553 Chester Harvey Rov.nell <^ ^ • J / ^^ / bb-B^ / bbb-E' / bbbb-Ai- / bbbbb-Di' / bbbbbb-G^ / bbbbbbb-C' G- D- A- E- B- F«- -% \ (A M 53 /^ Place clip here 1 P ! 11 3^ " 2 •s« .9- a c «u a£ Is en V •53 II !c "o ti ■3 0. 3 U lace clip here 0. \) - — F — E- .s. F - E- > ' — — -c— - 1 u •a e 3 be c 2. •§ e V 5 i U --e B —A- — G — F (r ~\ — B ^ — A-— 1 0) \^ ) — — j: — -F- \ (^ — — E S D 2 c c •S ' c E — D — — B— - Ji A ^ "in 1) bo rt a g 3 c -B A I Ca ^^•- — A g — G "i o -G V^ y • — — t-— s 1 F -E- ^^^ y — u r -2- u C a V 'i - e- B —A — - ^ 8 A- E CHART I '^ Place clip here lace clip here Va MUSICOLOGY A TEXT-BOOK FOR SCHOOLS AND FOR -:- -:- -:- GENERAL USE -:- -:- -:- By MAURICE S. LOGAN HINDS. NOBLE & ELDREDGE. Publishers. 31. 33. 35 West 15 th Street. New York City. Copyright. 1909. BV MAURICE S. LOGAN. r:^ '^ c/^^ Entered at Stationer's Hall, London. Ent^land. Printed in the United States of America. PREFACE The object of this work is to furnish a practical and com- prehensive text-book on the theory and philosophy of music, for schools and for general use. It deals with the science rather than the art, while most music text-books deal with the art rather than the science. For school use it is intended to be included in the science course, rather than in the music course. From the educational standpoint, either of mental training or useful knowledge, the science of music, as distinct from the art, is entitled to rank with the other sciences taught. M. S. L. iv;58?947 TABLE OF CONTENTS PART FIRST. ELEMENTARY. PAGE Common Terms and Signs Used in Music - - - - 17 Notes, 1— Rests, 2— Staff, 4— Clefs, 7— Score, 14— Sharps, Flats, and Naturals, 20 — Signature, 21 — Accidentals, 22. Rhythm - - ------ 22 Accent, 1 — Bars, 3 — Time, 5 — Duple. Triple, 6 — Simple, Compound, 12 — Double, Triple, Quadruple, 12 — Broken Pulses, 14 — Tataing, 16 — Counting Time, 22 — Measures, 3 and 23 — Tempo or Rate of Movement, 28 — Standard of time value, 29 — Applying Words to Music, 30 — Interrupted Rhythms, 34 — Change of Tempo, Pause, Syncopation, 34 — Triplets, 36 — Repeats^ 37. Expression - - ------29 Terms Relating to Power, 2 — Terms Relating to Quality, 3 — Terms Relating to Movement, 4 — Embellishments, 6 — Turn, 6 — Trill, 7 — Mordent, 8 — Grace Notes, 12 — Appog- giaturas, 13 — Acciaccaturas. 14 — After Tones, 15. Keys --------- 31 Chart I, 2— Key Patterns, 4— Major, 5— Minor, 6— Forma- tion of Signatures, 15 — Intermediate Tones, 17 — Keyboard, 18— Chart II, 19— Diatonic Scale, 22— Chromatic Scale, 23. Re.\ding Music - - - -- - -35 Running the Scale, 1 — Mental effect of each Key tone, 2 — Transposition, 5 — Key-Tonality, 7 — Hearing with the eye, 8 — Seeing with the ear, 8 — Solfaing, 12 — Laing, 13. Tonic Solfa Notation - ----- 37 Interrelationship of Major Keys - - - - 38 Key Building, 1 — Major Key-Table, 5 — Analysis of Table, 6 — Locating Key-note, 11 — Enharmonic Change, 13 — Key Circle, 15 — Relation of Sharp and Flat Keys on same letter, 16. Interrelationship of Major and Minor Keys - - - 43 Relative Minor, 1 — Tonic Minor, 6 — Combined Major and Minor Key-Table, 12 — Analysis of ' Table from relative and tonic views, 16 — Double sharps and flats, 41 Melodic Minor Scale - - - - - - 52 Old Minor Mode - - - - - - -53 12 TABLE OF CONTEXTS PAGE Ancient Greek Modes - - - - - - 54 Distinction between modes and keys, 3. Harmonic Scale Names - - - - - - 55 Dominant and Sub-Dominant sides of the scale, 1. Intervals - - ------ 55 Standards of measurement, 2 — Major and perfect intervals, 2 — Analysis of the scale, 2 — Simple and compound intervals, 2 — Minor, diminished, and augmented intervals, 3 — Inverted intervals, 6 — Diatonic and Chromatic intervals. 7 — Concordant and discordant intervals, 12 — Formation of the Diatonic and Chromatic Scales, 14. PART SECOND. STRUCTURE OF MUSIC. Harmony - - - - 59 Chords and Discords, 1 — Building in 3rds, 2 — Harmonic Scale, 3 — The Triad. 4 — Four Part Harmony, 7 — Major, Minor, Dim. and Aug. Triads, 11, 12 — Analysis of the triads of the major and minor modes, 13, 14 — Characteristic Harmonies, 19— Laws. 21. 23. Positions and Forms of Triads - - - - - 64 Thorough Bass Figuring of Triads ' - - - - 66 Dissonant Chords - - - - - - -67 Dissonant Triads, 3 — Extensions of the Triad, 6 — Chords of the 7th, 11 — Classification and analysis of chords of the 7th. 11 — Positions and Forms of Chords of the 7th, 17 — Thorough Bass Figuring of Chords of the 7th, 18 — Chord of the 9th, 20 — Chords of the 6th, 23. Interrelationshii' of Chords in General - - - - 74 Common Key Relationship, 2 — Root Relationship, r, — Com- mon Tone Rel.\tionship. 6. Interrelationship of the Triads of a Key - - - 75 Progression --------76 General principles, 1 — Progression of Triads, 6 — Voice Leading, 6 — Skips, 14 — Progression of the Bass, 17 — Con- secutive .")ths or Octaves, 19 — Hidden Consecutives, 24 — False Relations, 26 — Motion of the Parts, 28 — Preparation of Dissonances. 29. Resolution . . - . - . _ _ g2 General principles, 1 — Tendency of Dissonances to resolve, 6 — Rules of Resolution. 8 — Ornamental Resolution. 12 — Resolu- tion of Dissonant Chords, 14 — Treatment in upper part. 19 — Treatment in lower part, 19 — Formulas for the Resolution of 7th Chords, 19— When the rules of Resolution are set aside, 22. TABLE OF CONTENTS 13 PAGE Suspension and Anticipation - - - - - 87 Object, 3— General Rule, 5— Principal Suspensions, 6. Passing Notes ..-----87 Pedal Passage - - - - - - -83 The Cadence -------89 Authentic: Plagal, 1— Half Cadence, 2— Deceptive Cadence, 4 — Cadenza, 5. Sequence --------90 Modulation .-------90 General principles, 1 — 11. Methods of Modulation - - - - - - 93 Modulation by Means of Connecting Triads, 2— Table of the triads common to different keys, 3 — Common triad form- ulas, 7 — Confirming the Modulation, 12— Modulation by Means of Connecting Tones, 15 — General principles, 15 — Major, minor, and dim. triad formulas combined, 20 — Applica- tion of the Combination Formula, 21 — Modulation with the Dominant 7th Chord, 24 — The modulating properties of the Dom. 7th Chord, 25 — The modulations resulting from forming the Dom. 7th Chord on the different tones of a key, 28 — When the modulation leads to major, and when to minor keys, 42 — Resolving the Dom. 7th Chord of one key to the Dom. 7th Chord of any other key, 4.3 — Modulation by chromatic runs, 44 — Modulation with the Diminished 7th Chord, 46 — The modulating properties of the Dim. 7th Chord, 47 — Enhar- monic modulation through the Dim. 7th Chord, 54 — The modulating capacity of the Dim. 7th Chord, 60 — Practical limit of modulation in either direction, 62 — Modulation with the Augmented 6th Chord, 63 — Italian, French and German 6th Chords, 64 — Their common resolution, 68 — The natural seat of the Aug. 6th Chord, 70 — Their modulating tendency, 71 — Modulation by enharmonically changing the German 6th Chord into a Dom. 7th Chord, and vice versa, 72 — Modulation by Inversion (over to page 120) — Modulation by Imitation (over to page 125). Summary of Modul.\tion ------ 107 Diatonic and chromatic tones, 1, 2 — Diatonic and chromatic chords, 3 — Modulation without or in advance of accidentals, 5 — Three stages of modulation. 7 — Classification of Modula- tion, 9 — When to modulate, 14. Transposition - - - - - - - - 109 Distinction between Modulation and Transposition, 1 — Transposition merely a change of pitch, 2 — Singing in a dif- ferent- key from that in which the music is written, 7 — Play- ing in a different key from that in which the music is written, 8. 14 TABLE OF CONTENTS PAGE Counterpoint - - - 112 Distinction between Harmony and Counterpoint 2 — Origin of Counterpoint, 4 — Outline, 7 — Simple Counterpoint, 8 — Contrapuntal Rules, 12 — Double Counterpoint, 28 — Table of Inversions, 29 — Analysis of Inversions, 31 — Modulation BY Inversion^ 44 — Counterpoint Invertible in Various Intervals, 53 — Triple and Quadruple Counterpoint, 56. Imitation -------- 123 Kinds of Imitation, 4 — Modulation by Imitation, 13. Contrapuntal Music - _ - . _ - 126 Fugues, 2 — Canons, 10. Melody - - - - 130 Interrelationship of Melody, Counterpoint, and Harmonj^ 1 — Chords and Alelodic Figures, 5 — Harmonic, melodic, and rhythmic elements of figures, 7, 11 — Simple and compound figures, 11 — Development, 12 — Thematic Treatment, 13. Linguistic Character of Music - . _ . . 133 The Period ----.-_. 134 Rhythmic structure, 3 — Thematic structure, 9. Form - - - - 137 Strophe Form, 5 — Art Song Form, 6 — Unity. Contrast, and Symmetry, 10 — The Song Forms, 16 — OutHne. 18 — The Scherzo Minuet or Applied Song Form, 19 — Outline — Rondo Form, 23 — Outline — Sonata Form, 25 — Outline — Suite Form, 28 — Outline — Contrapuntal Forms (back to pp. 126-129) — Introductory, Intermedi.\te, and Concluding Parts, 35— Outline, Classification of Music - - - - - - 144 PART THIRD. ACOUSTICS. Acoustics - - - - - 147 Cause of sound, 1 — Range of sound, 2 — Sound waves, 4 — Laws of Vibr.vtion, 7 — Compound Tones, 16 — Compound vibration of strings, 16 — Harmonic Series, 19 — Simple Tones, 24 — EfTecl of overtones, 26 — Chord Formation, 35. Scale of Nature ----_.. 157 Interval Ratios, 2 — Standards of Pitch, 8 — Application of ratios, 13 — The Enharmonic Scales, 16. TABLE OF CONTENTS 1 5 PAGE Equal Temperament - - - - - - - 162 The Equal Temperament Scale, 1 — Application of logarithms, 9 — True and tempered intervals compared : 1st, through their logarithmic values, 18; 2d, through their semitone values, 20. Consonance - - - - - - - - 167 Blending of overtones, 1 — Blending of vibrations, 10. Beats - . - . - .... 170 Imperfect Unison Beats, 5 — Imperfect Consonance Beats, 9— Beat Tones, 16. Differential and Summational Tones . _ - - 174 Orders of Differentials, 5 — Complimental relations, 8 — Theories, 16. Dissonance -_--._-- 179 As due to beats, 1 — As due to combinational tones, 6, Sympathetic Resonance - . . . . _ igo PART FOURTH. PRINCIPAL SOURCES OF MUSICAL SOUND. Outline - . _ . . - - - 185 Strings, Rods, Pipes, Reeds - - . . . ige Transverse Vibrations of Rods, 3 — Longitudinal Vibra- tions OF Rods, 7 — Pipes, 14 — Resonance Boxes and Sounding Boards, 27— Reeds, 28. Membranes, Plates. Bells --.-.. 192 Membranes and Plates, 2 — Bells, 14. PART FIFTH. APPENDIX. History of the Diatonic Scale - - _ . . 197 Ancient Greek Modes ---... 202 The Earlier Greek Modes, 4 — The Later Greek Modes, 7 — The Early Church Modes. 11. Table of Common Musical Intervals _ . . . 2O6 Temperament - --__.. 207 Equal Temperament, 3 — Mean-Tone Temperament, 5. Tuning - -.-_.._ 2II Calculation of Pitch Numbers - . . . . 214 Character of the Different Keys _ . _ _ 217 PART FIRST ELEMENTARY COMMON TERMS AND SIGNS USED IN MUSIC 1 . Notes are characters used in music to represent tones. As to relative time value, there are six kinds of notes in com- mon use : the whole note («> ), the half note (J), the quarter note (J), the eighth note (^^), the sixteenth note ( 1^), and the thirty-second note ( ,s ) : each being half the time value of the preceding one. The stems may extend either up or down ; and the hooks, either right or left. Two or more notes are frequently attached to one stem. When necessary, the direc- tion of the stem is made to show the part to which the note belongs. When a note has two stems in opposite directions, it belongs to two parts. 2. Rests are characters to represent silence. There are six rests, corresponding in time value to the six notes: the whole rest (— ■), the half rest (-"), the quarter rest {"<*, I, or ^), the eighth rest (•/), the sixteenth rest (^), and the thirty-sec- ond rest ( ^ ). 3. A dot after a note or rest increases its time value one- half; thus a J. is equal to jj j, a J. is equal to ^^ J^ J^, and a ^ . is equal to ^^^^^^ — and similarly as to rests. 4. A Staff consists of five lines with its four intervening spaces, on which the notes are written. It is a kind of musi- i8 MUSICOLOGY cal ladder to indicate the pitch of tones. When necessary, short hnes are added above or below. Each line or space of the staff is called a Degree. Treble Staff. Bass Staff. For Soprano and Alto voices. For Bass and Tenor voices. Fig. 1. (Base in music is usually spelled dass ; but dase is more in accord with its meaning, as it means the base or foundation.) 5. In music the treble and bass staffs are usually sepa- rated so that words of a song may be written between, but it must be remembered that their relation to each other (as shown in Fig. i) is not changed. Therefore, when thus sep- arated, the second added line above the bass staff is the same as the lower line of the treble staff, and the second added line below the treble staff is the same as the upper line of the bass staff. 6. Sometimes (in choir or other music) each voice, or part, is written on a separate staff. 7. The S clef is always placed on G, above middle C (encircling the line marked G), and is therefore called the G clef (or Treble clef). When this clef is used, middle C is always on the first added line below the staff. 8. The ^ clef is always placed on F, below middle C (starting on and encircling the line marked F), and is therefore called the F clef (or Bass clef). When this clef is used, mid- dle C is always on the first added line above the staff. It will ELEMENTARY I 9 be seen in Fig. i that middle C is exactly midway between the G and F clefs. 9. The jSj: clef, when used, is placed on middle C (center- ing on the line or space marked C), and is therefore called the C clef (or Tenor clef when used for the tenor; or Alto clef when used for the alto). The C clef| M: I is a movable clef, and may be used to locate middle C on any degree of the staff by centering it on the desired line or space. It must be fixed in the mind that middle C is the same tone wherever written. It is called middle C because it is the middle of the great vocal compass, and is therefore the center around which the different voices range : female voices ranging mostly above ; and male voices, mostly below. (See the average range of the different voices in Fig. i.) 10. The clefs are used for the purpose of adjusting the staff to the range of voices that are intended to sing from it. It will be seen from Fig. i that the G I ^ I clef, by locating G on the second line, adjusts the staff to the range of soprano voices ; and that the F ( ^' j clef, by locating F on the fourth line, adjusts the staff to the range of bass voices. But we see that the alto and tenor (especially the tenor) ranges ex- tend over onto both staffs thus adjusted: the alto, notes (written on the treble staff) would require added lines be- low the staff, or the tenor notes (written on the bass staff) would require added lines above the staff (supposing the staffs separated as we find them in music). Foi this reason the C I ^ I clef is sometimes used to adjust the staff to the alto or the tenor (especially the tenor) range by locating middle C (which is near the middle of both ranges) on any desired de- gree of the staff (usualh' on the third space or fourth line for tenor or on the third line for alto). 11. The tenor staff (middle C on third space) and the 20 MUSICOLOGV treble staff (upper C on third space) are both lettered alike and therefore read alike. For this reason the treble or G clef is sometimes used for writing the tenor; but the tenor is thus written an octave higher than sung. 12. A staff consists merely of five lines (and the spaces be- tween); its range being determined by the clef placed upon it. Observe that the staff is adjusted to the clef, and not the clef to the staff; for each clef represents a fixed tone. 13. As the C clef represents middle C, its natural position is exactly midway between the G and F clefs. 14. When the music for the different voices or instruments is written on separate staffs, it is called score music, and two or more staffs connected by a brace is called a Score. 15. In most song-books, however, we find the soprano and alto written together on the treble staff, and the bass and tenor together on the bass staff. In which case the so- prano is represented by the upper notes on the treble staff, and the alto by the lo\\'er notes on the treble staff; the tenor is represented by the upper notes on the bass staff, and the bass by the lower notes on the bass staff. 16. The tune, or melody, is usually in tlie soprano — the other parts being an accompaniment. 17. Female voices range one octave (eight degrees) higher than male voices — female voices being limited to the soprano and alto ranges, and male voices to the bass and tenor ranges — so that men in singing the soprano or alto from the treble staff sing an octave lower than written, and women in sing- ing the bass or tenor from tlie bass staff sing an octa\-e higher than written. This, howexer, occasions no incoiuenience, as singing in octaves produces the same musical effect ; for which reason, tones an octave apart are lettered alike. 18. The bass and treble staffs (shown in Fig. i) are fixed as regards their tone range, so that each line and each space represents a certain pitch of tone, each of which is named by one of the seven letters. A. H. C. 1). E, I*", G ; these seven ELEMENTARY 21 letters being repeated in each octave. (I'rom any letter to the same letter above or below is an octave.) 19. The names usually applied to the relative tones of a key (see p. 31:1) are the seven syllables Do, Ri\ J//, Fa, Sol, La, Ti, which are repeated in each octave. These syllables are a great assistance in reading music, and aid in producing good tones. They are of Italian origin, and must be given the Italian pronunciation by giving i the sound of e, e the sound of a, a the sound of a in "far," and o the sound of o in "no." For the sake of uniformity in the different coun- tries, the Italian language is recognized as the language of music. 20. A Sharp (ji) placed on a line or space raises its pitch a half-step. A Flat ( b ) placed on aline or space lowers its pitch a half-step. A Natural (^) destroys the effect of a sharp or a flat. 21. The sharps or flats at the beginning of a piece of music is called the Signature. They are placed on the lines and spaces that are to be sharped or flatted through- out the piece. They also affect the same letters in each octave. 22. Sharps, flats, and naturals are called Accid- entals when they occur during the course of the music. An accidental influences the line or space on which it is placed only to the end of the measure, unless the affected note is tied over into the next measure. 23. In reading music thus affected by acci- dentals, to give a sharp effect to any key syllable use the sound of i (pronounced c) ; thus chang- ing Do to Di, Re to Ri, Fa to /^/, Sol to Si, and La to Li. To give a flat effect to any key syllable use the sound of r (pronounced rt) — in Re use the sound of ^ (pro- nounced a) — thus changing Ti to Tc, La to Lc, Solt o Se, Mi to JSIc, and Re to Ra. b ~Do- -Ti~ « Te -La- Li Le Sol- Si Se -Fa- -Mi- Fi Me -Re- Ri Ra - Di -Do-^ Give Italian pro nur.ciation. F IG. 3. 22 MUSICOLOGV RHYTHM 1 . Accent is a stress of the voice on certain pulsations of the music. 2. Music has regularly recurring accents which cause it to throb or pulsate. This throb or pulsation is called the Rhythm of the music. 3. A light bar is drawn across the staff before each heavy accent, thus dividing the music into equal parts called Meas- ures. Measures therefore indicate the rhythm of the music. 4. Heavy bars across the staff are frequently used to mark the end of a line of words. A double bar across the end of the staff marks the close of the music or of a strain. 5. Time. Time refers to the number and time value of the pulses in each measure, by which the time value as well as the rhythmic nature of each measure is shown. 6. The time value to be given to each measure is shown by the figures at the beginning of the music, called the Frac- tion; the upper figure showing the number of pulses in each measure, and the lower figure showing the time value of each pulse. (| time is sometimes marked thus, g; | time thus, g; and \ time thus, ^ ^ or g g.) The lower figure of the fraction is usually 2, 4, or 8, according as each pulse is to be valued at a half, quarter, or eighth note. The upper figure is usually 2, 3, 4, 6, 9, or 12. If the upper figure is divisible by 2, it is called duple or even time (or rhythm); if divisible by 3, it is called triple or uneven time (or rhythm). 7. In duj)le time every other pulse is accented. In triple time every third pulse is accented. 6 and 12 naturally divide into groups of 3's by accents on every third pulse, and are therefore triple as to the groups, though even as to number of groups, in a measure. 9 is triple, both as to groups and number of groups. 8. The heavy accent in each measure is always on the first note of the measure; the other accents (if any) in each meas- ure are lighter accents. As every second or third pulse is ac- ELEMENTARY 23 cented, therefore, if a measure contains four or more pulses, it must contain one or more light accents; and the pulses will be divided by means of the accents into groups of 2's or 3's, according as the time is duple or triple. 9. Sometimes the upper figure of the fraction is 5 or 7. These are called peculiar rhythms, being neither duple nor triple, but may divide into alternate groups of 2's and 3's, and therefore alternately duple and triple as to the groups. 10. Two dissimilar rhythms are sometimes used at the same time (but in different parts) and called combined rhythm. 11. The usual method of keeping time in singing is by beating time with the hand. The usual manner of beating time is — down, up; down, left, up; or down, left, right, up — according as there are 2, 3, or 4 pulses to the measure, in- cluding one pulse to each beat. If there are 6, 9, or 12 pulses in a measure (divided into groups of 3's), the time is beat in the same manner by including three pulses to each beat. 12. If one pulse is included in each beat, the time is called Simple. If three pulses are included in each beat, the time is called Compound. As regards the number of beats in a measure, the time is commonly called double, triple, or quad- ruple (if simple) ; or, compound double, triple, or quadruple (if compound). In compound time, only the accents are beat. 13. The manner of beating time is not essential- — evenness being the essential feature. Beating time is not difTficult if the rhythm or throb of the music is felt : feeling the rhythm is beating time mentally. A common method of keeping time when playing is counting the pulses in each measure. 14. Broken Pulses. If the time value of a pulse is broken in any manner, it is called 7k broken pulse. The time value of a pulse maybe broken up into any kind of notes, or notes and rests, which together equal the time value of the pulse. On the other hand, a single note may cover the time value of one 24 MUSICUI.UGV pulse and part of anotlier or two or more pulses. Thus there is no fixed relation between the time values of the pulses and the notes. If their time values corresponded each to each, keeping time in music would be a very simple matter. This is the case in some very simple pieces of music. The diflfi- culty in keeping time correctly naturally increases as the rela- tion between the time values of the pulses and the notes becomes more and more complex. 15. Beating time serves to mark the time value of the pulses (singly, in simple time; by 3's or by accents, in compound time), so that the real art of keeping time consists in giving the proper relative time values to the different parts of the broken pulses. This involves a development of the sense of relative time. This sense is naturally stronger in some persons than in others, but it is usually more or less a matter of development. 16. Tataing is a method which has been found useful in developing the sense of relative time. It consists in giving time names to the pulse and its various divisions, which must become associated with time just as the key syllables are associated with pitch, 17. The following is a diagram of a system of Tataing: a h c d e f 9 1 1 1 1 1 Ti" 1 "3 1111 i 4 4 4 1 1 1 4 4 1 1 4 4 I 3 1 4 4 4- ^ T 4 Tii lii-la tii lii - tc ta-fa-tc-fc tii t(5 - fe ta - fa 15 tii - e - fe tii - fa - a (.Pronounce vowels as marked.) Fu;. 3. 18. Td is the name of the whole pulse regardless of the time value of the pulse note. If it extends over one pulse, the vowel is repeated (Ta-a) for each pulse over which it ex- tends. The sections n, b,- c, d, e, f, g represent broken pulses — each broken as indicated by the fractions. The value of the notes corresponding to these fractions depends, of ELEMENTARY 25 course, on the value of the pulse note. The fractions | rep- resent dottetl notes. 19. Observe that d is made up of the first half of a and the last half of r ; e is made up of the first half of rand the last half o{ a \ /is the same as d except that the consonant part of the middle syllable is dropped, since the first two syllables belong to the same note. Similarly g is the same as e except that the consonant part of the last syllable is dropped. 20. All of these combinations are equal in time value and must be pronounced in the same time, as each represents the time value of one pulse. Changing the value of the pulse merely changes the rate of movement throughout the music. 2 I. Only the time values (not the pitch) of the notes of the music are considered while tataing. 22. Counting Time consists in counting the pulses in each measure. Whole pulses may be counted thus, i, 2, etc.; half pulses thus, i and 2 and etc. ; triplets thus, i and a 2 and a etc. ; quarter pulses thus, i a and a 2 a and a etc. (the words borrowing their time from the counts). The shortest note in the music usually determines the manner of counting, and the music is then counted through (from beginning to end) accord- ingly. 23. Measures. The common method of reckoning the measures in music is from measure bar to measure bar, and when the music begins and ends with an incomplete measure, the two ends count together as one measure; but, evidently, the two ends cannot belong to the same measure. 24. It is, perhaps, more correct to reckon the measures- from like phase to like phase of the rhythm, beginning with the first note of the music. (If we measured music as we measure anything else, we would begin at one end and a[)ply the meas- ure.) There are thus no incomplete measures at the begin- ning and the end of the music, for the music always begins and ends at the same phase of the rhythm. The measure 26 MUSICC'tLOOV bar is used primarily for the purpose of indicating the princi- pal recurring accent, and may be in the middle or an}- other part of the measure, depending on the phase with which the music begins. However, the common method is more con- venient, as it appeals directly to the eye, and for this reason is generally adopted. 25. In dividing the music into measures, the composer would naturally (by singing or feeling the music) decide whether some accents were stronger than others, and thus determine the principal recurring accent and place a bar be- fore each note upon which it fell. He thus divides the music visibly into sections which ma}' be called measures. He then determines the number of pulses and the location of the lesser accents, if any, in each section ; and he also decides the time value of the pulse note to correspond to the value of the writ- ten note or notes sung to each pulse. The result is expressed in the form of a fraction and put at the beginning of the music — it being understood that every second or third pulse (accord- ing as the rhythm is even or uneven) is accented strong or weak. 26. Determining which are principal and which secondary accents is largely a matter of judgment. In march and in dance music the rhythm is so pronounced that there is no difficulty in determining the character of the accents ; but in some other music the distinction between the principal and the secondary accents is so slight that there is room for a dif- ference of judgment, and different persons would divide the music differently. Thus, whether a piece of music is written in | or | time, or in | or | time (as the case may be), depends on whether every or every other accent is regarded as the principal recurring accent. But, as a rule, the whole matter is left to the judgment of the composer. 27. Singing very slow tends to develop accents not other- wise observed, and at the same time the ear tends to divide each pulse into two, thus changing ", into ■*- time, etc. Sing- ing very fast has the reverse effect. ELEMENTARY 2/ 28. Tempo or Rate of Movement. The time values as in- dicated by the shapes of the notes are only relative (not ab- solute). The same tune may be sung as slow or as fast as the singer desires ; but the rate of movement with which the music begins should be maintained throughout, except where the movement is purposely accelerated or retarded for a special effect. However, the time values given to the written notes are intended to indicate the rate of movement within reasonable limits ; for, evidently, if the composer intended the music to be sung fast he would give shorter values to the notes than if he intended it to be sung slow. The actual time value of the pulse necessarily depends on the rate of movement, so that the rate of movement is also indicated within reasonable limits by the value given to the pulse note in the fraction at the beginning of the music. 29. It is desirable to have a standard of time value with which to compare. The standard time value given to the quarter-note is one second ; therefore the quarter as a pulse note usually represents about one pulse per second, or 60 per minute, written M, 60. [The Metronome (M.) is an instru- ment for counting pulses.] If the music is intended to be sung nearer M. 120 than M. 60, the eighth-note would be taken as the pulse note and the music written accordingly. If the music is intended to be sung nearer M. 30 than M. 60, the half-note would be taken as the pulse note and the music written accordingly. 30. Applying Words to Music. In applying the words of a song to the music, one syllable is applied to each note unless two or more notes are tied together. Notes may be tied by having their hooks united, or by the tie (- — -) or slur {^ — ). One syllable is usually applied to all the notes thus tied. 31. Poetry as well as music has regularly recurring accents, producing duple or triple rhythmical movement. Rhythm therefore is an essential part of both poetry and music. The 28 MUSICOLOGY blending of poetry and music in songs is due largely to their conmion rln-thniical nature. 32. The pulses of the music are all of the same length — being regulated by the value of the pulse note regardless of the written notes being of different lengths, some greater than one pulse and others less than one pulse. T,],. The rhythm of the poetry is not thus restricted. But as one note is applied to each syllable, therefore the time value of each syllable depends on the value of the note ap- plied to it ; so that a syllable may cover more than one pulse of the music, or a pulse may cover more than one syllable. Therefore the duple or triple character of the rhythm of the music is not necessarily determined by the duple or triple character of the rhythm of the poetry; each rhythm restrict- ing the other only to the extent of causing accented syllables to come on accented pulses and unaccented syllables on un- accented pulses. 34. Interrupted Rhythms. The regular rhythm of the music is often purposely interrupted for a short time in order to heighten by contrast the rhythmical sense of the music. There are three principal ways of doing this: first, by tem- porarily quickening or retarding the time — indicated by such words as accelerando, ritardando, etc. ; second, by stopping the rhythm for a short time — indicated by a pause ( '^ ) over a note, which shows that it may be prolonged at pleasure, us- ually about double the regular time of the note (in beating time the hand should be held during the pause, as the rhythm stops); third, by temporarily displacing the regular accent — called Syncopation. 35. The principal ways of syncopating are: 1. Beginning a note on an unaccented part of a meas- ure and extending it over the following accent, as at a (J'ig- 4) ; 2. Reversing the accent by placing accent marks over the usually unaccented parts of a measure, as at b\ ELEMENTARY 29 3. Writing rests on the accented parts of a measure and notes on the unaccented parts, as at c; 4. Tying unaccented notes to accented notes, as at d\ 5. Writing notes between, instead of on, the usual pulses of a measure, as at c. abbe d d e .^ggiggEg^g^jgi J: Fig. 4. 36. Triplets. A numeral placed above or below a group of notes affects their normal value. The most common in- stance is a 3 placed above or below a group of three notes, which indicates that they are to be sung or played in the time of two. The notes thus affected are called triplets. They do not, however, affect the rhythm of the music. 37. Repeats. Dots between the lines at the left of a bar mean to repeat, and dots at the right of a bar show zvJiere a repeat begins. D. C. {Da Capo) means repeat frojii the begin- niiig. D. S. {Dal Segno) means return to the sig)i (S or -S'). The word fine means the end of a repeat. EXPRESSION 1. Expression marks affect the character of music as regards Power, Quality, and Movement, for the purpose of bringing out the sentiment. 2. Terms relating to Power. Pianissimo {pp), very soft; Piano (/), soft; Mezzo {m), medium; Forte {/), loud; For- tissinio {ff), very loud ; Crescendo (— ==^ ), increasing tone or tones ; Dinmuiendo { I^:^==~ ), diminishing tone or tones ; Sivell { ^=^^^:==- ), increasing and diminishing; Forzando {fz, or>-), a sudden burst of tone; Staeeato (im ), short and distinct; Semi-Staeeato { ••• ), less short and distinct; Legato (opposite of staccato), smooth and connected — sometimes indicated by a slur (x*— ) placed over the notes. 30 MUSICOLOGY 3. Terms relating to Quality. Somber, tones of reverence, sadness, or fear; Clear, tones of courage, joyfulness, or gay- ety ; Maestoso, loud and majestic ; Affettiioso, soft and sad ; Dolce, soft and sweet ; Con Spirito, with spirit ; Con Dolore, with grief; Giojoso, joyfully. 4. Terms relating to Movement (or tei/ipo). The chief of these from slowest to fastest are : Grave, Largo, LargJietto, Adagio, Lento, Andante, Andantiuo, Moderato, Allegretto, Allegro, Presto, Prestissimo. 5. For other musical terms see Musical Dictionary at back of book. 6. Embellishments. A Turn (««) placed over a note iz?:z) means that it is to be sung or played thus. * * • ^ > involving the notes above and below, beginning with the high- est. An Lnverted Turn {^o ox%) includes same notes in an inv^erted order. A turii after a note \ — »— ) or dotted note ( ifzii ) is made by striking the note first antl making the turn afterward. A sharp or flat above the turn affects the upper auxiliary note; but if below, it affects the lower aux- iliary note. 7. A Trill (tr) placed over a note means a rapid alterna- tion with the note above to the time value of the note. A trill usually ends with a turn, and may also begin with a turn. There are also trill chains (succession of trills), and double trills (two trills at the same time). 8. The Mordant (w) is a short trill, involving usually but one beat. 9. These signs are used to abbreviate the writing of music : 10. A line drawn through any sign, thus, f^, inverts it. 11. Any two signs combined give the combined effect of both ; thus iw, or ^, is a mordant followed by a ///;-;/. ELEMENTARY 3 1 12. A Grace Note is a small note written before or after the note it is intended to embellish and from which it bor- rows its time. 13. The Appoggiatura (usually written as a grace note) is an embellishing note prefixed to an essential note on an ac- cented pulse of a measure. When written as a grace note, it receives the accent and borrows half or more of the time value of the note, and is therefore made more prominent than the note to which it is prefixed. Its effect is languishing and sorrowful. 14. The Acciaccatnra is a grace note a half-step above or below the note to which it is prefixed. It is very short and never accented. It takes as little of the time value of the note to which it is prefixed as possible (opposite of the appog- giatura). Its effect is crisp, bright, and joyous. 15. An After Tone is a grace note which follows an essen- tial note. KEYS 1. Music is divided into what are called Keys — tones hav- ing a sympathetic relation to each other. A key therefore is a family of related tones. Every key is made up of seven steps, or intervals, which are repeated in each octave. 2. See Chart I. at front of book. Place key patterns in position (see instructions on Chart). 3. The lines on the face of the Chart represent the lines of the music staff, and the dotted lines represent the spaces of the music staff. These are marked with the scale letters A, B, C, D, E, F, G, and are so spaced as to represent the natural tone steps, or major key intervals; these seven letters are repeated in each octave. Observe that there is a half- step between each E and F, and B and C, and whole steps between all the other letters. The short marks show the half-steps between the letters. Each short mark is the sharp («) of the letter below it or the fiat (I?) of the letter above it. 4. Next observe that the movable key strip (marked 32 MUSICOLOGY major) also has lines drawn across it, which are also spaced to represent the major key intervals, or tone steps, and are marked with the seven syllables Do, Re, Mi, Fa, Sol, La, Ti, and the seven numerals i, 2, 3, 4, 5, 6, 7, which are re- peated in each octave. Observe also that there is a half- step between each Mi and Fa (or 3 and 4), and each Ti and Do (or 7 and i). 5. There is a natural sympathy existing between the tones thus related. Any group of tones thus related is called a Jiiajor key. This major key strip may therefore be regarded as the pattern, or mode (meaning mode of arrangement), for all the major keys. 6. There is also a minor key pattern, or mode (see key strip marked minor). Observe that there is a half-step be- .| tween each 2 and 3, 5 and 6, and 7 and i, and an augmented \ interval (equal to one and one-half steps) between 6 and 7. Sol being sharped is changed to si, and the key-note (i) is on La instead of Do. It is formed by sharping 5 (Sol) of the major key pattern and changing the key-note to La. This is the pattern, or mode, for all minor keys. 7. The major and minor key patterns are placed side by side in Fig. 10 (p. 43) for comparison. The major mode is of major (greater) importance, and the miiior mode is of minor (less) importance. 8. To adjust Chart I. to the key in which any piece of music is written, first count the sharps ( #'s) or flats ( !? 's) at the beginning of the music (the signature), then find the same number in the key table at top of Chart : the letter opposite will be the key letter. 9. Now set the major key pattern so that tiic kej'-note i (Do) will be opposite the key letter. The syllables on the key pattern will match all the letters on the staff except the letters which are to be sharped (raised a half-tone) or flatted (lowered a half-tone). Observe also that the sliarps or flats in the signature of any piece of music are on the same lines ELEMENTARY 33 and spaces that the key pattern (wlien set to the same key) shows are to be sharped or flatted. 10. Take a music book and test the signatures in all the different keys by Chart I., by setting the major key pattern to the desired key and comparing the signature with it. 1 1. When a piece of music has no signature it is in the key of C. Now set the key pattern so that i (Do) will be oppo- site C. It will be seen that all the letters and syllables match, so that none of the letters have to be sharped or flatted to conform to the key pattern. For this reason the key of C is also called the natural key. 12. It will also be noticed from the Chart, that when any letter is sharped or flatted it is affected the same way in each octave. Therefore the sharps or flats in any signature affect the same letters in each octave. It is customary to place the signature once on each staff. 13. Set the major key pattern to the keys of one, two, three, four, five, six, and seven sharps in succession (writing down the letters sharped in each case). Observe that each key is formed by adding one new sharp to those of the preceding key. 14. Examine the flat keys in the same way. 15. As each new key is formed, the last sharp or flat is placed in the signature on the line or space affected, a little in advance to the right of the others. 16. Fig. 5 shows the signa- tures of the key of seven sharps and the key of seven flats. These include all the other signatures, by taking them from left to right as they come. 17. The intermediate tones, between the fixed tones rep- resented by the letters of the staff, are indicated on the Chart by the short marks between the letters. They are indicated in music by sharps and flats. They arc indicated on the key- board of the piano or organ by the black keys. Fig. 5. 34 MUSICOLOGY Fig. G. -jjy •r_" 1 8. Fig. 6 shows the relation between the music staff and the piano or organ key-board. Observe that the letters on the staff match the white keys of the key-board, and the short marks between the letters match the black keys. Each black key is the sJiarp of the white key next below, or the fiat of y — s- ft — -1^ I the Avhite key next above. r^ j» " p """"- — ^ (When referring to the key- \^_y "E— "iLai — I \^Q-^x<\, the word key refers to one of the white or black finger- bars ; elsewhere the word key refers to a family of related tones.) 19. See Chart II. at back of book. Place key pattern in position (see instructions on Chart). 20. A general knowledge of music involves also a knowl- edge of the principles of the key-board. It is only necessary to remark here that all observations on Chart I. apply also to Chart II. in a similar manner. 21. As the signature of any piece of music shows what tones are to be sharped or flatted throughout the music, and as the black keys represent sharps or flats, therefore the sig- nature of any piece of music shows what black keys are used in playing it. As there are only five black keys in each oc- tave, therefore six or seven sharps or flats in the signature show that E and H or V and C are, one or both, sharped or flatted, as the case may be. But as there is no black key between E and F or B and C (they being only a half-step apart), one is used as the flat or sharp of the other. 