc. Now, by repeating these tones in a higher and a lower pitch we obtain a series of pei'fed Octaves of each, thus of Z-^, d\^\^ and c simulta- neously, all three being one and the same tone as to sound — and 80 on of the rest of the tones. In this way the tuning is — at least, approximately — equally tempered* throughout the whole key- board ; and although the Seconds, Thirds,f Fourths, Fifths, etc.^ lose somewhat of their mathematical exactness and absolute acous- tical purity, yet the effect is, on the whole, not offensive even to the most sensitive ear. The great point gained is the absolutely necessary series of perfect Octaves. Moreover, modern music de- rives from this system a very decided advantage, in tlie Enhar- * We shall see further on that we obtain by transposition (see Chap. XIII) repetitions of the model scale of C, each one in a dififerent pitch — in fact, 11 additional scales. Now, we might time perfectly all the intervals in one par- ticular scale, say that of C ; but then the other scales would be out of tune, some more, others less so. By Equal Temperament, however, the various scales are tuned alike — or very nearly so, the acoustic impurity of the inter- vals (which is sometimes called the "wolf") being equally dispersed through- out all the scales, by which dilution it loses its offensiveness. f Unlike the Fifths, the Thirds must be tuned slightly sharp. For instance, I tune \.o e \is perfect major Third, e ; then to e, its perfect nuijor Third, gdt ; and to gt, its perfect major Third, ht. Now, this ht has to serve also for c', the Octave of the starting-tone, c ; moreover, the same tone, W, has to serve, as c', for the major Third to ah. But bt, as perfect major Third to gt, cannot serve as c' . Why not ? Because, c', as perfect Octave of c. vibrates twice whilst c is vibrating once (45), whereas ht would vibrate too slowly for this. But we must absolutely have c' exact (perfect Octave), and since we cannot get this by three ^e?'/«'^ major Thirds from ^ , viz. : e, gt,ht,'we must needs sharp each Third a little. Thus the major Third to gt will actually sound d exact, giving the desired perfect Octave to c, and at the same time the tem- pered major Third (as ht slightly sharped) to gt, and (as c') to ab, the tone accepted as identical with gt. MODERX TOXALITY. 37 monic notation of tones, a subject to be considered in the next chapter. Eemaek. — The introduction of the tempered system of tuning dates from about the year 1700: its universal acceptance in Ger- many is ascribed chiefly to the exertions of the organist Werck- meister, of Halberstadt, and of the immortal John Sebastian Bach. The 48 Piano Prehides and Fugues of the latter, published under the title: ''Das wohltemperirte Clavier" (The Well-tempered Clavichord), were composed to show the practicability of the tem- pered system. QUESTIONS. 43. Are all the Intervals of the Scale tuned, in keyed instruments, per- fectly, i. €., with mathematical exactness ? What is the technical name of the modern system of tuning?. . . .44. What principle has been followed ia the selection of the 13 tones of modern music ? When two tones form aper- fect Oct'ii:e,v>\iai do you say of their relationship? What, when two tones form a perfect Fifth ? . . .4o. When do two tones form a perfect Octave ?. . . . 40. When do two tones form a perfect Fifth ? How can we hear a perfect Fifth ? (Remark.). . . .47. If we multiply a tone by the perfect Octave, do we obtain new tones, strictly speaking ? What, if we multiply a tone by the per- fect Fifth?. .. .48. What do we lose in multiplying atone by the perfect Fifth? Explain this. . . .41). How does the modern system of tuning meet this difficulty ? Our modern tonal system consists, then, of what ? By what means has this reduction of 35 tones to 12 been effected? What, in partic- ular, is done in tuning by Fifths? Give an example. What kind of Fifths, then, exclusively, will admit of our having perfect Octaves? {Ans. Tempered^ i. e., slightly flatted, not perfect Fifths.) When two tones — say bi and dbb—- have, according to acoustical laws, almost, yet not exactly, the same vibra tions as a third tone — say c, supposing this third tone already tuned, what i- the actual process of reduction by which the 3 tones are sounded by one keyr {Ans. Instead of It and dbb exact (implying a separate key for each), we acc^p approximations, suflBciently near for practical purposes, i. e., the c-key serves for both these tones, the slight discrepancies of vibration being ignored.) Give instances of other reductions of the same kind. Does oven a sensitive ear require that all the intervals be tuned with absolute (mathematical) exact- 38 MODERX TONALITY. ness? What is the great point gained by the tempered system of tuning? What other advantage does modern music derive from this system ? What is the date of the general introduction of this system ? CHAPTER YII. The Modern Enharmonic Scale. 50. In comparing the descending with the ascending Chro- matic Scale of C (Fig. 9, p. 21) we notice that some of the tones take tivo names. Thus, the second tone ascending, c^, corresponds, in sound, exactly with the last but one descending, yet the latter is called d\^. Similarly, d1^ in the ascending series appears as e\} in the descending; again, f^ in the ascending as g\^ in the descend- ing — and so on. These are illustrations of tiie effect of Equal Temperament, as explained in the previous chapter. Strictly speaking, c| and d\^ are really different tones, one being generated by more vibrations than the other ; and the same is to be said of d^ and e\^, of f^ and ^[?, and so on. But, as we have seen, the modern practice in tuning keyed instruments ignores these mathematical differences (which are, after all, not very great), and makes one single sound answer for three sounds with different names, while yet retaining the nmiies, to be differently applied to one and the same tone, as the case may require. We have an analogy to this in spoken language, as, for .instance, when the 07ie sound produced in pronouncing "r-i-t" (i Jong) is variously writ- ten "rite," "right," "write," "wright," according as different ideas are to be conveyed. In the same way. one and the same tone will have to be written in various notation, according to circum- stances — and this is what constitutes musical oiihography, the knowledge of which is so necessary to the musician. For a few MODERN TO XA L I T T. 39 practical hints on this important subject we refer the reader to Appendix II, p. 86. 51. The twelve tones of our modern system, written in Tarious notation, form what is called the Enharmonic Scale, an illustration of which, derived from Fig. 9, here follows : Fig. SI. Db Gb A B 52. In the above figure we see that one and the same piano- key, sounding Cj and Ih, gives modifications for two tones, viz. : C'J for C, D\^ for I) — and so on of the other short keys. The Enhar- monic Scale above illustrated is, however, imperfect, as not exhib- iting all the modijications which are practicable with the tones. For instance : just as D'^ is a half-step above D, and A^ above A, so is Fin regard to E, and Cin regard to B, yet this figure does not relate E to F, nor C to B. Now, according to the theory of Enhar- monics, each and every tone may, under certain circumstances, be regarded as a modification of the tone on the decree next above, or below. The tone sounded by any C-key of the piano, and generally written C, may therefore sometimes be considered a raising of B (on the degree below), and be Avritten B^. for it is the only B^ the piano gives us. So, too, the tone sounded by any 5-key will some- times have to serve as CJ?, and Ije written accordingly; ami (he tone generally written F will sometimes have to be Avritten E^ ; and similarly, the tone E will sometimes appear as F\y. We can therefore enlarge our p]nharmonic Scale, as follows : 40 MODERN TO NA LITY. Fig. 82. c« DJt Fjf GJ AS Db Eb Gb Ab Cb m Fb Elf Cb U c D E F G A B c 53. But neither does Fig. 33 represent a complete Enhar- monic Scale, for the modification of a tone embraces double rais- ing and double depression (see 18. p. 19.), implying the use of the K and \f\^. In fact, it often happens that when we wish to raise a tone already sharped, or to lower one already flatted, musical orthography requires us to keep the tone on the same staff- degree, and to use the x or [^[j.thus raising or depressing the tone hy two half-steps. In this way, the tone whose primary name is D will serve as C double-sJmrp, and reciprocally, the primary tone will serve as D double-flat. So, too, JS will appear as the double raising of Z) (Z> x ), and Z) as the double depression of U {^\}\}), F as G\^\^, G as Fy. and as A\^\^, A q,s, Gx and as B\^\^, and 5 as yl X. Moreover, among the intermediate tones, C^ will ap- pear as i? X , ^[7 as F\^\^, F^ as Ex, and B\f as C\^\^. The complete Enharmonic Scale will then appear as follows: Fiff. .5.5. Db Fbb Gb Ab Cb Eb F« Ex GS Bb Alt Dbb Ebb ^M Fb Gbb 1^ la- Abb m Cb Dbb C D E F G A B C Bit Cx Dx Elf Fx Gx Ax B« MODERN TO NA LITY. 41 54. The modern Enliarmonic theory may thus be summed up : Each tone may be regarded, 1st, ahsolutdy (i. e., without relation to the tones on the adjacent degrees), as in the Diatonic and Chro- matic Scales ; 2d, as a raising, single or double, of the tone on the degree next beloiv ; and 3d, as a depression, single or double, of the tone on the degree next above. Hence, each tone (except Gj^*}, takes a triple name, viz. : 1st, its primari/ name, either simple or with a qualification added ; Jd, the name of the degree next beloiv, properly qualified ; and Sd, the name of the degree tiext above, properly qualified. 