22. A Diatonic Scale consists of the regular tones of a key in successive order, as shown by the major and minor key patterns. ELEMENTARY 35 23. A Chromatic Scale consists of both the regular and in- termediate tones in successive order (see Fig. 2, p. 21). 24. Although in music the lines of the staff (and therefore the letters) are spaced equally, yet it must be fixed in the mind that there is always a half-step in tone between E and F, and between B and C, and a whole step between all the ^ other letters as shown on the Charts ; so that the key of C con- forms to the pattern without any sharps or flats. In all other keys, the sharps or flats in the signature show what lines and spaces must be raised or lowered in tone to make the key conform to the pattern. * READING MUSIC 1. The first step is learning the letter names of all the lines and spaces of both the bass and treble staffs so that they can be named without hesitation. The second step is learn- ing to run the scale with the voice, using the key syllables. Do, Re, Mi, Fa, Sol, La, Ti, Do : this will have to be learned from some musical instrument or the voice of a teacher. 2. Each of the key-tones, represented by the seven syl- lables, produces a certain mental effect. The character of the key-tones may be described as follows: Do, "the strong or firm tone"; Re, "the rousing or hopeful tone"; Mi, "the steady or calm tone"; Fa, "the desolate or awe-inspiring tone"; Sol, "the grand or bright tone"; La, "the sad or weeping tone " ; Ti, " the piercing or sensitive tone." The mental effect of any tone is called its key-tonality and is due to its relation to the key-note. 3. After learning the key-tonality of each tone the key-note may be pitched with the voice to any key, and the other tones will adjust themselves to it by reason of their key-tonality. The key-tonality of each tone does not change ; it is only the pitch that changes. This may be illustrated by shifting the key pattern (Chart I.) to bring Do opposite any letter: it is evident that the syllables all move together and that their relation to each other and to Do does not change. 36 MUSICOLOGY 4. A piece of music (being made up of key-tones) can be changed to any key just the same as the key pattern can be shifted to any key (being merely a change of pitch), but the music should be written in that key which will bring all the notes within the average range of the voice as shown by the staff. 5. Changing the key of a piece of music is called Trans- position. We may transpose any piece of music to suit the voice, as the key is only a question of pitch. 6. There can be no intelligent understanding of music until this key-tonality, or sympathetic relation of the tones of any key, is understood ; and this is most readily accomplished by the use of the key syllables. 7. The mental effect (key-tonality) of a tone is due to a sort of unconscious mental measurement of the interval, or distance, from the point of repose (key-note previously es- tablished in the mind), and its association with the tone or tones immediately preceding, the effect of which has not yet passed out of the mind. A tone entirely unassociated with other tones has no key-tonality by which its key can be rec- ognized, as it may belong to different keys. 8. The ability to recognize the key-tonality of tones (and also rhythm) is so developed through practice as to enable one to mentally hear a piece of music by merely following the notes with the eye, as distinctly as if it were sung or played ; and also to mentally see a piece of music (how it is written) by merely hearing it sung or played, and thus be able to write it. 9. The ability to thus hear with the eye is essential in order to read music readily, while the ability to see with the ear is essential in order to write music; both are the natural growth of practice, exactly as in reading and writing spoken language. We understand what we read without reading aloud, thus applying the principle of hearing with the eye. In writing what wc hear spoken, we apply the principle of seeine with the car. ELEMENTARY n 10. Beating Time is indicating the pulses of the music with the hand. 1 1. Tataing the music is singing the time names (p. 24 : 17, 18) of the notes on one tone. This is more especially for the purpose of fixing in the mind the relative time value of the parts of broken pulses, after which beating time is sufficient. Both exercises are intended to develop the faculty of feeling the rhythm of music. 12. Solfaing the music is singing the music with the key syllables. 13. Laing the music is singing the music with the open syllable La, the object of which is to overcome the depend- ence on the key syllables which solfaing tends to produce. 14. These are the usual stages of practice in learning to read music, before applying the words. Beating time and solfaing are the essential stages, the others are side helps. THE TONIC SOLFA NOTATION 1. There are two principal methods of notating, or writing music, in use. 2. The Staff Notation consists in representing the music by notes on a staff, as already explained. 3. The Tonic Sol/a Notation consists in representing the music by the first letters of the key syllables (instead of notes) written in a straight line. The first octave above the key- note (called the unmarked octave) is written thus — d r PI f S 1 t; the octave above the unmarked octave, thus — d' r' Hi' f S' 1' t' ; the octave below the unmarked octave, thus — d| Ti Wi f| S| 1| t|. The figure 2 may be placed above or below if a further extension is needed in either direction. 4. The time is indicated by punctuation marks between the letters. A bar ( I ) is placed before each principal accented pulse, thus dividing the music into measures. A short bar ( I ) is placed before each lesser accented pulse, and a colon (:) before each unaccented pulse, thus dividing each meas- ure into pulses. A period ( . ) divides a pulse into halves; a 38 MUSICOLOGV comma (, ) divides a half-pulse into quarters; a period and comma close together divide a pulse into three-quarters and one-quarter — the period being on the side of the greater part.* Inverted commas divide a pulse into thirds. A macron ( — ) indicates a continuance of the preceding tone. Rests are in- dicated by vacant spaces. A horiJ:ontal bar ( — ) over or under two or more letters ties them together the same as a tie (' ") or slur ('^) in other music. 5. The time valufe of each pulse, or the rate of movement, is usually shown by a metronome mark, thus, M. 60 (or any other number), placed over the beginning of the music. The key is also shown by the key letter being placed over the be- ginning of the music. Also, whenever the key changes dur- ing the music (modulation, p. 90) the new key letter is placed over the point where the new key begins, and ihe^ Bridge To?ie has its old name placed above its new name, thus, ^. and takes its new name at its old pitch, and the following tones in the new key are pitched accordingly. 6. The following example is intended merely to illustrate each of the foregoing principles: Key of D. M. 60 A.t. 1 d :ijc-; and a double flat thus, !?). When thus written on a line or space already sharped or flatted in the signature, it necessarily includes the sharp or flat on the same line or space in the signature, as otherwise the note affected would be triple sharped or flatted. 43. A double sharp raises the tone a whole step. A double flat lowers the tone a whole step. 44. Keys which involve an extension of either table are used only in modulation, and will be referred to again under " Modulation." MELODIC MINOR SCALE 1. The key-note (i) of any key is the controlling tone of that key, as it is the tone of complete repose, or home feel- ing, and therefore the point of reference with which each tone is compared in its tonalit}'. 2. The ton<" a half-step below the key-note is called the ELEMENTARY 53 leading tone, because it tends to lead to the key-note, there- by emphasizing the importance of the key-note. 3. This leading tone is now generally regarded as an essen- tial feature of all keys, major or minor, in order to give due prominence to the key-note (the trained ear seeming to re- quire it). This is the reason for sharping the 7th in relative minor keys, as otherwise the minor key would have no lead- ing tone a half-step below the key-note. 4. Sharping 7 leaves an augmented interval between 6 and 7. This can only be remedied by sharping 6 also, thus: fc==j^fg^ •wi This would give the scale too much of a major character, as will be seen later. 5. As a leading note is not needed descending the scale, we may (after sharping both 6 and 7 ascending the scale) make both natural descending, thus: This is called the Melodic Minor Scale, but cannot cor- rectly be called a key (family of related tones), as it is one family ascending and another family descending; nor a mode, as it is one mode of arrangement ascending and an- other mode descending. In minor keys the ^6 and tl 7 are only used as melodic or passing notes (p. 87), but are never used in chord formations as harmony notes. 6. Harmonic Minor is the general name for both relative and tonic minor, to distinguish from Melodic Minor. THE OLD MINOR MODE I. The original or old viijior pattern, or mode, differs from the major pattern only in the key-note being La instead of Do. It is sometimes found in old tunes, as the following: 54 MUSICULOGY Idumea. Observe that the music begins and ends on La, and that the /th (Sol) is not sharped. 2. If in the minor key table (p. 46) the figure 7 is made natural where sharped and flatted where natural, the table would show all the minor keys according to the old viinor mode. Observe that each key would differ from its relative major only in the position of the figures on the staff, and from its tonic major by flats or naturals on 3, 6, and 7. 3. The modern minor mode (which is the old minor Avith the 7th sharped) is most generally adojitcd, yet the old minor mode is still occasionally used ; while the melodic viinor scale is a sort of compromise. THE ANCIENT GREEK MODES I. Observe that in the major pattern each octave is made up of five whole steps and two half-steps, the mode of arrange- ment placing the half-steps between 3 and 4, and 7 and i ; I z i 4 5 (5 7 bi-it the steps and half-steps may be ar- ranged in seven principal ways, as shown in h^ig. 13. These were all used in the music of the Ancient Greeks, and hence are called the Ancient Greek Modes. 2. After the science of Harmony (combining of tones) began to develop, it was found that two of these, the 1st and tlie 6th (corresponding to our major and old minor modes), were best suited ''"■■ '^- for Jiarmoni;:ation. The others, there- fore, have become obsolete. El.E.MEMAKV 55 3. Observe that the word iiwdc refers to the mode of arranicnted triad is made up of a major 3d and augmented 5th (or two major 3ds, one above the other). 13. Analyzing the triads of the major key pattern, counting the steps and half-steps in each interval, we find that the Tonic triad ( » jj^ ) is made up of a major 3d (2 steps) andperfect 5th (3i steps), and is therefore a major triad ; Super-tonic triad ( % \^ ), made up of a minor 3d (i i steps) and perfect 5th, is a minor triad ; Mediant triad (5801 ), made up of a minor 3d and perfect 5th, is a minor triad; Sub-dominant triad (|La), made up of a major 3d and -1 — Do 7 — Ti-\ 6 — La -5— Sol -i—Fa ■Z—Mi- 2 — Re- -1 — Do- 7—Th 6 — La 5— Sol -'i—Re -1 — Do u — Th 6 — La- ■5— Sol A— Fa 2— i?e- 4 — Do- 62 MUSICOLOGV rl — La-i -Si- -Q—Fa 5—Mi- -i—Ee- -La perfect 5th, is a major triad. In the same way, we find that the Dominant triad(7£) is a ;//<'?;'i^r triad ; the Sub-mediant (11;°) is a vii)ior triad; and the Sub-tonic ( s^f ) is a diminished triad (made up of a minor 3d and dim- inished 5th — 2|=3steps). 14. Analyzing the triads of the minor key pat- tern in the same way, we find that the Tonic triad ( 3 »° ) is a minor triad ; the Super-tonic ( f i^ ) is a dim- inisJied triad ; the Mediant ( g |' ) is an augmented triad (made up of a major 3d and augmented 5th — 4 steps); the Sub-dominant (|};^) is a minor triad; the Dominant ( | ^|i ) is a. major triad; the Sub-medi- ant ( J ^° ) is a major triad ; and the Sub-tonic ( | :jf ) is a diminished triad. 15. Representing the major triads by large h—Mi- roman numerals, the minor triads by small roman numerals ; the dinmiished triads, same as minor with (°) added ; and the augmented triad, same as major with (') added, all the triads of both major and minor keys w^ould be represented thus: Triads of major keys — I, 11, ni, IV, V, vi, vii'. Triads of minor keys — i, h°, III', iv, V, VI, vu°. Or analyzed thus: I V" n VI 5 — Mi -i—Re 3— Do- 2 — Ti- ■1 — La -7 — SH 6—Fa- 5—Mi- 4 — Ee -3 — Do- -2 — Ti- -1 — ia-1 Do Mi Sol Ti Re Fa La Do Mi Triads of major keys — i 3... 5 7... 2... 4. . . .C. . . i 3 IV VI La Do Mi Si Ti Re Fa La Do Triads of minor keys — i... 3 5 7... 2... 4... 6 1...'3 III' vn° IV Dots show number of half-steps between; . . . . = majnr 31I, ... = minor 3<1. STRUCTURE OF MUSIC 63 16. We see that major keys have three major triads (I, IV, V), three minor triads (II, III, VI), and one diminished triad (Vll°); and that minor keys have two minor triads (I, iv), two major triads (V, VI), two diminished triads (ll°, Vll°), and one augmented triad (HI'). 17. Major triads have a bold, aggressive character, while minor triads have a subdued, tempering, plaintive character. 18. We have seen (p. 58:14) that the octave, perfect 4th, and perfect 5th are the principal divisions of the scale; and evidently, therefore, the divisional points i, 4, 5 are the prin- cipal points of the scale. We noticed also the prominence of these intervals in the interrelationship of keys as shown by the major key table (p. 40) ; that in the ascending direction (left to right), the dominant (5) of any key is the point around which the next key forms (called therefore the key of the dominant); and in the descending direction (right to left), the sub-dominant (4) of any key is the point around which the next key forms (called, therefore, the key of the sub-dom- inant). In the first instance, raising the key-note (i) a per- fect 5th or lowering it a perfect 4th; and in the second in- stance, raising the key-note a perfect 4th or lowering it a per- fect 5 th. 19. Naturally, therefore, the perfect intervals (octave, 4th, and 5th) are the most prominent intervals, and the tones i, 4, 5, the most prominent tones, and the triads I, IV, V the most prominent triads, of a key. For this reason, the triads I, IV, V are called the characteristic harmonies, as they not only determine the character of the key (whether major or minor), but also form the background or framework of all music. Any tune, or melody, can be harmonized by these three chords alone, as the three together contain all the tones of the key. These chords (with the Dominant 7th) are the ones commonly used in extemporizing accompaniments on piano, guitar, or other instruments. 20. These characteristic chords (I, IV, V) are the only 64 MUSICOLOGY major triads in any major key, and determine its major char- acter ; I and IV are the only minor triads in any modcrti minor key, and determine its minor character. In the old minor mode (p. 53), all three characteristic chords (I, IV, V) are minor, thus makin<^ it more distinctly minor (being sym- metrically the opposite of the major mode). But the leading tone (i step below the key-note) is now generally considered essential in order to give due prominence to the key-note, thus making it necessary to raise the 7th of the old minor mode a half-step. This makes the dominant triad (V) major. 2 I. Hence the law : TJic Dominant Chord must he major in both major and minor keys. 22. If we should also raise the 6th a half-step (see p. 53: 4), to avoid the augmented interval between 6 and 7, then the Sub-dominant chord (iv) would also become major, and the minor key (two of its three characteristic triads being major) would have too much of a major character. 23. Hence the law: The Tonic (l) and the Sub-dominant {\\') triads must both be minor in minor keys. 24. Since these laws determine the form of the character- istic triads of minor keys, they necessarily fix the mode, or pattern, of minor keys. POSITIONS AND FORMS OF THE TRIAD 1. On the principle that harmony builds in 3ds, the natural or direct position or form of any chord is that in which its notes are arranged in 3ds; and the lowest note is therefore regarded as the root or foundation of the chord. The chord, however, may be taken in various positions and forms, but as long as the same tones are used (counting octaves as merely replicates of the same tone) it is regarded as the same chord. 2. The triad has three positions with reference to the so- prano, or highest note, thus: if the root of the triad is in the soprano, the triad is in 1st position ; if the 3d is in the soprano, the triad is in 2d position ; if the 5th is in the soprano, the triad is in 3d position. STRUCTURE OF MUSIC 65 3. The triad has three forms with reference to the bass, or lowest note. The lowest note is always the bass (whether root, 3d, or 5th). The bass (base), as its name implies, is the base or support of the entire chord. When the root of tlie triad is the bass, it is called root bass and the triad direct. If the root of the triad is not the bass, the triad is said to be inverted. If the 3d is the bass, it is called the ist inversion; if the 5th is the bass, it is called the 2d inversion. 4. Thus we see that \.\\q position (ist, 2d, 3d) is determined by the soprano or highest note, and \.\\q forjii (direct or in- verted) by the bass or lowest note (bass and soprano being the outside and therefore most prominent parts). A triad is strong- est and most independent when in ist position with root bass, the root of the triad being thus in both bass and soprano. 5. The treble staff has to do mainly with position, and the bass staff with form. In playing the piano or organ, the right hand has to do with position ; the left hand, with form. 6. Position and form are entirely independent of each other, so that any form of the triad (root, ist inv.or2d inv.) may be used with any position of the triad (ist, 2d, or 3d). To determine the position, we need look only at the highest note; to determine the form, we need look only at the lowest note. 7. The different positions and forms of chords relieve the monotony which would result from using only one form and position of each chord. They are also the necessary result of voice leading, as will be seen later. (To avoid the incon- venience of using both words, we may use either of the words — position or form — in a general sense ; but a chord is usually referred to as direct or inverted.) 8. Fig. 14 shows the different positions and forms of the tonic triad in the key of C major. 9. When four-part harmony is written so that the three upper parts are within the compass of an octave, so they can be played with one hand, it is called close harmony; other- wise, it is called open harmony. In open harmonv, tones 66 MUSICOLOGY proper to a cliord arc omitted between two or more of the three upper parts. pos. ISl pos. 2d pos. I St pos. 2d 3d pos. pos. m 2d pos. id 1st pos. pos. 1 — ll -^ ~~^ — r — K— .' • \ M W'l t— -i — -5 —3 — »■ — T — u_- — r — 3^- Close Harmony 1 -•- 1 Open Harmony -0- -0- • Open Harmony 1 J /^"\ • ^ 1 J 1 L # ~ 1* ! d 1 i^J» ^ j |i 1* m 1 ^ V. V "i m m « ■ 5 . — 1 — 1 f ~ • •' 6 6 6 6 6 6 irect Form ist Inversion Fig. 14. 4 4 2d Inversion 4 10. In V'v^. 14 the figures on the staff at the left show all the tones proper to the tonic triad in the key of C 1 1. Close harmony is generally used for male or for female voices alone ; open harmony, for mixed voices. 12., Close harmony is usually best adapted for loud, full expression, and open harmony for soft, delicate expression. As a general rule, however, it is best to keep the parts as nearly as possible equidistant. Not more than an octave should ever intervene between any two contiguous parts, exc.ept the bass and tenor. THOROUGH-BASS FIGURING OF TRIADS (Fipuriiig from the bast-.) 1. In the direct form of any triad, as | (taking the triad I for example), the intervals figured from the bass are ^'^. Therefore, the figures ^ (representing the intervals of the triad) stand for the direct form of the triad. When no fig- ures arc expressed, this form is understood. 2. In the 1st inversion (.5) the intervals figured from the bass are :1,i\ Therefore, the figures |, or simply 6 (3 being understood), stand for the ist inverted form of the triad. 3. In the 2d inversion ( ? ) the intervals figured from the bass are 5!h- Therefore, the figures | stand for the 2d inverted form of the triad. STRUCTURE OF MUSIC 6^ 4. Again (taking the triad IV for example), the direct form (root bass) would be \ — intervals-^:,*", istinv. | — inter- vals I? ; 2d inv. 4 — intervals lit. All triads are figured the same way, therefore the figures 6 and % are merely to show the inversions of the triad, and when used, are usually writ- ten under the triad (see Fig. 14). Observe that the triads are figured as if the notes were all in the same octave (octaves being regarded as merely replicates of the same tone). 5. The key figures and thorough-bass figures should not be confused, as they do not correspond except where the chord intervals and the key intervals happen both to be figured from the key-note; as the \ of the I triad, the \ of the IV triad, and the % of the VI triad. 6. It should also be fixed in the mind that thorough-bass figuring is merely a system of measurement from the lowest note, and has nothing whatever to do with the generic nature of a chord. Thorough-bass figuring is useful in training the mind to measure all intervals from the bass. 7. As a matter of convenience, we may speak of the 3 chord the 6 chord, the % chord, etc., bearing in mind that these ex- pressions refer to the form, and not to the nature of the chord. DISSONANT CHORDS 1. Dissonant chords (or discords) are not always unpleas- ant, but often quite the reverse in relieving the monotony of too much concord. They make the concords more effective by contrast, and increase the means of expression, therefore they are very important in music ; however, they must be short and always resolve into their proper concords. 2. The major and minor triads are the only concords; since in their different positions, while containing only consonant intervals, they contain all the possible consonant intervals. 3. The Dissonant Triads. Diminished and augmented triads are dissonant, because they contain the interval of a dissonant 5th — the dim. triad containing a dim. 5th, and the aug. triad containing an aug. 5th (hence their names). 68 MUSICOLOGY 4. The dim. triad is a diatonic triad on 7 in major keys, and on 7 and 2 in minor keys (see Fig. 10, p. 43). The aug. triad is diatonic only on 3 of minor keys. These triads are used in all their positions and inversions — the dim. 5th in- verted becomes an aug. 4th, and the aug. 5th inverted be- comes a dim. 4th, and therefore still dissonant. 5. All chords not in their diatonic position in a key are chromatic chords, as they involve accidental or chromatic tones. 6. Extensions of the Triad. The triad is the foundation of all chords, but on the principle that harmony builds in 3ds, we may extend the triad by adding 3ds ; but any extension of the triad will involve dissonant intervals ; thus, by adding a 3d to the triad we get the interval of a 7th from the root, adding another 3d we have the 9th together with the 7th, etc. 7. A chord containing the interval of a 7th is called a chord of the 7th ; a chord containing the interval of a 9th is called a chord of the 9th ; etc. Thus continuing to add 3ds, we would get the chords of the i ith, 13th, etc. 8. A full chord of the 7th contains four distinct tones; a full chord of the 9th contains five distinct tones, etc. If in the chord of the 7th either the ist or 7th be omitted, the chord loses its identity by becoming a triad ; or if in the chord of the 9th cither the ist or 9th be omitted, the chord loses its iden- tity by becoming a chord of the 7th. The 5th or middle tone of a chord of the 9th is usually omitted in four-part harmony. 9. Some theorists explain these chords as combined triads (two triads combined through their common tones). Thus the triads ?_,? combined through their common tones would 4 ^ 6 « give 2 — a chord of the 7th ; or g—o combined would give | 5 5 6 — a chord of the 9th. A chord of the iith would be two complete triads, one above the other. But the simplest method of treatment is that of extended triads. 10. As the triads arc represented by the roman numerals, the extended triads may be represented by simply attaching STRUCTURE OF MUSIC 69 a figure corresponding to the distinguishing interval; thus, ll^, Vy, or Vg, etc. II. Chords of the 7th. A 7th chord contains six inter- vals, which in the direct close form of the chord are : three 3ds one above the other, two 5ths interlocked, and a 7th spanning all, thus: (J) . But a closer analysis (see major and minor patterns, p. 43) will show that there are seven kinds of 7th chords, outlined as follows: Class 7th Chords in Direct Close Form Analysis 1st Major triad with major 7th — It and IVt in major keys, and VI7 in minor keys. major 3d + minor 3d + major 3d, two perfect stlis inter- locked, major 7th spanning all. 2d Minor triad with minor 7th — Ht, HI? and VI7 in major keys, and iv, in minor keys. minor 3d + major 3d + minor 3d, two perfect 5ths inter- locked, minor 7th spanning all. 3d Major triad with minor 7th — ■V^ in both major and minor keys. major 3d-4-minor 3d + minor 3d, perfect 5th and dim. 5th interlocked, minor 7th span- ning all. 4th Minor triad with major 7th — It in minor keys. minor 3d + major 3d-l-major 3d, perfect 5th and aug. 5th interlocked, major 7th span- ning all. 5th Dim. triad with minor 7th — vn'- in major keys, and ii"? in minor keys. minor 3d + minor 3d + major 3d, dim. 5th and perfect 5th interlocked, minor 7th span- ning all. 6th Aug. triad with major 7th — III't in minor keys. major 3d + rTiajor 3d+minor 3d, aug. 5th and perfect 5th interlocked, major 7th span- ning all. 7th Dim. triad with dim. 7th — viT- in minor keys. Three minor 3ds, one above the other, two dim. 5ths in- terlocked, dim. 7th spanning all. Class— 7th Chords in Major Keys ist I; 2tl 11; 2d ist IVt 3d Vt 2d VI7 5th VI T- Class- 7th Chords in Minor Keys 4tl. It stli t,th III't 2d IVt Vt ist VIt 7th VII T 70 MUSICOLOGY 12. Observe, as to the number of dissonant intervals con- tained in each class : the 1st contains one (major 7th); the 2d, one (minor 7th); the jd, two (dim. 5th and minor 7th); the 7///, two (aug, 5th and major 7th); the jM, two (dim. 5th and minor 7th); the 6tJi, two (aug. 5th and major 7th); the ytJi, three (two dim. 5ths and a dim. 7th) — thus giving a comparative idea of their dissonant character, although some dissonant intervals are more dissonant than others. 13. To become more familiar w^th the structure of these chords by comparisons, we may make the following observa- tions : First, that the jc/ and 5///, also the ^tJi and 6t]i classes are composed of the same intervals, but in reverse order. Second, that in the ist and 2d classes the intervals that are major in one are minor in the other. Third, that major 3d + minor 3d = perfect 5th ; minor 3d + minor 3d = dim. 5th ; and major 3d + major i([ = ''^>^'g- 5th. Fourth, that IV, and VI., vil° and II7, II . and IV,, are, each pair, one and the same chord in relative major and minor keys (Fig. 10, p. 43), and that V, is one and the same chord in tonie major and minor keys. h'ifth, that the stroke under Vll° in minor keys is to distin- guish it from Vll° in major keys. Sixth, that the ist and 2d classes occur more than once in any major key, and the other classes occur only once in any key. Seventh, that the form of the vii, is not changed by inverting it; as the 3ds arc equal, and each equal to the augmented 2d (i^ steps), and together equal an octave, and therefore form the same intervals in any order (see minor pattern). 14. The inverted forms of these chords (except the VII^) would, of course, involve different combinations; as 3ds invert into 6ths, 5ths into 4ths, and 7ths into 2ds (major into STRUCTURF, OF MLTSIC 71 minor and vice versa, dim. into aug. and vice versa, and per- fect into perfect), but dissonant intervals invert into disso- nant intervals, and consonant into consonant, so that the dis- sonant character of the chords is but little changed. 15. The intervals do not all invert at once, but in the order shown in Fig, 15. Fig. ir,. 16. The chords of the 7th, with their inversions, are all more or less used, but some are much more prominent and important than others. Those of special importance are the V, (called the Dominant 7th) and the vil° (called the Dimin- ished 7th) chords. These are the principal chords used in modulation, and will be more fully described under ''Modu- lation." 17. Positions and Forms of the 7th Chords. A chord of the 7th has four positions, as each tone may be in the highe.st voice; also, four forms (direct, and three inversions), as each tone may be in the bass. 2d 3d 4th pos. pos. pos. 1 1 -_ __ I St pos. 3d 4th pos. pos. ist 2d pos. pos. 1 4th pos. ist pos. 2d pos. 1 3(1 pos. 1 rP" ' *' i i d m ' 1 J J 1 * ' J I .J S 1 CS— — 3 a r — —'r — ^# — » — -^# — U — i~ — «— — a— -i i Direct form ^WT- ist Inv. -0- 2d Inv. 1 1 __l t • 3d Inv fm\' P P P 1 ) 1 1 ' • ' ' ' \fij' ill 1 ' J J ^ ^ r 1 CS' J ^ ^ 1 1 1 1 1 1 1 1 5 5 S3 3 3 Showing the positions and forms of the V7 chord in the key of C. Fir.. 10. 72 MUSICOLOGV 1 8. Thorough-bass Figuring of 7th Chords. Taking the V, chord as an example and figuring the inter\-als from the bass, we find that when the bass is the root I f I the intervals of the other parts from the bass are ^'h — the figure 7 repre- sents the chord in direct form. In the ist inversion (i) the intervals would be |;|' — 5 shows 1st inversion. In the 2d inversion 1 1 \ the intervals would be fj — f shows 2d inversion. In the 3d inversion ( 1 1 the intervals would be4t'h — ^, or simply 2, shows 3d inversion. 19. It will be noticed that the abbreviated figuring of each form represents the intervals peculiar to that form ; the other intervals, which are always built in 3ds and therefore implied, are omitted for the sake of brevity. All 7th chords are fig- ured the same way ; therefore, these figures are only to show the form of the chord (whether direct, ist, 2d, or 3d inver- sion). (See I^^ig. 16.) 20. The Chord of the 9th (figured ?). The chords of the 9tli, iith, etc., are used practically only on the dominant. (When on any other note than the dominant, Qths, i iths, etc., are usually regarded as suspensions.) 21. The full chord of the 9th in its direct close form con- sists of ten intervals, anah'zed in major keys, thus: /f"v\. j Major 3ci + minor 3d + minor 3d + major 3d. (a. f\\ I Two perfect 5ths linked together by dim. 5th. Vg fvi?// I Two minor 7ths interlocked. Vr// 1 ■■^ major 9th spanning all. It cont.iins, therefore, four dissonant intervals (dim. 5th, two minor /ths, and a major 9th). In minor keys it would contain five dissonant intervals (two dim. 5ths, minor 7th, dim. 7th, and a minor 9th). It is used in its ist, 2d, and 3d inversions. 22. If the root of the Y ,j chord is omitted, it becomes either the Vii! or vii°. chord, according as the 9th is major STRUCTURE OF MUSIC 73 or minor. As these two chords have no perfect 5th, and on the principle that every fundamental chord should have a perfect 5th, some theorists regard them as V^ chords with root omitted, thus including them in the general class of dominant harmony. 23. Chords of the 6th. From Fig. 15 (p. 71) we see that a chord of the 7th (like the triad) inverts into a chord of the 6th (spanned by a 6th), and the first inversion has the appear- ance of being a triad with a 6th added, so that Fig. 15 may be regarded either as a chord of the 7th with its inversions or as a chord of the 6th with its inversions (the direct form of the 7th chord being the 3d inversion of the 6th chord); but on the principle that harmony builds in 3ds, it is more cor- rect to regard this as a chord of the 7th. However, the 1st inversion of the 7th chord, when it consists of a major triad with a major 6th (as in the ist inv. of the 2d class, p. 69), it is often called a chord of the Added 6th. 24. In the analysis of the 2d class (p. 69), we see that the three upper parts of the 7th chord form a major triad as against the minor triad formed by the three lower parts (re- garding the 7th chord as two overlapping triads) ; and in view of the stronger and more aggressive character of the major triad, the ist inversion, which places it below as the founda- tion, may therefore be regarded as the fundamental position of the chord ; hence, the justification of the added 6th chord. Observe that this is the case only in 7th chords of the 2d class (formed by adding a minor 7th to a minor triad). 25. The 1st inversion of a major triad is sometimes called the Neapolitan 6tJi. It consists of a tone with its minor 3d and minor 6th. 26. The Augmented 6th chord is a chord containing an aug. 6th. By examining the major and minor patterns, it will be seen that the aug. 6th is not a diatonic interval of either. It is formed only by augmenting a major 6th, and is therefore a chromatic interval, and the aug. 6tli chord is therefore a 74 MUSICOLOGY chromatic chord in any key. It exists in three forms: Ital- ian 6th, French 6th, and German 6th. (These are fully de- scribed on p. 104.) All other chords are diatonic in some key, and when used in other keys as chromatic chords, they have a tendency to lead to the key in which they are diatonic, or natural, thus tending to induce modulation (change of key). INTERRELATIONSHIP OF CHORDS IN GENERAL 1. Chords may be related in three ways: first, through be- longing to the same diatonic scale (common key relationship) ; second, through the relationship of their roots (root relation- ship); third, through common tones (common tone relation- ship). 2. Common Key Relationship. A key is a family of tones related through belonging to the same diatonic scale ; there- fore, chords belonging to the same diatonic scale, or key, are related as being composed of interrelated tones. Naturally, also, those chords based on the most prominent divisional points of the scale (1,4, 5) are most prominently related to each other through their common relationship to the key. 3. Chords thus related through belonging to the same dia- tonic scale are naturally associated together; hence their ten- dency (when used in other keys as chromatic chords) to lead to the key in which they are diatonic. 4. But certain keys (especially those closely related) have a number of tones in common, and have therefore certain chords in common, but occupying different positions (with different roman numeral name) in the different keys. These common chords naturally form connecting links between the keys containing them, and are convenient stepping-stones in passing out of one key into the other. They arc called equiv- ocal chords, because, belonging to different diatonic scales, they lead to no particular key, but to any one of the keys in STRUCTURE OF MUSIC 75 which they are common, while those chords which belong to only one diatonic scale always lead to that particular key. 5. Root Relationship. In this sense, chords are related in proportion to the consonance of the interval between their roots. If the interval is a perfect concord (octave, perfect 5th, or perfect 4th), we have the 1st degree of relationship. If the interval is an imperfect concord (major or minor 3d ox 6th), we have the 2d degree. If the interval is a dissonance, we have the 3d degree. When the interval is an octave, the relationship is so close that the chords are regarded as one and the same. 6. Common Tone Relationship. Keys are naturally related in proportion to the number of tones in common ; and, on the same principle, chords, whether belonging to the same key or different keys, are related in proportion to the number of tones in common. 7. In view of the different kinds of relationship, the grade of relationship between chords depends upon the number of points of relationship they contain. INTERRELATIONSHIP OF THE TRIADS OF A KEY 1. As the triads are the foundations of all chords, it is well to notice their relation to each other in the same key. 2. The triads 1 1, III I, V?, vii°t ill, IV S, vi ?, U, have 1 3 5' 7 a ,4 6 1 each two tones in common with those on either side (form- ing a complete circle of the triads), and may be called double- connected (abbreviated dc) triads. Thus we see that triads whose roots form the interval of a 3d are iff triads. 3. The triads IL V?, ut, vi ?, ml, vn°i IV J, I g, have ^ 15263 741 each one tone in common with those on either side (also forming a complete circle of the triads), and may be called single-connected (abbreviated sc) triads. Thus we see that triads whose roots form the interval of a 5th are sc triads. 4. Triads whose roots form the interval of a 2d have no tones in common. 76 MUSICOLOGY 5. The common tone relationship of the triads may be illustrated as in Fig. 17, showing the circle of triads with the dc triads connected by double lines, and the sc triads by single lines. These common tones are called con- necting or binding tones. 6. We see that dc triads are more /iqi closely related from the view of com- mon tone relationship, and the sc tri- ads are more closely related from the ^ ^S view of root relationship (except be- FiG. ir. tween diminished or augmented triads and the triad on the 5th above, where the interval between the roots is a diminished or an augmented 5th); and therefore the actual relationship in the two cases is about equal. The very close interrelationship of the triads I, IV, V, is mainly due to their occupying the places of special prominence in the same key. 7. The triads of minor keys are also related in a precisely similar way. PROGRESSION 1. Progression is passing from one chord to another. Mere change of position or inversion of the same chord is not pro- gression. A chord may pass through several positions or in- versions before progressing to the next chord. 2. The interrelationship of chords forms the natural means of progression, as chords naturally lead to related chords; therefore the closer the relationship between chords the more natural the progression from one to the other, but va- ried relationship is necessary to varied expression. 3. If we analyze a sentence we find that all the words are grammatically related ; similarly in a musical sentence, the chords should be musically related. For special effect, chords are sometimes used without musical connection, just as, for the same purpose, words are sometimes used without gram' STRUCTURE OF MUSIC jy matical connection ; but in cither case, this is the exception and not the rule. 4. The common or connecting tones are the natural step- ping-stones in passing out of one chord into another, and nat- urally, the more tones in common the less motion involved and therefore the smoother the progression. On the other hand, the more motion involved the livelier the progression. 5 . When a chord progresses according to its natural tendency, it is said to resolve. The words progression and resolution are therefore, to a certain extent, synonymous; but in so far as we may draw a dJ\'s,\\x\.Q.\\ov\.^ progression applies to concords, or tones free to move in any direction, having no special ten- dency ; and resolution^ to discords and leading tones, which have a tendency to move in a certain direction. Therefore, progression deals more especially with the triad, or consonant part of chords ; and resolution, with the dissonant part of chords and leading tones. 6. Progression of Triads. Each tone of a triad or other chord is regarded as a voice ; and Voiee Leading consists in leading each voice of one chord to its proper place in the next chord. 7. General Rule for Progression, or Voice Leading : (i) Keep the connecting tones, if any, each in the same voice in both triads; (2) Lead the other voiee or 7'oices one degree up or down {as the case may be), each to the nearest tone i)i the sec- ond triad. 8. It will be seen that the rule involves the least motion possible. If the rule is strictly followed, we have strict voice leading; otherwise, free voice leading. 9. It may be shown that from any triad we can reach any other triad without moving any tone more than one degree. Thus, taking any triad for example, as 1 1 15^ ist position I. Now if all three of the tones ascend one degree, we have the triad II I zbt ist pos. j; if all three descend one degree, we have triad Vll"^ ( riz ist pos. I; if the tones % ascend one de- 78 MUSICOLOGY gree, we have the triad IV (ij- 3tl pos. 1; if the tones f de- scend one degree, we have the triad V ( ^5^ 2d pos. ); if the tone 5 ascends one degree, we have the triad V^I ( zhz 2d pos, j; if the tone i descends one degree, we have the triad lll/i]z 3d pos. \. Thus we hax-e passed from I direct to each triad of the key. 10. We notice also that the different positions of chords are the necessary result of voice leading where some of the parts arc held as connecting tones. 11. Progression by sc and dc triads difTers only in abrupt- ness. Thus the progression I to V may be made direct, as ifigi, moving both tones at the same time ; or through the intermediate (^c triad III, as ifi£r2z, moving the same tones one at a time. The progression IV to V may be made direct (moving all three tones at the same time), or through the sc triad I, or through the ^/c triads II and Vll°. 12. The characteristic triads I, IV, V, on account of their prominence, form the greater part of any piece of music, and the progression from one to the other is frequently direct ; but we see that dc triads may be used between to make the progression smoother. 13. It should be remembered that the rule for progression applies only in passing from one triad to another of different name, but does not apply in passing from one position to another of the same triad (which is not progression). 14. Rule for Skips : Consonant intervals may progress by skips, but (iisso)iant interi'als should be approaeJied and quitted witJioiit skips. The naturalness of a skip is in proportion to the consonance of the interval taken. 15. The different positions of the same triad ma}- be used freely, as only consonant intervals are involved ; but in pro- STRUCTURE t)K MUSIC 79 crressins to a new triad with one or more tones dissonant to the first triad, it is best to do so with the least motion possi- ble, by holding the connecting tones and moving the others without skips to the nearest tones of the new triad. There- fore skips most usually occur in repeating same triad in a dif- ferent position. When chords progress by skips, without any definite connection, they are called leaping or jumping chords. ffi ^s £: T=^ :e=J: Fig. 18. 16. Fig. 18 illustrates the rule of progression. It will be observed that the rule applies only to the three upper parts (triad proper), but does not apply to the bass. 17. The Progression of the Bass involves the same prin- ciple (relationship of intervals) as the root relationship of chords. The natural tendency of the bass is to progress by consonant skips ; the more perfect the interval of the skip, the more natural the progression : therefore the most natural progressions of the bass are the perfect concords (octave, per- fect 5th, and perfect 4th), after these the imperfect concords (major and minor 3ds and 6ths), and lastly, the single dia- tonic degree or 2d. 18. The bass is usually a doubling of one of the tones of the triad above it. Its progression is therefore influenced by both considerations. Besides, if the bass progressed like the three upper parts according to the rule, we would find it im- possible to avoid consecutive octaves and perfect 5ths, while the tendency of the rule, as between three parts, is to avoid them. 8o IMUSICOLOGY 19. Rule Regarding Consecutives : Tzuo perfect ^t/is or octaves should not occur co)iseciitively between the same parts. Consecutive Fifths Consecutive Octaves Fig. 19. 20. They are not considered consecutive unless they occur between the same parts. Neither are repeated perfect 5ths and octaves (see Fig. 20) con- f—~\ -- |— -| sidered consecutives, as they are merely repetitions of the same 1i=t p,(- 20. tones and equivalent to holding two tones. 2 1 . The general reason for the objection to consecutive perfect 5ths or octaves is due to the sense of perfectness which each carries in itself, and therefore want of connection with each other, so that the sense of relationship between the consecutive chords is lacking. 22. The special reason for the objection to consecutive per- fect 5ths is due to the tendency of the perfect 5th, when used consecutively, to carry with it its key limitations, thus chang- ing the key with each progression, often giving to the music the effect of being in more than one key at the same time. 23. The special reason for the objection to consecutive octaves (also unisons) is that they destroy the indi\iduality of the parts moving in octaves (or unison), since two parts moving together in octaves (or unison) blend so as to sound like one part ; and when the individuality is thus destroyed for a few chords only, it gives the impression of an accidental bad progression. This is not the case when the parts do not pretend to vary, as when two or more voices sing the same part in unison or octaves ; thus when a man and woman sing the same part together, they naturally sing in octaves with evidently no disagreeable effect. They could not, however. STRUCTURE OF MUSIC 8 1 sing together in perfect 5ths without giving the effect of each being in a different key. 24. Consecutive perfect 5ths and octaves are sometimes hidden (heard but not seen). Hidden consecutives occur when two parts move by similar motion to a perfect 5th or octave, as we mentally pass through the intermediate tones which involve the perfect 5th or octave. 25. Consecutive faults can occur only in similar or parallel motion, hence oblique motion should be used as much as pos- sible to avoid them. They are most noticeable betw^een the outside (most prominent) parts ; between the inside, and be- tween inside and outside parts, they are sometimes allowed. 26. Rule Regarding False Relations : Fa/se relations should be avoided. A false relation is where a false (imperfect) prime, octave, or 5th is formed between two difTerent voices in consecutive chords; thus, if a voice in one chord sings C and another voice in the next chord sings C i|, a false prime or octave relation is formed ; or if a voice in one chord sings B and another voice in the next chord sings F, a false 5th relation is formed — but the same voice may sing both notes. False 5th relations are forbidden only between the outside parts. 27. A false prime or octave relation involves two keys, since the two voices are not in the same key; but if the same voice sings both notes, the second note is merely of the na- ture of a passing note. The same in effect is also true of the false 5th relation. 28. Motion of the Parts. The progression of chords with their changes of position produce what is called motion in the parts. There are three kinds of motion as between any two parts : Contrary motion (when one part ascends and the other descends) ; Oblique motion (when one part moves horizontally and the other ascends or descends) ; Similar motion (when both parts ascend or descend together). Similar motion is also called parallel motion when the parts move parallel to each other. 82 MUSICOLOGV 29. Preparation of Dissonances. A dissonant tone is said to be prepared when it appears (in the same voice) as a con- sonant tone in the preceding chord. 30. As a rule, every dissonant tone, except the do)niiiant jtJi and diiiiinisJied jtJi {zvJiich juay or may not be prepared) and passing notes, should be prepared by being- first heard as a eon- sonant tone before it is heard as a dissonajit tone. Dominant /ths and diminished 7ths have become so familiar as to cease to require preparation. It is difficult for the voice properly to intone a dissonance, but when first intoned as a consonance it is easy to hold ; hence the rule. RESOLUTION 1. Resolution in its general sense applies to any tone or chord that progresses according to its natural tendency. Resolution therefore is merely natural progression. Any chord (if it cannot remain stationary) tends to move to the chord that occasions the least motion, using the common tones as connecting links. The rule for progression (p. /J:/) is based upon this tendency; therefore chords which pro- gress strictly according to the rule may properly be said to resolve. 2. The key-note, or tonic (i), is the tone of complete re- pose or home feeling in any key, and is therefore the tone to which all other tones tend for final resolution. Next to the tonic in restful character are, first the 5th, then the 3d. All other tones are restless and tend to resolve to one of these 13 5 13 5 three restful tones (Do, Mi, Sol, in major keys; La, Do, Mi, in relative minor keys"). 3. The Tonic triad being composed of these three restful tones is the chord of final resolution of all other chords, and is the natural beginning as well as ending of all music ; hence all music should begin and end with the Tonic triad, as well as frequently return to it (to prevent weariness) during the music. STRUCTURE OF MUSIC 83 4. As the bass is the foundation, it should begin and end on the tonic. If the soprano also ends on i (in octave or octaves with the bass), it is called a perfect close ; but if the soprano ends on 3 or 5, it is called an imperfect close. The soprano may begin on i, 3, or 5. 5. As already observed, resolution in its strict sense ap- plies to discords and leading tones; and progression, to con- cords (discords resolve, concords progress). 6. Dissonant intervals have a determined resolution (ten- dency to move in a certain direction), while consonant inter- vals are said to be free (without tendency to move in any particular direction). 2ds, 7ths, Qths, etc., all diminished and augmented intervals, and perfect 4ths (when just over the bass)* are classed as dissonant intervals. 7. A dissonance, or discord, is caused by a tone forming a dissonant interval with some other tone, and the tone thus causing the dissonance is called a dissonant tone. 8. All rules for Resolution are based on natural tendency, and the natural tendency is to resolve in the easiest way. A dissonant interval is, in a sense, an unnatural interval, and most naturally tends to resolve (or melt as it were) into the nearest consonant interval as being within easiest reach. For the same reason, a dissonant interval is easiest reached from the nearest consonant interval. Hence the general rule: Rule I. Dissonances should be approached and quitted con- junctly {by the single step of a 2d ; i.e., without skip). 9. It is usually easier for the voice to descend than to ascend, as descending tends toward relaxation, while ascend- ing tends toward tension. For which reason, it is also a gen- eral rule : * The dissonance of the perfect 4th is only relative, as the perfect 4th in itself is always a consonant interval. But its natural relation in the key is as an inverted per- fect 5th ; and when used just over the bass, it is, in a sense, as a substitute for the per- fect 5th, and thus antagonizes the natural key relationship, and is therefore relatively dissonant to the key— having the effect of a tone foreign to the key, and tending to in- duce change of key. Therefore, when thus used, it must be treated as a dissonance. ^4 MUSICOLOGY Rule II. Dissonances tend to resolve doivn, unless by resolv- ing up they ean do so by a smaller interval {half-step instead of a zvhole step). As a result of this rule, 7ths, 9ths, etc., diminished intervals, and perfect 4ths (when just over the bass) tend to resolve down, while leading tones and aug- mented intervals tend to resolve up. 10. In view of the tendency to maintain the key relation- ship, it is also a rule : Rule III. Dissonances tend to resolve by steps and half- steps of the diatonic scale of the key. In a general sense, how- ever, dissonances tend to move by half-steps as the smallest interval in use, and therefore by ignoring the key boundary they may easily be made to resolve chromatically into a new diatonic scale, or key, when by so doing a proper chord is formed in the new key, 11. Rule IV. A dissonant to7ie should not be heard at the same time ivith the tone to luhich it resolves. This rule gen- erally causes the interval of a 2d to expand (its upper limit ascending or its lower limit descending). It is also the princi- pal reason why the 7th descends, as otherwise it would resolve to the octave of the root (which is harmonically the same). 12. A chord to which a discord resolves is called its chord of resolution, and the note to which a dissonant tone resolves is called its note of resolution. If the resolution of a disso- nant tone is retarded by one or more notes intervening, it is called an ornainental resolution. 13. Dissonant tones are pleasing if short, not too frequent, and properly resolved. Dissonant tones, by calling for reso- lution, express energy and progress, and therefore give vigor to the music. 14. In the extensions of the triad (chords of the 7th, etc.), the added tone or tones often form dissonant intervals with more tlian one tone of the triad (see p. 69) ; but it is only necessary to apply the rules of resolution to the dissonant STRUCTURE OF MUSIC 85 tone which is the cause of the several dissonant intervals, as the different tones of the triad are, in general, subject to the progression of the triad. 15. In the diminished and augmented triads (p. 67: 3) the dissonant 5th will require resolution. 16. Applying the rules of resolution and progression to the Dominant 7th chord /V,, ^ ): 7, being the leading tone of the key, ascends to i ; 4 (minor 7th) descends to 3 ; let 5 remain stationary and 2 ascend or descend (5 and 2 being free), and we have the Tonic chord. If we let 5 ascend to 6, we will have the Tonic chord of the relative minor (see Fig. 10, p. 43). If in the V, chord of the relative minor we let 5 ascend to 6, we have the IV chord of the relative major. These are the principal resolutions of the Dominant 7th (V,) chord. Thus a dissonant chord may have several resolutions, owing to the free nature of some of its parts, the dissonant parts resolving the same in each case. 17. In a similar manner, the other dissonant chords may be resolved by applying the rules of resolution to the disso- nant parts. 18. In the I chord (2d inversion of a triad) the interval of a 4th being just over the bass tends to partake of the nature of a dissonance, and calls for resolution. The most satisfac- tory resolution of the | chord is to the direct (§) chord on the same bass. 19. It is possible, though not so usual, instead of resolving the dissonant part, to hold it and bring the other parts, which form dissonant intervals with it, into consonant positions; or, which amounts to the same thing (in the case of the extended triad), we may either regard the lower three tones of the direct chord as the triad proper and the upper tone or tones as dissonant to be resolved (called treatment in the upper part), or we may regard the upper three tones (especially if forming a major triad) as the triad proper and the lower tone MUSICOLOUV cr tones as dissonant to be resolved (called treatment in the lower part). Thus (taking the I, chord for example): (Trcatmenl in the upper part) 13 5 7 13 5 7 \/ i \/ / 13 6 14 6 V (Treatment in the lower pan) 13 5 7 13 \/ I I \ \. . 2 5 7 2 4 7 It will be noticed that the last two forms are symmetri- cally the reverse of the first two. In the first two, the upper note descends by reason of Rule IV; in the last two, the lower note ascends for the same reason. 20. These forms as general patterns show the most natural resolutions of all /th chords, except the VII^ and VII°, , in which the root of the chord being the leading note ascends and the 5th (being diminished) and 7th descend, giving for their primary resolution the following form (see major and minor patterns) : 7 2 4 6 \ \/ / 1 3 5 21. As a general rule, the resolving parts resolve the same in all the different positions and inversions of chords, as the different positions and inversions are merely the same tones in different orders (counting octaves as replicates of the same tone). 22. The rules of progression and resolution are not to be regarded as absolutely binding at all times. They aim only at a smooth, connected, natural flow of music ; therefore to express effort and excitement or any unusual effect they are temporarily set aside. STRUCTURE OF MUSIC 8/ SUSPENSION AND ANTICIPATION 1. Suspension is the holding of a tone of one chord over into the next chord in which it is dissonant, thus producing a momentary discord. 2. Anticipation is taking a tone belonging to the following chord and holding it until the other parts follow (reverse of suspension). 3. The object of suspension and anticipation is the closer binding of chords by thus, in a sense, linking them together. 4. Three things are to be considered in suspension : its preparation, its entrance, and its resolution. 5. The general rule is: The preparation takes place on an unaeeented pulse ; the entrance, on the following accented pulse ; a)id the resolution, on an unaccented pulse — in other words, the oit ranee is accented ; the preparation and resolution, unac- cented. The resolution may be ornamental (one or more notes intervening), and the chord accompanying the suspen- sion may progress in the meantime to any chord containing the note of resolution. 6. The principal suspensions are : 9 to 8 (a suspended 9th resolving into the consonance of an octave), 4 to 3 (a sus- pended 4th resolving to a 3d), 7 to 8 (a suspended 7th re- solving to an octave). 7. Suspension and anticipation may also be double or triple (occurring in two or three parts at the same time). PASSING NOTES 1. Music is made up of Harmony notes and Passing notes. Notes belonging to a chord formation are called Harmony notes. Notes added to a triad forming chords of the 7th, 9th, etc., though dissonant, are included with harmony notes. Suspended and anticipated notes are merely harmony notes held over or taken in advance. These are all essential to the harmony and are classed as Harmony notes. 2. Notes used merely in passing from one harmony note 88 MUSICOT^OGV to another, but which do not belont^ to the prevailing chord, are called Passing notes. Notes used merely for embellish- ment (p. 30:6) are also classed as Passing notes. 3. Passing notes a 3d, 6th, or octave apart may be used simultaneously in different parts. The chords thus formed are called passing chords, being too transient to be harmon- ically recognized. 4. The true Passing note (used in passing from one har- mony note to another) comes in and goes out without skip. Two passing notes are thus used in succession when the inter- val between the harmony notes requires. These increase the smoothness of the melodic flow, and for this purpose are freely used in music, especially in the melody part. 5. Other Passing notes come in by a skip and go out with- out skip. These are more of the nature of embellishing notes. 6. Other Passing notes come in without skip and go out by a skip of a 3d in the opposite direction (often going to another passing note, which latter, of course, belongs to the preceding class). 7. Other passing notes leave and return to the same har- mony note. In this case, and also in the case where the passing note comes in by skip and goes out without skip, if the passing note is below the harmony note, it tends to a half-step below, thus often requiring an accidental ^ or tt in writing it where the diatonic interval is not already a half- step. Before the 3d of a chord or the 3d or 5th of a V, chord, the passing note may be a whole step below. 8. Passing notes are not prepared ; in short, are not bound by any of the laws of harmony except that being dissonant they require to be resolved. PEDAL PASSAGE I . A Pedal Passage is a passage in which the bass sus- tains the tonic or dominant (sometimes both), while the other parts move independently of it. The tone thus sustained is called the pedal point, or tone. The passage should begin . STRUCTURE OF MUSIC 89 with the harmony of the pedal tone, frequently return to it, and finally end with it. 2. During the pedal passage, the part next above the pedal becomes the real base as to chord relations. 3. There should not be more foreign harmony in the pas- sage than the pedal tone is capable of counterbalancing. The tonic and dominant (having the most sustaining power) are best adapted to form the pedal point. 4. If the sustained tone is held by one of the upper parts, it is called a stationary tone, and has not the sustaining power peculiar to the bass. THE CADENCE 1. The Cadence is the close of a musical strain; its object is to form a satisfactory ending. It consists, in its regular form, of the Tonic as the final chord preceded either by the Dominant (V or V^) or Sub-dominant chord; the first is called Authentic; the second, Plagal. 2. If the cadence ends with the Dominant or Sub-domi- nant, instead of the Tonic chord, it is called a Half Cadence. 3. The Authentic is the strongest and most common form of cadence. It is also called a Perfect Cadence when the final Tonic is in direct forni and ist position, thus having the root in both bass and soprano ; otherwise, it is called an Imperfect Cadence. 4. A Deceptive Cadence is where the Dominant resolves to some other chord than the Tonic, thus deceiving our expec- tations. 5. The cadence is sometimes ornamented by runs, trills, etc., in which case it is usually called a Cadenaa. 6. In music the notes arc grouped into musical phrases, the phrases into musical clauses (called sections), the clauses into musical sentences (called periods). These are arranged rhyth- mically as in poetry. Every musical clause or sentence ends with some kind of cadence. The cadences should be varied, reserving the most complete cadence for the final ending. 90 Mrsicoi.OGY 7. In poetry or prose the divisions arc shown by punctua- tion marks. In music the punctuation marks are not written, but are, in a sense, understood ; thus a cadence calls for a semicolon, colon, or period (understood), according to its de- gree of completeness. 8. The phrases of a musical clause are separated from each other by a slight unwritten pause called a caesura, which in poetry is usually marked by a comma. This, however, does not, as a rule, involve a cadence. SEQUENCE 1. A Sequence is a succession of similar movements. 2. A Phrase Sequence is a musical phrase repeated at a higher or lower pitch. 3. A Chord Sequoice \^ 3. swcccssion of similar movements of chords or groups of chords. 4. Symmetry is the controlling object in a sequence ; there- fore the rules of resolution are suspended when necessary to preserve the symmetry. MODULATION 1. Modulation is passing from one key to another during the course of a piece of music. If the modulation is very brief, it is called a digression. 2. An accidental sharp ( ^ ) or flat (t?) or natural (3) usu- ally means a change of key if a diatonic chord in a new key is thus formed. Sometimes, however, the accidental is only on a passing tone, and is called a chromatic tone. 3. An accidental necessarily introduces a new tone (foreign to the signature key), and therefore tends to lead to that key which contains the new tone while retaining as many tones of the old key as possible. As keys are related to each other in proportion to the number of tones the\' have in common, therefore the nearer two keys are related, the easier is the modulation from one to the other. STRUCTURE OF MUSIC 01 4. The Combined Key Table (p. 46) shows all the keys (major and minor) in the order of their relationship to each other; so that (in major keys) each key is related in the first degree to the nearest key on either side of it, and in the sec- ond degree to the second key from it on either side, etc. The connecting lines between the major and minor key tables show the order of relationship between major and minor keys. 5. Modulation from any major key to the next key on either side or to its relative minor is called natural modula- tion, because easy and natural (the keys differing by only one tone). 6. Modulation to a remote key is either gradual (when made by passing through the intermediate keys) or abrupt (when made by stepping over the intermediate keys). In any case, however, the general rule applies, that the last sharp {ivJicthcr in signature or as accidental — major or minor keys') is ahvays on the jtJi of the key {key letter, first letter above) ; and the last flat is alivays on the ^th of the key in major keys {key letter, fourth letter belozv), or on the 6th of the key in minor keys {key letter, third letter above). This rule applies also to naturals by remembering that a natural has the effect of a flat in sharp keys, and the effect of a sharp in flat keys. 7. The rule already given for reading accidentals — giving a sharp effect to the key syllable by using i (e), or giving a flat effect by using e (a) or a (see p. 21 :23) — is the common rule, and the most convenient ; this, however, is treating the acci- dentals as passing tones. It would be more correct, there- fore, when a modulation occurs, to use the regular key sylla- bles readjusted to the new key letter, returning to the signa- ture key as soon as the modulation ends (usually at the end of the measure, unless the accidental is repeated in the next measure). 8. To illustrate the general principles of modulation, first find an accidental in a piece of music, then take the major 92 MUSICOLOGY key table and start from the key in the table corresponding to the old key in the music, and at the point on the staff at which the accidental occurs in the music, pass horizontally (to the right in ^'s, to the left in t^'s) till you come to the ji| or b (treating a I; as a t?, in sharp keys ; and as a ^ in flat keys) ; however, do not pass a 5 (to the right), as it will in- dicate the relative minor of that key ; also, do not pass a 6 (to the left), as it will indicate the tonic minor of that key. If more than one accidental occurs in the music at the same time, take the first key that will satisfy them all ; in which case, it will be observed that the last sharp or flat (which de- termines the new key) will be in the farthest key from the starting key. 9. It is evident that sometimes the accidentals involved in a modulation are not all marked in the music, as there may not always be notes on all of the affected lines or spaces to mark, but are understood if not marked ; the last sharp or flat of any key necessarily involving all the other sharps or flats of that key. 10. Wiien the modulation extends beyond the limits of the key table, we may continue around the circle (Fig. 9, p. 42) by making the enharmonic change (p. 42:13); but it is preferable, however, to extend the table in the direction of the modulation when doing so will occasion fewer accidentals in writing the music. Extending the table is merely for the purpose of avoiding the enharmonic change, and thus avoid- ing the accidentals which would be involved; but extending the table will involve double sharps or flats, as already shown (p. 52: 41, 42) 11. In a harmonic sense, an accidental does not by itself produce modulation, but only when with other tones a dia- tonic chord of some other key is formed' the chord thus be- ing a chromatic chord of the old key, but a diatonic chord of the new key ; the modulation being the result of the ten- dency of chromatic chords to lead to the key in which they are diatonic, or natural. STRUCTURE OF MUSIC 93 METHODS OF MODULATION 1. There are many ways of modulating, but noticing a few principal methods will be sufHcient to illustrate the general principles involved. 2. Modulating by Means of Connecting Triads. Examin- ing the key table (p, 46) and comparing the triads of the different keys, we would find that all triads (except the aug- mented triad)are common to several keys (same tones though the triads have different names in the different keys). These may be called connecting chords, since they connect the keys. 3. It is evident that if a chord is common to any two keys it may resolve into either key. Therefore we may modulate from one key to another by first striking a chord common to both, then resolving it into a distinguishing chord of the new key. .' E^ f B^ c F R c d G a D e A b V III V VI I II III I IV VI VI IV II I IV 7 V V VII° I III IV V VI VIT° I II VI IV I IV IT° VI Fig. 21. 4. Fig. 21 gives the triads of the key of C and the corre- sponding triads of each in other keys (though differing in name — roman numeral — yet composed of the same tones). 5. By similar comparisons, we would find that the triads of any other major key than C would have the same corre- sponding triads in other similarly related keys ; so that the key letters at the top of Fig. 21 may be omitted, thus mak- ing the figure general in character, as it is applicable to all keys similarly related. 94 MUSICOLOGV 6. We notice also that the 1st, 4th, and 5th formulas (Fig, 21) are similar (composed of the same w^?y'(r^;' triads in the same order); and that the 2d, 3d, and 6th formulas are similar (composed of the same viiiio)- triads in the same order). Now omitting the repetitions of the same formulas we would have Fig. 22. _m^ V I IV VI III YI n I IV vir 11° Fig. 'ii. 7. Making the figure general by omitting the key letters (since these formulas apply equally to any other keys simi- larly related), we will have three general formulas, which may be expressed thus: (Major triad formula) (Minor triad formula) (Diminished triad formuki) If^ — t— ^^ IH— Y^— ^r V-^^^ (1) ^~~^I (2)^-A-^V (The curved lines compare with the curved lines in the key table, p. 46.) Fig. 23. 8. These formulas may be copied (exact dimension.s) and applied to the key table (between the major and minor keys) so as to show the keys related to any key through any given triad. By placing the given triad in line (vertically) with the given key (using the proper formula), the other triads will fall in line with the related keys (the roman numerals showing the relation of the triad to the different keys). As formulas 1,2, and 3 are very simple, they can easily be ap- plied mentally instead of applying a copy. 9. We may also observe that as every major key has three major triads (I, IV, V, p. 63 : 16) and every minor key has two major triads (V, VI), therefore every major triad is I, IV, STRUCTURE UF MUSIC 95 and V in three different major ke}-s, and V and VI in two different minor keys (relationship shown by formula i). As ev^ery major key has three minor triads (ll, III, Vi) and every minor key has two minor triads (l, iv), therefore every minor triad is ll, ill, and VI in three different major keys, and I and l\' in two different minor keys (relationship shown by for- mula 2). Also, as every major key has one diminished triad (vil°) and every minor key has two diminished triads (ll° and VII°), therefore every diminished triad is Vll° in one major key, and 11° and Vll° in two different minor keys (re- lationship shown by formula 3). 10. Therefore, formulas 1,2, and 3 cover the entire ground of modulation by common (or connecting) triads in both ma- jor and minor keys. Thus, if any major triad is struck in any key (major or minor) it may be regarded as belonging also to any one of four other keys related as shown in formula i, and if any minor triad is struck in any key (major or minor) it may be regarded as belonging also to any one of four other keys related as shown in formula 2, and if any diminished triad is struck in any key (major or minor) it maybe regarded as belonging also to either of two other keys related as shown in formula 3, the modulation depending entirely upon the resolution of the triad in each case. 11. To modulate beyond the reach of any one of the for- mulas, we may reach to any convenient key through one of the formulas (or any part of it), then move to another triad of same key, then, with the formula containing this new triad, reach (as before) to another convenient key, and so on until the desired key is reached. 12. To confirm the modulation (if not already confirmed), on reaching the desired key we should resolve to a distin- guishing chord of this key. The distinguishing chord should include the 7th of the key when modulating in the direction of sharps (or the relative minor), or the 4th of the key (the 7th is also desirable, as it is the leading tone) when modulat- 9^ MUSICOLOGY ing in the direction of flats; since the last sharp is on the 7th and the last flat is on the 4th. The YA^\ chord (contain- ing both 4 and 7) is best adapted for this purpose. 13. Hence the rule: A key is usually confirmed by some form of domifiant harmony. Other chords may lead to or introduce the new key. 14. The V, chord is a very restless chord and calls strongly for the Tonic (I) chord, the modulation not being completed until the Tonic, or chord of home feeling, is reached. 15. Modulation by Means of Connecting Tones. As already "observed, keys are related to each other in propor- tion to the number of tones they have in common ; by the same principle, chords (whether in the same key or different keys) are also related in proportion to the number of tones in common. These common tones are naturally connecting tones in passing from one chord to another (evidently, the more tones in common the less motion required"). 16. A diatonic chord is one which does not require any change in the natural intervals of the key (involving only the sharps or flats of the signature) ; all others are called altered or chromatic chords. 17. There are only four kinds of diatonic triads in a key: the major triad (major 3d + minor 3d), the minor triad (minor 3d + major 3d), the dim. triad (minor 3d + minor 3d), and the aug. triad (major 3d + major 3d). 18. Observe that there are two ways of changing a triad of one kind into another kind — one by flats, the other by sharps — since moving one tone one half-step in one direction or moving the two other tones one half-step in the other direction will have the same effect on the intervals involved, but of course the modulation will be in opposite directions. 19. Thus, for example, we may change the major triad | into a minor triad either by flatting the 3tl or sharping the ist and 5th, as b? or ^7. In either case we have changed STRUCTURE OF MUSIC 97 the intervals of the major triad (major 3d + minor 3d) into the intervals of the minor triad (minor 3d + major 3d), (See major pattern.) However, neither b? nor ! 7 is now a diatonic (but a chromatic) chord of the old key, but is a dia- tonic chord of a new key. As they are now minor triads they may each exist (under different names) in five different keys related as shown in formula 2, but the modulation would most naturally be to the one nearest related to the key from which we started. See key table and observe that if we start from ? in key of C major, then b? will be bl (ll) in key of F, and $ ? will be H (III) in key of E. rv vii' Fig. 24. 20. Formulas 1,2, and 3 may be combined, as in Fig. 24, by repeating the formulas in order, thus forming an unbroken chain. The ends of the chain may be lapped, forming a circle (corresponding to the key circle). A copy of this com- bination will be found at back of book, which may be cut out and applied to the Combined Key Table to show the modu- lation effected by changing any triad of one kind into a triad of any other kind. 21. Thus, starting with the formula corresponding to the nature (major, minor, or diminished) of the given triad, and placing the given triad in line with the key we start from, then, if the change is by flats, the formula to the left corre- sponding to the nature of the changed triad will locate all the keys containing the changed triad. If the change is by sharps, the formula to the rigiit corresponding to the nature of the changed triad will locate all the keys containing the changed triad. Though the modulation in either case leads 98 MUSICOLOGY first to the key nearest related to the starting key (as differ- ing by the fewest tones), yet the changed triad may be re- solved into any one of the different keys in which it is com- mon (as already shown in the method by connecting chords). 22. If the change is to an augmented triad, then III' to the right or left (as the case may be) will locate the new key. 23. However, the principal chords used in modulation by connecting tones are the Dominant 7th (V,) chord, in modu- lating in major keys and major to minor, and the Diminished 7th (vii°) chord, in modulating in minor keys and minor to major. 24. Modulation with the Dominant 7th Chord. The Dom- inant 7th (V-) chord is peculiarly adapted for modulating pur- poses for several reasons. 25. First, the V, ( f jchord contains the three key limiting points: 7 (toward the left), 4 (toward the right), and 5 (toward the relative minor). See key table (p. 46) and ob- serve that we cannot move from 7 of any key to the left (on same degree of staff), nor from 4 to the right, nor from 5 to relative minor, without chromatic change ; but all four tones of the chord may be moved to the tonic minor without change. Hence the V. chord limits the key in all directions except toward the tonic minor. 26. Second, the V^ chord calls strongly for' the Tonic chord (7 being the leading tone one half-step below i, 4 being one half-step above 3, and 5 being a connecting tone). 27. Third, there is no other chord built like it (major 3d, perfect 5th, minor 7th), so that there is no other chord like it in the same key (the same V, chord is common only to one major key and its tonic minor — see formula i). The V, chord, therefore, has a peculiar locating power and always calls for a certain major key or its tonic minor, whichever is near- est related to tlie key from which we are modulating. 28. It is evident that u e can form the interx-als of the V, STRUCTURE OF MUSIC 99 chord on any tone of any key by using the necessary sharps or flats, and that each chord thus formed will be the diatonic V, chord of some key and will call for that key. 29. I, and IV, differ from V, only by the 7th being major instead of minor (see major key pattern), therefore I, and IV, can be changed to a V, either by flatting the 7th or by sharping the ist, 3d, and 5th. 30. II,, III,, and VI, differ from V, only by the 3d being minor instead of major, therefore II,, III,, and \i., can be changed to a V, either by sharping the 3d or by flatting the 1st, 5th, and 7th. 31. VII°, differs from V, by the 3d being minor and the 5th being diminished, therefore Vii° can be changed to V, either by sharping the 3d and 5th or by flatting the 1st and 7th. 32. Thus we can form the V, chord in twelve different ways in any one key, each of which would be a diatonic V. chord in one of the twelve major keys and its tonic minor, and which would modulate to one of the major keys except where the tonic minor comes in first as being nearer related. (These chords would be chromatic chords viewed from the old kev, but diatonic chords viewed from the new key.) 33. We may begin at any major key, but the key of C (being without sharp or flat) is the most convenient for pur- poses of comparison. Compare the following results by re- ferring to the key table (p. 46). 34. Forming the V, chord on i by flatting the 7th, thus, ^, we have the V, chord of the key of F or /, which there- fore modulates to the key of F as being nearer related. Sharping the 1st, 3d, and 5tli, thus, j^, we have the V. chord o{ /^ or F ^, and modulation is to/|^ as being nearer related. !j3 3 35. Forming the V, chord on 4. thus, I or jl, we have the * J* V, chords of BI? or /'^ and /; or B. lOO MUSICOLOGY 36. Forming the V, chord on 2, thus, ^ or ^l, we have 3 b2 the V, chords of G or^, and G^ or ^'■). 2 b2 37. Forming the V^ chord on 3, thus, 4 or ^4, we have the V, chords of a or A, and A 7 or a'?. 5 bs 38. Forming the V. chord on 6, thus, jf or "j, we have the 6 be V, chords of D or ii, and D 7 or ^/?. 6 be 39. Forming the V, chord on 7, thus, j* or i, we have the 7 b- V, chords of e or E, and E !^ or r ?. 40. We have thus modulated to twelve difTerent major or minor keys: the key (major or minor) given first in each case being nearest related to C. 41. If we had taken any other major key than C as a start- ing-point, we would have obtained similar results, the only difference being that we would have had to use naturals to sharp the flatted tones in flat keys, and to flat the sharped tones in sharp keys. 42. Observe, in the key table, that a minor (relative of C major) is nearer related to C by two tones (3d and 6th) than its tonic A major. Therefore, all the minor keys to the right of a are also nearer related by two tones (3d and 6th) than their tonic majors. The result will be similar if we start from any other major key than C. Therefore, in modulating with the V, chord we cannot go directly beyond the nearest two major keys to the right of the starting key. after which the modulation leads into the minor keys (being nearer related than their tonic majors); but toward the left the modulation leads to the major keys (being nearer related than their tonic minors). In cither case the final modulation is determined by the resolution of the V, chord, which may resolve to the Tonic chord of either the tonic major or minor key. 43. Another method of modulating with the V, chord is STRUCTURE OF MUSIC lOI by resolving the V, chord of the old key directly into the V, chord of the new key by moving the tones that are not com- mon each to the nearest tone of the new V, chord and using the common tones, if any, as connecting tones. Comparing the V, chords of the different keys (see key table) will show the moves necessary in each case. 44. We may also modulate by making a chromatic (half- step) run up to 7 or down to 4 of the desired key, then taking the rest of the V, chord of that key. 45. Observe (see Fig. 10, p. 43) that sharping the root (5) of the V, chord of the major key gives the Vil° (Diminished 7th) chord of the relative minor key. Also observe (Fig. 1 1, p. 44) that flatting the 7th (6) of the Vll° chord of the major key gives the Vll° chord of the tonic minor. Therefore, we may modulate to any minor key by first modulating to the V, chord of its relative major, then sharping the root (5) — by making both moves at the same time the modulation would be direct; or, first modulating to the VII° chord of its tonic major, then flatting the 7th (6) — or direct, by making both moves at the same time. The two methods when direct (in one move) are necessarily the same, as they result in the same chord — Diminished 7th (vn,) of the minor key. 46. Modulation with the Diminished 7th (viJt) Chord. This chord is also peculiarly adapted for modulating purposes for several reasons. 47. First, it is only found as a diatonic chord in minor keys (on the 7th), and therefore calls for a minor key. 48. Second, it contains all four of the key limiting points of minor keys (see pp. 47: 17-48: 18); 7 and 4 from the , tonic view, and 2 and 6 from the relative view. Observe, from the key table, that we cannot move (on same degree of staff) from 7 either to the right or left, nor from 4 to the right, nor from 2 to the left, nor from 6 to the right, nor from 7 to relative major, nor from 6 to tonic major, without chromatic changes. So that the vn° chord limits the key in 102 MUSICOLOGY all directions (toward the right by 7, 4, and 6, toward the left by 7 and 2, toward the relative major by 7, toward the tonic major by 6). 49. Third, it can easily be converted into a V, chord. 50. Fourth, no other chord is built like it (minor 3d, dim. 5th, dim. 7th), so there is no other chord like it in the same key. 51. Fifth, three of its members are capable of an enhar- monic change, each change converting it into the same chord of another key. (Modulation by means of enharmonic changes is called eiihannoiiic modulation.') 52. Comparing any minor key with its relative and tonic majors (see key table), we will see that the VIT° chord of any minor key can be changed into the V, chord of its relative major by simply lowering the root (7) one half-step (see Fig. 10, p. 43), or into the Vll° chord of its tonic major Sy rais- ing the 7th (6) one half-step (see Fig. 11, p. 44V Thus we have a simple method of modulating from any minor key to its relative and tonic majors, and vice versa. 53. By referring to the minor key pattern we see that the VI r^ chord is made up of three minor 3ds, one above the other, and that the augmented 2d between 6 and 7 (the extremes of the chord, harmonically speaking) is equal to a minor 3d (U steps). Therefore the VIl" chord is practically a circle of minor 3ds, and the aug. 2d may take the place of any one of the minor 3ds without affecting the tones, though causing a refiguring of the tones, since the aug. 2d must come between 6 and 7, and involving also an enharmonic change in one or two of its members. 54. We observe also (from relative minor pattern") that three of the members of the vii° ^ chord are capable of an en- harmonic change (change of name without change of tone) ; thus, Jt7 toh, 2 to 1? 