55. The distinguishing characteristics of the Diatonic, the Chromatic, and the Enharmonic notation, may be thus abbrevi- ated, for memorizing : DiATOXic : difierent tones, names, and seats on the staff. Chromatic : alteration of a tone, same name and seat on the staff. ExHARMOXic : one and the same tone, with three-fold name and seat on the staff. 56. When one and the same tone is repeated, the second time in a different notation (as frequently occurs), this is called an Enharmonic change, as in the following example : * Why (t $ is alone excepted is very easily exjJained. Every tone can be related to (regarded as a modification of) a primary tone distant a half step, or even two half-steps, hut not farther. Tlius — to compare the other altered tones with Gt — the tone C% is not only a single depression of B and raising of C, but also a double-raising of the tone two half-steps distant, viz., B ; similarly, Di is related, not only to D and E, but also to F ; Ft, not only to F and G, but also to E ; and Ai, not only to A and B, but also to C. But Gt cannot be related to the primary tone F, for tliis tone lies three half-steps distant, nor to the primary tone B, for the same reason. Hence, Gt can be related only to the primary tones G and A, i. e., as the single raising of the former, and single depression of the latter ; it therefore cannot have more than two names. 42 MODERN TONALITY. Fig, 34. 53: Ell. Change. 57. It should be added, in concluding this chapter, that in our modern tempered system of tuning, the real significance of the word " enharmonic," as employed in ancient times, is lost ; for a genuine enharmonic scale would be, for instance, c, c^, d^, d, d^, e\^, e, 4 /, yj, (j]^, (J, y% «l7, a, a^, Z*^, h, l'^, c'—m all, 19 diferetit tones ; whereas, by considering every pair of tones thus connected - — ■ as one and the mine tone variously named, as in our modern system, we have only 12 different tones, not counting the Octave. Hence the Enharmonic Scale proper" has disappeared, being re- duced to our Chromatic Scale, and the expression "enharmonic" denotes merely a varying notation for one and the same sound. QUESTIONS. 50. What illustrations of Equal Temperament do we find in Fig. 9, p. 21 ? Exjilain how it is that ct and ch, for instance, are played by one and the same key. Is there anything in spoken language analogous to the princi- ple of making one musical sound serve for three tones ? For instance . In what does " musical orthograi)hy " consist?. . . .51. What is the so-called "Enharmonic" Scale? . 5'2. Explain Fig. 31, and show why it does not represent a complete Enharmonic Scale. . . .53. Explain why the Enharmonic Scale represented by Fig. 33 is not yet complete. . . .5-4. Give a summary of the modern Enharmonic theory. Why does Ot alone take hut one additional name {Ah)t {Ans. Oi could not be related to F but by being written Fx raised {FW), nor to B, but by being written Bbh lowered {Bbhb). Now, we do not go further than double raising or depression, in modifying a tone ; hence, Gt can be related only to O and A. See JS/ote.) 55. What are, in a few words, the cliaracteristics of Diatonic notation ? Of Chromatic ? Of Eriha/r- monicf. . . .5(i. What is the " Enharmonic change"? 57. What is the difference between the ancient Enharmonic system and the modern ? What is the real significance of the word " enharmonic" in modern music? MODERy TONALITY, 43 CHAPTER VIII. Modification of Intervals by Chromatic Alteration. 58. As we have already seen, the tones of the Diatonic Scalft may be modified, i. e., chromatically altered; from this follows the practicability of modifying the Irdervals of the Diatonic Scale. 59. It must, however, be well understood that in the case of the Interval, implying a relationship befiveen two tones, chromatic alteration in the broad sense does not necessarily involve wluit we mean by modification, i. e., a change in the nature of the interval — it may effect only transposition. In Fig. 9, page 21, are several instances of chromatic alteration, e. g. the Fifth, c-g, into (-'^-g'^y the Fourth, d-g, into d'^-g^^ etc., yet in each case the nature of the interval is unchanged, (^-g^^ for instance, forming the same bind of Fifth (major) as c-g. What has taken place is not exactly chromatic alteration but rather Transposition, wliich, as applied to an interval, changes the pitch of both tones equally and in the same direction, i. e., raises or lowers each tone by the same number of half-steps, steps, or degrees, so that the interval remains the same in kind as V)efore transposition. 00. Different from the effect of trans]iosition is that of chro- matic alteration in its usual and strict sense, upon an interval — it changes the interval -as to kind, and in some cases even creates additional kindx, besides the two normal kinds — minor and major — found in the Diatonic Scale. f»l. This chromatic alteration or modification of intervals is limited in extent to a chromatic half -step, and it effects, 1st, the- change of minor intervals into major, and contrariwise ; or, 3d, the diminution of minor intervals; or, 3d, the avgmentation of major intervals. The latter two processes give rise to two new kinds of intervals. 44 MODERN TONALITY. Minor Intervals changed into Major, and contrariwise. G2. Since the Formula of a majo7' interval invariably gives one chromatic half-step more than that of a minor- interval of the same denomination (see Figs. 21 — 30, pp. 31, 32), it follows that to change a minor into a major interval we have only to make the former greater by such a half-step, which is done by chromatically raising the tipper tone or lotveritig the loiver tone. In cbanging, •on the contrary, a major into a minor interval, the upper tone is chromatically loivered, or the lower one is chromatically raised. Remark. — The chromatic alteration whereby minor intervals are changed into major, and contrariwise, effects the transposition of intervals, in this sense, that every pair of staff-degrees which is the normal seat of a minor interval — for instance, a-c (minor Third), e-a (minor Fourth), h-f (minor Fifth), etc. — may be used also for a major interval of the same denomination, whereby major intervals may be said to be transposed to new seats on the staff. But, while the chromatic alteration of minor into major, and con- trariwise, accidentally effects transposition in this way, yet its primary effect is to change an interval as to kind, Avhich is not done in transposition proper, as we have seen. Exercise.* Change all the intervals of the Diatonic Scale of C (from c to a, inclusive), in the following manner: 1. Every minor into major, in both ways. 2. Every inajor into minor, in both ways. * In writing this and the subsequent exercises, it would be well to leave a blank staflF under that on which the exercise is written, for such corrections ■as may be necessary, so that these may stand immediately under their several corrigenda. M D E R y TO XA L IT Y. 45 Diminution of Minor Intervals, 63. To diminish a minor interval (make it less than minor), is to bring the two tones nearer together by a chromatic half-step. This is efiFected by chromatically lowering the upper or raising the lower tone. Fiff. 33. Minor Third. Diminished Third. -=p— :>- idra; 1 Remark 1. — The Formula of the diminished Third is the same as that of the major Second, viz.: 3 (see Fig. 20, ji. 31 . But the two intervals, having a different harmonic and melodic sig- nificance, are differently written. It will be noticed that whereas the major Second is formed by a cliromatic and a diatonic half- step, the diminished Third is formed by two diatonic half-steps. Remakk 2. — Since to diminish, in the present sense, is to ren- der a minor interval smaller, it follows, 1st, that there cannot be a diminished Second^ the minor Second being the smallest interval used ; and 2d, that the term " diminished " should be used oid y of an interval proximately derived — by chromatic cdteration — from a minor interval of the same denomination. An illogical exception is^ however, commonly made of the small Fifth — as, b-f, d-ak, which is called "diminished." Now the Fifth of this construction (see 36, p. 28) is a normal interval in a diatonic scale, and never appears as a minor Fifth made smaller. A (jenuine diminished Fifth, on the other hand, is essentially a chromatically uttered Fifth, and represents the interval in its smallest possible measure- ment, as, for instance, s J, equal to a mino'* Fourth. 46- MODERN TO NA L I T Y. Augmentation of Major Intervals. 64. Augmentation is the contrary of diminution — to augment an interval which is by supposition already major, is to make it still greater by a chromatic half -step. 65. Augmentation, like diminution, may be practised in two ways ; but, unlike diminution, is effected by chromatically raising the upper or loivering the loiver tone. Fiff. 30. Major Second. Augmented Second. \=^ ^ t »'=* — g=' ^z=[|:rfez^ ^^b a^:r:g 13 3 12 3 4 1 3 3 4 Remark 1. — The Formula of the augmented Second is the same as that of the viinor Third, viz. : 4 (see Fig. 21, p. 31). But, as in other similar cases, the two intervals have a different har- monic and melodic significance, and are therefore differently writ- ten. The minor Third is formed by two diatonic half-steps and one chromatic, the augmented Second by tivo chromatic half-steps and one diatonic. Remark 3. — To augment, in the present sense, is to render a major interval greater. Hence, the common practice of calling a Fourth like/-6 "augmented " is inconsistent and confusing. For the Fourth of this construction (see 35, p. 28) is normal in the diatonic scale, and never appears as a major Fourth enlarged. Wliereas, a genuine augmented Fourth is essentially a chromati- callg altered Fourth, and represents the utmost possible extension of the interval, as, for instance, ]t, equal to a major Fifth. 66. From what has been said in this chapter we sum up the following : \st. That the chromatic alteration of intervals in the Diatonic Scale impl'es either simple transposition, or modification MODERN TO KA L I T Y. 47 — chromatic alteration in the usual and strict sense — the latter changing the interval as to kind, Avhich the former does nt>t. 2d. That anv minor interval of the Diatonic Scale may be chromat- ically changed into major, and contrariioise. Sd. That some minor (and only minor) intervals may be diminished, by bringing the two tones nearer together by a chromatic half-step. 4fk- That .-ome major (and oiili/ major) intervals may be augmented, by puttin^'j an additional (chromatic) half -step between the two tones. Half-step Formulas of all the Intervals, including those ch/i- maticallv altered. The Formula of the Minor Second is 3. Major Second is 3. Augmented Second is 4- Dwiinished Third is 3. Minor Third is 4. Major Third is 5. Augmented Third is 6. Diminished Fourth is 5, Minor Fourth is 6. Mujoi' Fourth is 7. Augmented Fourth is 8. Diminished Fifth is G. Minor Fifth is 7. Major Fifth is S. Augm,ented Fifth is 9. Diminished Sixth is S. Minor Sixth is 9. Major Sixth is 10. Augmented Sixth is 11. Diminished Seventh is 10, Minor Seventh is 11. Major Seventh is IB. 48 3I0DERN TOXALITT, Exercises. 1. Diminish, in both ways, the following minor intei'vals : Thirds, cl-f, 5-^, c^-e, h\l-g, e-g, a-c, /||-«, <^i^-f- — Fourths, g-c, e-a, /If-^, g-d, c-f, a-d. — Fifths, d-a\z, b-f, a^'e, g^-d, «-el2.— Sixths, c-e, J'-a, d-F^, ^-b. — Sevenths, d-c, A-g, <-'!^-b, e-d, f- Gy a-b, c-V^. 2. Augment, in both ways, the following major intervals : Seconds, d-e, ^-f, f-g, Th-c, c-d, e-f^, a-b, g-a. — Thirds, c-e, «-/ft ^-^) f-a.— Fourths, Ak-d, J-B, e-c/jf, d-y^, e^-a.— Fifths, G-d, d-a, c-g, f^c^, e-b, f-c, e]z-b\i, (lf-^|f- — Sixths, d-b, f-d, c-a, «-/ft ^i2-^> ^-ej, e\z-c. QUESTIONS. 58. Are the Intervals, as found in the Diatonic Scale, unchangeable ? . . . , 59. Can an interval be chromatically altered without being modijied? What is the effect of Transposition on an interval ?. . . .OO. What effect has chromatic alteration, in its usual sense, on an interval ?. . . CI . What is the limit of the chromatic alteration or modification of an interval ? What are the effects? .. (i'2. How is a minor changed into a major interval? A major into a minor interval ? How does this modification accidentally effect the transposition of intervals? (Remnrl:). . . Aiii. What is it, to diminish a minor interval ? How is it done ? What . minor interval is never dimin- ished ? Give the reason. — To what species of interval should we exclu- sively apply the term "diminished"? What do you say of the usual practice of calling the minor Fifth "diminished"? What is a genuine diminished Fifth? {Bemark -?.) . - 64. What is it to augment a major inter- ral ? . . . . 05. How is augmentation effected ? — What do you say of the tommon practice of calling the major Fourth " augmented " ? Explain this tiatter, and state what is a genuine augmented Fourth. {Bemark 2). . . .66* Give a summary of this chapter. MODERN TONALITY. 49 CHAPTER IX. Inversion of Intervals.* 67. To invert an interval is to transpose one of its two tones by the Octave, the result being that the same two tones form an- other intei'val as to denomination and hind. Simple change in the order of the two tones, as in Fig. 37, does not effect inversion, but merely makes a descending out of an ascending interval, or vice-versa. Fiff. 37. 1 6 1 6 68. The rule of inversion is this: transpose the lotver tone an Octave\ higher, or the upper tone an Octavef lower, &o that the tone which was the upper is now the lower. The following are, examples : Fiff. :is. Ma.j. Sccniid: inverted. Mill. Third: inverted. Dim. Third: inverted. 13 16 Aii,s,nn. P'ifth; inverted. 16 15 :|« * The Inversion of intervals is in some works improperly styled the "Transposition" of intervals. But w.o have seen that transposing doos not affect the 7mti'rr. of th: intcrml as such, whereas Inversion does. •f- Or two Octaves, three Octaves, etc , higher or lower— the number of Oc- taves does not in the least affect the inversion. See first Note, p. 24. 3 50 MODERN TONALITY. 69. From the foregoing examples we see that the jnajor Second c'-d' is changed into the minor Seventh d'-c", the minor Third e'-g into the major Sixth g'-e", the diminixlied Third d\-f' into the augmented Sixth f'-d '^, and the augmented Fifth f -c"| into the diminished Fourth c'^/"- Inversion, therefore, elfects the chan2"3 of an interval both in denomination and in kind. 70. The changes of denomitiation are expressed in the formula: 7 P) T 4 S 2 r, o A ► ^ ^ , which may be read thus, from the lower row of figures upward: the Second, inverted, becomes a Seventh; the Third, a Sixth — and so on of the rest. 71. The changes of kitid effected by the inversion of intervals are thus summed up: minor becomes major, Siud cotitrariwise ; diminished becomes augmented, and contrariioise. Exercise. Invert, giving two or three examples of each, all the intervals (a full enumeration of which is given under the heading, "Half- step Formulas," etc., p. 4'7), for instance, the minor Seconds, e-f, d!-e\, d\-e; the major Seconds, A-B, d''^-e'\^, B-c^ ; the aug- mented Seconds, F-G^, c-d^, d'\^-e, etc., etc. QUESTIONS. 67. What is it, to invert an interval? Wliat is tlie result? What is the effect of merely changing the onhr of tlie two tones of an interval ? 68. State the rule of inversion. (Does the expression "an Octave higher" or "an Octave lower" necessarily mean only one Octave higher or lower? gee Note 2.) 69. What double change in an interval does inversion effect ? 70. State the changes of denomination 71. State the changes of kind. MODERN TO N A. L IT Y. 51 CHAPTEE X. Intervals as Symphones.— Consonances and Disso- nances. 72. We shall now consider the interval from that point of view which supposes its two tones to be sounded simultaneously. 73. A simultaneous sound of two or more tones constitutes a Symphoxe (German, '-der Zusammenklang").* We are at present concerned with the two-voiced] Symphone, in other words, that formed by the two tones of an interval, sounded together. It is classed as either a Co:n^soi^axce or a Dissoxance. Each of these terms has relation not exactly to the absolute sound of the Symphone, considered as agreeable or disagreeable, but rather to the necessity or non-necessity of its connexion with another Sym- phone. 74. A Consonance is a Symphone which is self-sufficient and independent ; either of its tones may progress to another tone, but * We have absolutely no word in oyr language corresponding to the gen- eric German word " Zusammenldang " (literally, a " together-sound "), except " Consonance," which, indeed, expresses it exactly, but is used in a specijie sense only. I have therefore ventured to introduce the new word " Sym- phone," derived from the Greek. The expression is a comprehensive one, including what are called "Chords," "Triads," "Consonances," "Disso- nances," etc., — in short, any combination, agreeable or otherwise, of musical tones. The Interval has been hitherto considered from a melodic stand-point and indicated horizontally (so to speak), as for instance, c-c, c-g : regarded as a Symptione, in other words, from a harmonic stand-point, it is indicated ver- tically, for example '-.,., etc. c c f The expression " voice " is often used as synonymous with " tone," irre- spectively of the tone producing agency. Thus, a " two-voiced" Symphone is one composed of two tones, whether sung or played. 52 MODERN TONALITY. need not, as far as any incompleteness of the Symphone is con- cerned. 75. A Dissotiance, on the contrary, is a symphone essentially transitory in character, pointing to what is called its resolution into a consonance, by the 2Jrogression of one or both of its tones. The resolntion need not follow at once — a dissonance may lead into another, and so on, leaving only the last one to be really re- solved into a consonance. Fig. 5.9.