3, and 6 to ^5. Applying these changes 6 6 6 55 to the VI r, chord, we would have *■ 2. ba- '^"^1 | (in each ^ J7 tti bi V STRUCTURE OF MUSIC IO3 case changing a minor 3d to an aug. 2d), but the last three will have to be refigured, so that the aug. 2d will come be- 6 4 2 3t7 tvveen 6 and 7 ; this would give |, ,, ^,1, and ". Thus we 87 be b4 (a have the VIl^ chord in four different minor keys (made up of common tones, though involving different accidentals in writing it in the different keys). If the first key is a minor, the second will be ^ minor, the third r> minor, and the fourth /If minor. (The Vll, of tonic minor would give similar results). 55. We may observe that every third minor key (see minor key table, p. 46) is thus related through the common Vli, chord, and also that these same keys are linked together through their majors (the relative major of one being the tonic major of the next, etc.). 56. In this sense the Vll° chord, like the major, minor, and dim. triads, is an equivocal chord because it does not point to any particular key, since it may resolve into the Tonic chord of any one of four different minor keys (the modulation depending on the resolution in each case). 57. The Vll° may evidently be regarded as the V^ chord, with root omitted, of a minor key (see p. 72 : 22), thus being classed as dominant harmony, and therefore included in the rule that a nciv key is usually co)ifirmcd through sonic form of the Dominant. 58. Observe, from the key table, that in resolving theVll° chord of one key to the vil° of the next key to the right, we raise each tone one half-step ; or to the vil° of the next key to the left, we lower each tone one half-step — there being no connecting tone. 59. It is evident that the intervals of the Vli'^ chord are the same in its several positions and inversions, since the members are all equidistant. 60. By combining the several methods of modulation by the Vll*^ chord, we may modulate from any key to any other. Thus, beginning at any minor key, we may modulate to any I04 MUSICOLOGY minor key having the vif^ common by simply resolving it to the Tonic chord of the desired key, or to any other minor key by first resolving it to the VII ° of the next key (right or left), then resolving to the Tonic of the desired key, or to any major key by changing the Vll° chord (before resolving to the Tonic) to the V^ of its relative major or the vii° of its tonic major; or, beginning at any major key, by first chang- ing the vil° of the tonic major or the V, of the relative major to the vil° of the minor, then proceeding as above in- dicated. We make the modulation abrupt (direct), in any case, by making the different moves at the same time. 6i. Beginning at the major key of F^, we may take its vii''. chord and make natural the 7th (6), which gives the VII° of its tonic minor /if , then Avith the common Vll° (by making the proper enharmonic changes) we may step over to e\^ minor, then flatting the root (7) we have the V, of its rel- ative major G^; thus we have modulated from F |^ to G^. But these keys are equivalent (interchangeable) keys ; there- fore, by using the extreme limit of our means of modulation, we have modulated entirely around the key circle to the starting-point ; or, in fact, we have not modulated at all, but simply made the enharmonic change and moved from vii'i to V, of equivalent key. 62. It is evident that we cannot modulate to advantage, in either direction or by any means, farther than half-way around the key circle, as it is then nearer to modulate in the opposite direction ; and, for the same reason, if we extend the table to avoid the enharmonic change, we cannot modu- late to advantage farther than the equivalent of half-way around the circle, as extending the table is merely adding overlapping keys in the key circle. 63. Modulation with the Augmented 6th Chord. The Aj(ginentcd 6th chord is a chord containing the interval of an aug. 6th (5 steps). It is used in three forms, called Italian 6th, French 6th, and German 6th. STRUCTURE OF MUSIC IO5 64. The Italian 6th chord consists of a major 3d and aug. 6th ; thus, ^ (in minor key). The French 6th consists of a major 3d, aug, 4th, and aug. 6th (same as Italian 6th, with aug. 4th added) ; thus, f (in minor key). The German 6th 6 consists of a major 3d, perfect 5th, and aug. 6th (same as Italian 6th, with perfect 5th added); thus, ^ (in minor key). If we omit the 5th in the German 6th, we have the Italian 6th ; or if we change the 5th to an aug. 4th, we have the French 6th. 65. The thorough-bass figuring of the Italian 6th is j|6, of the French 6th is ^l, of the German 6th is *§. These chords 3 are only used in direct form. 66. The Italian 6th is a three-toned chord, but in forming four-part harmony the 3d only must be doubled. 6y. Observe that the intervals of an aug. 6th and minor 7th are equal. Therefore the German 6th and V, chords are composed of the same intervals (as to measure), and one can be converted into the other by enharmonically changing one member; but the two chords resolve differently — the aug, 6th ascends, while the minor 7th descends, 68. The Italian 6th, French 6th, and German 6th, being dissonant chords, require resolution. The primary resolu- tion of each is the same (V, with root doubled), and there- fore (to save space) may be treated together. Thus, ? 2) re- 6 solves into 57 (minor key). It will be seen (see minor pat- 5 tern) that the 6th (j| 4) ascends one half-step to 5, and the other three members descend each one half-step. In the case of the Italian 6th — the 5th (3) being omitted — the 3d (i) may either descend one half-step to || 7 or ascend one step to 2 (or if the 3d is doubled, it may take both parts). In the case of the French 6th the 4th (2) will remain stationary. 69. In the case of the German 6th, if the ist and 5th both descend one halt-step at the same time, we would have con- Io6 l\IUSICOLOGV secutive 5ths. To avoid this we may retard the resolution of the 3d and 5th until after the other parts resolve P|, f, J j, thus forming the | (2d inversion) of the minor Tonic between ; or the 5th may ascend one half-step / |, *J, jf i, forming the | of the major Tonic between; or we may preresolve the 5th I ^f' ^1- jj? )• forming the French 6th between. 70. The natural seat of the aug. 6th {It., Fr. , or Ger.) chord is on the 6th of a minor key, as only in that position is it in the same key with its chord of resolution, as will be seen by trying it on all the degrees of the major and minor patterns. 71. The aug. 6th chord is often used merely as a chro- matic chord, but when used as a means of modulation it nat- urally leads to that minor key where it is based on the 6th of the key and is immediately confirmed by its chord of resolu- tion. (In the aug. 6th chord the last sharp is on the 4th of the new key and is an exception to the rule that the last sharp is always on the 7th of the key.) 72. Since the German 6th and the V, chords may be en- harmonically changed the one into the other, they may, in a sense, be regarded as a common chord of different keys, the modulation being determined by its resolution. Observe, from the key table, that the German 6th chord of any minor key, as (for example) a minor, is composed of the same tones (enharmonically changing the 6th to a 7th) as the V. chord of the key of B 1? major (or its tonic d I' minor). Therefore, in enharmonically changing a German 6th to a V, chord, we modulate from a certain minor key to the second key to the left of its relative major, or conversely, when enharmonically changing a V, to a German 6th chord. Modulation by Inversion. (Seep. 120.) Modulation by Imitation. (Seep. 125.) STRUCTURE OF MUSIC IO7 SUMMARY OF MODULATION 1. The diatonic tones of any key are its natural tones un- affected by accidentals (involving only the sharps or flats in the signature), except the 7th in minor keys, which is dia- tonic, though marked by an accidental. 2. If a diatonic tone of a key is chromatically altered by an accidental, it is called a chromatic or altered tone. A tone may be chromatic in one key and diatonic in another. As already observed (p. 92: 11), an accidental does not always produce a change of key, as it may sometimes indicate only a chromatic passing tone or chord. 3. h. diatonic cJiord (p. 96: 16) is one that contains only diatonic tones. A chromatic chord is one that contains one or more chromatic tones, A chord may be chromatic in one key and diatonic in another. A modulating chord contain- ing accidentals is a chromatic chord as viewed from the old key, but a diatonic chord as viewed from the new key; and modulation is only the natural result of a chromatic chord calling for that key in which it is a diatonic or natural chord. If a chromatic chord does not follow up its suggestion of a new key, but returns immediately to a distinguishing chord of the old key, it should be regarded merely as a chromatic chord of the old key. 4. Sometimes a chromatic chord is followed by another chromatic chord, the two suggesting keys in opposite direc- tions, in which case they counteract each other and no mod- ulation is produced. 5. We see, therefore, that an accidental does not always produce modulation. On the other hand, modulation some- times takes place without an accidental or some distance in advance of the accidental (or sign of the new key), as modu- lation takes place whenever the home-feeling changes to a new tonic, whether an accidental occurs at that point or not ; but in most cases the modulation will sooner or later be confirmed by the appearance of the accidental. (This will be referred to again under Modulation by " Inversion," and by "Imitation.") I08 MUSICOLOGY 6. It is evident, especially between any two closely related keys where the tones are nearly all common, that the music may continue some distance on chords common to both keys; the music being, as it were, on a balance between the two keys, so that a slight cause or suggestion (or even a slight mental effort) may turn the balance without necessarily in- volving an accidental. Therefore, the new key may be taken at any convenient point (when the music is thus on common ground), as, for instance, the beginning of a line of words, or the beginning of a musical phrase or figure, or any other nat- ural division of the music, or after a rest, or when the arrange- ment of tones or chords seems to suggest the new key. 7. There may be three stages of modulation — the new key being first suggested, then confirmed, then completed. The suggestion and confirmation may or may not take place at the same time. We accept the new key Avhen first suggested, but the suggestion must sooner or later be con- firmed, usually by some form of Dominant harmony of the new key ; we are still not at home in the new key till we reach the Tonic, or chord of home-feeling, thus making a complete cadence in the new key. 8. If a modulation is very short and incomplete it is called a transient modulation or merely a digression ; but if a com- plete cadence (Dominant followed by Tonic harmony) is made in the new key, it is called a eadential niodiilation. 9. Modulation may be classified as follows: \ Xaiural (to a closely related key). As to Distance -^ Remote (to a distant key). . ,, j Gradual (passing through the intermediate keys). ^ ° "^ ' I Abrupt (stepping over the intermediate keys). {Diatonic (by means of common chords). Chromatic (by means of chromatic chords). Enharmonic (by means of enharmonic changes). ( Digressive (when short and incomplete). As to Completeness -! Cadential (when making a complete cadence in ( the new key). STRUCTURE OF MUSIC IO9 10. Duration, location, the form and position of the Dom- inant and Tonic chords, the place in rhythm, etc., — all have a bearing on the importance of a modulation. 1 1. The different methods of modulation already explained bear only on the question of " How to modulate." " When to modulate" is an equally important question. 12. The art of modulating does not consist merely in the ability to ramble through various keys, but consists in know- ing when, as well as how, to modulate. 13. The purpose of modulation is to enlarge the means of musical expression by placing the combined harmonies of all the keys at our disposal instead of confining us to the limits of one key. 14. We may enter a new key whenever the old key proves insuflficient to produce the musical effect we wish ; also, to avoid too much uniformity (especially in the longer pieces of music), and to secure variety and contrast. Just as disso- nances heighten by contrast the effect of consonances, so modulation heightens by contrast the sense of key tonality ; but either should be used only when it gives a sense of justi- fication in the effect produced. Modulation should thus be the result of necessity rather than a display of skill. 15. Our sense of symmetry and proper balance requires that the music should end in the same key in which it begins ; otherwise, it would seem incomplete. TRANSPOSITION I. As already seen, modulation consists in passing from one key to another during the course of the music, thus in- volving only accidentals. On the other hand, transposition consists in moving the entire music bodily into another key, thus involving a new signature. However, if the transposi- tion be by the interval of an octave, the signature (and there- fore the key) will not be affected, since the scale is not changed. But if the transposition be by any other interval than the no MUSICOLOGY octave, the music will evidently be in a different scale, or key, requirini^ the corresponding signature. 2. Music may be transposed, or moved bodily, to any key exactly as the key pattern (Chart I.) may be moved bodily to any key (the music being made up of key tones). And as moving the key pattern does not affect the relation of the key tones to each other, so transposition does not affect the interrelationship of the tones of the music. Transposition, therefore, is merely a change of pitch of the entire music. 3. Any modulations of the music before it is transposed will, after it is transposed, be similarly related to the new key as formerly to the old, and involving therefore the same accidentals. 4. Writing the transposed music will evidently involve both a new signature and the raising or lowering (as the case may be) of each note on the staff a distance equal to the in- terval of transposition. 5. Transposition does not affect the reading of music nor the figuring of chords, as each note has the same position (and therefore the same key syllable name and numeral) in the new key as formerly in the old, the entire music remain- ing intact though in a new key. 6. Music may be transposed in the singing or playing of it without rewriting it. As the reading of the music is not changed, it is only necessary, therefore, to apply the key sylla- bles or numerals with the key-note pitched to the desired key. 7. When several persons sing together it is necessary that they use the same key; but when a person sings alone the music may be pitched in any key that best suits his voice, regardless of the key in which it is written. In singing, the key is merely a question of pitching the voice at the start, the music naturally following in the key thus established. 8. In a similar manner, playing on the keyboard in a dif- ferent kev from that in which the music is written consists STRUCTURE OF MUSIC III merely in moving the key-note to the right or left (as the case may be) and playing the music accordingly with reference to the new key-note. This will involve both reading and play- ing the music with reference to the syllable or numeral name of each note regardless of its degree on the staff. A study of Chart II. (at back of book), by moving the key pattern to the different keys and remembering that the pattern repre- sents the music to be transposed, will make the principle clear. 9. Each position of the pattern shows the white and black keys used in playing in that key. In playing in the key of C, only white keys are used, and the only difference in play- ing in other keys is that certain black keys are substituted for the white keys to the right or left (according as the music is in flats or sharps), and therefore requires a mental note of what keys are thus substituted. Also, E and F, and B and C are, in certain keys (with six or seven sharps or flats in sig- nature), substituted for each other, as explained on p. 34: 21. 10. We may also observe that each interval includes a cer- tain number of finger-bars (taking the white and black suc- cessively), and that the same interval includes the same num- ber of finger-bars at any place on the keyboard, and there- fore the same hand-span spans the same interval at any place on the keyboard. This involves the development of what is called the "sense of location," by which one may uncon- sciously measure the intervals on the keyboard by hand-spans, just as in singing he unconsciously measures the intervals by their key tonality or mental effect. 11. Transposition to the tonic minor involves merely the flatting of the 3d and 6th of the key, but transposition to the relative minor involves lowering the entire music a minor 3d and sharping the /th of the key. 112 MUSICULUGY COUNTERPOINT 1. Counterpoint (meaning the point opposite or counter j refers to writing notes (formerly called points) opposite or counter to others, or the art of writing music in parts. 2. Harmony is the accompaniment of melody or tune with chords, while counterpoint is a combination of melodies, or an accompaniment of melody with melody, the chief aim in counterpoint being to give a melodic flow to each part. 3. Counterpoint is much older than harmony, and there- fore precedes harmony from the view of priority ; however, the same general principles are involved in both, which are most naturally treated under harmony, for which reason har- mony is usually treated first. 4. The earliest part-music consisted of melody accompa- nied in the octave, 5th, or 4th. In time, the singers began to vary the monotony of these intervals by adding (extempo- raneously) what are now called passing notes. It was also discovered, about this time, that several independent melo- dies could be sung together with good effect by adapting them to each other in rhythm and pitch and making slight changes where the dissonances were too apparent. The pop- ularity of these new methods naturally led musical composers to combine the principles involved by adding melodic parts to their principal melodies (also to existing melodies), hence the origin of counterpoint. 5. At first the whole interest was centered in the melodic movement of the parts, the harmony being merely the inci- dent result of avoiding dissonances. On tiie other hand, in modern harmony the whole interest (aside from the single melody) is centered in the harmonic combination of tones. 6. Naturally the highest development of music is in the combination of these two elements of interest, or the har- monic combination of melodies. Counterpoint must there- fore conform both to the requirements of melody and to the requirements of harmony. STRUCTURE OF MUSIC 113 7. Counterpoint may be outlined as follows: 3 as 6 > ■ Analysis I - Species Subject (or Canius Firmus). Added part or parts (the counterpoint proper). 1st species (one note of the counterpoint to each note of the C. F. — note against note). 2d species (two notes of the counterpoint to each note of the C. F — two against one). 3d species (four notes of the counterpoint to each note of the C. F. — four against one). Syncopated (each note of the counterpoiiU syncopated with each note of the C. F.). Florid (a mixture or combination of the foregoing species). (Inversion means the upper part placed below or the lower part above.) [Double (two parts which also make correct music when inverted). Triple (three parts which also make correct music in any order). Quadruple (four parts which also make correct music in any order). Manifold (any greater number of invertible parts). Inversion in the octave. Inversion in the gth (octave + 2d). Inversion in the loth (octave + 3d). Inversion in the nth (octave + 4th). Inversion in the 12th (octave + 5th). Inversion in the 13th (octave + 6th). Inversion in the 14th (octave + 7th). Inversion in the 15th (double octave). Classes Interval of . Inversion 8. In Simple Counterpoint a melody — called the subject ox cant us fir urns — is taken, to which one or more melodic or flowing parts are added, the part or parts thus added being called the counterpoint . 9. The cantus firuius maybe in any part, and therefore the counterpoint may be written above the cantus firuius, or be- low, or both. 10. Simple counterpoint includes five species. In the 114 MUSIC()LO(]Y first species one note is written in the counterpoint to each note of ihcitnitus Ji}-iiins; thus, j, called note against note. In the second species, two notes are written in the r I I --■ counterpoint to each note of the r^/////^-_/?>w//^, thus, P — ^ — ^^ \j ^-~~ — zy ; called two against one. (In triple time three notes are writ- ten against one — the notes of the r^r ///■// .vyfrw/zjr being dotted.) In the third species, four notes are written in the counter- point to each note of the caiitj/s Jiniius, thus, f J J # ■ J \ called four against one. (In triple time six notes are written against one — the notes of the cantns firinus being dotted.) In syncopated counterpoint two notes are written to each note of the cantiis firmus, but syncopated, or tied over, thus, ■i — ^/H d — H In Jiorid counterpomt ^ any or all of the foregoing species are combined. When more than one part is added, they ma\- be of different species. 11, In counterpoint, as in harmony, each combination of tones (except passing tones) represents some definite chord (complete or incomplete). All intervals are classed as con- cords or discords, the concords being the octave and per- fect 5th (called perfect concords), and major and minor 3ds and 6ths (called imperfect concords). All other intervals are treated as discords, including the perfect 4th when between the lowest and one of the upper parts (see foot-note, p. 83). 12. Contrapuntal Rules. Contrai)untal music was origi- nally for vocal performance; and the rules of counterpoint were for the purpose of making the i)arts easih' singable, as well as to avoid disagreeable effects. Progressions that were found to be awkward or difificult for singers to take in con- nection with other voices, or combinations that produced dis- STRUCTURE OF MUSIC II5 agreeable effects, were naturally forbidden — hence the origin of the rules. 13. In modern counterpoint the rules are not so rigidly observed, as the ears of modern singers are more accustomed to dissonant chords ; while in instrumental counterpoint the players are not dependent on the guidance of the ear. How- ever, the influence of the rules is always beneficial. 14. (i) In strict counterpoint of the first species {note against note) otily consonant combinations should be used, as consonant combinations are easier to sing than dissonant ones, as ivell as more agreeable. When two or more notes are written against one, a consonant combination may change conjunctly (with- out skip) into a dissonance, as a dissonance thus taken is not difficult to intone. 15. (2) Accented pulses should, as a rule, be consonant and so far as possible complete chords, while unaccoitcd pulses may be either consonant or disso)iant. In syncopated counterpoint the first note of each syncopation must be consonant, the second note being eitJier consonant or dissonant. The rules for the resolution of dissonances (pp. 83, 84) are applied. 16. (3) Skips. Consonant intervals are free to move by skips. Dissonant intervals should, as a rule, be approached and quit- ted ivithout skips. A skip from a discord on an unaccented pulse to a concord — called a changing tone — is sometimes used. Concordant skips (3ds, 6ths, perfect 4ths, perfect 5ths, and octaves) are freely used. Discordant skips (/ths, aug- mented and diminished intervals) should be avoided. A succession of wide skips (or more than two skips — even of a 3d) in the same direction should be avoided. After using three or four notes alphabetically a skip, even of a 3d, in the same direction should be avoided ; but the skip may be made at the beginning of the passage. 17. (4) The Tritone (aug. 4th = 3 steps). This interval occurs naturally between the 4th and 7th of major and minor keys (see major and minor patterns). It is not only to be avoid- Il6 MUSICOLOGY ed as a skip, but also hcti'.'coi parts of successive chords, thus, |?y— #^ -^^ ; or hetu'ccn parts of chords on successive accents; or bettveoi the first and last notes of an ascending or descend- ing passage, thus. ^^ 1 8. (5) Consecutives. Similar perfect concords {octaves and perfect jths) should not occur successively nor on successive accents, especially betzveen outside parts. The same applies, though less rigidly, to hidden consecutives (p. 8 i : 24). Tivo suc- cessive Jiiajor jds {unless taken by a half -step, as on the jth and 6th of minor keys— see mitior pattern) should be avoided. Successive jds and 6ths should, so far as possible, be alternate- ly major and minor. 19. (6) False Relations. False relations should be avoided (p. 81 : 26, 27). 20. (7) Doubling. Double major jds should be avoided, es- pecially on the accents. The same applies less rigidly to the 6ths and minor jds. Doubled leading notes should especially be avoided. 21. (8) Unison. The u)iison may occur in the first and last chords and occasionally on unaccented pulses, but should be used sparingly. 22. (9) Crossing. 'The parts may occasionally cross for the sake of a more melodious flow, but should not eross on the accent. 23. (10) Pedal. J\dal passages (p. 88) are occasionally used. 24. (11) Sequence. Sequences (p. go) are always effective, and may be freely used. 25. (12) Modulation. Modulation to closely related keys is not restricted. 26. (13) Motion. Oblique and contrary motion (^p. 81 : 28) should be used as much as possible, thus giving more individu- STRUCTURE OF MUSIC II/ ality to the parts and lessening the liability to faults. All the parts slumld rarely move in the same direction at the same time. 27. F"aults are less observed between inside parts or be- tween inside and outside parts, also on unaccented pulses, and also as the parts increase in number. On the other hand, increased number of parts necessitates increased free- dom of movement to secure the necessary flow to all the parts. Therefore, rules are less rigidly applied as the parts increase in number, or between inside parts, or between in- side and outside parts, or on unaccented pulses. 28. Double Counterpoint consists of two invertible and equally important parts. 29. The following table shows the intervals of inversion, and the intervals within each and their inversions. TABLE OF INVERSIONS Inversion in octave j i 2 3 4 5 6 7 3— intervals ( S 7 6 5 4 3 2 I — inversions Inversion in the Qth | 1 " £ 4 5 ^ 7 ^ 9-intervals (98765432 I — inversions Inversion in the loth I ' 1 ^ 4 5 6^8 9 lo-intervals ( 10 9 S 765432 I — inversions Inversion in the nth .^_[ 2 3^ 4 ^ 6 T S^ 9 i^ n-intervals (11 10 9S7 6 5432 I — inversions , . ., ,\l234s67Snioii 12 — intervals Inversion in the I2th ■ _ " _ __ (12111098765432 I — inversions , . . , ,(12345678 9 10 II 12 13 — intervals Inversion in the 13th - __ _ ( 13 12 II 10 9 S 7 6 5 4 3 2 I — inversions 3 T . . , , \ I 2 3 4 5 6 7 8 9 10 II 12 13 14 — interval Inversion in the 14th -' „__ _ / 14 13 12 II 10 9 8 7 6 5 4 3 2 I — invers. nversi.jn inthe 15th i i 2 3 4 5 6 7 S 9 10 11 12 13 14 15 —int. ( 15 14 13 12 II 10 9 8 7 6 5 4 3 2 I — inv. Il8 MUSICOLOGY 30. Observe that any interval added to its inversion is one greater (the central note on which the inversion turns being counted twice) than the interval in which the inversion is to be made. This suggests how the inversion of any interval may readily be found. 31. In the above table the dissonant intervals are marked with a stroke over, thus, — . Observe tliat in the octave all con- sonant intervals invert into consonant intervals and dissonant intervals into dissonant intervals, except 4 and 5 ; the same in the lOtJi without any exceptions ; the same in the 12th, except 6 and 7 ; and that the /J//^ (double octave) corre- sponds to the octave (iiths and I2ths being compound 4ths and 5ths). These are the intervals of inversion most used. It will be seen that the others invert so contrary as to give less satisfactory results. 32. Two parts intended for inversion should not at any point exceed the compass of the interval of inversion ; other- wise, the parts will cross when inverted (^the portion in ex- cess remaining on the same side as before inversion, and there- fore not inverted but merely contracted). Likewise, the two parts intended for inversion should not cross at any point, as the reverse of the above will happen when inverted (the por- tion crossed expanding instead of inverting). Therefore, the greater the interval of inversion, the greater the range allowed to the parts. It is evident that when parts are inverted as a whole, each interval between corresponding notes is inverted according to the interval of inversion. 33. Special attention must be given to those intervals that invert contrary (as regards consonance), the other intervals practically taking care of themselves (if forming correct pro- gressions, the inversions naturally forming correct progres- sions also). Consonant intervals that invert into dissonant intervals must therefore be treated as dissonant as regards preparation anil resolution, the treatment, however, being in the lower part, which becomes the dissonant part when in- STRUCTURE OF MUSIC 119 verted. Thus, in the octave the perfect 5th, though conso- nant, inverts into the perfect 4th, which is dissonant, and requires resolution and usually preparation, especially on the accent; the perfect 5th must therefore conform to the same requirements (in the lower part) in order to form correct pro- gression when inverted. The same applies to inversion in the 1 2th as regards the 6th and 7th ; however, Dominant /ths and Diminished /ths do not usually require preparation (see p. 82:30). 34. It should be observed in the lotJi that consecutive 3ds and 6ths must be avoided, as they become consecutive octaves and 5ths in the inversion. Also, if these intervals are ap- proached by similar motion, objectionable hidden octaves and 5ths are apt to occur in the inversion. It is apparent, there- fore, that oblique and contrary motion must generally be employed. 35. In the i2tJi, 3ds invert into loths and vice versa; these intervals, therefore, may freely be used even in similar motion ; but for the other intervalsoblique and contrary motion will gen- erally be found necessary. 36. In the other intervals of inversion (9th, iith, 13th, and 14th, which are but little used) it is evident that conjunct move- ment (without skip) will generally be necessary so that every dis- cord may be approached and quitted conjunctly. 37. To more closely analyze the intervals and their inver- sions, see Fig. 25. Each dia- gram represents a section of the major key pattern, in which the major intervals are measured from each end (the natural intervals upward from i of major pat- (12 til) rl3 ^1-| (lOti^) 11^2 10 — 1- 10 — - 3 -9—4 (8^e) -9—2 r8 1-1 -8—3 -8 — 5- f1 7 — 4 7 - 2 - - 6- G -6—5 6 3 - 7- -5—4- -5— G -5 — 8- "4—5" -4 — ;- -3 — 8 -4 — 9 -3 3 - 6 - - 10- 2 -2—9 -2—11- ■I 8- -l-^« -1—12- Fig. 25. I20 MUSICOLOCV tern arc taken as the standard and called major; see p. 56:2). 38. Observe that in the octave perfect intervals invert in- to perfect intervals, major into minor, and vice versa, and diminished into augmented, and I'ice versa. 39. Observe that in the lotJi, major intervals invert into major intervals (including the perfect intervals, which are sometimes called major), minor intervals (including the diminished perfects, which are sometimes called minor) invert into augmented intervals, and vice versa. The other diminished intervals, being equivalent by enharmonic change to major or perfect intervals, invert accordingly ; thus, the diminished 7th, being equivalent to a major 6th, inverts into a perfect 5th (otherwise a double-augmented 4th, which is not a classified interval). 40. Observe that in the J2tJi, 3ds, 6ths, /ths, and lOths invert as in the octave (major into minor, and vice versa, aug- mented into diminished, and vice versa), while the other in- tervals invert as in the lotli. 41. Inversion in the i^th (double octave) is practically the same as in the octave. 42. The other intervals of inversion may be analyzed in a similar manner. 43. It is evident that inversion may sometimes involve augmented and diminished intervals, which are usually chro- matic. The available augmented and diminished intervals (p. 57 : II) are the aug. 2d, aug. 4th, aug. 5th, and aug. 6th, and the dim. 5th and dim. 7th. The aug. 6th cannot be used in the octave, as it inverts into the dim. 3d. Of course these intervals are dissonant, and usually require treatment, and also involve accidentals which tend to induce modulation. 44. Modulation by Inversion. If in\ersion in the octave takes place by the upper part being placed an octave lower, or the lower part an octave higher, or both parts an octave in opposite directions, no modulation occurs, as the scale is STRUCTURE OF MUSIC 12 1 not changed ; but if both parts arc moved in opposite direc- tions by steps that involve a new scale, modulation will be effected to the key corresponding to the new scale. Thus, if the lower part is placed a perfect 5th higher and the upper part a perfect 4th lower, they will evidently both be in the scale, or key, of the dominant; or if the lower part is placed a perfect 4th higher and the upper part a perfect 5th lower, they will be in the scale, or key, of the sub-dominant. Modula- tion may thus be effected to any key by simply moving each part bodily to the scale of that key (moving the parts in such direction as will produce inversion). 45. Inversion in the iot]i can only be effected in four ways: first, the upper part may be placed a lOth lower; sec- oud, the lower part may be placed a loth higher; third, the upper part may be placed an octave lower, and the lower part a 3d higher; fourth, the lower part may be placed an octave higher, and the upper part a 3d lower. Of course the mod- ulation will not be through the part moving an octave ; but the part moving a 3d or loth (octave + 3d) will evidently be in a new scale a major 3d above or below the old key, accord- ing as the modulating part is placed above or below. If the old key is C, the new key will be E or A 7 (as the case may be). 46. If the inversion were by any other than the four ways mentioned, both parts would modulate, but to different keys. Thus, if the lower part is placed a perfect 5th higher and the upper part a major 6th lower, then one part would be in the key a perfect 5th above, and the other in the key a major 6th below, the old key. If the old key is C, then one part would be in the key of G and the other in the key of E k; but the music cannot be in two keys at the same time. 47. Modulation by inversion in the 12th corresponds in every point to that in the loth by simply substituting the per- fect 5th for the major 3d. The modulation will be to the key of the dominant (perfect 5th above) or sub-dominant 122 MUSICOLOGY (perfect 5tli below), according as the modulating part is placed above or below. 48. Modulation by inversion in the isth corresponds to that in the octave. 49. Modulation by inversion in the ^///, /////, ijtJi, and i^th is similar to that in the loth and I2t]i (differing only in the modulating step). 50. In any case, the modulation is caused by one or both parts moving by some other interval than an octave. Though the modulation may be only in one part so far as the acci- dentals involved are concerned, yet, of course, both or all the parts are included in the modulation, since the music cannot be in different keys at the same time. The modulation will naturally begin with and continue through the entire section of the music thus inverted, and may therefore begin some distance in advance of the accidental, or sign of modula- tion. 51. If modulation occurs in the original parts (before in- version), then, of course, the modulation by inversion will be reckoned (during the continuance of that modulation only) from that key instead of from the original or signature key. 52. It sometimes occurs (where the keys are closely related) that the tone in which the keys differ (and therefore the acci- dental) does not happen to be used, yet still modulation may result through the tendency of the part inverted to retain and carry with it its key individuality (if strong), since the inter- relationship of its tones remains intact. Thus modulation sometimes occurs without accidentals. Modulation is effected whenever the feeling of modulation is produced. 53. Counterpoint Invertible in Various Intervals. Coun- terpoint in the octave may also invert in the lotJi, 12th, or 75///, since its compass is within the others. For the same reason, counterpoint in the loth may also invert in the 12th or iStJt ; and counterpoint in the i2tJi may also invert in the STRUCTURE OF MUSIC 1 25 i^th. In each case, however, the parts should be constructed with reference to each inversion. 54. Counterpoint may also be made invertible in different intervals by adding 3ds. This consists in duplicating one or both parts in the 3d above or below, thus giving three or four parts. If in counterpoint in the octave 3ds be added above the upper part or below the lower part, the new part thus added will be in the loth with the other outside part, and the counterpoint is thus invertible in both the octave and lotJi. If 3ds are added both above the upper part and below the lower part, then the inner parts will be in the octave, the inner and outer parts in the lotJi, and the outer parts in the i2tJL. If the original parts invert in the I2t]i and 3ds added both below the upper and above the lower part, the result will be the same. 55. In applying this method of added 3ds, the following additional rules will be necessary : (i) Use dissonances only as passing notes. (2) Use consonant intervals alternately as much as possible. (3) Use only oblique or contrary motion. 56. Triple and Quadruple Counterpoint. These consist of three or four distinct parts, each standing to each in the rela- tion of a double counterpoint in the octave. 57. Triple counterpoint will give six different combinations, and quadruple counterpoint will give twenty-four different combinations, according to the principle of Permutation. IMITATION 1. Counterpoint is made up largely of Imitation. 2. Partial hnitation is where a part of a preceding melody is imitated. Canonical Imitation (see Canon, p. 128) is where the preceding melody is imitated throughout. 3. Imitation may take place in the unison or at any other 124 MUSTCOLor.Y interval, in any part or number of parts, aiul at any place. The part imitated may be called the model. 4, Imitation may be outlined thus: f Strict. Free. Retrograde. Imitation. \ By Contrary Motion. By Augmentation. By Diminution. By Reversed Accents. 5. In Strict Imitation the intervals of the model are not changed — half-step answering to half-step, whole step to whole step, and therefore major to major, and minor to minor. This is the usual character of imitations in the unison or octave, since the scale is not changed. 6, In Free Imitation some of the intervals of the model are changed — the imitation being by similar degrees of the staff regardless of the half-steps, and therefore major often answering to minor, and vice versa. This is the usual char- acter of imitations in any other interval than the unison or octave, since the scale is changed. 7. In Retrograde Imitation the model is imitated backward, _j from end to beginning (strict or free). 8. In Imitation by Contrary Motion the model is imi- tated upside down (strict or free). If strict, the imita- tion must be in accordance with the pattern shown in Fig. 26, which represents a section of the major key pattern (and may be extended into the full pattern). 9. Observe that the reverse scale begins a 3d above the key-note (i) of the direct scale, and also that the intervals of both direct and reverse scales are the same. Therefore, if the imitation corresponds in the reverse scale to the model in the direct scale, it will be strict (half-step answering to half-step, and whole step to whole step). Of course, this pattern, like the key pattern, may be set to any key. 3- — I- •2- -3- -1- -3 •7- -4 6- -5 5- -6 ■4- -3- — 1- 2- -2 1— -* Fig. 2G. STRUCTURE OF MUSIC 125 f« — h 4 — 2^ 3 -3 — 4 -1—5^ -2 — 4 — 5-" Fig. 27. 10. Imitation by contrary motion in minor keys is best made by the pattern shown in Fig. 27, which represents a section of the minor key pattern (and may be ex- tended into the full pattern). It will be seen that when the 3d of either scale is used, the imitation ceases to be strict, or else involves a chromatic tone. 11. In Iinitatioii by Augmentation the time value of each note of the model is increased or augmented (usually doubled). 12. In Iinitatioji by Diminntio)i the time value of each note of the model is diminished (usually one- half). 13. In Imitation by Reversed Accents the accents are reversed (unaccented for accented, and vice versa) by the imitation entering on an opposite phase of the measure from that on which the model entered. 14. Modulation by Imitation. If the imitation takes place in the unison or octave it is evident that no modulation will be involved, since the scale is not changed ; or if the imita- tion be free, it may take place at any other interval without involving modulation (the imitation being by similar degrees of the staff regardless of the half-steps) ; but if the imitation be strict and at any interval involving a new scale, modula- tion will naturally result. The interval between the old and new key letters will correspond to the interval at which the imitation takes place ; thus, if the imitation takes place at the interval of a perfect 5th above or a perfect 4th below, the modulation will be to the key of the dominant, etc. 15. If the imitation be by contrary motion in accordance with the pattern shown in Fig. 26, no modulation will be in- volved, since the intervals of the reverse scale correspond to the intervals of the diatonic scale and therefore in the same key. Otherwise modulation may occur, since any other re- verse scale would be in some other key. In any case, 3 of reverse scale will be on the key-note as in pattern. 