— Consonances. a be d e f 1^=^ ff C) Fig. 40. — Dissonances. etc. 76. Each of the Consonances, as in Fig. 39, appears as an in- dependent symphone, not pointing to another as its complement. The Dissonances (Fig. 40), on the contrary, severally tend to their resolutions, which are indicated in small notes. 77. Some dissonances— viz. : the harshest — reqnire what is called preimration also ; i. e., just as they must resolve into con- sonances, they must likewise, as it were, grow out of preceding symphones, so as not to be introduced too abruptly.* Thus, a dissonance is said to be " prepared " if one of its tones was heard in the symphone immediately before. In the following Fig- f ure, at A, the harshness of the dissonance J , is softened by the e having been heard in the previous consonance ^, (as may be proved * Of course, a composer is perfectly free to introduce a dissonance abruptly in order to produce some particular effect. In general, the rule of preparation of dissonances is not now so strictly adhered to as formerly. MODERN T O XA L IT Y. 53 by taking, against g', some other tone than e', as in the same Figure, at B) : the dissonance, thus prepared, then resolves inl ♦ f the consonance -S/ . Fig. 41. A B P 1^ :j=d=e=t^=, I 78. The difference between the Consonance and the Disso- nance may, for practical purposes, be stated thus: the Consonance may be introduced abruptly, and may cease sounding as abruptly, or may l'»e indefinitely repeated, or may pass to another consonance or to a dissonance. The Dissonance is not generally introduced abruptl}', and, once introduced, is not perfectly free as to its dis- continuance or its next progression, as has been explained. 79. The Consonances are the following, an example of each being given in Fig. S9 : the Octave (a) ; the major Fifth* (b), and its inversion, the ininor Fourth* (c) — the latter is, however, under * It has already been stated that in many works the minor Fourth is called the " perfect " Fourth, and the major Fifth the " perfect " Fifth. We may now- state that the expression " perfect " is used to mark a peculiarity of tbese two denominations of intervals as distinguished from the other consonant inter- vals, viz. : tluit the former cannot be altend without ceasing to be conso- nances. The Tlnrd and the Sixtti may be changed from major into minor, and contrariwise, and yet remain consonances ; but if the Fourth — for in- r' Stance, , be made major, or otherwise altered, it becomes dissonant, so ajso the Fifth — say ^. , if made minor, or otherwise altered. Hence the predicate •'perfect" is applied to the minor Fourth and the major Fifth, to denote that single condition in which they are consonances. Nevertheless, many of the best modern musical theorists use the expressions "minor "and "major" instead, deeming the significance of the word "perfect," as just explained, not important enough to justify an exception of the Fourth and the FiftJi in that simjjle and consistent system wliich classes the normal intervals as minor or major, according to their measurement. 54 MODERN TONALITY. certain circumstances a dissonance (as explained in the study of Harmony) ; the mcyor Third (d), with its invereion, the tninor Sixth (e) ; and the minor Third (/) , with its inversion, the major Sixth {(/). All the other intervals are dissonances, viz. : the Second, of any species ; the diminished and augmented Tliird ; the diminished, augmented, major, and sometimes the minor. Fourth ; the diminished, augmented, and minor Fifth ; the diminished and augmented Sixth ; and the Seventh, of any species. (See Appendix III, p. 91.) 80. Some dissonances do not, of themselves, {. e., without the addition of a third voice, strike the ear as dissonant, in the sense already explained. This is the case with some of those chromat- ically altered intervals whose formulas severally coincide wi;h those of consonances (see p. 47). Thus, the dissonant angmented Second sounds exactly like the consonant minor Third (for instance, ^v like J J ), the dissonant diminished Fourth like the consonant major Tliird (for instance, ^^^ like ^*'\ etc., etc. Other disso- nances, on the contrary, at once strike the ear as such — a progres- sion of one of the" tones seems necessary, if the ear is to be satis- fied. Such dissonances are the minor Second, the major Second^ the diminished Third, the major Fourth, the minor Fifth, the awpnented Sixth, the ininor Seventh, and the major Seventh. 81. It should be Avell understood that although extremely harsh dissonant combinations are possible, yet the expression "dissonance" does not necessarily or invariably imply harshness. Some of the most charming effects in harmony are produced by means of certain dissonant combinations, and it might surprise one not learned in music to be told, after listening with deliglit to the performance of a composition, that the composition abounded in dissonances, or, as some say, "discords." Indeed, without the intermingling of dissonances a succession of consonances would soon be found insipid, just as repose is not thoroughly enjoyable except after exertion. Now, as Dr. Macfarren observes, the disso-* MODERN TO NA L IT T. 55 nance, as being inconclusive, and, however prolonged, requiring resolution, "is thus the musical exponent of unrest, activity, aspiration." Hence the best preparation for the repose expressed by the consonance is the activity typitied by the dissonance.* QUESTIONS. 73. "What is a Symphone ? How is the two-voiced Symplioue classed ? "What does each expression properly signify ?. . . . 74. What is a Consonance ? . . . c 75. What is a Dissonance ? Must a dissonance always be resolved into a consonance in the follomng progression V. . . .77. What is generally required in the case of the harshest dissonances ? When is a dissonance prepared ? (Is this preparation always and absolutely necessary? See Note!).... 78. State the practical dilFerence between the Consonance and the Disso- nance.... 71). Name the Consonances. (Why is the expression "perfect'* sometimes applied to the minor Fourth and the major Fifth f See Note.) Name tlie Dissonances. .. .80. Do all the dissonances, of themselves, strike the ear as such ? Which are those that do not ? Which are tliose that do ? ....81. Does the term " dissonance " necessarily imply harshness ? What is the practical use of dissonances in a piece of music ? * An acoustical phenomenon in connection with this subject is thus described by Professor Macfarren in his " Lectures " : "Vibrations are more or less rapid in proportion as the sound is higher or lower of which they are the utterance, and combinations (of tones) are more or less conmnant in pro- portion to the greater or less number of coincident vibrations of the two sounds — thus, two sounds in unison (the most perfect consonants) vibrate simultaneously ; the upper note of an 8th has two vibrations for every one of the lower ; while the major Seventh (one of the harshest discords) has fif- teen vibrations of its upper note for every eight of its lower, the coincidences of which vibrations are as rare as the dissonance of the combination is ob- vious." 56 MODERN TONALITY. CHAPTER XI. The Triad, or Three-voiced Chord. 83. Under the general term " symplione " is included what is called the Chord, which is a simultaneous sound of several tones* combined according to certain laws. The particular species of chord which we must now discuss — though only at such length as is necessary for the purposes of this work — is that which is called the Triad. 83. The Triad is a symphoue of three tones, so notated that the uppermost tone is a Fifth of some kind, and the middle tone a Third of some kind, to the lowest tone, or Fundamental. In other words, a Triad consists of a tone assumed as a fundamental, with its Third and Fifth above, sounding together with it, as in the following Figure : Fi(j. 4:2. a h c g=pEj=iOl 84. The Triads in general are distinguished by the hind of Third only, or by the ]ci7id of Fifth only, or by the kind of Third and of Fifth, peculiar to each. 85. In Fig. Jf2, at a and at h the Fifth is in both cases major, while the Third at a is major, and at I is 7ninor. The Triad at a is a major Triad, that at ^» is a minor Triad. Each takes its dis- tinctive name from its different kind of Third only, the same kind of Fifth (major) being common to both. * At least three different tones are requisite to form a complete cliord. Hence, certain two-voiced combinations are chords in a limited sense only. MODERN TOXALITY. 57 Remark. — The major and the minor are consonant Triads; all the others are dissoncaif Triads. SG. At c the Triad differs from both of the two preceding ones — it has the minor Tliird in common with the Triad at b, but not the major Ft'ffh J while it differs from that at rr in its kind both of llilrd and of Fifth. The Triad at c, being characterized by a minor Third and Fifth, may be appropriately named '"donble- minor." though it is generally called '■ diminished." * 87. Let it then be well understood that the expressions *' Major Triad " and '• Minor Triad," without any additional desig- * The author of this work respectfully submits that the expression " double-minor " (?<;^rtf» the Triad concerned, and instantly suggests its com- position, /. ('., its kind of Third and Fifth — which can hardly be said of the common designation " diminished." But cherished traditional names must be defended at all hazards, and so it is said that the Triad of which we are speaking is called '■ diminished " because its compass is " smaller by a half- a step than the minor Triad." Now the altered Triad /, for instance, called by d$ the author of the present work " diminished-minor " (diminished Third and minor Fifth), is "smaller by a (chromatic) half-step than a minor Triad," yet does not represent the so-called " diminished " Triad. On the other hand, in this example : ^^H^iS the symphone at b does represent the " diminished " Triad ; and while it undoubt- edly is smaller by a half-step than the minor Triad at a, it just as truly is smaller by a half-step than the innjor Triad at c ! So tliat tiie reason for the use of the word " diminished " in tiiis connection is hardly satisfactory. Should any one object to the proposed substitution of " double-minor " fo^ "diminished" through unwillingness to recognize the "minor" Fifth, the objection is met by the suggestion that the Triad whose name is in dispute may properly be called " double-minor " aa being composed of two minor Thirds, one above the other. 58 MODERN TONALITY. nation, always imi^ly the major Fifth ; and that when the Fifth is other than major, a second designation is necessary. SS. The tliree normal Triads exemplified above may be modi- fied by the chromatic alteration of their tones. At present we notice, in passing, only the raising of the Fifth in the major Triad, whereby the Fifth is augmented, rendering an additional designation necessary, to distinguish this Triad from the major Triad. Fig. 4^3. 3iE 89. We may call the above Triad " major-augmented " — thft first designation (major) showing the kind of Third, the seconcj (augmented) the kind of Fifth, as is the case with the names of the other altered Triads. Eemaek.— In this chapter only so much is said of the Triad as seemed necessary as a preparation for the remaining chapters. To the Remark to S5, that the only consonant Triads are th© major and the minor, we add, that every possible consonant com- bination in music, how many-voiced soever, is reducible, in the last analysis, to either the one or the other of these simple three-voiced chords. Of these two Triads the major sAohq is strictly a primary, purely natural formation, as being evolved from a fundamen- tal by the natural acoustic law of generation. Thus, the piano- string C, for instance, being made to vibrate, produces also its Octave c, then successively g, c, e, g, etc.,* in wldch series we recognize the major Triad. This law of acoustics, by which a tone evolves its harmonic over-tones, has been compared, by way * Tones tlius produced fonn what is called " nature's harmony," the word " harmony " being used in the narrower sense of an euphonious combination of sounds. Some writers find in this natural harmony the starting-point of the historical development of the Diatonic Scale. M B E R X TONALITY. 59 of natural analogy, to the optical law whereby light is resolved into the colors of the prismatic spectrum. QUESTIONS. 82. What is a " Chord " '? (Wheu is a Chord incomplete ? See Note.) 83. Describe the Triad 84. How are the Triads distiugiiished ? ... 85. Give au example of a major Triad. Give an example of a minor Triad. What constitutes the diflference between the two? Wliich are the consonant Triads? (Remfirk.) 86. Describe the double minor Triad. ...87. What kind of Fifth is always implied in the major and the minor Triad? ... 88, 89. What is the "major-augmented" Triad ? F^yevy \)oss\h\e consonant chord may he reduced, in its simplest form, to what ? (In what sense is the major Triad alone a purely natural harmony ? Give an illustration. Doess nature furnish any analogy to the acoustic law of the harmonic over-tones 'i (Remark.) CHAPTER XII. The T^ATo Modes, or Keys, of Modern Tonality.— The Major and the Minor Scale. 90. The most ancient musical system of which any trust- worthy accounts have come down to us — that of the Greeks — com- prised three distinct tone-genera, viz. : diatonic (see 9, ]>. 15), chromatic, and enharmonic (see 57, p. 42). We may form an idea of some kind as to what their melodies* of the diafonic genus may have been like, from the structure of tlie different species of Octaves or Scales illustrated in Fig. 7, p. 17. But as to the kind * No Greek melodies have come down to our time — at least, none whose authenticity is satisfactorily established. If is an interesting question, whether harmony, as w(; understand tlic expression, was known to the Greeks. The general opinion among the learned is that it was not. 60 MO D E R X T O XA L IT Y. ■of melody involved in the cliromatic and enharmonic genera * we can scarcely form an intelligible idea, in spite of the explanations of Greek writers on the subject. Now, modern music recognizes the diatonic genus as the essential basis of its tonality, dividing it into TWO hinds, modes, or keys, viz. : the major and the minor, and adopting for each, one of the ancient diatonic scales as a model. When, however, we say that modern music is essentially diatonic, it is by no means meant that it is exclusively so — on the contrary, both the modes, or keys, are, in modern practice, tem- pered by the employment of chromatic tones, and allow the appli- cation of enharinonics, in the modified sense which has been ex- plained. (See Chap. VII.) 91. We may observe here that the chromatic element in mod- ern music is to be regarded from either a purely melodic or a har- monic point of view, which latter, of course, does not exclude the former. Chromatic tones, from the former standpoint, are em- bellishments of the melody, i)assing-notes, etc., not necessarily regarded in the harmony ; from the latter standpoint they imply enrichment of the harmony as a means of greater warmth and intensity of expression, etc. The enharmonic element is very prominent in modern music, the most striking and agreeable har- * In the Greek system the Tetrachord played almost as important, a part as the Octave (in its broad sense) does in ours. The Tetrachord is a gradual succession of four tones within the compass of a minor Fourth, as e-f-g-a, which is a Tetrachord of the diatonic genus. Tlie Tetrachords of the chro- mntic genus consisted of a succession of two half-steps and a minor Third, as a-ih-bn-d'. In the enharmonic genus, the Tetrachords progressed by what .we may call — though not with strict SiCcwMy— quarter -steps, followed by a leap of a major Third, somewhat like this : e-eir^f-a. The three genera, or kinds, ■>az. : diatonic, chromatic, and enharmonic, were employed both sepa- rately and inconjimction : though we can hardly form any idea of the practical use of either of the two latter independently. It has been humorously ob- served that traces of the enharmonic genus are found in the practice of those singers in whose vocal method the "portamento " plays so prominent a part. MODERX TOXALITY. 61 monic surprises being effected by means of the " enharmonic change." 93. The principal elements of our modern tonaHty may, then, be summed up thus : we have only tioo modes, or keys, the one major, the other minor ; each one admits the employment of the diatonic genus, independently, as also the mixture of the chromatic element with the diatonic: hence each of the two keys is provided with its diatonic and its chromatic-diatonic scale, as models for multiplication by transposition, so that either key may be made available throughout the entire tone-compass. Moreover, the modern system of tuning (Equal Temperament) brings witli it the enharmonic notatio7i, whereby (as is explained in tlie science of Harmony) the most manifold and intimate tonal relationships are easily established. 93. Before proceeding to examine separately the major and the minor key, Ave must briefly consider the significance of the expression "' key," in general. 