126 MUSICOLOGY 16. In minor keys if the imitation by contrary motion be in accordance with pattern shown in Fig. 27, no modulation will be involved, for same reason as given above (3 being free or involving a chromatic tone). Otherwise modulation may occur. In any case, 5 of reverse scale will be on key-note as in pattern. 17. Of course, in any case, the modulation is usually indi- cated by the accidental following. The modulation naturally begins, however, with the imitation and therefore frequently some distance in advance of the accidental. It sometimes happens (when the keys are closely related) that the acciden- tal involved in the new key is not required, yet modulation may result through the tendency of the model to retain and carry with it its key individuality (if strong and the imitation strict). CONTRAPUNTAL MUSIC 1. Any combination of melodies may be regarded as con- trapuntal music ; but the most important contrapuntal forms of musical composition are the Fugue SiWd the Ca?ion. Points of contrapuntal imitation are frequently found in other music, especially in choruses. 2. The Fugue is a composition in which a musical phrase called the Subject is given out by one part and imitated by the other parts in succession, thus: ( Soprano Answer \ } Alto Subject Counter-subject f ") Tenor Answer Counter-subject Free Counterpoint f ( Bass Subject Counter-subject Free Counterpoint Free Counterpoint ) (Any part may begin the Fugue.) 3. The A?is2c>er is the subject imitated in the key of the dominant, either above or below. The Counter-subject is the continuation of the subject or answer during the imitation in the next part. 4. The counter-subject need not appear in the voice which enters last. After the counter-subject each part is continued by adding free counterpoint. The subject and counter-sub- ject should be in double counterpoint so that they may be STRUCTURE OF MUSIC 127 inverted, since they are to be interwoven in all possible com- binations in the subsequent treatment. 5. The subject, answer, and counter-subject are called the Exposition, as they form the subject-matter from which the fugue is to be developed. The structural form of the fugue may be outlined thus: I, The Exposition f Subject. - Answer. [ Counter-subject. Fugue Form -j Episode. 2. The Contrapuntal r> • j • . I '1 Repercussion and episode. I Development "1 c. .. '- ^ I Stretto. [ Final episode. 6. The Episodes are the free modulations which connect the parts and lead from one key to the next. 7. The Repercussion consists of new combinations of the sub- ject and answer in different keys ; each combination being followed by an episode leading into another key. 8. The Stretto consists of combinations in which the sub- ject and answer overlap. Where there are several strettos, they should come in the order of their closeness, the closest coming last. Sometimes the repercussion is made up of strettos. 9. The principal kinds of fugues may be classified thus: i Two-voiced fugues. As to number of voices < Three-voiced fugues. ( Etc. As to number of subjects As to kind of imitation As to character of subject Single fugues. Double fugues. Etc. Augmented fugues. Diminished fugues. Inverted fugues. Diatonic fugues. Chromatic fugues. A . 1 . . . \ Strict fugues. As to general treatment \ ^ ( Free fugues. 128 MUSICULOGY 10. The Canon is a composition in which the parts enter one after the other, each imitating the melody of the first part (called the subject) throughout. 1 1. An imitating part may enter at any point, and at any interval, up or down. The difTerent kinds of imitation may be employed and the different parts may employ different kinds of imitation ; or some of the parts may be in Canon, and the other parts free. There are also canons with more than one subject. . 12. Canons are usually described by figures showing the number of parts and subjects, the first figure showing the number of parts, or voices, and the second figure showing the number of subjects, thus: Canon 2 in 1,3 in i, 4 in 2, etc. 13. K Finite Canon is one in which each part is silent after completing the melody, or subject; or one which ends with a regular close like other compositions. 14. An Infinite Canon is one in which each part begins again after completing the subject, the canon thus being with- out end. 15. A Circular Canon is one which modulates from one key to the next around the key circle, the subject recom- mencing each time in the key a 4th or 5th higher or lower; or the subject may recommence a tone higlicr or lower, thus modulating by alternate kc)'s. The conclusion of the sub- ject should lead naturally each time into the new key. The circular canon is also endless. 16. An Open Canon is one in which the different parts are written out in full. 17. A Close Canon is one in which the principal part only is written out, and the number of parts and their places of entrance are indicated by the sign Jj. 18. The Enigma (or riddle) Canon is a kind of musical problem in which the places of entrance of the succeeding parts are not indicated, but left to be solved. STRUCTURE OF MUSIC 129 19. The canon may be classified as follows: As to interval of imitation Canons in unison. Canons in 2d. Canons in 3d. Canons in 4th. Etc. Mixed Canons I f Canons 2 in i. As to number of | Canons 3 in i. parts and subjects j Canons 4 in 2. I Etc. \ the parts imitating at ( different intervals. As to strictness of imitation As to style of imitation As to limitations As to score I Strict canons (see Strict Imitation). 1 Free canons (see Free Imitation). I Canons in similar motion. ! Canons in contrary motion (see pp. 124:0 ; 125:10). Canons in augmentation (see p. 125:11). j Canons in diminution (see p. 12512). [Retrograde canons (see Retrograde Imitation). i Finite canons (limited). -I Infinite canons (endless). ( Circular canons (endless). ( Open canons. I Close canons. Enigma canons. 20. In the fugue the subject is sometimes only a short musical phrase. In the canon the subject is an entire melody. 130 MUSICOLOGY MELODY 1. Music is divided into three general classes: MonopJionic (single sounding), one-part music (melody) ; Polyphonic {vad.v\y sounding), combined melodies (counterpoint) ; Harvionic (united sounding), accompanied melody (harmony). 2. AhmopJionic music (melody) is the original form of all music. The ancients, though sometimes having choirs of many voices, sang only in unison (or naturally in octaves, when men and women sang together). Melody is still the prevailing music of the Turks, Greeks, and most oriental nations, who seem to have a distaste iox part music. 3. Polyphonic music (counterpoint) dates back about seven hundred years, when the early attempts at part music began to assume shape. 4. Harmonic music (harmony) is based on principles of acoustics. It is also, in a sense, a gradual outgrowth of counterpoint, and dates back only about two hundred years as a recognized distinct system. 5. Melody is a succession of tones, while harmony is a 1 9 concord of tones. V^'^^W-^ m -|~T~a--] At i we have a concord of tones, or tones heard simultaneously in harmony ; while at 2 we have the same tones, but heard successively in melody. The first is called a chord; the second, a melodic figure; and just as harmony is made up of chords, so melody is made up of figures. 6. A chord and a figure composed of the same notes con- tain, in a sense, the same musical idea; but the figure con- veys the additional idea of motion, while the chord conveys the idea of rest, or completeness in itself. 7. The i)rinciples of harmony as applied to chords, natu- rally, in a general sense, apply also to figures ; for harmonic I STRUCTURE OF MUSIC I3I combinations naturally make melodic successions. In this sense a figure is a resolved chord and melody is resolved har- mony. But, as already observed, the figure conveys the ad- ditional idea of motion, so that it has a melodic element com- bined with its harmonic clement ; the harmonic connection be- tween the successive notes is in proportion to the consonance of the interval between, while the melodic connection is in proportion to the closeness of the interval between. 8. The melodic importance of the 2d is due to its being the shortest melodic step. It is freely used in melody, since the notes forming it do not sound simultaneously. It is thus often used as a passing note between the harmonically con- nected tones to increase the smoothness of the melodic flow. It is the only diatonic step with a purely melodic value (be- ing harmonically dissonant) ; the other diatonic steps used have both a harmonic and melodic value. 9. Melodic Progression involves relationship between the consecutive notes (grouped into figures), just as harmonic pro- gression involves relationship between the consecutive chords, and just as speech involves grammatical relationship between the different words. 10. Melody is more than a mere succession of related tones. For a succession of tones to be melody it must have melodic flow and rhythmir movement. Melody, therefore, is a melo- dious, rhythmical succession of tones. 1 1. l.^h.Q figJire is the smallest complete rhythmical division of a melody. The figure, therefore, has a rhythmic element in combination with its harmonic and melodic elements. The figure usually covers one or two measures, called, respectively, simple and compound figures. The figure may begin in one measure and end in another, in which case the melody begins with an incomplete measure. 12. Development. A figure may also be defined as a series of notes containing a musical idea, and which may be taken as a subject, or theme, for development. When thus used 132 MUSICOLOGV the figure is called a theme, and the treatment involved in development is called tJuniatic trcatmoit. 13. The principal methods of thematic treatment may be outlined as follows : Transposition (placing the figure at a higher or lower pitch). Expansion (expanding one or more intervals of the figure). Contraction (contracting one or more intervals).. Augmentation (increasing the time value of each note). Diminution (diminishing the time value of each note). Repetition (repeating fragments or certain notes of the figure). Omission (omitting fragments or notes). ' Reversion (the figure written backward from end to beginning). Contrary Motion (the figure written upside down). Change of Order (irregular change of the order of the notes of the figure). Rhythmic Change (change of the rhythm of the figure). Ornamentation (with various passing notes, turns, trills, etc.). Simplification (reverse of ornamentation). Combination (any two or more of the above combined). Some of these methods are much used, and others but little used. 14. Thematic treatment modifies the character of a figure, or theme, without entirely destroying its individuality. Of course, if a figure is so changed as to lose its individuality it becomes a new figure. The figure may be recognized through any one or more of its three elements (harmonic, melodic, and rhythmic) ; but if all three are changed there is no means of recognition, and the figure loses its individuality. 15. By means of thematic treatment an entire melody is often developed from a single figure, but more usually a number of figures, or themes, are used: the first, being most prominent, is called the principal theme ; and the others, secondary themes. The greater the number of themes, the greater the variety and contrast ; but too many themes tend to destroy the unity of the melody. A principal idea must pervade the entire melody ; otherwise, it will have no definite meaning; just as a literary composition without a principal subject, or theme, would ha\c no tlefiiiite meaning, and STRUCTURE OF MUSIC 1 33 therefore lack interest by being difficult to understand. In either case, the unity or oneness of the composition is through a principal idea pervading the whole. On the other hand, without variety and contrast the composition would lack interest by being monotonous. 16. In melody variety is largely attained by thematic treatment (development) of the same theme, just as in literary compositions variety is largely attained by presenting the same theme in various lights (also thematic treatment) ; but in either case, the sense of the principal theme is often most clearly brought out by using contrasting themes. LINGUISTIC CHARACTER OF MUSIC 1. Music should not be looked upon merely as a pleasing combination of tones. Properly understood, music is a lan- guage through which ideas and thoughts, sentiment and emo- tion, are expressed. It differs from spoken language in that it is natural and universal (in the form of melody), while spoken language is artificial and local. 2. Musical perception is to some extent a natural gift pos- sessed by all, though in different degrees, and which is highly susceptible of cultivation. 3. The basis of spoken language is the alphabet ; the basis of musical language is the music scale. 4. Just as spoken language is made up of letters combined into words, words into phrases, phrases into clauses, clauses into sentences, sentences into paragraphs, and paragraphs into compositions, so musical language is made up of tones combined into figures (or chords in harmony), figures into phrases, phrases into sections, sections into periods, periods into movements, and movements into the higher forms of musical composition. 5. Melody (being regulated by rhythm) may be called the poetry of music, and the recitative (musical declamation), its prose. 134 MUSICOLOGY 6. Melodic motion imitates mental motion. Thoughts are mental motions. Our thoughts may move slow or fast ; they may revel in pleasant fancies or wander through sad memo- ries ; they may be aimless or energetic, restless or deter- mined, calm or excited. Melodic motion is capable of ex- pressing by imitation every species of mental motion in the most delicate and exact manner. 7. Melody, therefore, is more than merely a melodious, rhythmical succession of tones ; it is sentiment and emotion expressed in tones instead of words. THE PERIOD 1. The Period is a musical sentence. It consists of several phrases so related as to produce the sense of completeness. 2. The Phrase is a musical expression having a well marked repose. A Passage is a series of figures having no well marked repose. 3. Rhythmic Structure. The /rr/(3 — STRUCTURE OF MUSIC 141 25. The Sonata Form. This is the climax of musical form. A sonata is an instrumental composition consisting usually of three or four distinct movements (differing in tempo or rhythm). The first movement determines the character of the whole, and is therefore called the Sonata niovonoit. 26. The sonata form in four movements may be outlined thus : r I. Theme. . , . 1 I n l^^'' '■^^- niajor, 2. hpisode, usual! v ; .^ ., . . I . ^ •' I II theme is in a ] in Doni. key ' 1st part 1st, or sonata movement {allegro) 3. Final group Middle part Repetition 2d, or slow movement (usually, grave, lai-go, ada- gio, or andante) Generally in a related key to that of ist movement [ minor key. consisting of one or more closing epi- sodes, alsoin Dom. key. 1 1, 2, and 3 are now repeated. f Development (usually contrapuntal) of the theme and episode of ist part I (called the Free Fantasia, because not I confined to any form), modulating through several keys and returning to [Tonic key. Theme. Episode — now in Tonic key. Final group — now in Tonic key. (Sometimes an appendix, or finale, is \ added.) fThis movement has no fixed form but may be in any of the following forms: 1st — the sonata movement form (as above). ( A series of pieces the variation j in which the theme 2d- form j is changed [ varied. 3d — the second rondo form. and 3d movement — a scherzo or a minuet. 4th movement ("Usually in the rondo form; but some- I times in the sonata movement form I or the variation form. IJ P pT y Hi •Jl n' r/3 3 P < 5' n p H :■" in (fl ^< ' 3 "H- \^ 3 cr Ul n ^ n 3 3 c n i/i n n' ^ 142 MUSICOLOGY 27. If an entire sonata were in one movement it would be- come very wearisome, but the different movements are so contrasted that each comes in as a relief to another; thus the slow movement comes in like a lull in a storm, and the 3d movement is preparatory to the final climax in the last move- ment. Sometimes the scherzo or minuet is placed before the slow movement. In three-movement sonatas the scherzo or minuet is usually omitted. 28. Suite Form. The old suite is a succession of old dance movements arranged with reference to contrast, and which gradually assumed the following order: (Old Dances.) c fi. Allc7naude — a cheerful movement in even rh\'thm. o I 2. Courantc — a rapid, running movement in triple rhj'thm. ^ • 3. Sarabande — a dignified, stately movement in triple rhythm. .ti 4. Bott7ree — a bright, quick, hearty movement in even rhythm. CO [ 5. Gigue — a very rapid movement, possessing a rough heartiness. 29. Other old dance movements sometimes used are: chaconne, a slow movement, generally major, and usually in triple rhythm ; passacaglia, a rather bombastic movement in triple rhythm, closely resembling the chaconne, but usually minor; viiniict, a slow movement in triple rhythm, danced with mincing, dainty steps, hence the name ; gavotte, a genial skipping movement in even rhythm, usually beginning on the third beat, thus producing a mild syncopation; pavane, a slow, stately movement, similar to the sarabande, but in even rhythm ; rigandon, a lively movement in even rhythm, beginning on the third or fourth beat, and is sometimes sung as well as danced. 30. In the modern suite, more modern movements (not always dances) are frequently substituted for the old dance movements. 31. The suit e \% 'OiXQ. oldest form in which different move- ments are combined, and is therefore the origin of the sonata and other modern forms. STRUCTURE OF MUSIC I43 32. Contrapuntal Forms. (See pp. 126-129.) 33. The foregoing are the principal recognized forms, and with slight changes practically cover nearly all varieties of music, and are therefore sufficient as a general analysis of form. 34. The common practice of calling nearly every slight distinction or difference a form leads to a great deal of con- fusion, and any attempt at outline is vague and unsatisfac- tory. 35. There are also certain introductory, intermediate, and concluding parts or passages, which arc not generally regarded as an essential part of the form of a composition, as they are sometimes used and sometimes omitted. They may be classified thus : I Prelude 1 . • i • . j .• I A musical introduction to a more Introduction ■ . •.■ T . , . „ important composition, or a piece Introductory - Overture '- , . . used as an opening to an oratorio, opera, or concert. Intrada Voluntary, etc finterlude lA piece of music placed between Intermediate I Intermedium [more important compositions or be- [ Intermezzo J tween the acts of an opera. f Postlude ] I Conclusion I The supplementary ending of a corn- Concluding jcoda ("position. I Finale J 36. The general definitions given above sufficiently de- scribe the general character of each division without going into distinctions, as the aim here is merely outline. 37. The terms of each division are sometimes used inter- changeably. They are not, as a rule, confined to any fixed form. They should, in general, be proportioned in length to the composition to which they are attached. 144 MUSICOLOGY CLASSIFICATION OF MUSIC I. Music may be classified as follows Sacred Vocal (with or without in- strumental accom- paniment) j Hymn, Psalm, Chant, Choral, Anthem, / Antiphony, Motet, Mass, Oratorio, etc. Instrumental j Ballad, Glee, Madrigal, Opera, Opera ( Bouffe, Grand Opera, etc. Sacred or j Song, Recitative, Aria, Arietta, Chorus, Secular I Cantata, Cantatilla, etc, ( Sonata, Sonatina, Symphony, Concerto. ■\ Concertino, Suite, Overture, Nocturne, ( Fantasia, Capriccio, Scherzo, etc. Secular Classical Dances Old Dances Modern Dances As to number of voices j Duet, Trio, Quartet or instruments ( Quintet, Sextet, etc. Allemande, Courante, Sara- bande, Bourree, Gigue, Cha- conne, Passacaglia, Minuet, Gavotte, Pavane, Rigaudon, etc. r Polonaise, Mazurka, Polka, I Schottische, Waltz, March, I Quickstep, Galop, Fandango, L Reel, Quadrille, Cotillon, etc. I Called.Chamber Music I when written for ■\ stringed instruments, j or for piano and other [ instruments. 2. Sacred Vocal Music. The Hyjnn is a song of praise or thanksgiving to God. A Psa/in is a metrical translation of one of the Psalms set to music. A Chant consists of words recited to musical tones without musical measure. The Clioral is a psalm or hymn sung in unison by the congregation. An Antheni is a setting of scriptural texts to music. The Antiphony is the most ancient form of church music, and con- sists of responsive singing, one part or choir answering or re- sponding to another. The motet is also a species of hymn with a scrij)tural text or texts, designed for chorus or for choruses interspersed with solos; it is usually livelier and more brisk than other religious music and is sometimes used as STRUCTURE OF MUSIC 1 45 synonymous with anthem. The Mass is a musical composi- tion used in the CathoHc Church in celebrating mass, hence the name. The Oratorio is a sacred musical drama, and con- sists of recitatives, choruses, solos, duets, trios, etc., accom- panied by orchestra. 3. Secular Vocal Music. A Ballad is a narrative told in song. The Glee is a composition of a light character, usually sung by one voice to a part, and generally unaccompanied. The Madrigal is a composition for voices without accompani- ment, each part being supported by several voices. The Opera is a secular musical drama, and consists of an overture, arias, choruses, recitatives, duets, trios/ etc., accompanied by scenery and dramatic action. If the character is serious it is called a Grand Opera. If the character is comic it is called an Opera Douffe, or comic opera. The Operetta is a short, light opera. 4. Sacred or Secular Vocal Music. Song is a general term applicable to all vocal music, whether of human beings or of birds, but most usually suggests a simple melody re- peated to each stanza of a sacred or secular poem. The Reei- tative is a musical declamation imitating declamatory speech with the singing voice, usually with instrumental accompani- ment. The Aria, in its general sense, is an air, or melody ; more strictly, it is an accompanied solo, usually similar in form to a minuet. The Arietta is a small aria. The Chorjis is a composition for numerous voices. The Cantata is a composition mixed with recitatives, arias, and choruses, with instrumental accompaniment. The Cantatilla is a small can- tata. 5. Classical Instrumental Music. The Sonata (see p. 141 : 2$). The Sonatina is a small sonata. The Symphony is a sonata for full orchestral accompaniment. The Concerto is usually a sonata in three parts, in which one or several concerting instruments play the principal parts accompanied by an orchestra. The Coneertino is a small concerto. The 146 MUSICOLOGY Suite (see p. 142 : 28). The Overture is a composition used as an opening to an oratorio, opera, concert, or drama, and is usually in the form of the Sonata Movement. The Concert Overture is an independent composition for concert perform- ance. The Nocturne is a composition of the character of a calm night, hence its name. The Fantasia is a composition for solo instrument written without regard to the restrictions of form. The Capriccio is a composition written in a capri- cious or free st-'le — a species of fantasia. The Scherzo (see p. 139: 19V 6. Old Dances. (See p. 142 : 28, 29.) 7. Modern Dances. The Polonaise and the iMacurka are Polish national dances. They are written in 4 time. The Polonaise is slow and stately and is more of a walking or step- ping than a dance. The Mazurka is lively and of a senti- mental character. The Polka is a skipping \ movement of Bohemian origin. The Schottische is similar to the polka, but somewhat slower. The Waltz is a circular whirling dance in 4 time. The March is a movement suited to guide the walk- ing of masses. The Quickstep is a quick, lively march. The Galop is a quick dance tune. The Fandango is a lively Spanish dance tune in g or § time. The Reel is a lively Scottish dance tune. The Quadrille is a French dance tune performed by four couples placed in quadrangular position, hence the name. The Cotillon is a lively, animated dance tune, usually in % time. 8. Every dance has its own particular rhythm, which gives it its character. PART THIRD ACOUSTICS 1. Acoustics treats of the laws of sound. Sound is caused by vibrations. If the vibrations are regular a musical tone is produced, while noise is the result of irregular vibrations. 2. The range of sound is from about i6 vibrations per second to about 38,000 per second (about eleven octaves), as these limits represent the capacity of the ear to respond. The range of the seven-octave piano is from about 32 to 4100 vibrations per second. 3. The velocity with which sound travels may be deter- mined by the simple method of observing the time between seeing a gun fired and hearing the report (the time for light to travel being inappreciable) ; then by calculations based on the distance and time we may determine the velocity, which is thus found to be about 1 100 feet per second, being influenced somewhat by the state of the atmosphere. It has also been found by tests that all sounds travel with the same velocity. 4. Sound travels through the air in waves similar to waves of water, except that the waves are to and fro instead of up and down. As the pressure of the air is in all directions there is only room for compression and expansion, thus producing to-and-fro waves, which circle outward in all directions. 148 MUSICOLOGY The vibrations of the sounding body are transmitted to the air, producing similar vibrations in the particles of air, each impulse passing onward from one particle to the next. Each particle of air passes through every possible phase (position) of vibration during one complete vibration. 5. The length of a sound-wave is the distance between any two particles in exactly the same phase, including between them every possible phase of one complete vibration. The waves thus advance, while the particles of air merely vibrate. 6. Since the velocity of sound is about 1 100 feet per second, if we divide 1 100 by the number of vibrations cor- responding to any pitch we would get the wave-length of that pitch. We would thus find that the wave-lengths correspond- ing to all the tones of the seven-octave piano would range from about three inches to about thirty-four feet. 7. The Laws of Vibration are most easily studied by means of taut strings. 8. If a taut string is plucked (as in the guitar), bowed (as in the violin), or struck (as in the piano), it will be seen to vibrate from side to side ; the tone produced depending upon the length, tension, thickness, and weight of the string. Three things will be noticed : loudness, pitch, and cjuality. The loudness depends on the force of the stroke (or the amplitude of the vibrations) ; \\\e pitcJi depends on the number of vibrations per second ; and the quality depends on the compound nature of the vibrations (all strings vibrate, not only as a whole, but also in parts which produce overtones upon which depends the quality of tone, as all simple tones of the same pitch sound alike). 9. A Sonometer is an instrument for studying the vibrations of strings. It consists of a long resonance box over which one or more strings may be stretched by means of weights and pulleys, and provided with a graduated scale and a movable bridge so that the exact vibrating length of the string may be measured. ACOUSTICS 149 10. By means of the sonometer the following laws have been demonstrated : 1 1. First Law. The shorter the string the faster it vibrates in inverse proportion to the length (shortening the string to I, \, \, \, etc., increases the number of vibrations 2, 3, 4, 5, etc., times). 12. Second Law. The tighter a string is drawn the faster it vibrates, but in direct proportion to the square root of the tension (thus, increasing the tension 4 times, 9 times, etc., in- creases the number of vibrations 2 times, 3 times, etc.). 1 3 . Third Law. TJie thicker the string the slozver it vibra tes in inverse proportion to the thickness (thus, doubling the diameter of the string diminishes the number of vibrations one-half). 14. Fourth Law. The greater the density, or weight, of the string the slower it vibrates in inverse proportion to the square root of the weight (thus, multiplying the weight by four divides the number of vibrations by two). Strings are sometimes coiled with wire to increase their weight. 15. Fifth Law. The intensity, or loudness, is in direct pro- portion to the square of the amplitude {extent) of vibration (thus, doubling the amplitude increases the loudness four times). The time of vibration is not affected by the ampli- tude of vibration, provided other conditions remain the same: just as the time of each swing of a pendulum is not affected by the amplitude, or extent, of the swing, provided the length of the pendulum remains the same. 16. Compound Tones. If a taut string be struck, its fun- damental tone is due to its vibrations as a whole, thus: tone. Now if the which we will suppose is this, f^^ string be lightly touched at its middle point (just enough to i;o -MUSICULUGV prevent its fundamental vibrations but not to push it aside from its straight position) and made to vibrate, it will vibrate in halves, thus: (called segments or loops, and the dividing point a node); each part, being half the length of the string, will vibrate twice as fast (Law i); the tone produced will be the octave of the fundamental tone, which is this, -^- A. Again, if the string be touched at one-third its length and the shorter part struck, the whole string will vibrate in thirds, thus: each part being one-third will vibrate three times as fast as the whole string, and the tone produced will be a perfect 5th above the first octave, which is this, -(^ Again, if the string is touched at one-quarter its length and the shorter part struck, it will vibrate in fourths, thus: and each part, being one-fourth the length of the string, will vibrate four times as fast, and will sound the second octave of the fundamental, which is this, -^' Again, if the string be touched at one-fifth its length and the shorter part struck, it will vibrate in fifths, thus: each part vibrating five times as fast as the fundamental, and sounding the major third above the second octave, which is this, rac 4 We may continue thus to divide the strine indefinitely. I ACOUSTICS 151 17. Now if the string be made to vibrate free, it will not only vibrate as a whole but will at the same time vibrate, more or less (varying somewhat, as to relative proportions, with character of string and place, manner, and force of its excitation), in halves, thirds, fourths, etc., indefinitely. 18. By trying to conceive of the string vibrating as a, b, c, d, and c at the same time (the whole like a large wave bear- ing on its surface smaller waves, and these still smaller, etc.), we may form an approximate idea of the compound nature of the vibrations and, in turn, of the compound nature of the resulting tone, which is a compound of the fundamental tone and the overtones produced by the vibrating segments — the overtones growing weaker as they ascend in pitch. Some of the lower overtone vibrations may be plainly seen on the lowest strings of almost any stringed instrument, but the higher overtone vibrations are too rapid to be seen. 19. Any fundamental tone with its overtones is called a Harmonic Series. Fig. 29 shows the harmonic series of great C. The series is the same for all tones, differing only in pitch. 20. Any harmonic series is called a chord of nature, as the compound tone thus pro- duced may be considered in Pm. 39. itself a natural chord or combination of tones. 21. By the pitch of a compound tone is meant the pitch of its fundamental, this being the absorbing or ruling tone. 22. The overtones produced by the uneven divisions of the string (thirds, fifths, etc.) are easier to hear than those pro- duced by the even divisions (halves and fourths, forming octaves, so completely blend into the fundamental as to be practically absorbed). The perfect 5th above the first octave s 152 MUSTCOI.OGY (produced by the string vibrating in thirds) is the easiest heard, and after this the major 3d above the second octave (produced by the string vibrating in fifths). Lightly sound- ing the note we wish to distinguish, previously to sounding the fundamental, will assist in hearing it in the combination. 23. It is not to be inferred that the overtones are weak because they are difficult to hear; the difficulty is due to the blending. The lower overtones are usually quite prominent in the best qualities of tone. They increase in prominence descending the scale. At below 40 vibrations per second the pitch is quite indefinite owing to the prominence of the lower overtones. 24. Siviple tones are those free from overtones, and due, therefore, to simple vibrations. Simple vibrations are those without segmental divisions, and are also called pendular vibrations because similar in character to a swinging pendulum. 25. It is nearly impossible to produce tones free of over- tones (simple tones). The nearest approach to simple tones is the tone produced by a tuning-fork, the lowest tones of the larger stopped pipes of the pipe-organ, and the lowest tones of a wooden flute. 26. The practical effect of overtones is to give quality or tone-color to sounds. A chord of nature (harmonic series) is a compound of its different tones, just as light is a compound of the seven colors of the rainbow, which blending together in their natural proportion produce white light, but any deviation from this natural proportion produces a shade of color which partakes of the character of the predominating color or colors ; so the quality, or tone-color, of tones de- pends upon the proportionate strength of its various over- tones. Decrease in overtones tends toward a dull, hollow, monotonous quality, while increase tends toward a bright, sharp, penetrating quality. The lower overtones tend toward a deep, mellow quality, and the higher overtones toward a thin, metallic quality. ACOUSTICS 153 27. The overtones are influenced largely by the character of the string. The harmonic series will extend as high as the thickness and stiffness of the string will permit. Thin flexi- ble strings will naturally vibrate in shorter lengths than thicker and stiffer ones, and thus produce higher overtones. The more elastic the string the stronger all the overtones will be. 28. The overtones are also affected by the manner in which the string is set in motion (whether struck, plucked, or bowed) ; also, soft broad hammers and picks tend to soften by preventing many of the smaller loops from forming, while hard sharp hammers or picks have the opposite effect — hence the felted hammers of the piano, which are softer and heavier toward the lower tones and harder and lighter toward the higher tones. The force of the blow naturally affects the extent of the harmonic series as well as the strength of all. 29. The place where the string is struck (or otherwise ex- cited) is also very important. No node can form at or very near the point struck, hence all overtones having a node at or very near that point will be missing ; thus, if we strike the string at its middle point, all the even numbered harmonics will be missing, producing a hollow, nasal twang. If we strike the string at one-third its length, every third harmonic will be missing. If we strike the string at one-fourth its length, every fourth harmonic will be missing, and so on ; and of the harmonics which remain, those are naturally the strongest which have the striking point midway between their nodes, the others being more or less modified. On the other hand, if we touch a vibrating string at any point, thus form- ing a node instead of a loop, we damp all the harmonics which have no node at that point, while those having a node at that point will remain. 30. Observe, from Fig. 29, that the harmonic series from the 7th up is inharmonic, as the overtones by their closeness 154 MUSICOLOGY begin at this point to form dissonant intervals with those nearest. However, the inharmonic part of the series is largely, though not wholly, drowned in the more prominent harmonic lower part ; so that the harmonic series is in the main harmonic. 31. In pianos the strings are struck by the hammers at between one-seventh and one-ninth of their length, thus de- stroying the 7th, 8th, and 9th harmonics (6th, 7th, and 8th overtones). The higher overtones, being weak and having nodes within the influence of the stroke, are practically de- stroyed also. 32. By thus removing the inharmonic part, the harmonic series is made more perfectly harmonic as well as more clearly ^ defined (being within definite limits), as shown in Fig. 30. ' This is all of the harmonic series , (chord of nature) that is of — •— practical use or application to IJ f^ S\, ' music. (Of course this same \J - series of intervals belongs to any ~ fundamental tone, differing only in pitch.) -©- 33. In Fig. 30 we observe Fig. 30. that wc have three C's, two G's, and one E, and that we have the major triad — C, E, G — di- rect on the second octave of the fundamental. Also, observ- ing the successive intervals from the fundamental up, we have the octave, perfect 5th, perfect 4th, major 3d, and minor 3d, by which we observe the relative prominence of these inter- vals in the music scale. 34. We also notice that the second octave is subdivided into a perfect 5th plus a perfect 4th; and that the perfect 5th above is similarly subdivided into a major 3d plus a minor 3d. The entire harmonic series being thus a series of diminishing similar subdixisions indcfinitclv extended. ACOUSTICS 155 35. Observe that any harmonic series limited to the fifth overtone, as in Fig, 30, consists of three distinct notes (or letters), which, when sounded together, form a major triad ; and the root of the triad is naturally the tone from which the triad is derived. The major triad is the only chord that has thus a root in the sense that a plant has a root. In the other triads the word " root " is used more in the sense of ground note, or lowest note, upon which the triad is built in 3ds. 36. Major triads are also called harmonic triads, because thus in perfect 'harmony with the harmonic series of their fundamental root notes ; to which fact is evidently due their greater relative importance. The fundamental note of the series from which a chord is derived is called the fundamental bass of the chord in distinction from the common bass, or lowest written note. 37. It is most convenient and usual to regard a minor triad as a major triad with an intruding 3d ; but by taking each note in turn as the intruding note, the minor triad may be regarded as derived from any one of three different harmonic series. 38. Taking the minor triad, E, G, B, Fig. 31, for ex- ample, we see that the two lower notes belong to the har- monic series on C, the outside notes belong to the series on E, and the two upper notes belong to the series on G ; so that in this sense the minor triad has three distinct but imperfect roots, or fundamental basses (re- garding each note of the triad in turn as the intruding note). LO- We see that the minor triad has Q-&- no perfect root with which all Fk,. 31. three of its notes harmonize. We notice that these three imperfect roots together form the major triad, C, E, G; the n / in / Da f J C» #- r m t) -• — • • ^^v • • • V • -G-^- 156 MUSICOLOGY Dim. Triad. Aug. Triad. middle root E, being the .same letter as the lowest note of the minor triad, is the usually accepted root, as it conforms to the simple uniform method of treating all chords as built in 3ds on the lowest note as root ; but all three roots may be taken interchangeably as the fundamental bass of the triad. All minor triads being built alike may be analyzed with sim- ilar results. We see that every minor triad is directly re- lated to the major triad formed by its three roots. 39. In Fig. 32 the diminished and augmented triads are analyzed. In either case, the outside note^ not forming a perfect 5th, can- not both belong to the same har- monic series, so that these triads have two imper- fect roots, or f u n d a me n t a 1 basses, instead of three (regarding the upper and lower notes in turn as the in- truding note). 40. Observe that the roots of the diminished triad, B, D, F, are G and B b, neither of which belongs to the triad ; for which reason it is frequently treated as a dominant 7th chord with root omitted. 41. The roots of the augmented triad, C, E, Gj|, are C and E, both of which belong to the triad ; but C being also the lowest note of the triad is the usually accepted root. 42. The 7th chords (see outline on p. 69) mostly receive their justification from the triads on which they are based (see exception, chord of added 6th, p. yi))' 43. The harmonic 7th chord consists of a major triad with 5 / VII •• Ill' (( ^ VL Jf. B» • b» C- — T ^a> • l^ > . ' • ^-^ > • J* -E-^- bOBly G^ GG Kig. 33. ACOUSTICS 157 the 7th harmonic added ; but the 7th harmonic is not ex- actly represented by any tone of the scale in common use (see Fig. 29, p. 151); its nearest representative is the minor 7th. The Vj chord being the only chord with major triad and minor 7th is therefore the nearest representative of the harmonic 7th chord, and may be regarded as a slightly de- fective harmonic 7th chord, hence the prominence of the V^ chord. 44. The VIIt and Vlli chords may be regarded as Vg chords with root omitted (see p. 72 : 22), for the same reason that the vil° triad (Fig. 32) may be regarded as a V^ chord with root omitted. 45. We see that the harmonic series is the foundation of harmony, or chord formation. THE SCALE OF NATURE 1. In Fig. 33 we have the harmonic series (chord of na- ture) on C, including the fourth C ; and on the G thus obtained we have based another series. The figures of the first series show the divisions of the string and therefore the comparative rate of vibration of the different tones of that series. The fig- ures of the second series are practically a continuation of the first, being based on G, which has three times the vibrations of the first base ; therefore each interval of this series will have three times the vibrations of the corresponding interval of the first series (the series of in- tervals in both being the same, differing only in pitch). 2. Between i and 2 we have the interval of an octave, the ratio of which (i : 2) is 2. Between 2 and 3 we have a per- •18 *15 •12 -•-9- -•-6- •4- s • 3 CO CO 00 4- 8 (_n CT> O^ -^ CO CD t^ O^ w M M to 8 8 8 I 18 IF 135 9 8 US F 5 7 5 5 4 3 84 II 1 8 6 76 5 18 225 75 3 "¥" T^S 15 2 c at Db D D« Eb E F F« Gb G G« Ab A A« Bb B c § M M en 10 4- <-n ^j 10 t>5 to 0^ ? LTK O^ ^J ^4 ^4 t-n (0 0> ^J 00 CO ^4 Fig. 37. 17. The ratios are also given decimally for more conven- ient comparison, from which it is seen that in the upper scale E^ is higher than Fl?, and Bjj: than Ct' (the sharps and flats at these points overlapping) ; and that in the lower scale the re- sults are all overlapping (If being greater than a half-tone), and the tones between E and F and between B and C are eliminated (|f being the exact interval between these notes). 18. We see that in the scale of nature there is a difference between the sharp of one tone and the flat of the tone next above, and also that there is a great lack of uniformity in the intervals of the scale as shown by their ratios. 19. If these intervals were equal, transposing or modulat- ing to different keys would be merely a shifting of the scale (all the intervals coinciding in each change) ; but in view of the lack of uniformity shown above, transpo.sing or modulat- ing would greatly increase the number of tones involved, were l62 MUSICOLOGY all the keys to be exact (some of the tones would coincide, but the majority would not). 20. The violin and the voice are capable of making these changeable intervals ; but the inadaptability of the scale of nature to keyed instruments (as the piano) is very apparent. 21. The difTerent methods of overcoming this difificulty are called Temperaments. EQUAL TEMPERAMENT 1. The scale in common use is called the Equal Tempera- ment Seale, because the octave is divided into twelve equal parts called semitones, or half-steps, thus conforming to the piano keyboard, which contains twelve keys (seven white and five black) to each octave. 2. Equal Temperament removes the distinction between major and minor tones and semitones, and also between sharps and flats (the same black keys being used for both). The enharmonic scale (a scale containing intervals smaller than a half- step) is thus changed to the chromatic scale (con- sisting of half-steps), and the enharmonic change (see p.41 : 13) becomes a mere change of name. 3. Since the octave (2) is divided into 12 equal semitones we may correctly regard the octave either as 12 semitones {s) added, thus, j' + .f+j'+^ + .J + J + ^ + ^+^ + J + J + .y = 12 .y, or as 12 semitones multiplied, thus, .yX^Xi'XJX s yi s y. s y. s 'yi s y. s y. s y^ s =^ s^' \ for, as already shown (p. 160 : 14), multiplying ratios is equivalent to adding dif- ferences. We must bear in mind, however, that the semi- tones are equal only in tone value, but not in number of vibrations, as the number of vibrations in any interval varies with pitch. 4. Regarding the octave as 12 semitones multiplied to- gether, therefore, the semitone ratio is that ratio which mul- tiplied by itself 12 times equals the octave (2), or the 12th root of 2 (]^2 = 1.05946 = If nearly); but for the sake of ACOUSTICS 163 simplicity, let s represent the ratio of the semitone. The chromatic scale will then be represented thus : S'- = 2 Vibration values — i j' s- s'^ j-* s^ s^ s'' s'^ s^ .f"^ Chromatic scale — C Clt D D3 E F F$ G G$ A At B C Consecutive ratios— i X s X s X s X s X s X s X s X s X s X s X s X J=2 Dropping the sharp tones, the diatonic scale will be thus: Vibration values — i .?'■' s* s^ j' j-' jH j^'-'=:2 Diatonic scale — C D E F G A B C Consecutive ratios— i X s^ X s- X s X s'^ X s'^ X s'^ X s =2 Fig. :38. 5. We see that each vibration value is the product of all the ratios up to that point (see p. 160 : 12). 6. As the semitone (s) is the common unit of measure, and as the exponent of j- in each vibration value shows the num- ber of times s is used as a ratio, therefore the exponents (as showing the number of semitones contained) are the measures of the intervals; thus, a major 2d is 2 semitones (s^), a major 3d is 4 semitones (s*), a perfect 4th is 5 semitones (/), etc. 7. It would appear, perhaps, that the number of semitones should be expressed by coefficients, thus, 2s, 4s, 5 j, etc., instead of by exponents, thus, J^ /, /, etc. The explana- tion of this is that the semitone is expressed in the form of a ratio, and not in the form of a difference. 8. Observe that (by the algebraic rules of involution and evolution) we multiply by adding exponents, divide by sub- tracting, raise powers by multiplying, and extract roots by dividing ; thus, s^ X / = .y ^ + '^ = / ; / h- / = / - ^ = / ; (^s^y _.^x4=nj.i2. ^ ^,12 __ ^12 H- G _ ^,2 . ^Q j.j^^j. ^j^g exponents of s are compared (added, subtracted, etc.) exactly as if they were coefficients. 9. The exponents of s may also be called logarithms. A logarithm of any number is the exponent of the power to which it is necessary to raise a fixed number (called the base) to produce the given number. In Fig. 38 the log. of G (perfect 164 MUSICOLOGY 5th) is 7, the log. of E (major 3d) is 4, etc. (on .y as a base). 10. We find these exponents, or logarithms, sufficient in com- paring intervals of the tempered scale with each other. How- ever, in comparing the tempered scale with the scale of nature, it is necessary to use a common system of logarithms. 11. A general table of logarithms (the computation of which belongs to algebra) gives the logarithms of all numbers up to a certain point, on an assumed base (usually 10), giving also the decimals involved in all intermediate numbers (be- tween exact powers), usually to five or six places of decimals, so that incommeasurable numbers are thus made sufficiently commeasurable for comparison. 1 2 . The value of a logarithmic table consists in its enabling us to multiply by adding, divide by subtracting, and especially to raise powers by multiplying and extract roots by dividing. 13. The ordinary process in using the table consists in first finding the log. of the given number, then adding, sub- tracting, multiplying, or dividing, as the case may require; then find the number corresponding to the resulting log. However, our use of the logarithms here is only as a basis of comparison, since they are the exponents of a common unit of measure (10), and, as already shown, may be compared (added, subtracted, etc.) as if coefficients. 14. The log. of I (10^ -^ 10^ = 10^-' = lo*^*, but any num- ber divided by itself = i, therefore 10"— i) is o, and the log. of 10 (10' = 10) is I ; therefore the log. of any number be- tween I and 10 is between o and i, and therefore wholly decimal. As the interval ratios within the octave are be- tween I and 2, their logs, will be wholly decimal. Logarithms of the ratios] of the intervals ot tne ;- en o lo ^i to ^ o Scale of Nature. j P,^^|J^^8tS Scale— C D E F G A B C Logarithms of the ratios | ^mOM-^M^o of the intervals of the ' « u> 4- o-jo o I ^i 4^ uo o "^ -f» <->> Tempered Scale. J Fig. ;J9. ACOUSTICS 165 15. The upper logs, in Fig. 39 are taken from a loga- rithmic table and are the logs, of the ratios of the scale of nature (subtracting the log. of the denominator from the log. of the numerator where fractions are involved, since we divide by subtracting). 16. To find the lower logs, we first divide the log, of the octave (.30103) by 12 (since we extract roots by dividing), thus, .30103 -^ 12 = .02509 = log. of semitone; now, multi- plying (since we raise powers by multiplying) the log. of the semitone by the number of semitones in each interval of the tempered scale, we get the log. corresponding to each. 17. The log. of any interval not given in Fig. 39 may be found by taking the difference between the logs, of those let- ters between which it is found. Thus, between E and G is a minor 3d, the log. of which (. 17609— .09691) is .07918 in the scale of nature, and (. 17560— . 10034, or .02509 X 3) .07527 in the tempered scale. 18. We have now a basis of direct comparison between the scale of nature and the tempered scale, by which we may de- termine the amount of deviation of each interval of the tem- pered scale from true pitch. Thus, the difference between the logs, of G (perfect 5th) is .17609— .17560= .00049; divid- ing by the log. of the semitone (.02509), we have -gffg^ = -gS" nearly : therefore the perfect 5th is about ^y of a semitone flat (the log. of the tempered G being the lower). The per- fect 4th (inverted perfect 5th) will of course be -^j of a semi- tone sharp. This deviation from true pitch is scarcely per- ceptible, and therefore the 4th and 5th are classed with the octave and called perfect intervals. 19. Comparing the logs, of the other intervals, we would find them more perceptibly out of tune, as follows: major 3d, about Y s sharp; minor 6th (inverted major 3d), \s flat; minor 3d, ^s flat; major 6th (inverted minor 3d), ^ s sharp; major 7th, \ s sharp ; minor 2d (inverted major 7th), J s flat. The whole tone has a double value in the scale of nature : 1 66 MUSICULOGY viajor (ratio |) between C and D, F and G, and A and B (log, .05115); and minor (ratio V) betweeji D and E, and G and A (log. .04576). In the tempered scale its log. is .05017, which is about g^j s flat as compared with the major, but about \ s sharp as compared with the minor tone. The major 2d is therefore 3V •$" A^t compared with the major tone, or \ s sharp compared with the minor tone ; the inversion (minor 7th), in either case, being the opposite. Our ears have be- come so accustomed to the tempered scale that we do not notice the slight deviations from true pitch. 20. By means of the fractions thus found we may easily find the value of the true intervals in semitones ; thus, from C to D is a major 2d (2 .f)and ^^s flatter than the true major tone, therefore the true major 2d is ^V ^ sharper (3^-^ = .04) or 2.04 s. From C to E is a major 3d (4 s) and | s sharper than the true major 3d, there- fore the true major 3d is -^ j- flatter (I = . 14) or 4 J- — . 14 .y = 3.86 .y. From C to F is a perfect 4th (5 s) and ^V ^ sharp (51= -02) ; the true perfect 4th is therefore 5 ^— .02 s = 4.98 s. From these the others may be found ; thus, C — F = G (see table), F + E = A and G -j- E = B : or, the difference between the values of any two intervals is the value of the interval between ; or, the sum of the values of any two intervals is the value of the interval of their sum. This gives the values to within one- hundredth part of a semitone, which is much beyond the limit of sensible error, as j\ s is regarded as the sensible limit. 21. In this way we get a more definite conception of the comparative values of the true and tempered intervals. True Equal Intonation Temperament Semitones Semitones c 12.00 12.00 b 10.88 11.00 A 8.84 9.00 G 7.02 7,00 F 4.98 5.00 E 3.86 4.00 D 2.04 2.00 C ACOUSTICS 167 CONSONANCE I. Naturally, the closer any two tones are related the more consonant the interval between. The relationship between any two tones depends in the first place upon the frequency of the blending of their overtones. In Fig. 40 we have the different consonant intervals within the octave compared in this respect : * n ^e •s •6 •! •6 • 6 J«5 •6 •6 ♦8 ••^ •^ .1 ) -5 •5 ,-K • 5 • 5 "4 •5 'S •6 'S •6 /\ •4 •2 .4 '^ • 4 • 3 •^ •4 #•5 '?«4 J»5 'I'S c ^ .3 1 .3 .3 '^ •* .3 •4 \ •2 -0 .^ '2 —♦2 Ol- -•^2 -^2 -«-2 -^2 '- -^2 \^* o\ ^^t — I'^i ZJr o\ N, .^'C?! o\ o\ 01 o\ 01 01 C?i Octave J Perfect sth ^ Perfect 4th | Major 3d } Major 6th § Minor 3d | Minor 6th Fig. 40. 2. Observe, in each case, that the frequency of the blend- ing of overtones is in exact accord with the ratio of the inter- val ; thus, in the octave (ratio f ) every harmonic of the upper note blends with every second harmonic of the lower note, in the perfect 5th (ratio |) every second harmonic of the upper note blends with every third harmonic of the lower note, and similarly as to the other intervals. If we were to extend each series indefinitely we would find the rule continuing to hold true. 3. These blending overtones mutually reinforce each other, the amplitude of the resulting vibration being practically equal to the sum of the amplitudes of the separate vibrations; and since the intensity of sound is proportional to the square of the amplitude, therefore the intensity of the blending over- tones is equal, not to the sum, but to the square of the sum of the separate intensities. This gives the blending overtones special prominence. i68 MUSICOLOGY cm 4. Observe, in Fig. 40, that in each interval the ratio of the number of harmonics in each series up to any blending point (reduced if necessary) is the ratio of the interval, or the vibration numbers of any two blending overtones (put into the form of a fraction and reduced when necessary) gives the ratio of the interval ; or, if Ave regard the blending overtones as dividing the series into sections, we again see the ratio of the interval duplicated in each section in the ratio of the number of harmonics on each side. 5. We also notice that the remoteness of the first two blend- ing overtones is also a measure of relationship. 6. Observe that in the octave the upper note with all its overtones blends with overtones of the lower note, the two notes thus blending into one without adding any new element, the effect being merely to bright- en the lower note by strengthen- ing every other overtone ; hence octaves are regarded as replicates of the same tone, as no new element is involved. 7. Similarly the interval of a perfect 12th (octave -{- perfect 5th) adds no new element, as every harmonic of the upper note blends with every third har- monic of the lower note. The same applies to any interval which corresponds to the inter- val between the lower note and any one of its harmonics. 8. If in Fig. 40 we were to ex- tend each series indefinitely, we would find that in each interval the blending overtones form in themselves a harmonic series, as their vibration numbers are 1,2, 3, 4, etc., times the number of the lowest blending ov^ertones. Major Cth 5 — / • • (( >) \ J tJ ^. > -y (^ Fig. n. ACOUSTICS 169 9. If we were to arrange the intervals in the form of a downward harmonic series, as in Fig. 41, the double notes would show the blending overtones of each separate interval. (The 7th harmonic, which is omitted in Fig. 41, as it does not belong to the scale, may be included.) By ex- tending Fig. 41 sufficiently we may include also the dis- sonant intervals, as the principle holds good for all inter- vals. 10. The relationship between any two tones depends in the second place upon the frequency of the blending of their vibra- tions. In the octave every vibration of the lower tone blends with every second vibration of the upper tone as indicated by the ratio f ; thus, [ ' I ' I ' I • (The spaces between the marks represent the comparative duration of the vibrations.) In the perfect 5th (ratio |) every second vibration of the lower tone and every third vibration of the upper tone blend ; thus, J ' , ' [ ' , ' [ . Similarly, in the perfect 4th (ratio I) every third vibration of the lower tone blends with every fourth vibration of the upper tone. In the major 3d (ratio I) every fourth blends with every fifth. In the minor 3d (ratio I) every fifth blends with every sixth. In the major 6th (ratio |) every third and fifth blend. In the minor 6th (ratio I) every fifth and eighth blend. In the major 2d (ratio I) every eighth and ninth blend. In the major 7th (ratio V) every eighth and fifteenth blend. 11. We see that the ratio of any interval is duplicated, both in the blending of the overtones and in the blending of the vibrations, and therefore the whole may be summed up in the general rule, that tJie consonance of intervals (and therefore the relationship of the including tones) is proportional to the simplicity of their ratios. We naturally conclude that sim- plicity is the essential element of consonance, and that the ratio of any interval is its characteristic stamp by which the same interval always gives the same impression as to degree of simplicity regardless of pitch. I/O MUSICOLOGY 1 2 , The blending overtones only blend perfectly when in per- fect unison ; and they are only in perfect unison when the interval between the generating tones is just (in accord with the scale of nature). But in the tempered scale all the inter- vals (except the octave) are slightly false, and hence the blending overtones do not blend perfectly, but sufificiently to still clearly define the interval ; and the effect, if percept- ible, is only a sense of inaccuracy. For, as already seen (p. i68: 4), the ratio of any interval is duplicated, not only in the blending vibrations of the tones forming the interval, but also in the frequency of blending overtones, in the har- monics as a whole up to any blending point, and in the har- monics of each section separately; so that thus the true ratio of the interval is clearly defined, even though the interval itself should be slightly false. There could be no sense of inaccuracy without a sense of the true interval as a basis of comparison, 13. Observe that all the consonant intervals of the scale of nature are a natural selection of those having the simplest ratios, and therefore most consonant. BEATS 1. Sound-waves (p. 147 : 4) increase in length as they dim- inish in number, for what is lacking in number must be made up in length, since all sounds travel the same distance in the same time. 2. When two notes of different pitch are sounded together, the longer waves of the lower note steadily gain on the shorter waves of the upper note; and the gain per second will equal the difference in the number of waves or vibrations per second in the two notes. 3. It is evident that the reverse phase of each wave gained must be met, while at other times both waves move in the same direction, so that they will alternately coincide and op- pose. When the)' coincide they strengthen each other and ACOUSTICS 171 increase the sound ; when they oppose they counteract each other and diminish or entirely destroy the sound ; thus pro- ducing an alternate swell and lull, which are called Beats. 4. The number of beats per second produced by any two notes sounding together is equal to the difference in the number of vibrations per second of the two notes. 5. Imperfect Unison Beats. Tones in perfect unison do not beat, as their vibrations coincide ; but an imperfect unison is marked by distinct beats, which increase in rapidity with increase of error. This is very important in tuning instru- ments ; thus, to tune two notes to a perfect unison it is only necessary to gradually bring them together till they cease to beat. 6. Beats which are due to imperfect unison are called Imperfect Unison Beats. 7. Beats may occur between overtones, which are therefore called overtone beats, while the beats between the prime tones are called prime beats. Prime beats and overtone beats often occur at the same time, producing more or less confusion. 8. Though on the one hand beats increase in number with increase of interval, on the other hand they rapidly decrease in strength with increase of interval (the reason is given under "Sympathetic Resonance"); therefore, overtone beats are sometimes stronger than the prime beats by reason of a closer beating interval. 9. Imperfect Consonance Beats. Wide-apart consonances (as octaves, perfect 5ths, perfect 4ths, etc.) when slightly false also produce distinct beats, which are called Imperfect Consonance Beats, and which also increase in rapidity with increase of error. 10. These are due in the first place to the imperfect uni- son beats of the lowest blending overtones. When an inter- val is slightly false, all the blending overtones become imper- fect unisons, and therefore beat because of their imperfect blendins. Of course the beats of the lowest blending over- lyZ MUSICOLOGY tones are the most distinct and determine the number; but these are strengthened by blending beats of the higher blend- ing overtones. 11. The number of beats in the higher blending overtones are respectively 2, 3, 4, etc., times the number in the lowest blending overtones; and therefore every 2d, 3d, 4th, etc., beats respectively blend with the lowest. 12. The strength of these imperfect consonance beats de- creases with decrease in consonance of interval, by reason of the increasing remoteness of the blending overtones (see Fig. 40). 13. It has been shown by experiments that imperfect con- sonance beats occur when the tones are simple (without over- tones) ; but it has also been shown by resonators (tuned jars which detect corresponding disturbances in the air) that these beats are formed within, and not outside the ear. These, however, correspond in number and therefore blend with and strengthen those formed by the blending overtones (when both exist). The internal beats are called subjective, as they have only a subjective existence ; while the external beats are called objective, as they have an objective existence. 14. Helmholtz explained the subjective imperfect conso- nance beats on the theory that simple toties when poiverful enough produce overtones zvitJiin the ear by reason of the pecu- liarly favorable construction of the ear-drum and connecting parts. Koenig claimed that in his experiments no overtones were audible, and explained the beats as due to the beating of the nearest multiples of the simple tones. As these multiples correspond to the overtones, the result is the same as if we as- sumed the actual existence of the overtones. It makes no difference, so far as the sensation (practical fact) is concerned, whether the beats are objective or subjective; and therefore the question as to the relative objective or subjective charac- ter of the beats is of no practical (only theoretical ) importance. 15. In tuning consonant intervals, as in tuning unisons, it ACOUSTICS 173 IS only necessary to raise or lower the dissonant note (as the case may require) till it ceases to beat or (in tuning to any degree of temperament) till the number of beats correspond- ing to the desired degree of temperament is heard. Beats thus furnish the most accurate means of tuning. 16. Beat Tones. When the rapidity of the beats exceeds 16 per second, they begin (owing to their regular periodic recur- rence) to form a tone, which is therefore called the beat tone. 17. As the rapidity increases the beating effect decreases, while the definiteness and pitch of the beat tone increases ; but during the transition of the beats into a tone, both beats and tone are heard, since the two different sensations are re- ceived by different parts of the ear. 18. As already observed, the strength of beats, and there- fore of the beat tone, decreases with increase of interval. But we may increase the number of beats without changing the interval by taking the same interval at a higher pitch, thus raising the pitch of the beat tone ; or, on the other hand, we may (ascending the scale) narrow the interval with- out changing the number of beats, thus increasing the loud- ness of the beat tone without changing its pitch. Also, the closer the interval, the lower the pitch of the beat tone as compared with its generators, and therefore the more easily heard by contrast. Thus we see that, other things being equal, beat tones increase in prominence ascending the scale. 19. If two notes (called generators) are sounded together, beats or beat tones will exist theoretically between all the harmonics of the two series; but taking into account the effect of distance and the tendency of the stronger to absorb the weaker, it naturally results that only the most prominent have audible existence. 20. When two beat tones are of nearly equal intensity, both may be heard at the same time and may in turn beat (if differing slightly in pitch). Also beats and beat tones, due to different parts, are sometimes heard at the same time. 174 MUSICOLOGY DIFFERENTIAL AND SUMMATIONAL TONES 1. These, like beat tones, are phenomena resulting from sounding two notes together. 2. Differential tones have a vibration frequency equal to the difference in the number of vibrations of their generators, hence the name. Summational tones have a vibration fre- quency equal to the sum of the number of vibrations of their generators, hence the name. Summational tones are there- fore always higher than both generators ; while differential tones are the same in pitch as the lower generator at the octave (the upper generator being double the lower), but lower than both generators when their interval is less than an octave, and between both generators when their interval is greater than an octave. 7, II, and 13 are only approxi- mate, as their true pitch is not exactly shown bv the staff. f^Octave 2/j Per. 5"> % P^. ithy^iiajorSrd% Uinor^rde/^ *"" ) y • 8 ^8 f? ) D • * ^-m J • 5 L-f5>s- L-^5-J L-1^^ -(©l ~0l /7.^.«3 03 G3 0^ • 3 k> \^G2 C?2 • 2 Jifa/W6"'-ys Jfi«or6'»% -^^\ -1 -•-1- FiG. 42. 3. Fig. 42 shows the differential and summational tones for the consonant intervals within the octave. They of course remain relatively the same for each interval, regardless of the position of the interval on the staff. Observe that taking the difference between the generator numerals gives the differential numeral, and adding the generator numerals gives the summational numeral. 4. Summational tones are usually weak and hence are of but little importance. Differential tones, however, are often ACOUSTICS 175 distinctly audible, and are the more easily heard the lower they are as compared to their generators. 5. Taking the difference between the generators themselves we get the differential of the first order. Taking the differ- ence between the differential of the first order and either generator we get the differentials of the second order. Tak- ing the difference between the differentials of the second order and either generator, or the differential of the first order, we get differentials of the third order. And so on, theoretically, till the harmonic series from i (root of interval) up to the generators is complete, which may then be con- tinued through the overtone differentials and summationals' indefinitely. 6. Fig. 43 shows the harmonic series formed by the dif- ferential tones of each of the consonant intervals within the octave ; but the principle applies similarly to all intervals. The intervals in the figure are arranged in their natural order in the harmonic series, and hence have the same root ; and the harmonic series formed in each case is the same as far as it goes. Otherwise, the intervals would have different roots and the differentials would form different harmonic series. Octave I n Per. 5th i Per. 4th 1 Major 3d f Minor 3d \ Major 6th i Minor 6th § ) / ®%.T (( \ V J . . _. .^-ic ■rf^ir «J -€^ -^A-:i -^ -*^ -^i ^■^' G^ GZ 'Z • 3 • 3 ©3 'S •3 (cJ- \ll/02 02 m2 • 2 • 2 • 2 •2 • 2 - - Oi * i ist (2 — 1 = 1) (Orders) -^h -»-fc —^1 —^1: — ►!: — ^> ist (3-2=1) ist (4-3=1) ist(s-4=i) 1st (6-5=1) ist (5-3=2) ist (8-5 =3) 2d (3-. = 2) 2d (3II-) .d (^1-3) ,d (11-4) .d (3--) 2d (^3-) 3d (5-3 = 2) 3d (6-4 = 2) 3d (5-1 = 4) ,d/'8-2=6\ 4th (5-2=3) 3CH^3_3^J Fig. 43. '''^8-1=7) 176 MUSICOLOGY 7. The different orders of differentials are indicated at the bottom of Fig. 43. It will be seen that the simpler the ratio of any interval the fewer the differentials formed, and there- fore the simpler the character of the interval. 8. Fig. 44 shows the interval of a major 3d and the three harmonic series involved, thus showing their relation to each other. The middle series is extended up to the summational tone of the generators. 9. Observe that the summational of the generators ex- presses also a differential relation indirectly as between either £4»- -•-25 generator and the harmonics of ~^^ the other, as shown at a\ or •1 5 as between the harmonics of each, as shown at b and c ; or • 10 . ' ■ X\ directly as between the har- -^7 -♦6 ^^- -•* monies above and below each ^-5 • blending point, as shown at d\ and is thus, in a sense, a focus- — ing point as between the two 75 " harmonic series. . 10. Also, the summational — of the overtones taken success- j-^ ively, thus, 8+10= 18, 12 + io-(s-4) = 9) ( i2-r 8-5) = 9 15 = 27, 16+20=36, etc., i5-(io-4) = 9r ^ Ji6-(i2-.s) = 9 i ^ 2o-(i5-4) = Qf ]2o-(i6-s) = 9 are in like manner hifrher focus- 25 — (20 — 4) - 9 ; ^ 24 - (20 - 5) = 9 ^ o ^^- ^^- *<^- *<^- ing points ; and these focusing , , , „ , , X ,^ o N points taken together form in (10-4) + ( 8- s>=9 (i5-4)-(io-8 )=9 ^ ^ ^'5-^^+("-^_°'=9 (2o-^8)-(i5-j2)=9 themselves a harmonic series, d thus, 9, 18, 27, ^6, etc., and 2oV'2ol 24-13 = 9 therefore harmonize and blend 6/ \ - 1 25 — 16 = 9 &c.'^ &c. together, forming a compound Fig. 44. tone. II. Also the differentials of the harmonics of the gener- ators taken successively, thus, 5 — 4= 1, 10 — 8 = 2, 15 — 12 ^3, 20 — 16 = 4, etc., form a harmonic series which is ACOUSTICS 177 always built on the first differential (which, however, is not always i), and if extended high enough will include the sum- mationals. Comparing this series with the series produced by the summationals, we see that they stand to each other, in whole or in part or in any similar combination of parts, in the ratio of the sum and difference of the generators ; so that differential and summational tones are, in a sense, comple- ments of each other, and one can scarcely exist without the other. 12. Observe, again, that the summational of the generators is the exact mean between the first overtones ; and each sum- mational successively is the exact mean between the even numbered harmonics of the generators. In each case the overtones being of equal rank are equal in strength, suppos- ing the generators are equal. The summationals are also exact means between the differentials of the odd numbered harmonics of one side and the generator of the other; thus, 15 — 4=11 25 — 4 = 21 mean = 9 mean ^= 18, 12— 5 =7 20—5 =15 etc. Evidently an exact mean can only thus be indirectly formed between those harmonics having an uneven differ- ence, since all elastic bodies must vibrate in whole numbers (exact number of parts). Other interesting relations might also be pointed out. 13. These relations are general and apply to all intervals, and therefore have a bearing on the nature of differential and summational tones. 14. The complemental relations between differentials and summationals indicate that they have a common origin ; and the relations pointed out indicate that they are the resultant effect of the influence which each generator exerts on the other. 15. Differential tones were discovered about 1745 by Sorge, a German organist, but were made more generally 178 MUSICOLOGY known by the Italian violinist Tartini, and are often called Tartini's tones. Summational tones were discovered by Helmholtz, about 1854. 16. Helmholtz showed that by mathematical theory, when the amplitude of the vibrations of two generators sounding together is great enough, other vibrations are also produced having a frequency corresponding to the differential and summational tones; and thus developed the theory that dif- ferential and summational tones were both due to vibrations produced by the combined action of the generators. He also showed that the construction of the drum skin of the ear and connecting parts is peculiarly favorable for magnify- ing these tones ; so that they are thus produced in the ear even when they are not sensibly produced in the air, and hence are more largely subjective than objective. 17. Previous to Helmholtz's theory and discovery of sum- mational tones, differential tones were generally regarded as due to beats, and therefore identical to beat tones. 18. Helmholtz does not recognize beat tones, evidently regarding them as differential tones. Koenig and others rec- ognize both beat tones and differential tones, but as differ- ing in origin. The fact that audible tones sometimes exist where beat tones would be inaudible by reason of the width of the interv^al, would also indicate that these tones were due to some other cause than beats. (Differential tones also dim- inish somewhat with increase of interval.) 19. Beat tones and differential tones both have a vibration frequency equal to the difference in number of vibrations of their generators ; therefore, \vhere they both exist as due to the same generators, they coincide, blend, and strengthen each other. This would appear to account for the degree of strength which neither alone would sometimes seem to war- rant ; especially if we apply the principle that tlie intoisity of blending tones is proportional to the square of the sum of their separate intensities. ACOUSTICS 179 20. Differential, summational, and beat tones are in gen- eral called combinational tones, because produced only by the combination of two or more generating tones. They are thus distinguished from harmonics. DISSONANCE 1. Beats bear directly on the subject of dissonance. In- tervals increase in dissonance with number of beats up to about 33 per second, after which their roughness diminishes as they merge more and more into the beat tone. 2. The number of beats in the major 2d (whole tone) be- tween middle C of 256 vibrations and D of 288 vibrations (288-256) is 32, and hence near the point of greatest dis- sonance. If taken an octave higher it would contain 64 beats, and the minor 2d (semitone) would contain 32 beats. Or if taken an octave lower it would contain 16 beats, and the major 3d (two whole tones) would contain 32 beats; but here the increase of interval counteracts the effect of the beats, as the audible limit of beats is about a minor 3d, otherwise, any interval if taken low enough would be at the point of greatest dissonance. However, the same interval increases perceptibly in roughness descending the scale till the point of about 33 beats is reached, and vice versa. 3. We may regard the interval of the 2d (major or minor, according to pitch) as representing about the maximum de- gree of dissonance. 4. Observe, in Fig. 40 (p. 167), the interval of a 2d be- tween the second and third harmonics in the perfect 4th and major and minor 6ths, and between the third and fourth har- monics in the perfect 5th and major and minor 3ds. In the perfect 5th, however, it is above the first blending overtones. We observe, also, other more remote dissonant intervals. 5. We see that all intervals (except the octave) contain both a consonant and a dissonant element, and that the disso- nant element increases as the consonant element decreases, so l80 MUSICOLOGY that the boundary between consonance and dissonance is merely a question of predominance. In consonant intervals the consonant element predominates, and in dissonant inter- vals the dissonant element predominates. Fig. 40 shows only those within the octave that are classed as conso- nant. 6. Where more than two generators are involved, disso- nance is due in a large measure to combinational (especially differential) tones. In Fig. 43 we observe that where only two generators are involved the differentials belong to the same harmonic series with the generators and therefore form consonant intervals with them (except in the higher parts, see 7, in the case of the minor 6th). But if we had three or more instead of two generators, they would form three or more different intervals with each other, and unless these in- tervals were in their natural harmonic order (as only in the major triad) the differentials of each interval would form separate harmonic series, which would form a greater or less number of dissonant intervals with each other (producing beats); but naturally the more harmonic the combination of the generators the fewer the dissonant intervals produced by their differentials. 7. If Fig. 43 were extended to include the dissonant in- tervals, we would find that the differentials soon begin to form dissonant intervals with the generators. 8. In Fig. 42 we sec that the summational tones form dissonant combinations with the generators, except in the octave, perfect 5th, and major 6th ; but as summational tones are generally weak, if audible at all, they may usually be dis- regarded. SYMPATHETIC RESONANCE I. Sympathetic Resonance is the tendency of any resonant body to vibrate in sympathy when a note is sounded near it having a vibration frequency corresponding to its own. ACOUSTICS l8l 2. The sympathetic vibration is produced by the repeated strokes of the vibrations of the sounding body being trans- mitted to the sympathizing body. The effect, however, is not instantaneous, as the first stroke in itself is not sufficient, but must be magnified by repetition ; just as to ring a very heavy bell requires repeated pulls on the rope at regular, rightly timed, periodic intervals before the bell begins to ring. The principle in each case is exactly the same. 3. Strings are very sympathetic when attached to a sound- ing-board (as in the piano) or a sounding-box (as in the violin or guitar) ; but the strings themselves have not sufficient sur- face to receive direct very much of the force of the trans- mitted strokes. 4. If we were to sing a note into a piano against the sound- ing-board (after raising the dampers) we would hear the note repeated by the instrument (the note should be sustained for a time to obtain a good result). The vibrations of the voice striking the sounding-board are transmitted by it to the strings; and those strings having a corresponding vibration frequency will vibrate in sympathy. If the note sung repre- sents a compound tone (containing overtones), then all those strings which are capable of sounding the whole or any part of the compound tone will sound proportionately. Thus the piano is, in a measure, capable of analyzing the compound tone. 5. Light and pliable bodies, such as strings, membranes, and enclosed air, being easily movable, are easily set into sympathetic vibration, which, however, as readily die away when the cause ceases. On the other hand, heavy or rigid bodies, as bells, tuning-forks, etc., respond less readily, and require a greater number and more accurately timed strokes and therefore longer sustained effort to produce sympathetic vibrations, which, however, die away as slowly as they are produced. 6. Easily sympathetic bodies may be made to vibrate by 1 82 MUSICOLOGY tones which do not exactly correspond in vibration frequency (pitch), but the ampHtude of the vibrations diminishes as the difference in pitch increases; and naturally the more easily sympathetic a body is, the greater the limit of difference in pitch by which it can be influenced. 7. The ear is wholly a sympathetic instrument. The end of every fibre of the auditory nerve is connected with small elastic appendages which vibrate sympathetically with the sound-waves. Each of these elastic appendages corresponds to a certain pitch of tone, but, being extremely sympathetic, is influenced within certain limits of pitch on either side ; the sensible limit being ordinarily somewhat more than a semi- tone on either side, or together about a minor 3d. 8. A string or other easily sympathetic body, intermediate in pitch between two sounding bodies within influencing dis- tance, will be made to beat by being set in sympathetic vi- bration by both sounding bodies at the same time ; its vibra- tions being alternately strong (when the vibrations of the two bodies coincide) and weak (when they oppose), as may be seen by a vibration microscope. 9. This illustrates how beats are produced in the ear. Thus, beats are produced when two sounding notes are near enough to each other in pitch so that the same clastic appen- dages of the auditory nerve will be set in sympathetic vibra- tion by both notes at the same time ; the combined effect focusing, as it were, at the point where both notes produce equal effect, which is nearly midway between (depending on the relative strength of the two notes). Naturally the combined effect diminishes as the distance from the focus increases, by increase of interval. This explains why beats decrease in strength with increase of interval. 10. When the number of beats is sufUcient to produce a beat tone, a different elastic appendage, corresponding in vibration frequency to the number of beats, is brought into sympathetic vibration ; thus, the beats and the beat tone, ACOUSTICS 183 being due to the sympathetic vibration of different parts, may be heard at the same time. 11. We noticed, on p. 181:4, that if a compound tone were sounded into a piano, those strings having a vibration frequency corresponding to the separate simple vibrations of which the compound sound-waves are composed will vibrate in sympathy. This roughly illustrates the principle by which the ear analyzes compound sound-waves ; but, unlike the piano, the ear is capable of distinguishing the slightest dif- ferences of pitch, and its sympathetic sensitiveness is incom- parably greater. 12. The principle is based on two laws: first, Fourier's law, the substance of w^iich (in a musical sense) is, tJiat every compound vibration is the sum of a certain mimber of simple vibrations, and is tJierefore capable of being analyzed back into those simple vibrations; second, Ohm's law, the substance of which is, tJiat the ear perceives only simple vi- brations and hence does not recognize the compound sound-wave as a single ivhole, but o)ily the simple elements of ivJiicJi it is composed. PART FOURTH PRINCIPAL SOURCES OF MUSICAL SOUND OUTLINE ■ Laws of Vibration How excited ^\ f Rapidity varies inversely as the length, inversely J as the thickness, directly as the square root of 1 the tension, and inversely as the square root of I the weight. f Loudness, depending on amplitude of rStrikine vibration. \ Plucking \ Pi'^.'^h' depending on rapidity of vibra- t owing I Quality, depending on form of vibra- [ tion. Application— Piano, Guitar, Harp, Violin, etc. ■ r.no-;tiiri;nai i Laws of Vibration— Same as in transverse vibrations. i/ihraHV.n< -^ How excited-By rubbing lengthwise with resined cloth, ^/ibrations ( ^^^ ^^^^ -^^ music. Similar to a stout string (rigidity answering to tension). Not used in music. 1^ Laws of (Rapidity varies directly as the J v;vi-raf;r.n '^ thickuess and inversely as the 1 Vibration | square of the length. [Application- Music-box, etc. f Supported in I . ,■ .■ n> c , I the middle ( Application-I uning-fork. 1 Supported at /,,■.- „ Nodal Points ( Application — Harmonicon, etc. f Law of Vibration— Rapidity varies inversely as the length. (Series corresponding to the odd figures, i, 3, r Transverse I Vibrations ! { Both ends fixed I I .Transverse I Qne end fixed I Vibrations i Both ends free .