94, We have said that only two of the ancient seven diatonic scales {Fiy. 7, p. 17) have survived to our day, viz. : that of A and that of C, and that these are the model scales of our modern tonal- ity. A Scale, it m\\ be borne in mind, represents the essential tone-material — melodic and harmonic — of a key* A key consists of a number of different tones having in their totality a deter- minate relationship to a certain fundamental tone called the key- \one, or Tonic ; and a key is either major or minor, according to the predominance of certain melodic or harmonic characteristics, as shall presently be explained. Now, as we shall soon see, the scale of A represents the minor key, and that of 6' the major — hence we call the former the model minor, the latter the model major scale. Every composition of modern times is in one or the other of these keys, i. e., draws its essential tone-material from a * The Scale represents tlie key me.lodieaUy ; if we add to the Scale its appropriate Triads, these represent the key iMrmonically. iJ'Z MODERN TONxiLITY. scale identical in structure with that of A or of C. But a scale may be ideutical in structure with that of A or C, and yet differ from it in pitch — thus we talk of the scale of B, for instance. The scale of B will be found, on examination, to have the same inte- rior structure as that of either A or U: in the former case we have the scale of B minor, in the latter that of B major j and as a scale represents a kei/, we speak of the ke^ of B major, or of B minor. Similarly, we may speak of the key of I), major or minor, of F, G, B\,, etc., etc., major or minor. Strictly speaking, however, there are, as we have said, only two really different keys, viz.: the major, and the minor, the former represented by the normal dia- tonic scale of C, the latter by that of A ; so that every other key than that of C and of A can be but a transiwsition, implying change in the pitcli only, not in the structure of the scale of the key. Hence it appears that the expression " key " has a twofold sense : 1st, a strict one, referring to interior structure only, os, when we say that a piece is in the major as distinguished from tJie minor key ; 2d, a sense referring to difference of pitch also, as when Ave speak of the various keys, as, for instance, A\^ major, B minor, D major, etc., etc., meaning the various transjwsitions of the two original model scales. 95. We must call attention, before going further, also to cer- tain additional names applied to the more important tones of the Diatonic Scale, whether major or minor. 96. The 1st degree of the scale is the seat of the key-tone, or Tonic* The Tonic, as the base of the tone-ladder, is of the first importance; it is the source of that unity and consistency with- out which a piece of music would not be in any particolar key — in other words, would be but an incoherent, unmeaning series of sounds, without tonal form. Hence, the close of every mel- ody is fittingly made either on the Tonic or on one of the * An exceptioi; to tliis rule seems to be made by C. F. Weitzmann, in the case of the minor Scale. See Note, p. 78. 31 D E R N TONALITY. 63 two tones which with the Tonic form a consonant Triad, \\z.: its Tuird (major or minor) and its major Fifth ; whilst in the harmony of the close, consisting necessarily of these same tones iu combination, the Tonic itself, by way of asserting its importance, its fundamental character, mast form the lowest tone, or bass, of the chord.* 97. The tones of the scale which are next in importance are : that on the Vth degree, called tlie Dominant, and that on the IVth degree, called the Subdomiiiant. Moreover, the tone on the de- gree next above the Tonic is called the Supertonic ; on the third above, the 3Iediant j and on the sixth above, the Submediant. Remark. — The Triads of the Dominant and Subdomitiant, to- gether with the Triad of the Tonic, form the character isfic har- monies of the key,\ whether major or mmor. The Tonic har- mony at the close of a composition most frequently takes imme- ■(liately before it either the harmony of the Doniinant or that of * "In modern music, all coherence, botli of melody and of harmony, all relationship, all principle, is involved in the arrangement of notes vrhich con- stitutes a Key. This arransrement refers to any note that may be arbitrarily chosen as the key-note. The key-note is in a piece of music, to speak com- paratively, as the point of sight is in a perspective drawing, whence all the lines diverge, and which regulates the proportions of all the objects in the picture." — Lictureson Harmony, by G. A. Macfarren, London, 1807. f For instance, the major key is characterized ly a wKJor Triad on the Tonic, Subdominant, and Dominant. It ought to be observed, however, that the characteristics of the major as distinguished from the minor key are not so constant as to exclude interchange to a certain extent. Thus the tiutjor key occasionally admits a minor Subdominant Triad— especially in the Plagal Cadence ; whilst the minor key, though it necessarily excludes a major Snih ^5^^^eI^1 . ypt admits. dominant Triad in the Plagal Cadence, e. g IV especially in its characteristic Plagal Cadence, a close on the Tonic friad with the Third raised— m oth(ir words, on a vvijur Tonic Triad. Moreover, the minor key frequently borrows from the major tlie Dominant Triad (major) ot the latter, especially for Cadences. 64 MODERN TONALITY. the Suhdominant, thus forming what is called tlie final Cadence. If the Touic harmony is immediately preceded by that of the Dominant, this is called the Authentic Cadence (expressed thus : V-I) ; if by that of the Subdominant, the cadence is called Plagal (expressed thus: IV-I). 98. Whenever, in a diatonic scale, the tone on the Vllth de- gree is a lialf-stei) below that on the Vlllth (or Octave), thus" forming a major Seventh to the Tonic, it is called the Leadinrj- tone, ou account of its natural general tendency to the Octave of the scale,* i. e., to the Tonic, as represented by its Octave. The following are examples of the most important tones of the scale of C. Fifj. 44r. Tonic. Subdominant. Dominant. Leading-tone. -^ IV t •"■ V VII \^ * » 1 • 1 » I IV V VII Remaek. — In marking the degrees of the Diatonic Scale by the Roman numerals, every degree is numbered exclusively w/y^mrc? from the Tonic. Hence, as in the above Figure, F, although a Fifth below c, is marked IV, being the lower Octave of degree IV, and G, though a Fourth below c, is marked V, being the lower Octave of degree V, and B, though a Second below c, is marked VII, being the lower Octave of degree VII. QUESTIONS. 90. Wliicli were the three tone-genera, or modes of melodic progression, of the Greek musical system ? Can we form any idea of the nature of the an- cient Greek melodies ? What is the essential basis of modern tonality V How • * This tendency is most strongly exhibited when the Seventh is contained (as major Third) in the Authentic Cadence, and especially in the "Dominant Seventh-chord," as explained in the study of Harmony. M D E R X TO KA LITY. 65 •mviD.y modes, OX keys, are there in modern music, and whicli are they? Is modern music exclusively/ diatonic? Explain how it is not?. ...Ol. From what two standpoints may the chromatic element in modern music be re- garded ? What are chromatic tones, from the melodic standpoint ? What do they imply from the harmonic standpoint ? What do you say of the enharmonic element in modern music? 92. Sum up the chief ele- ments of modern tonality. . . .04. We speak of the key of E winor, for instance, of Bi major, D minor, etc. ; yet there are, strictly speaking, oMy tico keys, viz. : the major and the minor, represented by the scales of C and A, respectively ; every other key, therefore, than G (major) or ^1 (minor), can only be — what ? {Ans. Either C or ^-1 in another pitch, i. e., transposed.) The word " key " has, then, a two-fold sense : what is the strict sense ? What is the secondary, usual sense ?. . . .90. What important tone of the Scale is seated on the 1st degree ? What is said of the importance of this tone ? On what tone of the Scale does every melody close ? How does the Key-ton* (Tonic) of a piece assert itself in the harmony of the close ?. . . .97. What ii^ the tone on the Vth degree of the Scale called ? That on the IVth ? That on the lid ? That on the Illd ? That on the Vlth ? Which Triads form the characteristic harmonies of a key ? How is the final Cadence formed ? When is this Cadence called " authentic " ? When " plagal " ? (Remark.) .... 98. When is the Vllth degree of a diatonic scale called "Leading-tone"? CHAPTER XIII. The Major Key.— Transposition of the Model Major» Scale. 99. Of the two keys of modern tonality, that one in whose characteristic harmonies the major Triad predominates, is called the major hey, and is especially adapted for mnsic of a cheerful, brilliant and triumphant character. Its -epresentative scale is, as has been said, our old acquaintance, the Diatonic Scale of C. 100. In examining the melodic structure of the model scale of C, we find that the tivo half -steps lie between degrees III-IV, and VII- VIII, respectively, all the other progressions being steps. 66 MODERN TONALITY. Fig. 45. m I II III IV V VI VII VIII 101. Again, by adding the Tliinl and the Fifth to the Tonic, the Suhdominant, and the Dominant — the three most important degrees and the seats of the characteristic key-harmonies — we have in each case a major Triad ; hence this scale is called "major," and serves as the model scale for the major key in general. Fig. 46. tfes^E^l^^ IV 102. But we are by no means confined, in the major key, to the pitch of the model scale — we may obtain varieties of pitch by means of transposition. If now we take each one of the remain- ing eleven tones, viz. : c^ (or d\f), d, d^ (or c\i), etc., as a new start- ing-point or Tonic, and by proper use of the chromatic signs reproduce each time a diatonic series exactly corresponding in structure to the above model major scale (Fig. 45), we shall obtain eleven additional major scales, each one representing the major key in a different compass, or pitch. 