^ ,. I I One end fixed \ 5 Longitudina' I /go, etc. Vibrations Sound-wave four times length of rod. Both ends fixed |. gound-wave two times length of rod. Both ends free I Q Air Columns Reeds Not used in music, f Vibrations— Longitudinal. Law of Vibration— Same as in longitudinal vibrations of rods. p- ) Stopped pipes— Analogous to rods fixed at one end. fipes-j Open pipes— Analogous to rods free at both ends. , Application— Pipe Organ, Wood Wind, and Brass Wind Instruments, i Tongue-shaped— Application— Reed Organs, Clarinets, etc. '' Mpinhrane * ^'Ps in playing Brass Instruments. Membrane y y^^^j^j c^rds in Singing. Vibrations — Radial, Circular. (Membranes-^ pi„^„ j With definite pitch— Application— Ke , I i^iasb ^ Without definite pitch— Application— D Kettledrum. rum, Tambourine, etc. f Vibrations— Radial, Circular. „, ^ It f.jr-K,.„^;^r, S Other things being equal, rapidity varies directly Plates ■! Laws of Vibration -^ ^^ ^j^^ thickness, and inversely as the area. [ Application— Cymbals, Gongs. \ ( By strokes— Application— Church Bells, Clock Chimes. ! How excited < etc. ,, . , ^, 1 I By radial friction— Application— Musical Glasses. I 1 2i 4 6i 9 tSeries-C D C G» D etc. [Bells 1 86 MUSICOLOGY STRINGS, RODS PIPES, REEDS 1. Only in strings, the narrower organ pipes, and the lon- gitudinal vibrations of rods are the conditions wholly favor- able for forming the natural harmonic series with vibrations proportional to the simple arithmetical series i, 2, 3, 4, 5, 6, etc., already described. 2. Most other sounding bodies produce overtones which are more or less inharmonic with their prime tones. 3. Transverse Vibrations of Rods. In the case of rods fixed at both ends, we have conditions somewhat similar to a stout string (rigidity answering to tension). 4. In the case of rods fixed at one end, an entirely new set of conditions prevails, involving new laws of vibration, which are given in the outline. Ordinarily, only very high inhar- monic overtones are produced when the rod is struck, which do not blend with the prime and soon die away ; but if the rod is very strongly agitated, some of the lower harmonic overtones are faintly present. These last are present only when the amplitude of vibration is sufficient to produce a sen- sible flexure in the rod. In the music-box the teeth of the music-comb are relatively thin as compared with their length, and being made of very elastic material, they admit of con- siderable amplitude of vibration, and therefore tolerably good musical tones are produced. (The weights on the teeth also tend to develop harmonic overtones.) 5. In the case of a rod free at both ends and supported in the middle, each half is of the nature of a rod fixed at one end. A tuning-fork is a rod of this kind bent on itself in the middle and provided at this point with a stem or handle. Its principal use is as a standard of pitch, for which it is specially fitted, as when lightly struck and held close to the ear only the prime tone is audible. The high inharmonic overtones, however, produce a sensible tinkling when the fork is first PRINCIPAL SOURCES OF MUSICAL SOUND 1 87 struck, but soon die away ; and as they do not fuse with the prime the ear easily separates them. 6. In the case of a rod free at both ends and supported at the two nodal points, the section between the points of sup- port will be of the nature of a rod fixed at both ends, and the end sections will be of the nature of rods fixed at one end. 7. Longitudinal Vibrations of Rods. If a rod fixed at one end and free at the other receive a blow against the free end lengthwise of the rod (or rubbed lengthwise with a resined cloth or the moistened finger), the impulse runs along the rod to the fixed end and is reflected back to the free end, but is now as a pulling force instead of a compressing force; and in this phase it runs along the rod to the fixed end and back to the free end, when it is again in its original phase, thus making a complete vibration, since from like phase to like phase is a complete vibration : hence, the time of vibration is the time required for the impulse to travel four times over the rod, and the wave-length is four times the length of the rod. 8. If the rod be fixed at both ends, the effect will be as if each half were fixed at one end and the middle of the rod were the free end of each half; thus, a rod fixed at both ends vibrates in the same time as a rod ^, — ^ half the length fixed at one end ; therefore, the time of vibration is the ^ time required for the impulse to travel Ci " — l- — 1 — =^ — l twice over the rod, and the wave- ''" ^' length is twice the length of the rod. 9. Fig. 45 shows how a rod fixed at both ends will break up into segments producing overtones : a shows the rod vibrating as a whole ; d, in halves ; r, in 3ds. In like manner the rod will divide _ 7 -^ into 4ths, 5ths, 6ths, etc. ; the harmonic series CO o c , . ' Fig. 46. bemg the same as ni strmgs. 10. Fig. 46 shows how a rod fixed at one end will break up into segments: a shows the rod vibrating as a whole ; l^ shows 1 88 MUSICOLOGY it vibrating in two segments, the node forming at one-third the whole length from the free end, the upper segment vibra- ting as a rod fixed at one end and the lower segment as a rod twice as long fixed at both ends, both therefore vibrating in the same time (in all cases, the rod must divide so that each segment will vibrate in the same time) ; c shows the rod vibra- ting in three segments, the upper segment being half the length of the others — the upper node, therefore, will be one- fifth the Avhole length from the end, so that each segment may vibrate in the same time, 1 1. Since a rod fixed at one end vibrates as a rod twice the length fixed at both ends, therefore, if we regard b as the lower half, the whole rod would be divided into three seg- ments; or, in the case oi c, into five segments. So we see that the harmonics of a rod fixed at one end will be in the proportion of the odd numbers, i, 3, 5, etc. ; the overtones corresponding to the even numbers not being able to form. „ — ^ ^ "g^— 12. Fig. 47 shows how a rod free at both ends will break up into seg- ^ ' ' ments : a shows the fundamental C ^ ! ~ ^ — ! — '^^^^ — ^^^=^ vibration of the rod, which forms a ^'"'' '*'■ node in the middle — each end vi- brating as a rod fixed at one end ; b shows the rod with two nodes, each one-fourth of the whole length from the end, the end segments vibrating as rods fixed at one end and the middle segment as a rod twice the length fixed at both ends, so that the time of each is the same ; c shows the rod with three nodes, the outer segments being half the length of the others but vibrating as rods fixed at one end, while the inner segments vibrate as rods fixed at both ends, so that the time of each is the same. 13. The arrows in each figure represent one phase of the vibration ; reversing the arrows would represent the opposite phase. In each figure we must conceive of the rod vibrating as a, b, c, etc., at the same time to form a conception of the PRINCIPAL SOURCES OF MUSICAL SOUND 1 89 compound nature of the vibrations; as the rod not only vibrates as a whole, but also in segments forming overtones with the prime, similarly as already explained in strings, ex- cept that the vibrations are longitudinal instead of transverse. 14. Pipes. A pipe closed at one end (the mouth is always an open end) contains a rod of air fixed at one end, which therefore vibrates like a rod fixed at one end ; and, like the rod, its harmonic series will be in the proportion of the odd num- bers, I, 3, 5, etc., the effect of which is to make the tone more dull and hollow. 15. A pipe open at both ends contains a rod of air free at both ends, which therefore vibrates like a rod free at both ends. 16. The prime note of the pipe closed at one end will be an octave lower than the same pipe open at both ends ; as the open pipe, like the rod free at both ends, contains a node in the middle in sounding its prime note and therefore sounds the same note as a closed pipe half the length. 17. The pitch of pipes is mainly a question of length. The width, however, afTects the quality of tone, the promi- nence of the overtones diminishing as the width increases; so that wide pipes produce hollow tones, while narrow pipes produce bright, penetrating tones. 18. In cylindrical pipes only every other overtone is prom- inent, producing a mellow tone. Flutes and clarinets are made on this principle. 19. Conical pipes tend to make the higher available over- tones relatively more prominent, thus producing a more pene- trating tone. Oboes and bassoons are made on this prin- ciple. 20. A peculiarity of the pipe, or tube, is that its prime tone can be made to jump from one harmonic to the next successively by gradually increasing the blowing force, so that any harmonic can be produced as the prime by blowing with the proper force. The narrower the pipe the more 190 MUSICOLOGY prominent the overtones, and the more readily will the prime jump from one to another. This principle is employed more or less in the various wood-wind and brass-wind (especially brass-wind) instruments. 21. Brass instruments without keys are called natural in- struments, and are capable of producing only the tones cor- responding to their natural harmonics. 22. Keys on brass instruments are for the purpose of lengthening the tube, thus lowering the pitch of the instru- ment, thereby producing lower harmonic series. In three- keyed instruments, the first key lowers the pitch a tone ; the second, a semitone; the third, a minor 3d; the first and second together, also a minor 3d ; the second and third, a major 3d ; the first and third, a perfect 4th ; and all three to- gether, an augmented 4th or diminished 5th; thus lowering the pitch from one to six semitones. By a proper selection from among the tones of the several harmonic series thus at command, the entire chromatic scale may be produced. 23. Other brass instruments (sliding trombones) have slid- ing sections which are drawn into six different positions, each successively lowering the pitch of the instrument a semi- tone. 24. Tone is produced on most of the brass instruments by the vibration of the player's lips as he blows into the cup- shaped mouth-piece ; the pitcJi depending on the length of the tube, the force of blowing, and the rigidity or laxity of the lips ; the quality depending on the width of the tube and the shape of the mouthpiece (the narrower the tube and the shallower the mouthpiece the brighter and more ringing the tone, and vice versa), and also on the size of the bell and shape of the lips. 25. In certain instruments of the flute order, the tone is produced by blowing at a certain angle and with a certain force against the edge of the mouth of the tube. 26. In the organ pipe and some other instruments, a PRINCIPAL SOURCES OK MUSICAL SOUND I9I mouthpiece directs the current of air at the proper angle against the edge of the tube ; and still other instruments have reeds in their mouthpieces to produce the necessary vibration. Clarinets have single-reed mouthpieces ; oboes, bassoons, and English horns have double-reed mouthpieces. So we see that the column of air enclosed in a pipe is set in motion by various contrivances, but on two general princi- ples : first, direct, by forcing the air against the edge of the pipe at a certain angle, thus producing a flutter of air at one end ; second, indirect, by communicating the vibration of some other substance (as the lips or reeds) to the air of the tube. 27. Resonance Boxes and Sounding-boards are not strictly sources of musical sound, as they merely strengthen but do not originate sound. They are used mainly to strengthen the sound of vibrating strings, as in the violin, piano, etc. The resonance box (with the enclosed air) or sounding-board vibrating in unison with the strings strengthens the sound by reason of its greater vibrating volume. The vibrations of the strings are transmitted to the box or board, which in turn transmits them to the atmosphere. The direct vibra- tions of strings can be heard only a very short distance, so that the tones of the violin, piano, etc., are due mainly to the resonance box or sounding-board. 28. Reeds. A reed in musical instruments is a thin tongue of wood or metal fastened at one end to a slotted plate in such a manner that the tongue will vibrate either within or against the slot (the former called free, the latter beating reeds). 29. In free reeds the tongue exactly fits the slot, except that it is enough smaller to vibrate freely within the slot. In beating reeds, either the tongue or the slotted surface is slightly curved to permit the necessary play. 30. In either case, if a current of air be passed through the slot, the tongue will be pressed alternately into or against the slot by the pressure of the air and out by the reaction of ig2 MUSICOLOGY its own elasticity, so that the air -".vill pass through in inter- mittent puffs. The end of the tongue is thinned down to a feather edge for high notes and sometimes weighted for low notes, so that the rapidity of vibration (and therefore pitch of tone) depends on the weight, length, and thickness of the tongue. The tone produced is due to the intermittent puffs of air, and not to the vibrations of the tongue, as might be supposed ; the reed being merely a mechanical contrivance for rapidly opening and closing the passage-way of the air, 31. Owing to the varying size and shape of the opening, during each opening and closing, the pulses of air produced are compound in form, resulting in compound tones with prominent overtones. When reeds are used with tubes, the quality is much improved by the sympathetic resonance of the tube, which strengthens the tones corresponding to its own proper tones and damps the others. 32. The lips in blowing brass instruments and the vocal cords in singing are classed with reeds, as they are made to vibrate in a similar manner by a current of air. MEMBRANES, PLATES, BELLS 1. These may be classed together under the head of sur- face vibrations. The preceding division involved only a lengthwise dimension, and the vibrations were confined to this one dimension; but surfaces admit of vibrations in all lateral directions, which evidently makes the analysis, as well as the series of overtones produced, more complex. 2. Membranes and Plates being similar in some respects may be considered together. Their vibrations may be di- vided into two general classes, radial and circular, which are usually more or less combined. 3. Fig. 48 may represent either a circular plate, or disk, or a stretched membrane as a drumhead. The circles within represent the nodal lines of the radial vibrations, and the diameters represent the nodal lines of the circular vibrations. J-KIXCIFAL SOURCES OF MUSICAL SOUND 193 4, If a drumhead be struck exactly in the center, vibra- tions will radiate from the center similarly as when a pebble Fig. 48. is dropped into the center of a circular pool of water; and when these radial vibrations break up into nodes they form nodal circles about the center (the signs + and — showing opposite phases of vibration) : a represents the vibrations of the prime tone ; and b and r, the vibrations of overtones. If the stroke is at one side of the center of the drumhead, other vibrations will also be produced, which tend to circle around the center and form nodal lines through the center, as at d and c. The vibrations circle both ways from the point of the stroke and meet at the opposite side, thus always dividing the membrane into an even number of segments by nodal diameters. Both systems of vibrations usually exist together, as at ./. 5. The overtones produced by either system are mainly in- harmonic w^ith the prime tone; and when the two systems are combined, many inharmonic overtones of nearly the same pitch are produced. 6. If we suppose Fig. 48 to represent a circular plate, or 194 MUSICOLOGY disk, then if this disk is supported at the rim, the conditions are somewhat similar to the drumhead (rigidity answering to tension) ; and if struck at the center, radial vibrations with nodal circles will be produced ; but if the disk is supported at the center (or suspended from the rim) and struck at the rim, the conditions are reversed, and circular vibrations with nodal diameters will be produced. 7. Under all other conditions, between these theoretical extremes, both systems of vibrations will be combined ; but, in general, the nearer the stroke is at the center the more the radial vibrations predominate, and the nearer the stroke is at the rim the more the circular vibrations predominate. 8. The overtones produced by plates differ from those of membranes, but are also mainly inharmonic with the prime, and involve also inharmonic overtones of nearly the same pitch (producing an empty tin-kettle quality) where the two systems of vibrations are combined. In both cases the series varies with varying conditions and cannot in any case be represented accurately. (The series produced by the cir- cular vibrations of plates, as in the gong, is similar to that of bells, p. 195 : 15.) 9. Fig. 48 represents only a few of the more simple forms of vibration of the simplest and most symmetrical form of membranes or plates. Any more irregular form of mem- branes or plates would evidently give only more complex re- sults. I-ong narrow plates would more properly be classed as rods. 10. Plates are not much used in music. Cymbals, how- ever, are used in military bands to mark time. 11. Membranes as used in music are of two classes: those w^ith definite pitch, as in the kettledrums; and those with- out definite pitch, as in the bass and tenor drums and the tambourine. 12. The kettledrum consists of a kettle-shaped shell of thin brass or copper, over the mouth of which is stretched PRINCIPAL SOURCES OF MUSICAL SOUND I95 the membrane called the drumhead; the. tone being rein- forced by the associated air-chamber thus formed. Kettle- drums may be tuned within the compass of a perfect 5th by varying the tension of the membrane. They are generally used in pairs of unequal sizes, and are usually tuned to give the tonic and dominant. Sometimes three are used to give the tonic, dominant, and sub-dominant. 13. The bass drum, the tenor drum (called also side drum, snare drum, and military drum), and the tambourine are used chiefly to mark the rhythm of music without any regard to their pitch ; they also excite the ear to a more acute ap- preciation of other sounds. The quality of tone and also the pitch, to a limited extent, may be altered by changing the tension of the drumhead. The pitch depends mainly on the size; thus, the bass drum, being larger, gives a lower pitch than the tenor drum. 14. Bells. A Bell is practically a bowl-shaped plate; the shape tending to eliminate the radial vibrations and empha- size the circular vibrations. In large bells the rim against which the clapper strikes is thickened (called the sound-bow). 15. In sounding its deepest tone a bell divides into four segments. It may also divide into any even number of seg- ments. In bells of uniform thickness throughout, the vi- bration numbers of the series produced are proportional to the squares of the even numbers, 4, 6, 8, 10, 12, etc., into which the bell divides itself. The squares of these numbers would be 16, 36, 64, 100, 144, etc. Reducing by dividing through by 16, we get i, 21, 4, 6^, 9, etc., which would produce the following series : I 2J 4 6i 9 C^, D,, C3, G«, D,, etc. (The lower figures indicate the octave within which the tone is found.) From Fig. 33 (p. 157) we find that the vibration number 4 corresponds to Q; and 9, to D^; their ratio, | or 2i, to D^. Sharping G^ by multiplying its 196 MUSICOLOGY vibration number (6) by |f (the minor semitone), we get 6|. None of these tones are harmonic to C except C3. 16. Overtones are more prominent in thin than in thick bells ; also in shallow than in deep bells. The deeper tones may be made mutually more harmonic by giving the bell a certain empirical shape. The tone also depends on the elasticity of the material and the thickness of the sound-bow. The body of the bell gives a deeper tone than the sound- bow, but not so loud. 17. The principal elements, then, which determine the tone of a bell, are weight, size, shape, thickness of the sound-bow, and elasticity of the material. If the bell is not cast perfectly homogeneous and symmetrical, irregularities naturally result. 18. Musical glasses may be classed with bells, as being similar in shape, though excited by friction instead of by strokes. When a bell is excited by a stroke the vibrations are necessarily transverse in character; but in musical glasses the vibrations are produced by passing the moistened fingers around the rim of the glass, and therefore, from the direction of the motion producing the vibrations, we see that they must be longitudinal (in the direction of the rim) in character. We would also judge, from the musical quality of the tone, that the series produced is harmonic. i PART FIFTH APPENDIX HISTORY OF THE DIATONIC SCALE 1. The earliest stages of music were naturally such as we still find among savage tribes, consisting of a few sounds dif- fering in pitch but without any system as to their relations and the mode of using them. But in the growth of music from this rude state different systems have developed, many of which, besides our own, are still in use, especially in orien- tal countries. 2. Most of these systems recognize the octave, and some also the 4th and 5th. The Chinese, Japanese, and some other countries use chiefly a scale of five tones, called the Penta- tonic Scale. The dividing points vary, however. The Hin- doos divided the octave theoretically into twenty-two parts, but practically into seven degrees changeable into distinct modes in a manner somewhat similar to our own. The Per- sians divided the octave into twenty-four parts, each corre- sponding to half our semitone. The Arabs divided the octave into sixteen or seventeen parts (according to different author- ities). Other systems might also be mentioned. 3. As to the ancient music of the Egyptians, nothing is clearly known regarding the scale except that the octave was largely subdivided. The music of the Chaldeans, Babyloni- ans, Assyrians, Phoenicians, and Hebrews is supposed to have been of a similar character. 198 MUSICOLOGY 4. Our own musical system can be traced back directly to the Greeks, and still farther back through the early migrations into Greece, to the Persians and Hindoos. 5. In the ancient four-stringed Greek lyre, the outside strings were tuned to the interval of a 4th, and the middle strings were more or less varied. The lyre thus gave a series of four notes, which was called a tetrachord, and which has ever since remained a prominent element of the music scale. 6. About 670 B.C. Terpander increased the number of strings to seven, making two tctrachords with the middle string common to both. The tuning depended entirely upon the ear; there was yet no means for accurately measuring or recording the pitch of tones. 7. This seems to be the history in brief of the development of the scale up to the time of Pythagoras, about 550 B.C., who is regarded as the father of musical science. 8. Pythagoras discovered the relations between the length of strings and the pitch of tones, and that intervals could be given definite numerical values. He found that the simplest division of the string (into two equal parts) gave a tone which his ear told him was the most closely related to the funda- mental, thus fixing the interval of the octave. He next found that two-thirds of the string gave a tone which formed a natural subdivision of the octave, and also that three-fourths of the string gave another natural subdivision of the octave. He no doubt observed the relationship of these two intervals (now called perfect 5th and perfect 4th) to each other; that either reckoned upward and the other downward gave the interval of an octave, and that the octave therefore was equal to their sum. Thus these three intervals were estab- lished as the most important intervals in music. 9. The tones of the scale thus fixed (taking the octave on C) are C F G C, leaving a perfect 4th between C and F and between G and C to be filled up. Since the 5th and 4th are natural intervals, their difference APPENDIX 199 (now called a tone) is also a natural interval, as it occurs naturally between F and G. This is taken as the most con- venient interval to subdivide the perfect 4ths.* This led to an unequal division, as the perfect 4th contained two tones, and a semitone (nearly) over, and admitted of three arrangements, namely, the semitone may be below, in the middle, or above, all of which were used. Thus the octave was made up of two tetrachords with a separating tone in the middle. 10. Having established the scale of the octave, it was easy to extend the scale by adding octaves of the notes already fixed. In this way the scale was extended to two octaves. 1 1. The scale thus established has remained practically un- changed for more than 2000 years to the present time. How- ever, a slight correction was found necessary. The difference between the perfect 4th and perfect 5th was a major tone (ratio I) ; using this as a measure, the Pythagorean scale would be as follows : T 9814 S27248/, CDEFGABC The two major tones together form the interval of a Pythag- orean major 3d having a ratio | X | = f j, which is inhar- monious ; but we find very near it the simple harmonious ratio |f=f, which is called the true major 3d. The differ- ence between the true major 3d (|) and the major tone (I) is J -f- I = \'*, which is called the minor tone. The difference between the Pythagorean and the true major 3d (f^ -^ ||) is |i. This added to the semitone gives |f-| X f i = t|. The corrected scale then is as follows : T » E 4 a SlSn CDE-FGABC T V » vioviov * V i0v'9v i« — o I A g A -j-AxsA g A -5 Aft- A y? — 2 ♦Another mode of construction is by perfect sths and octaves. Thus, beginning at C and taking sths upward, the tirst will give G, the next D; dropping down an octave, the next will give A, the next E; dropping down an octave, the next will give B; then, taking a 5th downward from C, we get F. This is practically the same as alternately taking sths up and 4ihs down, since a 5th up and an octave down is the same as a 4th down. 200 MUSICOLOGY which we see by the simpler ratios is more harmoni- ous. 12. The major 3d in the lower tetrachord could be corrected, either by shortening the interval between C and D or between D and E ; but D is an important note, as it forms a perfect 5th to the G below, G being second in importance to the tonic C ; the correction is therefore made by shortening the interval between D and E. In the upper tetrachord the cor- rection is made by shortening the interval between G and A, thus correcting the major 3d between F and A as well as the major 3d between G and B. This puts the I, IV, and V triads in perfect tune; but if A were major, the IV triad (F A C) would be out of tune. 13. Observe also that the minor 3d between D and F (ratio V X if = If) is not a true minor 3d, while all the other minor 3ds are true minor 3ds (ratio | X if = t) ! also that the ratio of the 5th between D and A is VX if X|XV = ij, while the ratio of the perfect 5th is |; therefore the li triad (D. F. A.) is still not in perfect tune — its minor 3d and its 5th being too flat by the difference between the major and minor tones (f -^-V = ll), called a Comma, or a little more than one-fifth of a semitone (log. of |J- is .00539; ^^S- of semitone is .02509 : /^W = i nearly). This is a necessary evil, since it can be remedied only by shortening the interval between C and D, instead of between D and E, which would put the V triad out of tune. 14. The correction of the Pythagorean major 3d to a true major 3d was first suggested by Didymus about the beginning of the Christian era, but was not fully adopted till toward the close of the middle ages. 15. These distinctions do not apply to the equal tempera- ment scale, all of the intervals of which, except the octave, are more or less out of tune; temperament being merely an attempt to conform the scale to a practical simplicity in keyed instruments. The tempered scale has come gradually into APPENDIX 201 use with the growth in popularity and prominence of keyed instruments. 1 6. The Greeks named the fifteen tones in the two octaves by different Greek words. The Romans adopted the Greek scale but named the fifteen tones by the first fifteen Roman letters from A to P. Gregory, about 600 A.D., adopted the system of repeating the same letters in each octave, but used capitals in the first octave, small letters in the second octave, and double small letters in the third octave ; which distinc- tions were finally dropped. 17. Guido of Arezzo, about the beginning of the eleventh century, discarded the tetrachord method of grouping the tones and arranged the twenty notes then in use into groups of six, called hexachords ; and to facilitate the singing named the six notes Ut, Re, Mi, Fa, Sol, La, which were suggested by a verse of a hymn to St. John, in which they occurred in order as the first syllable of each successive line, each line be- ginning a note higher. Ti has been added to fill out the octave ; and Ut, being a bad syllable to sing, has been changed to Do. Guido also introduced the system of nota- ting the scale on lines and spaces. 18. Franco of Cologne, about the middle of the twelfth century, is credited with introducing the system of represent- ing the relative time value of notes by their shape. The sys- tem of dividing the music into measures by drawing bars across the staff can be traced to about the year 1574. 19. The characters ff, b, and < originated during the eleventh and twelfth centuries. The y and the S came from different forms of the letter J3 and were used originally to show the position of that letter only- — 1? showing the lov/er, and t| the higher position. The H originally was a St. Andrew's cross. 202 MUSICOLOGY ANCIENT GREEK MODES 1. Mode refers to the octave form, or mode of arrange- ment, of the steps and half-steps in the octave. On page 198 : 9, the octave was shown as made up of two tetrachords with an added (separating) tone between ; the added tone was sometimes placed below or above, instead of between. 2. The tetrachord may be arranged in three principal ways, as follows (letting — represent the step and ^ the half-step) : — ^ ^ (Lydian tetrachord) ; _ ^ — (Phrygian tetra- chord) ; ^ _ _ (Dorian tetrachord). The names refer to three people or provinces of Greece, from whom they are sup- posed to have originated. 3. The modes or octave forms were termed Lydian, Phryg- ian, or Dorian, according to the kind of tetrachords involved. If the added tone was below, the term Hypo was prefixed; if the added tone was above, the term Hyper was prefixed. There was, however, some variation in the names, as shown below. 4. The Earlier Greek Modes. These were seven in num- ber, as follows : 1. Lydian C—^D— £ ^ F - t — J\ —B - C 2. Phrygian J) - E ^ T - C - 71 - B ^ C - J) 3. Dorian JS -- T - C - Jf -S'^^^-D-S 4. Hypo-Lydian (Syntono-Lydian) T — C ~ ^ ~ -B ^ Q — J) — E ^ T 5. Hypo-Phryjfian (Ionian or lastian) C— J\ — S "^ C — Jf — E ^ F — w 6. Hypo-Dorian (.^^olian or Locrian) Ji — ^^ C — J) — £1 '~ F — G — Ji ^ j Hyper-Dorian ^ j ^^^l^-^n/ - F- C - A - B \ Mixo-Lydian S [ £^ C-J)JJ^f — C'^^W^ B^ {€) (See also p. 54, Fig. 13.) 5. We see that all these modes, if starting on the proper letter as shown above, exactly fit into the scale without in- 1234567 f# 1 4i . 1 1 i 1 - APPENDIX 203 volving accidentals (corresponding to our keys on the same letters, with the signatures omitted). In this sense, each mode represents a certain section of the scale, as indicated in Fig. 49. It should not be inferred, however, that they were used only in this sense. They may each be taken at any pitch, which would fig. 49. thus involve accidentals to express them in our notation (remembering that ^'s, b's, and also the staff and roman letters were not used by the Greeks). In either sense, the modes represent different octave forms. 6. The Greek music (as all ancient music) was monophonic. The melodies were at first confined to the compass of the tetrachord ; later, two tetrachords were combined and the scale thus extended to the octave ; and still later, the scale was extended to two octaves. In the meantime, the Hypo-Dorian (corresponding to our old minor mode, see p. 53: I, 2) became the mode in most common use, and thus acquired prominence over the others in a manner analogous to our major mode, 7. The Later Greek Modes. About the fourth century B.C., at the time of Aristoxenus, the scale had been ex- tended to two octaves, and there were thirteen modes in use, which were afterward increased to fifteen, as shown on p. 204. 8. These might also be represented on the staff, similarly as in Fig. 49, by placing before each a signature correspond- ing to the number of sharps or flats in each. 9. It will be seen that the modes are named according to the character of the octave between the dotted lines in which all the modes stand parallel. It will be seen also that as be- tween the extremes of each they are all the same, differing only in pitch, and are merely transpositions of the Hypo- Dorian. In this sense, therefore, they are merely transposing 204 MUSICOLOGY PQ Ft; Pl^^ ) I ) I I ) ft; I J I "^ S '=^ I > ^)> ) cq ft:, cq -— -V-vS«J f*^ fs fs K I J I J- i<, to t I ) •« 1 •i I I I ,1 ■ I <0 <0 < "^ ) J < I { .RjJ.-B^ } I J I ) J J -I -Rt; p^..^g...e^._f5^- ) > 4 I a ►2 o U3 ^ S a cij q^ o:, *T3 O.C c c $.2 c .2 2 •^ OJL rt -3 'bb "s S* "o ^ ^ 6 O o 1— ' a. APPENDIX 205 scales (corresponding to our fifteen old-minor keys). How- ever, they may have been used largely in a parallel sense as distinct octave forms. 10. There has been much dispute as to their real character, some holding the theory of transposing scales, and others the theory of distinct octave forms. 11. The Early Church Modes. In the fourth century A.D., Bishop Ambrose of Milan, in view of the existing con- fusion, established four scales for church use, corresponding to the original Greek Phrygian, Dorian, Hypo-Lydian, and Hypo-Phrygian, and established also their character as dis- tinct modes, or octave forms, by fixing the lowest note of each as the tonic. A little later Pope Gregory added to these four others derived from them by ranging the scale from a 4th below the tonic to a 5th above. These were called P/a^d/, while the Ambrosian ones were caWedAut/u'^iftc. 12. In time, much confusion again existed, and in 1547 Glareanus proposed six authentic and six plagal modes, mak- ing twelve in all, which he called the Dodecachordon. Of these, the ten which remained are given in the following out- line : Church name Name given by Original Glareanus Greek name fist mode (Authentic) Dorian Phrygian 2d " (Plagal of 1st) Hypo-Dorian 3d " (Authentic) Phrygian Dorian 4th " (Plagal of 3d) Hypo-Phrygian •g 1 5ih " (Authentic) Lydian Hypo-Lydian S I 6th " (Plagal of 5th) Hypo-Lydian U I 7th " (Authentic) Mixo-Lydian Hypo-Phrygian [sth " (Plagal of 7th) Hypo-Mixo-Lydian Secular ^ Our Major Mode Ionian Lydian Modes "( Our (old) Minor Mode ^olian Hypo-Dorian Observe that by some confusion Glareanus misapplied the original Greek names. 13. The eight modes as established by Ambrose and Greg- ory are known as the Early Church modes. The other two, 2o6 MUSICOLOGY corresponding to our major and minor modes, were used only in secular music ; but these last are the only ones that have survived through the development of polyphonic music, since they are capable of more harmonic combinations than the others. TABLE OF COMMON MUSICAL INTERVALS Intervals Ratio Semitone value Intervals Ratio Semitone value ( Perfect \ 1 r Minor 1* 8.14 Unison •; . \ (Aug. -^ minor semitone S6 * .70 6th ■ ^^J*^^ s * s 8.84 Minor - major semitone 1 6 * 1. 12 [Aug. ("Dim. 7th 1 Minor 8SS 9.76 9.26 2d Major •] minor tone major tone V * 1 * 1.82 2.04 ¥ 9.96 [Aug, II 2.74 [ Major 1 5* "8" 10.S8 ( Dim. 3d \ Minor ( Major m 1* 1* 2.24 3.16 3.S6 /-> . S Dim. ^^'^-"^ \ Perfect 1 11.29 12.00 ( Dim. 4th "I Perfect (Aug. 4.2S 4.98 5.90 ■ Comma I Pythagorean Comma IIHIi .22 .24 ( Dim. 5th ^ Perfect (Aug. 3 * 6.10 7.02 Enharmonic Diesis 1 S8 .42 If 7.72 Schisma »S80B .02 1. The derivations of the ratios marked with a * are given on pp. 157:2; 159:11; 160:15. From these the others are easily found. 2. If we subtract ( -^ ) the major semitone E to F from the major 3d E to G||, we get the aug. 2d F to G^ (f -^ j| = -g^). The divi. 7th is tlTe inversion of this (i--^1t=-'t/)' 3. If we add ( X ) two major semitones we get the dim. 3d (If X 11= III). The aug. 6th is the inversion of this /8 .i. 256 — 885N APPENDIX 207 4. If we add two major 3ds we get the aug. 5th (| X |=f|). The dim. 4th is the inversion of this (f -r- y|= it)- 5. The aug. 4th (tritone) from F to B equals the major tone F to G plus the major 3d G to B (| X f = f|). The dim. 5th is the inversion of this (t ^~ If = ID- 6. The minor 7th is the inversion of the major 2d (f -j- 1 = V")- The major 7th is the inversion of the minor 2d 7. The dim. octave is the inversion of the aug. unison /2 _L_ 85 . iH\ \J -ST — Sj)- 8. The Coninia is the difference between the major and minor tones (f -^ V- = |f ). This comma is always meant unless otherwise designated. 9. The Pythagorean Comma is found by taking twelve 5ths ap and seven octaves down, thus: Sv/SvSv 3v3v3v3v3v-'5v3V-''V3v1 VlV^V ^ A 3 A ^ A 3 A 2^ A 3 A -g^ /\ g- /\ ^" A g' /N 3 /\ -g- /^ 2 /S 2 /N 2 /\ iXiXiXi = i^= fliHi- It is also equal to the sum of the comma and schisma (.22 + .02 = .24). 10. The Enharmonic Diesis is the difference between the major and minor semitones (If ^It^^tH)- 1 1. The Schisma is found by taking eight 5ths and a major 3d up and five octaves down, thus: ix|x|x|x|x|-x|x|xfxixixixix i=|fm. It is also equal to yV of a Pythagorean Comma (.24-7- 12 = .02). TEMPERAMENT I. Temperament is an attempt to conform the scale of na- ture to the keyboard. From the earliest invention of the keyboard it has contained, in general practice, twelve keys or notes to the octave. A few attempts have been made to make the keyboard more nearly conform to the scale of na- ture by increasing the number of keys; but such instruments were too complicated and never came into general use, so that twelve remains as the most practical number. Hence 208 MUSICOLOGY the object of the twelve-note systems of temperament is to limit the scale to twelve notes to the octave, so tuned as to give the best practical results. 2. There are only two systems of temperament that have been used to any extent : the Equal Temperament, which is the one now in most general use ; and the Mean- Totie Tem- perament, which was the one formerly in most general use, and is still at least of historical interest. 3. Equal Temperament, however, is the older. It was very early observed that twelve 5ths minus seven octaves equaled the Pythagorean comma (about | of a semitone), and Aristoxcnus (fourth century B.C.) is said to have advo- cated the idea of distributing this difference among the twelve 5ths (the 5ths thus tempered are called equal 5ths). This is the basis of equal temperament; for, taking twelve equal 5ths up, thus, C, G, D, A, E, B, F #. C #, GJt, D^^, A t, E # or F, B jj or C, or twelve equal 5ths down, thus, C, F, B1^, Eb, AI', T)^, GK Ci' or B, Fb or E, A, D, G, C, we get the entire chro- matic scale. In either case, tAvelve 5ths takes us once around the key-circle (which is 7 octaves) to the starting-point ; but if the 5ths were true, we would have landed a Pythagorean comma past the starting-point. If we had dropped down an octave whenever we exceeded that interval, or had alternately taken 5ths up and 4ths down (two successive 4ths down from B to C^), we would have gotten the chromatic scale direct. 4. The great advantage of equal temperament is that all the keys are equally in tune and may all be used. The principal objection is the sharp major 3ds which result from this tem- perament ; to remedy which, the Mean-Tone Temperament w^as invented, 5. Mean-Tone Temperament can be traced back to Zarlino and Salinas, two Italian authors of the sixteenth century, and in 1700 was in general use and continued in general use till about 1840, when the practice in tuning keyboard instru- ments began to return to Equal Temperament. APPENDIX 209 6. Mean-Tone Temperament is based on the principle that four 5ths up and two octaves and a major 3d down (I Xf X 2 X |XiXiX| = |o-) give the comma; and hence, if in tuning by 5ths up and octaves down we use 5ths diminished by i of a comma (called mean-tone 5ths), we would get true major 3ds. The difference between the major tone (f) and the minor tone (V) also equals the comma (§H-V' = |i-) ; hence, if we flat the major tone or sharp the minor tone by | of a comma, we would get the mean-tone, or tone midway be- tween. Now if we begin at C and tune by mean-tone 5ths, the first 5th up will give G, which will be i of a comma flat; taking a 5th downward from C above will give F, which will be :f of a comma sharp : therefore the interval between F and G is contracted |- of a comma and becomes a mean-tone. Now a 5th up and an octave down is the same as a 4th (in- verted 5th) down, and tuning an octave down after every two 5ths up is the same as alternately taking 5ths up and 4ths down ; and therefore every whole tone thus fixed is neces- sarily equal to the difference between the 4th and 5th, since no other intervals are used. Therefore tuning by mean-tone 5ths necessarily makes all the tones mean-tones; from which fact the temperament takes its name. 7. The comma (|^) is about 1 of a semitone, and -\ of a comma would be about aV of ^ semitone, and therefore the mean-tone 5th is about ^V of ^ semitone flat ; while the equal 5th of the equal temperament is about --V of a semitone flat. However, the error is still quite imperceptible in the chord. 8. The minor 3d is also gV of a semitone flat (being the difference between the true major 3d and the mean-tone 5th), while in the equal temperament it was J of a semitone flat, and is therefore much improved. 9. In the following table we have mean-tone tempera- ment, true intonation, and equal temperament compared. The second and third columns correspond to the table on p. 166. 2IO MUSICOLOGY Mean-Tone True Equal Temperament Intonation Temperament Semitones Semitones Semitones c 1 2. GO 12.00 12.00 R 10.83 10.8S 11.00 A 8. 89 8.84 9.00 G 6.97 7.02 7.00 F 5-03 4.98 5.00 E 3-86 3-86 4.00 D 1-93 2.04 2.00 C 0. 0. 0. 10. To find the values in the first column, first find the value of the comma, thus: 3.86 — 2.04 = 1.82, equals minor tone between D and E; 2,04 — 1.82 = .22, equals difference between major and minor tones, equals comma ; .22 ^- 4 = .05 = ;f of comma, equals amount of temperament of the mean-tone 5th. 1 1. 