103. We begin this reproduction by transposing the model major scale of C a chromatic half-step above and below, obtaining the following major scales of C'| and C\^. Fig. 47. Sionatiire. I II III IV VI VII VIII f MODERN T O y A L IT Y. 67 Fig. 4S. ^^ Signature. bm !7 g ^;=w=^ ^z^g ^ ^ ^^ I II III IV V VI VII VIII 104. In each of the above two transpositions every degree of the scale is chromatically altered. The alterations are indicated, in the case of every transposed scale, once for all, at the beginning of the staff", immediately after the Clef, the altered degrees being affected by this Signature, as it is called, throughout the whole of a piece, or until the signature is changed. Thus, the major scale of C!^ has a signature of seven sharps, that of C\,, a signature of seven flats, as in the examples immediately above. 105. Xow, in transposing a model scale so as to give all the 12 scales in a systematic order, according to their relationship, we do not begin Avith scales having the greatest number of alterations, as above, but follow a contrary course. That is, our first transpo- sition takes only one alteration, the next will have but tivo {i. e., one additional alteration), and so on — we pass, in other words, from a given scale to that one 7nost nearly related to it by commu- nity of tones. Two major scales are related in the fl7'st degree when they have the same tones in common but one — in the second degree, when they have the same except tivo, and so on of the third and fourth degrees. Or, to put it more practically, two major scales are related in the first degree if the Tonic of one forms with tliat of the other a major Fifth, or its inversion, a minor Fourth. Thus, the major scale of C is related in the first degree to that of Gy ^), and to that of F{p-\); each of these latter scales therefore takes htt one alteration. The scale beginning a major Fifth above, or a minor Fo^irth below G — that of D — will take one additional alteration, and so will that begin- ning a major Fifth below, or a minor Fourth above F — the scale of B7 ; therefore the major scale of D is related to that of C in the 68 MODERN TONALITY. second degree, as is also that of B\^, and so on of the other degrees. This is ilhistrated in the following Figure ; the model scale stands in the middle, above and below it are two transpositions by major Pifths or minor Fourths : Fi(j. 49. — Majob Scale of JD. $ =«*= i«»- MaJOR SCAIiE OF G. -»' 9- Model Ma joe Scale of G. Major Scale of F. ^ s Major Scale op J5b. _ ^ « — bm^ S Sicniatiire. i W 'm 106. As the above figure shows, in transposing the scale by the major Fifth above (or by the minor Fourth beloiv), degree VII of each new scale is chromatically raised; in transposing by the major Fifth below (or by the minor Fourth above), degree IV of each new scale is chromatically lowered. Now, this raising and lowering can be carried on with single sharps and flats np to the scales of Cj^ and C\^ (inclusive), but 7io farther, since in these scales every degree is already sharped or flatted (see Figs. Ji.7 and Jf8). Plainly, then, in continuing to transpose as before, raising the Vllth or lowering the IVth degree of each new scale, we should have to use the double-sharp or the double-flat. Thus, MO D E R X TO NA L I TY. 69 starting anew fi'om C^ and transposing by the major Fifth above, or from Cj? and transposing by the major Fifth heloiv, we obtain in the former case the scales of G'^ major, D^ major, etc., and in the latter those of i^j? major, B\f\f major, etc., written thus : Fig. 50. Si-ale of Gt ma,ior. j(^ Scale of Z)f major. f ji:3^^^ ^: ^^^^m==;^^E ^^ 3^"^^^ ^ j{ ^ ~y a- ^ Scale ot Kh nin.1or. , i bcale ot if^r* mainr. , . l(,_ 5^ lOT. The excessive chromatic alteration in the aboTe scales (which would increase with each new transposition) renders them practically useless. We may, however, use their key-tones for scales h)^ enliarmonically changing their notation. Thus, G^ being written A\f, the scale of the latter tone (signature, 4 flats) is sub- stituted for that of the former; D^ being written ^[?, the scale of the latter (signature, 3 flats) is used instead of that of the former ; and similarly, the scales of F\^ and B\f\} severally give way to those of E (signature, 4 sharps) and A (signature, 3 sharps). 108, The following figure gives a table of signatures of major scales transposed by major Fifths above and below from the model major scale. Some of these scales are inter cha^igeable at pleasure, viz. : ^ and C't? major; F'^ and G\f major; C'^ and D\^ major. Explanation of Fig. 51. — Starting from C we proceed to the right till avb come to a signature containing a x, when the scale must be exchanged for that whose signature stands imme- diately underneath : again, starting from C at the right we pro- ceed to the left till we find a signature with a \f\y, when the scale Avith that signature gives way to the scale whose signature stands immediately above. 70 MODERN TONALITY. Fiff. 51. C G D A E B Ft Ct Gt Fh Cb Gh Db Ab M Bo F C Exercises. 1. Transpose the model major Scale of C into A] E; B\ F^\ Gv, ; A\^; D; E\,; B\y- G\^. 2. Transpose the Chromatic Scale of C (Fig. 9, p. 21) into Cjj^; B; Ay, /'jt; Gfy, E; E\^. QUESTIONS. 99. Which one of the two keys is called " Tnajor " ? For what kind of music is the major key best adapted ? Which scale represents this key ?. . . . lOO. What peculiarity do we find in the melodic structure of the model scale of C? ...101. Why is the model scale of (7 called "major"?.... 102. How may we have the major key in a 'pitch other than that of the scale of C? How may we obtain 11 additional major scales ?. . . . 104. In the case of a transposed scale, how are the chromatic alterations indicated t What is this indication called'? What is the effect of the Signature ?.... 105. In transposing a model scale according to a systematic order, how do we begin ? When are two major '^csAes, related in the 1st degree? In the 2d degree? How is the first degree relationship of two major scales expressed more prac- tically?. ..106. Why may we not transpose with single sharps and flats beyond the scales of Ct and Ch ? What follows from this impossibility, for further transpositions .?.... 1 07. What is to be said of scales having in their signature the x or the bb ? In what way, however, may we utilize tlieir key- notes ? Give examples. . . . 108. Which major scales are interchangeable ? MODERN TO NA LIT V. 71 CHAPTEE XIY. The Minor Key. 109. The eminent modern musical theorist, C. F. Weitz- mann, speaking of the minor key, says: "This key, which exhibits minor Triads as its principal features, is of a gloomy and melan- choly character, and in melodic and harmonic respects the con- stnnt antithesis of the bright and aggressive major key," etc., etc {Harmoniesysffm, Leipzig.^ The minor key is represented by the diatonic scale of A, which is therefore the ?nodel minor scale. 110. In tlie model minor scale of A we find that — unlike th^ structure of the major scale of C — the half-step lies between de- grees II-III and V-VI. Fin. .->2. I II III IV V VI VII VIII Remakk. — As we shall presently see, a iltird half-step (between degrees VII and VIII) is frequently found in the minor scale, im- plying, of course, chromatic alteration. 111. Again, by adding the Third and the Fifth to the ToniCy ISabdominant, and Dominant of the scale oi A, we obtain in each case a minor Triad ; for which reason this scale is called "minor," and represents the minor key, in general. Fifj. /7.'i. I IV V 72 MODERN TO NA L IT Y. Remark. — The Triad of the Vth degree in the minor key ia often changed into major — a point to which we shall return. 112. The model scale of A may, like that of C, be transposed, any one of the remaining eleven tones beiug taken as starting- point or Tonic. Here also, as in the case of the major scale, we adopt the systematic order of transposition, i. e., by the vnajor FiftI/, nhoxe and below, in following which beyond a certain limit we find that some tones are not suitable for key-tones, as they are ivritten, inasmuch as the notation of their scales would be too complicated. We therefore enharmonically change the names of such tones, as explained in the preceding chapter. 113. The following table of signatures exhibits such trans- positions of the model minor scale of A as are practically useful, the scales of G^ and A\^ minor being interchangeable, as are also those of D^ and E \^ minor : Fig. 54:. G D 114. It is seen from iliQ above figure that the scale of E minor takes the signature of G major, B minor that of D major, F^ minor that of A major, D minor that of F major, G jninor that of B\; major, and so on. In -fact, the scale of E minor has the same tones as that of G major, the scale oi Dininor the same tones as that oi F major, and so on of the others. These and similar cases are examples of so-called parallel or related scales or keys. Every major scale has its relative minor scale; the two scales have tones and signature in common, but the minor scale starts from the MODERN TO NA LITY. 73 minor TJiird * below the key-note of its relative major. Thus, A minor is the relative minor scale to C major, and conversely, C major is the relative major to A minor. 