7.02 — .05 = 6.97 = G ; 12.00 — 6.97 = 5.03 = F ; we may take E directly from the middle column, as it is a true major 3d; 3.86 ^ 2 = 1.93 = D, or mean-tone ; 5.03 + 3.86 =8.89= A; 6.97 + 3.86=10.83=6. Or by taking 5ths up and 4ths down, thus, G — F=D, D + G = A, A — F = E, E -j- G = B ; or by taking 5ths up and octaves down (whenever the octave is exceeded) ; or we might use still other combinations. 12. In the preceding table we have the natural scale of C, corresponding to the white notes of the keyboard. As there are only five black notes in each octave of the keyboard we can extend the temperament to only five other keys, since in mean-tone temperament the sharps and flats do not coincide as in equal temperament. We might take the first five sharp keys or the first five flat keys; but the first three sharp keys (G, D, and A) and the first two flat keys (F and Bl?) are the most used. 13. The first three sharps are FH, Cit, and G#; the first two flats are Bi? and Ek To tune these we must make F|^ a true major 3d to D (1.93 + 3.86 =: 5.79) ; C|?, a true major 3d to A (8.89+ 3.86 — 12.00 = .75); G it, a true major 3d to E (3.864-3.86 = 7,72); bI', atrue major 3d below D (12.00 + 1-93 — 3-86 = 10.07); Ei?, a true major 3d below G(6.97 APPENDIX 2 I I — 3.86 = 3. 1 1). These, with the white notes, would give us all the notes of the six keys C, G, D, A, F, and B 1^, all equally in tune. The other keys are more or less out of tune, accord- ing to the number of notes missing- In the key of E, D^ is missing and we would have to use E "? in its place, which differs by the enharmonic diesis (see table, p. 206) ; this is what is called a wolf (comparing the effect to the howling of a wolf). Similarly in the key of E 1^, we would have to use GJI for A 1?. The other keys contain more wolves. 14. In some old organs the black keys between D and E and between G and A were divided and tuned to give both the sharp and flat notes, thus bringing the keys of E and E ^ into tune. 15. The merit of the Mean-Tone Temperament is the greater harmoniousness of the most useful keys, and its de- merit is in its limitation to those keys. On the other hand, equal temperament has the freedom of all the keys ; for which reason, the balances finally turned in favor of equal temperament. TUNING 1. Tuning in any temperament is done as far as possible by 5ths, 4ths, and octaves, till all the notes in the tuning octave (usually near the middle of the keyboard) are tuned. These are called the bearings, from which the other notes above and below are tuned by octaves, 2. Only the 5ths and 4ths are required to beat, as the octave is always a true interval and should not beat. The 5ths and 4ths are first tuned true, then the 5ths contracted and the 4ths expanded till the required number of beats are heard. 3. The usual method of tuning the bearings consists in al- ternately taking 5ths up and 4ths down. If we take the octave above middle C as the tuning octave, the notes would be tuned in the following order (C being first tuned by a standard tuning-fork or other standard). ,. A 2 12 MUSICOLOGY EQUAL TEMPERAMENT 1.2 1.3 1.4 1.5 1.6 1.7 .9 .g i.o I.I I.I 1.2 C Cf D D« E F F« G GJf A A« B C 72 94 II 61 83 10 5 12 Fiu. 50. 4. The figures below the letters show the order in which the notes are tuned: first, taking a 5th up to G; second, a 4th down to D; third, a 5th up to A; etc. The figures above the letters are the number of beats per second in each 5th up or 4th down (the number varying with pitch). First, calculate the number of beats in the 5th and 4th at a certain pitch, then, to find the number at any other pitch, multiply or divide by the ratio of the interval at which the 5th or 4th is taken above or below the pitch at which the beats are already found. 5. Let i5 = number of beats; iV= number of vibrations in the lower note; 71/= number of vibrations in the upper note; — = the ratio of the true interval in its lowest terms: then the algebraical formulas for finding the number of beats would be (i) ^= Nni — Mn if the interval is flat (less than the true interval), or, {2) B =^ Mn — Nni if the interval is sharp (greater than the true interval). 6. It was shown on p. 167: 2 that the blending harmonics were in exact accord with the ratio of the interval ; thus, in the 5ths (ratio |) every second harmonic of the upper note and every third harmonic of the lower note blend, therefore multiplying the lower note by the upper figure of the ratio or the upper note by the lower figure would give the lowest blending harmonics, and similarly for any interval ; so that the number of vibrations in the lowest blending harmonics of two notes may be found by multiplying the vibrations of the lower note by the upper figure of the ratio or the vibrations of the upper note by the lower figure of the ratio. If the interval is true, these products will be equal (.Vw ^^ Mii) and APPENDIX 213 the lowest blending overtones will be in unison; but if the interval is not true, the products will differ, and the lowest blending overtones will beat by not being in perfect unison. 7. It was shown also (p. 171 : 10) that the difTerence between the lowest blending overtones determined the number of im- perfect consonance beats ; therefore, Nm — Mn or Mn — Nm (according as the interval is flat or sharp) equals number of beats. 8. The amount of temperament of the equal 5ths is jV of the Pythagorean comma (see p. 208 : 3) ; this is called the Schisma, the ratio of which is ||41l (see table, p. 206). 9. If middle C = 256 vibrations, G = 256 X | = 384. Flat G by the schisma, thus, 384 -^ Iff f| = 383.56 +. Therefore N = 256, M= 383.56, ;;/ = 3, ;; = 2. Substitu- ting these values in formula ( I ), we get ^= .88. If middle C = 270 vibrations (the highest standard), we would get, by figuring from this standard, B = .92. Therefore .9 (which is the mean) may be taken as the number of beats in the 5th above middle C for all standards of pitch. 10. If middle C = 256, F = 256 X | = 341.33. Sharp F by the schisma, thus, 341.33 X ||4f| = 341 .72. Therefore A^=2 56, i/= 341.72, ;;/ = 4, ;/ = 3 ; and from formula (i) we would get ^=1.16; or if middle € = 270, wt would get B =^ 1.23. Therefore, 1.2 may be taken as the num- ber of beats in the 4th above middle C for all standards of pitch. 11. From these values we may obtain the number of beats at any pitch ; thus, the 4th from G down to D is a major 2d higher than the 4th from C to F (1.2 X -| — 1.3), which gives 1.3 for D. The 5th from D up to A is a major 2d higher than the 5th from C to G (.9 X | = i.o), which gives i.o for A, etc. 2 14 MUSICOLOGY MEAN-TONE TEMPERAMENT 3-3 3-6 30 41 3-3 4-<5 2.5 2.5 2.7 4.4 S-" C C« D £!> E F F« G G5 A Bb B C 72 4 6183 5 3 I 2 Fig. 51. 12. In Mean-Tone Temperament, tune the notes in the same order as before up to Gjlf, then begin again at C and tune a 4th up to F, then another 4th up to Bb, then a 5th down to 13. In the table on p. 206, the value of the comma is .22 s. i of .22 s = .055 s, which is the amount of temperament of the mean-tone 5th. Also the value of the schismais .02, which is the amount of temperament of the equal 5th (.055 -^ .02 = 24), from which we find that mean-tone is 2| times equal \.^xn- perament. Therefore the simplest method of finding the beats over the letters in Fig. 51 is by multiplying the number over the corresponding letters in Fig. 50 by 2|, except F, Bl?, and El', in tuning which we again started at C; but we found the number of beats for C to F in equal temperament was 1.2, therefore 1.2 X 2f= 3.3 = number for F in mean- tone; F' to Bt? is a 4th higher, therefore 3.3 X f = 4.4 = number for Bl?; B 1? to E 1? is a minor 3d higher than the 5th from C to G, therefore 2.5 X | = 3.0 = number for E b. 14. In a similar manner, we might calculate the beats for any other tuning octave in either temperament. 15. It is necessary to count the number of beats in 10 seconds to remove the decimals; 3.3 beats in one second equals 33 beats in 10 seconds. CALCULATION OF PITCH NUMBERS I. From formula (i) v^= Nm — Mn we get formulas (3") ^= ^^ '"^"^ (4) M= ^1^. From formula (2) B = Mn - Nm we get formulas (5) iV = ^^"~ ^ and (6) M = '^'"\'^ ^ ' APPENDIX 2 I 5 2. Formula(3) is used for 5ths down ; (4) for 5ths up; (5) for 4ths down ; and (6) for 4ths up. 3. In taking 5ths down or 4ths up -5 is + because the tun- ing note is raised to contract the 5th or expand the 4th, while in taking 5ths up or 4ths down B \s — because the tun- ing note is lowered for the same reason. 4. In the table on p. 216 the number of vibrations is cal- culated for each note of the octave above middle C for either temperament. The number of the formula which applies stands opposite each section, and the value of each letter of the formula is indicated. 5. The figures over the letters of the equal tempered scale are practically accurate for all standards of pitch ; but owing to the greater amount of temperament of the mean-tone 5th, it is necessary to divide the range between the extreme stand- ards of pitch into three sections, using the figures over the letters of the mean-tone scale for the middle section, and adding .1 to each for the higher section, and subtracting .1 from each for the lower section. 6. In Mean-Tone Temperament, middle C ranges in the different standards from 252.7 (Handel's pitch) to 283.6 (Durham's pitch). The pitch (264) given in table is called Helmholtz's pitch, and is nearly midway between, 7. By means of beats we may also find the number of vi- brations in any note, without the known vibrations of some other note from which to figure. 8 . From formula ( i ) we get B = [m — n — ) ^, or A^= r^ N D From formula (2) we would get iV= r,; so that ^ is + or — ^ ^ m — n izL N according as the interval is flat or sharp. 9. Now — = the actual ratio of the interval (being the vi- brations of the upper note divided by the vibrations of the lower), and — = the ratio of the true interval. Let it be re- Equal Temperament 1.2 1.3 1.4 1.5 1.6 1.7 .q .9 i.o l.i I.I 12 C CJ D D$ E F FJG G« A A« B C 7 20 4 II 6 18 ^lo 5 12 Mean-Tone Temperament 3-3 3-6 30 41 3-3 46 2.5 2.5 2.7 4.4 3.0 CC5D Eb E FF«GG«AB!JBC 72 4 6183 5 3 I 2 fN = I m = J 1" = L n = fM = B = 256.0 3 768.0 2 )767.1 .383-6 II 50. 8 = 4I114Q.5 fN m B n = 287.4 3 = 862.2 1.0 2)861 2 f M n Lm f N m = 430.6 3 = I2QI.8 IS Z 4)1200.3 322.6 3 967.8 ■M = B = B F| . m = 4)1084.8 CS GJ = D$ = AJ fN = ; ni ■-= 4 ■ B = 264.0 3 792.0 2-S I n = 2)789.5 394.8 3 Mb = 1184.4 3-6 lm = 4)1180.8 1 m = 295.2 3 I n = (M = n = 885.6 2.7 2)882.9 441.4 = 3 s J |B = 1324.2 4 I 4)1320.1 (-N = m = 4 B = t n = 'M = n = 330-0 3 990.0 3.0 2)987.0 4Q3.S 3 S J » B = m = .M = n = 1480.5 4.6 4)1475.9 36S.9 3 5 |B = lm = I 106.7 3-3 4)1103.4 m = 275-9 3 4 J B = 827.7 2.5 n = 2)825.2 B . m = 4)1366.7 FJ C« rN Jm 412.6 =» GJ 264.0 B = 1 B In (M J" 1056.0 3.3 3) '0.S9.3 353-1 4 1412.4 4-4 3) 1416.8 472-3 2 944.6 30 3 '947 -6 3 '5-9 Bb Bb APPENDIX 217 quired to find the pitch, or number of vibrations, in Tune three true 5ths up and then a true major 6th down (f X f X I X I — li), which will give an imperfect octave above the given note, the ratio being |^ instead of f ; AT hence - = |4 and ^ = f- Count the beats made by this imperfect octave and suppose them to equal 32 in 10 seconds, or 3.2 per second, then ^ = 3.2 ; but as \\ is greater than f the interval is sharp, so that B is — 3.2. Therefore N =^ zzj\ = 128. (See Poles, Philosophy of Music, p. 3 1 5 — Fourth 4 Edition.) CHARACTER OF THE DIFFERENT KEYS 1. There is a noticeable difference in the effect produced by the different keys, which may be described in a general way as follows : 2. The key of C major expresses earnestness, resolution, and decision. Its relative, a minor, is tender, melancholy, and plaintive. 3. G major (signature i sharp) expresses feeling from quiet and calm to bright and joyous, and is a favorite key. Its relative, e minor, is mournful and persuasive. 4. D major (signature 2 sharps) expresses grandeur, triumph, animated feeling, and lofty purpose. Its relative, b minor, is very melancholy and bewailing. 5. A major (signature 3 sharps) expresses sincerity, hope, confidence, love, and cheerfulness. Its relative, f\ minor, is dark, mournfully grand, and full of passion. 6. E major (signature 4 sharps) is the most brilliant key, and expresses joy, splendor, and magnificence. Its relative, c% minor, is the most melancholy key. . 7. B, or C I? major (signature 5 sharps or 7 flats), when loud, expresses pride and boldness; when soft, expresses purity and clearness — not a favorite key. Its relative, g% or 2l8 MUSICCLOGY a? minor, has a peculiar, sad, tender pathos, suitable for funeral marches. 8. F major (signature i flat) is contemplative and ex- presses peace, joy, and religious sentiment. Its relative, d minor, expresses anxiety, solemnity, and grief. 9. B b major (signature 2 flats) is bright and clear, though not specially energetic or grand. Its relative, g minor, is sad, dreamy, and romantic. 10. Eb major (signature 3 flats) is full and soft and capa- ble of a great variety of expression ; its general character is serious, solemn, firm, and dignified. Its relative, c minor, is soft, earnest, longing, and passionate, 11. A i^ major (signature 4 flats) is sweet, unassuming, delicate, and full of feeling. Its relative, f minor, is dismal and gloomy. 12. D i? major (signature 5 flats) is sublime, deep, and tragic. Its relative, b'^ minor, is somber and mournful. 13. Gt? major (signature 6 flats) is very soft and rich. Its relative, c^ minor, is full of melancholy tenderness. 14. The character, however, of a piece of music does not depend wholly on the key, but also on the treatment as re- gards time, rhythm, etc., so that it is possible in this way to largely counteract the character of the key. But it may be observ^ed that the character of the same piece of music is sen- sibly altered by a change of key. Therefore the character given to each key only shows the general tendency of each independently of other considerations. 15. These characters cannot be due either to a difference in the intervals or to temperament ; for all keys, if equally in tune, regardless of temperament, are composed of the same intervals differing only in pitch. This seems to indicate that the real cause is the difference in pitch. This view is sup- ported by the fact that pitch affects the quality of tone as regards the extent and prominence of the harmonics. ABBREVIATIONS USED IN MUSIC. The followinof Is a list of the abbreviations in most common use Accel. — Accelerando. Accom. — Accompaniment. Adg:. — Adagio. Ad lib. — Ad libitum. Affett.— Affettuoso. Affrett.— Affrettando. Ag. — Agitato. AH.— Allegro. Allgtt.— Allegretto. Air Sva— Air ottava. Al seg. — Al segno. And. — Andante. Anini. — Animato. Arp.— Arpeggio. A tem. — A tempo. Brill. — Brillante. C. B.— Col basso, Contrab- basso. Cad. — Cadence. Cal. — Calando. Cant. — Canto. Cantab.— Cantabile. Ch. — Cboir organ. Col. ott.— Coir ottava. Col. vo.— Colla voce. Con esp.— Con espressione. Cresc. — Crescendo. D. C. — Da capo. Decres. — Decrescendo. Diap.— Diapasons. Dim. — Diminuendo. P. Mu8.— Doctor of Music. Del. — Dolce. D. S.— Dal Segno. Esp. — Espressivo. F.— Forte. Falset. — Falsetto. Ff. — Fortissimo. Fin. — Finale. F. O.— Full organ. Forz. — Forzando. F.p. — Forte- piano. G. O. — Great organ. Grand. — Grandioso. Intro. — Introduction L.— Left hand. Leg. — Legato. liO. — Loco. Maest. — Maestoso. Magg. — Maggiore. Marc. — Marcato. Met. — Metronome. Mez. — Mezzo. Mf. or Mff. — Mezzo forte. M. M. — Maelzel's metro- nome. Mod. — Moderate. M. P. — Mezzo piano. M. V. — Mezza voce. Obb.— Obbligato. Sva or 8a— Ottava, Octave. Sva alta — Ottava alta. Sva bftH.— ottava bassa. P. — Piano, pousise. Ped.— Pedal. P. F. or Pf.— Piano-forte. P. f.— Piu forte. Pianiss. — Pianissimo. Pizz. — Pizzicato. PP. — Pianissimo. 4tte — Quartet. Ball. — Rallentando. Becit. — Recitative. B.H.— Right Hand. Bitar. — Ritardando. S. — Segno, senza, sinistra, solo, subito. Scherz.- Scherzando. Seg. — Segno, segue. St. or Sfz.— Sforzando. Sos. — Sostenuto. Spir. — Spiritoso. Stacc. — Staccato. Tern. — Tempo. Tr.— Trillo. Trem. — Tremolando. U. C— Una corda. Unis. — Unison. V. — Verte, voce, vocl, vol- ta, volti. Var. — Variation. Viv. — Vivace. MUSICAL DICTIONARY. Abbreviations— (F) French; (G) German; (I) Italian; (old E) Old English; (S) Spanish; (L) Latin; (Gk) Greek. A. A ballata (I)— In the ballad style. Abbandono, con (I) — With self-aban- donment, passionately. A battuta (1) — In strict time. Abendlied ((i) — An evening song. A bene placito (I) — At pleasure. Ab Initio (L) — From the beginning. A rappelia (I) — (Ij In the Church style; (2) Church music, duple time. A capriccio (I) — At will. Accelerando or Accelerato (I) — Grad- ually increasing the pace. Acciaccatura (I) — A short grace-note. Accoiade (F) — A brace, uniting several staves. Achromatic— Not chromatic. A cinque (I) — In live parts. Adag:ietto (I) — A diminutive of Ada- gio ; slower than Adagio. Adagio (I) — Slowly. Adaffissimo (I»— Very slow indeed. Additato (I)— Fingered ; having signs to show what fingers are to be used. Addolorato (I I— In an afflicted manner. Ad libitunt (D— At will. Ad piacitum (L)— At pleasure. AlTabile (I)— In a pleasing manner. Affannato (I)— In a distressed manner. AfTanno, con (I )— Mournfully. AfTannoso (I)- Mournfully, with grief. Affetto, con (I)— With affection. Aflfettuoso (I )— Affectionately. Affezione, con (I)— With tenderness. Affrettando (I)— Hastening the time. A fior di lalibra (I }— Speaking or sing- ing very softly on the lips. Agevole (I (—With facility, lightness. Agilita, con (I)— With sprightliness. Agilite (F) — Lightness and freedom in playing or singing. AKitamento (I)— Restlessness. Agitato ri)— An agitated, restless style. A ia (F), Al. All', Alia (I)— Like, in, at, in the style of. A la nieme (F)— In the original time. A la niesure (F) — In time. Alia breve (I)— A direction that the notes are to be made shorter. Air «va alta (1) The octave higher. All 8va bassa (I)— The octave lower. .AlleKramente (I) .loyfully, cheerfully. .Vllegretto (I)— Slower than Allegro. .Vllegrettino (I)— Not SO fast as Alle- grctio. .Vllegrezza (I) .Toy, rejoicing. Allegrissimo (I) — Extremely quick. Allegro (I) — Literally, Joyful. Quick, lively. In music it is sometimes qualified as: Allegro assai (I) — A quicker motion than simple allegro. Allegro con brio (I) — Quickly and with spirit. Allegro con fuoco (I) — Rapidly and with fire. Allegro con moto (I) — With sus- tained joyfulness. Allegro con spirito (I) — Joyfully and with spirit. Allegro di bravura (I) — A movement full of executive difficulties. Allegro furioso (I) — Rapidly and with fury. Allegro ma nou troppo (I) — Lively, but not too fast. Allegro moderato (I) — Moderately quick. Allegro vivace (I) — Lively and brisk. Aliemande (F) — Alemain, AUemaigne, Almain. A dance in duple time. .\llentando (I) — (Gradually slackening the time. All' unisono (I) — In unison or octaves. .M Negno (I) — To the sign. Alt (I) — The notes in the octave begin- ning with G above the treble stave iire said to l)e in alt. .Mtieramente lit Troudly, grandly. AltiNsimo (It — The highest. The notes in the octave beginning with G on the fourth leger line above the treble stave are s;ud to be in altissimo. Alto clef— The C clef, placed upon the third line of the stave. A mezza voce (11 — (1) With half the strength of the voice; (2) The qual- ity lictween the chest and head v(")ice; (.'<) The subdued tone of instruments. Aniore, con (H — With love, affection. Andante (1) — Literally, Walking. Slow, graceful, distinct and peaceful. cantabile. Slow, and in a sing- ing style. con moto. Faster than Andante, and with animation. Andantino (I) — Slower than Andante. Angenebm (G) — Pleasing, agreeable. Anglaise (F), Anglieo (I)— The Eng- lish country dance. MUSICAL DICTIONARY. 221 Aniiuato (I) — Animated, lively. Animoso (I) — Lively, energetic. Anschwellen Hi) — Crescendo, Antispastus (L) — A foot, consisting of two long between two short sylla- bles. Antode (Ok) — Responsive singing. A piaeere (1) — (1) At pleasure; not strictly in time, ad libitum; (2) The introduction of a cadenza. A poco a poco (I) — More and more; by degrees. A poco piu lento (I) — A little slower. A poco piu ino8so (I) — A little faster. Apotome (Gk) — A major semitone, B to C. Appassionato (I) — With feeling. Appenato (I) — With an expression of suffering; with bitterness or grief. Appoggiatiira (1) — A note leant upon in singing or playing, as a grace note. A piinto (1) — In exact time, precise. A quatre mains (F), A quattro mani (1) — For four hands on one instru- ment. Arcato (I) — With the bow, as opposed to pizzicato, plucked with the finger. Aria (1) — An air, tune, song, melody. bulTa (1) — A song with some degree of luimor in the words, or in the treatment of the music. d' entrata (I) — The first or en- trance air sung by any character in an opera. Armarius — Precentor. Arpe8:s:io (I) In the style of a harp. Assai (Ij — Very. Allegro assai, very quick. A tempo (I) — In time. A tre (I) — For three voices, instru- ments, or parts. Attacca (I) — Commence at once, with- out a pause. A vista (I) — At sight; used instead of a prima vista, at first sight. B. Barearola (I), Barcarolle (F) — A sim- ple melody in imitation of the songs of Venetian gondoliers. Barre (F) — In guitar or lute playing, the pressing of the forefinger of the left hand across all the strings. Battere, 11 (I)— The down-stroke in beating time. Bellicoso (I) — Warlike, martial. Ben (I) — Well. Ben marcato, well and clearly marked; ben sostenuto or ben tenuto, well sustained. Benedictus (L) — Portion of a Mass. Berceuse (F) — A cradle song. Bindung: (G) — Suspension. Bis (L) — Twice. Bolero (S)— A Spanish dance in triple measure, accompanied with singing and castanets. Bouche fermee, a (F) — With closed mouth; humming. Bourree (F) — A dance-tune in com- mon time. Bravour (G), Bravura (I)— Dash, bril- liancy. Breit (G)— Broadly. Brillante (I and F)— Brilliant. BrIIlo (I) — Joy, gladness. Brio, con (I) — With spirit, vigor. Brioso (I) — Joyfully, vigorously. Brise (F) — Broken chords, arpeggios. BuflTa, feni.. Buffo, mas. (I) — Comic, Aria Iniffa, a humorous melody; opera buffa, a comic opera. Burletta (I) — A comic operetta; a farce interspersed with songs. c. C clef— The clef showing the position of middle C. Cabiscbol, Cabiscola— The precentor in a choir. Cachucha (S)— A Spanish dance. Cacophony — Harsh-sounding music Cadence — A shake or trill, run or divi- sion, introduced as an ending, or as a means of return to the first sub- ject. Cadenz (G) — Cadence. Cadenza (I)— (1) A passage intro- duced toward the close of the first or last movement of a concerto; (2) A running passage at the end of a Tocal piece. Calando (I)— With decreasing Tolume of tone and slackening pace. Calata (I) — Italian dance in 2-4 time. Calcando (I)— Hurrying. Calore, con (I) — With heat, warmth. Caloroso (I) — Warmly, full of pas- sionate feeling. Canon— A composition in which the music sung by one part is, after a short rest, sung by another part note for note. Cantabile (I) — In a singing style.. Cantante (I)— A singer. Cantata (I) — Originally a mixture of recitative and melody for a single voice, but now a short work In the form of an oratorio. Canticle — A song or hymn in honor of God, or of some special sacred event. Canto (I) — The upper voice-part in concerted music. Cantor— Precentor. 222 MUSICOLOGY. Canzona, Canzone (I) — (1) An old form of soug; (2) An iustrunieutiil composition in two, three or four parts, containing contrapuntal de- vices. Canzonetta (I) — A little short song, tune, cantata, or sonata. Capischol (L) — Precentor. Cappella, alia (I) — In the ecclesiastical style; in duple time. Capo (I) — Head, commencement. Capricoio (I) — A freak, whim, fancy. Caprlocioso (II — Whimsical. Caprice (F) — Capricclo. Castrato (I) — A male singer with a soprano voice. Cavatina (I) — A melody of a more simple form than the aria. Celere (I) — Quicls, swift. Chaconne (F)— A slow dance In % time. Chanson (F)— (1) A song; (2) A national melody; (3) A part-soug. Chansonnette (F) — A little song. Chef-d'oeuvre (F) — The master-work of any composer. Chlave (I)— Key or clef. Chica — A popular dance among the Spaniards and the South American settlers descended from them. Chromatic— Notes not belonging to a diatonic scale. A chromatic scale consists of a succession of semi- tones. CIna (F), Cinque (I)— A fifth part In concerted music. Coda (I) — (1) The tail of a note; (2) An adjunct to the ordinary close ol a piece or song. Col, Coir, Colla, CoIIo (1 1— With the. Coloratura ( 1 ) — Divisions, runs, trills, cadenzas, and other florid passages in vocal music. Comodo (I) — Easily, without haste. Con (D— With. Contra (I) — Against. In compound words this signifies an octave be- low, as Contra-gamba, a 16-ft. ganiba. Contrappunto (I) — Counterpoint. Contrapunkt (fJ) — Counterpoint. Contrapuntal — Belonging to counter- point. Contra tempo (I)— Against time. (1) The part progressing slowly while another is moving rapidly; (2) Syn- copation. Contrepartie (F)— Counterpart, oppo- site. The entry of a second voice with a different melody. Contretemps (F) — .\gainst time. Coranto (U — An Italian form of the country dance; a running dance. Counterpoint — "The art of adding one or moi-e parts to a given melody." Couplet — (1) A verse of a song; (2) Two notes occupying the time of three. Crescendo (I) — Increasing; a grad- ual increase in the force of sound. Crotchet — A quarter note, one-fourth of the value of a semibreve. Da (I)— From, by, of, for, etc. Da capo, or D. C. (I) — From the be- Da capo al fine (I) — From the begin- ning to the sign Fine. al segno (1) — Repeat from the beginning to the sign S. al segmo (I)— From the sign S. ecrescendo (I)— Decreasing gradu ;illv the volume of tone. ecupIet^.V group of ten notes played 111 tVio fim<^ rif aitrlit- or .f(inr pcuplet — .\ group or ren note in the time of eight or four. Diatonic— One of the three genera of music among the Greeks, the other two being the chromatic and en- harmonic. Diminuendo (I)— Decreasing in power of sound. Ditone — An interval of two major tones. Divertimento (I) — (1) An instrumen- tal composition of a light, pleasing character; (2) Pot-pourri. Dolce (I)— Softly, sweetly. Dolente, Doloroso (H In a plaintive, sorrowful style; with sadness. Dominant— The fifth degree of a scale. Droite fF)— Right; as main drolte, the right hand. Duple time — Even time. E. Echelle (F)— A so.ile; as, ^chelle chro- matique, chromatic scale; echelle diatonique, diatonic scale. Eclogrue (F)— A shepherd's song; a pastoral piece. Rlferig: (O)— Zealously, ardently. Kinfalt (O)— With simplicity and dig- nity. Einlgrem Pomp, mit (O)— In a some- what pompous manner. Einsclilafen (O)— To slacken pace and dimlnlsli the power. Kin wenlg lebendigr (01— R.ither lively. Blegant (F). Elegante (1)— With ele- gance of style. Emozione, con (I) — With emotion. Empflndung (O)— Emotion, passion. En badinant (F)— Scherzando. Energia, Energico (1) — With energy. Enfler (F) — To swell; to increase in sound. Ensemlile (F)— Together ; the whole. The general effect of a musical per- formance. MUSICAL DICTIONARY 223 Entr'acte (F) — Music played between the acts of au opera, drama, etc. Kntrante (I), JJntree (F)— Eutry, in- troduction, or prelude. KntuMiasmo (I) — Witli enthusiasm. Kntwurf (G) — A sketch. ICpiceclion ((Jli) — A dirge; elegy. Kpilenia ((ik) — Vintage songs. Kpithalamium (Gk) — A nuptial song. Epoile (Gk) — An after-song. Krhaben (G) — Exalted, sublime. Krn»itlicli (G) — Earnestly, fervently. Krotique (F) — A love-song. K8i>agnuolo (I)— In the Spanish style. Espressivo (I) — Expressive. E»oi)ular cliaracter introduced iiito it. Liedertafel ( C I- Literally, Song-table. A society meeting tor the practice of part-songs for men's voices. Linke Hand (<;)— Left hand. Lire (K) — A lyre or harp. Loco (1) — In its proper place. A direc- tion to return to the proper pitch after having played .•in oct:ive higher. Loure iir liOiivre (I'") .\ dtiiice. Liistig ((;> Merry, merrily. Lyre — One of the most ancient stringed instruments. MUSICAL DICTIONARY 225 M. M — Abbreviation of mezzo, mano, main, manual. Ma(lrig:al — A word of doubtful orijiin. It became a general term for secular compositions, of which there were various forms differing in style. Maestoso (I) — With dignity, majesty. Maestro (I) — Master. Maggriolata (I) — A song sung iu cele- bration of the month of May. Main (F)— The hand. 3Iaitre de chapelle (F) — Choirmaster. Malinconico (I) — With sadness, sorrow. Maniere (F) — As maniera affettata, an affected style; maniera languida, a languid, lifeless style. Maennerstimnien (O) — Men's voices. Mano (I)— Hand. Marzlale (I) — In a martial style. Mazurka — A Polish dance of lively, grotesque character Meistersinger (G) — A title given to the most renowned musician of a town- ship or district in Germany during the Middle Ages. 31elangre (F) — A medley. Meno (I) — Less. Mestoso (I) — Sad, pensive. Mezzo (I) — Half or medium. Minuet — The name of a graceful dance iu triple time. Miserere — The 51st Psalm sung in the Tenebrae service in the Roman Cath- olic Church. Mit (G I— With. Moderate (I> — Moderately. Moll (G)— Minor. Molto ID— Much, very. Monotone, to — To recite words on a single note without inflections. Morceau (F) — .\ piece; a small compo- sition of an unpretending character. Mordente (I)— A beat, or turn, or passing shake. Morgenlied (G) — Morning song. Mosso (I) — Moved. Motet — .\ vocal composition in har- mony, now generally set to sacred words. Motiv (G) — (1) The sort of movement indicated by the opening notes of a sentence; (2) A subject proposed for development. Munter (G) — Lively. Musette (F) — (1) A small bagpipe; (2) A melody of a soft, sweet character, written in imitation of the bagpipe- tunes; (.3) Dance-tunes and dances in the measure of those melodies; (4) A reed-stop on the organ. Muthig (G)— With spirit. N. Nacli Belieben (G) — Ad libitum. Naehspiel (G) — A postlnde. Nachtstueeke (G) — Night-visions. The name of four pianoforte pieces by R. Schumann. Naeh und naeh (G) — By little and lit- tle, by degrees. Naivement (F) — Artlessly, unaffectedly. Nobilmente (I)— With grandeur, nobly. Noel (F) — "Good news." A word used as a burden to carols at Christmas. Non (I)— Not. Notturno (I) — Originally, a kind of serenade; now a piece of music of a gentle and quiet character. Nuances (F) — Shades of musical ex- pression. Obbligato (I) — An instrumental part or accompaniment of such impor- tance that it cannot be dispensed with. Offertoire (F), Offertory— A piece of music performed during the offer- tory. Opera (I)— A dramatic entertainment of Italian origin in which music forms an essential part. Opus (L) — A work. Oratorio (I) — A composition for voices and instruments illustrating some sacred subject. Overture — An instrumental piece writ- ten as a prelude to an opera, ora- torio, or other work. Paean (Gk» — The ancient choral song addressed to Apollo. Sung before or after a battle. Part-sonff- A vocal composition, hav- ing a striking melody harmonized by other parts Passionato (I) — In an impassioned manner. Pastoral, Pastorale fl) — CI) A simple melody in 6-8 time in a rustic style; (2) A cantata, the words of which are founded on [lastoral incidents; (.'?) A complete symphony, wherein a series of pastoral scenes is de- picted by siiuiid-painting, without the aid of words. 226 MUSICOLOGY Pafetiro III, Fathetique (Fl — (1) Pa- tbetic; (J) In a pathetic iiiauiier. I'entatonir h«-ale -The iiaiiie jiiven to the ancient musical scale, which is easiest described as that formed by the black keys of the piano-fortp. I'etit choeur (F) — The chorus which originally consisted of three parts only. Peu a peu (F)— Little by little. Phrasing: — The proper rendering of music with reference to its melodic form. Piacere, a (11— At pleasure. Pianissimo (I) — Extremely soft. Pianississimo (I) — As softly as possi- ble. Piano (D— Softly. Piu (I)— More. Pizzicato (I) — A direction to violinists to produce the tone by plucking the string with the finger. PlagHl cadence — The cadence formed when a subdominant chord imme- diately precedes the final tonic chord. Poco (I»— A little. Poi (I)— Then. Poiaoca (I) — Polish. A title applied to melodies written in imitation of Polish dance-tunes. Polka A Rohemian dance of world- wide popularity, in l.'-4 time. I'omx»oso (1 ) — Pomi)ously. Portamento (1) — A lifting of the voice, or gliding from one note to another. Postlude — A concluding voluntary; a piece played at the end of service. Pot-pourri (F)— A medley; a collec- tion of various tunes linked together. Pralltriller (G) — A transient shake. Precentor — An official in a cathedral who leads and directs the choir, etc. Precipitato (I) — With precipitation. Preciso (I)— With exactitude. Prelude — A movement played before, or an introduction to, a musical work or performance. Premiere (F) — First. Prestissimo (I) — Very fast indeed. Presto (I)— Fast. Prima (I, fern.) — First. Prime (I, masc.) — First. Proscenium— (1) The quadrangular space behind the logeum or stage; (2) The stage front. Provencales — Troubadours of Pro- vence. Psalter— A book of Psalms. Q. Quadrille — A well-known dance, con- sisting of five movements. Quasi of. (I)— As if, or in the style Raddolcendo (I)— With gradual soft- ness and sweetness. Rallentamento (D— At a slower pace. Rallentando (1)— Getting gradually slower. Rapidita, con (I)— With rapidity. Recitative — Musical declamation; a kind of half-speaking and half-sing- ing; a composition without any de- cided or rhythmical melody. Reel (old E)—.\- lively rustic dance, popularly supposed to be Scotch, hut prol)ably of Scandinavian origin. Reveil (old E)- Music which wakens from sleep. .\ signal given by drum to soldiers at d.iwn. KImpsodie (G). Rhapsody— A compo- sition of irregular form, and in the style of an improvisitioii. Rh.vthm— The arrangement of niusicil phrases or sentences in regular met- rical form, as regards accent ■■md nuantity. Rigore (I) Strictness. Risoluto (1) With resolution. Kitardando (I)- With gradually in- creasing slowness of pace. Ritennto (I) — Holding back the pace. Robusto (I) — Robust, strong, powerful. Role (F) The part in a drama as- signed to an iictor. Romance (F. SI, Romanza (11 — Any simple rhytlimical melod.v which is suggestive of a romance. Romanesca (I) — .\n Italian dance. Komcra — .\ Turkish dance. Rondeau (F». Rondo (I) A composi- tion goner.ally in two parts, with the principal subject often repe.'ited. Roulade (F) — An embellishment; a flourish. Round — .\ composition in which sev- eral voices, starting at stated dis- tances of time from eacli ottier. sing each the s.ime music, the combina- tion of all the parts producing cor- rect liarmony. Roundel A rustic song; a dance in wliicli all .ioin hands in a ring. Roundela.v — i\) A poem. cert.\in lines of wliich are repeated at intervals; (2) The tune to which a roundelay was sung. MUSICAL DICTIONARY 227 Salto (I) — (1) A dance in which there is much leaping and skipping; (2) A leap or skip from one note to another beyond the octave; (.'{) Counterpoint in which the part added moves in skips. Sans (F)— Without. Saraband, Sarabanda (I), Sarabande (F) — A Spanish dance of Moorisli origin, for a single performer, ac- companied with castanets. Satz (G) — A theme, subject, composi- tion, piece, movement. Srherzando (I) — (1) Playful, lively: (2» A movement of a droll character. Scherzo (I) — Literally, a jest, applied to a movement in a sonata or sym- phony, of a sportive character. Srhlummerlied (G) — A slumber-song. Srhluss (G) — The conclusion; finale. .Srhlu!«sstueck (G) — Finale. Schmelzend (G) — Literally, Melting away. Dying away ; diminishing. Srhnell (G)— Quick. Srhottisehe (G)— Literally, The Scotch dance. A slow dance of modern in- troduction, in 2-4 time. Score — A copy of a musical work in whicli all the component parts are shown either fully or in a com- pressed form. Sdegno, con (I) — Scornfullv ; disdain- fully. Sec (F) — Dry, unadorned, plain. Seoondo (I) — Second. Segno (I)— The sign S. Segue (I) — Follows, succeeds. Seguidilla (S) — A lively Spanish dance, similar to the country-dance. Semplice (I) — Pure, plain, simple. Senipre (I) — Always, ever. Senza (I)— Without. Sequence — The recurrence of a har- monic progression or melodic figure at a different pitch or in a different key to that in which it was first given. Serenade — (1) Originally a composi- tion for use in the open air at niglit. generally of a quiet, soothing char- acter : (2) A work of large propor- tions in the form of a symphony. Sestetto (1) — A composition for six voices or instruments. Sforzando (I) — Forced. Siciliana (I) — A graceful dance of the Siiilian peasantry. Siegeslied (G) — A song of triumph. .Silhouettes (F) — Sketches; recollec- tions. Singspiel (G) — Opera. Slargando (I) — Widening, opening. Slentando (I) — Slackening the time. Soave (I) — Agreeably, delicately. Soggetto (1) — Subject, theme, motive. Solennel, le (F) — Solemn. Sol-faing — A vocal exercise in which the notes are called by the several names Do, Re, Mi, Fa, Sol, La. Ti. Sonata — A composition consisting of three or four movements, generally for a solo instrument, and in sym- phonic form. Sonatina (I), Sonatine (F) — A short sonata. Sopra (I) — Above, before, over, upon. Sorda, Sordo (I) — MuflJed with a mute. Sortie (F) — A voluntary played at the close of a service. Sostenuto (I) — Sustaining. Sotto (I) — Below, under. Sous (F) — Under. Spianato (I) — Smooth, level, even. Spirito, con (I) — In a spirited, lively, animated, brisk manner. Staccato (I)— Detached, taken off. Stanza (I) — A verse or subdivision of a poem ; a strophe. Stentato (I) — Forced, emphasized. Stimme (G) — (1) The voice; (2) Sound; (.'5) The sound-post of a violin or violoncello; (4) A part in vocal or instrumental music; (.5) An organ- stop or rank of pipes. Strepitoso (I) — Noisy, impetuous. Stringendo (I) — I'ressing, hastening on the time. Stueck (G) — A piece, air, composition. Suave (I) — Sweet, agreeable, pleasant. Sub (D— Under. Sub-bass — A pedal register in the organ, of Z2-tt. tone. Subdominant — Tlie fifth below or the fourth above any key-note. Suite (F) — A set, series, or succession of movements in music. Super (L) — Above, over. Suess (G)— Sweet. Symphony — (1) A composition for an orchestra, similar in construction to the sonata, which is usually for a single instrument; (2) Formerly overtures were called symphonies; (.'?) The introductory, intermediate and concluding instrumental parts of a song or other vocal piece are also called symphonies. Syncopation — Suspension or alteration of rhythm by driving the accent to tliat part of a bar not usually accented. T. Takt (G) — Time, measure, bar. Tanto (1) — So much; as much. Tarantella (I) — A rapid Neapolitan dance in triplets, so called because I it was thought to be a remedy against the supposed poisonous bite of the tarantula spider. Tardamente (I) — Slowly. Tardo (I)— Slow, dragging. 228 MUSICOLOGY Tedinik (G) — A general name for the systems, devices and resources of musical art. Tenia (I) — A theme or subject; a melody. Tenii)<> (II~Tiiiie or measure. Temps (I'')— Time; the parts or divi- sions of a l)ar. Tenu (I''), Teniito (I)— Held on; sus- tained; kept down for the full time. Terzetto (1) — A composition for tliree lierformers. Tetraehord — A scale-series of four notes. The word in its modern sense sijrnifies a half of tlie octave scale. Thesis (Gk)— (1) In metre, the heavy tone or vocal accent; (2) In rliythm, tlie non-accent, or up-beat. Threnody (Gk) — An elegy, or funeral song. Tie— (1) A curved line placed over two or more notes in the same position on the stave, to show they are to be played as one; (2) When two or more quavers, semiquavers, etc., are united, instead of being written with separate tails, they are said to be tied. Timbre (F) — Quality of tone or sound. Timoroso (I) — Timorous; with liesi- t.ition. Toccata (I) — (1) A prelude or over- ture; (2) Compositions written as e.vercises; ('.'<) A fantasia; (4) A suite. Tonfiiehrungr (G) — (1) A melodic suc- cession; (2( Modul.ition. Tonic, Tonico (1 I, Tonique (F) — (1) The key-note of .my scale; tlie ground- tone or basis of a scale or key; (2i The key-chord in which a piece is written. Tonltuenstler (G) — A musir'ian ; a niu- sii-al artist. Tremolo (I)— Trembling. Tranquillo (I) — Tranquilly, calmly. Tres (F) — Very. Troppo (I) — Too much. Tufti (I» — All. Every performer to take part in the execution o-f the passage or movement. Tyrolienne — (1) A song accompanied with dancing; (2) Popular songs or melodies in which the jodle (q. v.) is freely used. u. T'ebellilang phony. (G) — Discord, caco- Un (I)— One. In poco (I)— A little. V. Va (I)— Go on. Vaudeville (F)— A play with songs set to popular tunes. Veloce (I)— Rapid, swift. Versette (G)— Short pieces for the organ intended as preludes or vol- untaries. Verte (D— Turn over. Vibrante (I) — Vibrating, tremulous. Viel (G)— Much. VierBesanfi: (G)— Song for four parts. VierliaendiK ((J)— For four hands. Vif (F> Tiively, brisk, quick. Vigoroso (H— Vigorous, bold, forcible. Vistamente (I)— Briskly, quickly. Vivace (1)— Lively, quickly, sprightly. Vivarissimo (I) — Very lively. Vivamente (I), Vivement (F) — Lively. Voce (1). Voix (F), Vox (L)— The voice. Volante (I) — Flying. .\pplicd to the execution o-f a rapid series of notes, either in singing or pl.iying. Volkslied (G) — A popular song. Voluntary — An organ solo played be- fore, during or after any office of the rinirch ; lience. called respec- tively introductory, middle or con- cluding voluntary. Vorsaenger (G) — Precentor. Vorspiel (G) — Prelude; overture. w. Wecbselgesang (G)— Responsive antiphonal song. Wehmuth (G)— Sadness, sorrow. WieKPnlied (G) — .\ lullaby; a cradle- song. Wuerde, mit (G))— Witli dignity. Zaleo— A Spanish national dance. /aertlirh (G)— Softly, delicately. Zeichen (G)— A musical sign, note, or character. Zeloso (I) — Zealous, energetic. Ziemlirli (Gl — Moderately. Zurueckhaltungr (G)— Retardation. -lO — 1 •0—2- CO— I |-^ — a^ai. — I |--J hr O— lO— I )-io — S— CO— I 1-^ —ai-oi~\ l-o>-^---"-4 COMBINKD MAJOR AND MINOK KEY TABLE. I I I ; I I II" I I 1 1 1 I I I I I I ** (See pM;re ;>7 : L'O. lil) ria<'C (■()nil)ii)ati()ii triad foriimla (cut from page 220) on the ciirveil lines between the major and minor liey tahlfs: and liold in place liy clip?! as in Chart 1 Csee slit marks). c C =" •5 a (B r 5 S c g >> > 5^ -S 3 S >- c U !<1 C 01 V :i'2 2 " >> T3 2 O ti! 3 OS .5- as 14 DAY USE RETURN TO DESK FROM WHICH BORROWED MUSIC LIBRARY This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. DfO 2 iqRfl JAN 21963 fFRl-^lSf^ OCT 9 1972 DEC! 5 1974 LD21A-10m-5,'65 ,, . General Library (F4308slO)476 University of California Berkeley MT6.L88 C0371 40828 U C. BERKELEY LIBRARIES CD37mDfiEa DATE DUE Music Library University of California at Berkeley HS'-iiiiMiiiiiiiiiiiiiiiiiiiiii