115. Two minor scales are related in the first, second, or third degree, precisely as two major scales are under certain circum- stances, i. e., in the jirst degree when the two minor scales have the same tones hut one; in i\\Q second w\\q\\ they have the same but hoo, and so on — relationship of the first degree existing, as before, between two scales whose Tonics form a tnajor Fifth or a minor Fourth. In this way A minor is related in the first degree to E minor and to D minor, G minor in the first degree to D minor, and in the second to A minor, etc., etc. 116. When the tone on the Vllth degree of the minor scale is to be used as leading-tone (see 98, p. G4) it will be necessary to —^ I — 1-^ I I — I 1 ' i-^-. If the foreign tone has not the charactei^ of leading-tone as above explained, the decision as to its orthography will again de- pend upon what is intended. An example or two will illustrate this. -9^ — -VI.* '• Our Consonances e and g are also, like the key-tone, provided each with its leading-tone" Thus e has its d'^, and g its/||. " Our neutral is a tone which appears sometimes as a weak •dissonance, sometimes as a consonance admitting of being doubled. {Note. A dissonance progresses either upward or downward: a neutral may move both upioard and doumward.)" " Dissonances are leading-tones tending upwards or down- wards." In accordance with the above theory, there is but one absolutely consonant Triad in a key, major or minor, viz: the Triad of the Tonic— 1, III, V, all the other Triads of the key being, relatively to that of the Tonic, either dissonant or imperfectly consonant. * See the sclieme of the Chromatic Scale adopted by Yioceat, Appendix I p. 83 of this work. INDEX. CHAPTER I. Tones, and their Notation. PAGES 1. In wliat sense the -word " Tone " is to be understood . . .2. Our mod- ern tonal system embraces twelve tones, seven of whicli are classed as •' primary.". . . .3. How tones are expressed to tlie eye. The Staff. 4 Clefs 6. Repetitions of tones in a higher and a lower pitcli, called " Octaves." Secondary (broader) sense of the word " Octave." The " Counter Octave," " Great Octave," " Small Octave," etc., etc. Questions 9 — 14 CHAPTER 11. Scale and Key. The Diatonic Scale in General. 7. The "Scale." Its distinction as Diatonic or Chromntic. . . .8. "Key." ... .9. Every key represented, in its essential tone-material, by its Diatonic Scale. The chromatic-diatonic genus of composition. The pure diatonic genus 11. The piano key-board divided into "half- steps." The " Half-step " and the " Step." 13 Exact definition of the Diatonic Scale, in general. Significance of the word "diatonic." ... .13. Tlie seven different Diatonic Scales. . . .14. The only two of the Diatonic Scales used in modern music .... Questions 14 — 18 CHAPTER ITI. Modification or Chromatic Alteration of Tones. 16. Twelve tones, counting the intermediate ones, embraced within an Octave. . . .17. How the intermediate tones are "derivatives" of the 94 MODERN TONALITY. PAGES primary tones. . . .18. Modification of a tone by raising or by depres- sion. The Sharp, and Double-sharp. The Flat, and Double-flat. . . . 19. The Cancel. What it indicates. .. .20. Chromatic Signs, and what they effect, taken together. The chromatic nomenclature. How one staff-degree may be the seat of five different tones. . . . Questions 18—20 CHAPTER IV. The Chromatic Scale. 21. What constitutes a Chromatic Scale. This Sca^e a Diatonic Scale, with intermediate tones. . . .22. Usual manner of writing the chro- matic Scale. . . .23. Theory of the usual notation of the Chromatic Scale ...24. Two-fold distinction of Half-steps, and of Steps. The diatonic half-step. The chromatic half-step. The use of the chro- matic signs in writing a half-step not necessarily determinative of the latter as chromatic. Why the Scale generally called " cliromatic " is more properly called " chromatic-diatonic". . . .25. Two-fold notation of the Step. The diatonic notation the most usual. Composition of the diatonic Step. . . .Exercises. . . .Questions 20 — 23 CHAPTER V. Intervals in the Diatonic Scale. 26. Definition and full significance of the term "Interval." 27. The two elements of the interval — Denomination and Kind. The six Denominations of Intervals in the Scale 28. Two-fold Kind of each Denomination of intervals in the Diatonic Scale. Difference between a minor and a major Interval. Proof that this difference is specifically that of a chromatic half-step 30. Seconds. An "En- harmonic change" not a second 31. Seconds in the Diatonic Scale of C. and distinction of them as minor or major. Every Step {d\&- Xomc)ii major Second. .. .32. The minor Second identical vsdth the diatonic JiaJf-step. Not every half-step a minor Second. . . .33. Orad- v(d, as distinjruished from skipping progression 34. Thirds in the Diatonic Scale of C ; which are minor, which major. . . .35. Fourths in the Diatonic Scale of C ; with what single exception all are minor. MODERN TO NA LITT. 95 PAOES The minor Fourth often called " perfect," and the major, "augmented." ... .80. Fifths in the Diatonic Scale of C ; with what single exception all are mnjor. The minor Fifth often called " diminished," and the major, "perfect." Harmonic importance of the Fifth 37. Sixths in the Diatonic Scale of C ; wliich are minor, which major. . . . 38. Sevenths in the Diatonic Scale of C; with what two exceptions all are minor. .. .39. The "Half-step Formula" of Intervals.... 40. Application of this Formula 41. The Half steps, in using this Foniiula, le.'/thentic Cadence; the Plagal. , . . .98. Tlie Leading-tone. Manner of marking the degrees of the diatonic scale by Roman numerals. . . , Questions 59 — 65 CHAPTER XIIL The Major Key. Transposition of the Model Major Scale. 99. The Major Key, its character and representative scale. . . .100. Pecu- liarities of the melodic structure of tlie model scale of C . . . 101. Major Triads the characteristic harmonies of the key of C. . . . 103. Variations, in pitcJi, of the major key, obtained by transposinc MODERN TONALITY. 99 PAGES its model scale 103. First transposition by a chromatic half-step. The major scales of Ct and Cb 104. The chromatic alterations in the last-mentioned scales. The Signature 105. The systematic order of transposition. Relationship in the first and second degrees between two major scales. 106. Continuation of transposition by Fifths above or below, giving scales with the x and the bh in their si^'nature. . . .107. Practical uselessness of the last-mentioned scales. Their key-tones rendered available for new scales by enharmonically changing their notation. Illustrations 108. Table of signatures of major scales transposed from the model major scale. Inter- changeable scales. Exercises, Questions 65 — 7( CHAPTER XIV. The Mixor Key. 109. Weitzmann's characterization of the minor key. The representa- tive scale of this key. . . .110. Melodic structure of the model minor scale of A. . . .111. Min»r Triads the characteristic harmonies of tlie key of A. Occasional alteration of the Triad of the Vth degree. . . . 112. Transposition of the model scale of A. The same order followed as in transposing the model major scale. Enharmonic change in the names of some key-tones. .. .113. Signatures of transposed minor scales 114. Parallel scales. .. .115. First and second degree rela- tionship existing between two ndnor scales... 116. The Vllth degree in minor raised, to serve as leadinrj-Ume. . . .117. The doctrine that the Vllth degree in minor must always be raised, not advocated in this work. .. .118. Only three kinds of Triads found in the nor- mal minor scale. . . .119. Triads in the " harmonic" minor scale. . . . 120. Why this scale is called the " harmonic". . . .121. Rationale of the " melodic" minor scale 122. Dr. A. B. Marx's rejection of the '• melodic" minor scale and advocacy of the " harmonic." Our rejec- tion of the " harmonic," and advocacy of the " normal " minor scale. ...123. The Seventh in minor not ahoays used as leading-tone. Why the raised Seventh is never included in the Signature of a minor key... 124. Dr. E. KrQger on the characteristic Triads in minor. . . .125. Arrey von Donimer on the same subject. . . .126. C. F. AVeitzmann's theory. . . .127. No need of dispute about the minor 100 31 D E R N TONALITY. PAGES scale. Each different fonn good for its purpose, but one of them the normal form. .. .128. Proposed theory of the minor scale. Brief summary of the whole doctrine of this work. . . .Quesiions 71 — h\ APPENDIX 1 83 APPENDIX II 86 APPENDIX III... 91 \ UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. CO Mfi^-rip 1 3 19/J3 NOV 2119' n^^c'-o\m ^f C *- 6 W/3 DEC 2 1 iy/3 ftF / " 1974 FEB - 7 1974 QUAKip^OAN li ^ ^P 17 1981! SEP 1 7 iisaj '^fC'D MUS-LIB Form L9-39, 050-8, '65 (F6234s8)4939 UCLA - Music Library MT50C815p1904 miiMi iiiiiiiiiii mill II III II mil III L 006 961 341 2 nustc UBRART m 50 C8l5p 190h '!. 1 *? UC SOUTHERN REGIONAL LIBRARY FACILITY AA 000 544 732 i 1 >■■■