fyxntll WLuivmxty J itaeg BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Henrg W. Sage 1S91 A if] 1/3 iikfi^mi... Cornell University Library MT 52.C81 An easy method of modulation by means of 3 1924 022 380 988 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924022380988 AN EASY METHOD OF MODULATION BY MEANS OF UNIVERSAL FORMULAS BY J. H. CORNELL. NEW YORK. — G. SCHIRMER. 35 UNION SQUARE. COPYRIGHT — 1884. G. SCHIRMER. ? Copyright 1884 by G. Schirmer. PREFACE. Almost every teacher of Harmony can count among his amateur pupils a large (perhaps the greater) number, on whom he feels that his instructions are in great part wasted, at least in the sense that they are not applied to some practical pur- pose, as, for instance, in musical composition or improvisa- tion, for which such pupils have no talent. It is for this large class of harmony-pupils that I have prepared the present work, as offering to them some tangible fruit to be gathered from their studies. For, if such persons will study harmony, they must be assumed to have some aim or other "in so do- ing; and since they do not compose, the next best applica- tion of the study of harmony is undeniably the practice of Digressive Modulation — a purely mechanical thing, with- in the reach of all. Those, therefore, who desire to study harmony sufficient- ly to be able to pass correctly from one key to another — an accomplishment rarely met with in music-amateurs, even though advanced piano-pupils, — will find assistance in the present work. I have endeavored to combine in my system — IV — the utmost simplicity and facility. Simplicity of material, — the only chords used being major and minor Triads (the knowledge of other chords being dispensed with, and even the major and minor Triads appearing mostly in one— the simplest — form only); simplicity of chord-connection, — the Triads being, in each modulation, without exception, inter-connected from beginning to end, admitting the application of the very easiest kind of chord-connection, and rendering faulty progres- sions impossible. With all its simplicity, the method is, so far as it goes, very thorough and exhaustive, and forms an indispen- sable and solid basis for the most complete course of harmon- ic study. The key-harmonies being severally represented by numerals, the pupil has to translate these numerals . into the corresponding Triads, according to the key, and is thus taught to think for himself, — especially when, instead of this translation from given formulas, he has to construct the for- mulas for himself, on given principles. In fact, to go through this method thoroughly is to acquire, with positive certain- ty, an exhaustive knowledge of the principal Triads of all the keys, major aud minor; which knowledge is, as every one knows, the ABC of the study of Harmony. It should be . borne in mind that in this method every thing is professedly subordinated to the one great considera- tion — the greatest simplicity and facility possible ; a system of elegant and ornate modulation would presuppose on the student's part a much more advanced harmonic knowledge than is required for the present method. By way of Supplement to this book I have prepared three Tables * in which the 872 modulations are classified * Published separately under the title: "Tables of the 24 major and minor keys". G. Schirmer, N. Y. — V — under every possible aspect. They will be found useful for reference, and bave this advantage, tbat tbey are applicable to tbe practice of Modulation in general, having no exclusive bearing on the present or any other special method. The "Primer" frequently referred to in this work is the author's "Primer of Modern Tonality", 2 d edition, 1877, New York, G\ Schirmer. I cannot conclude without acknowledging my deep obli- gations to the admirable theoretical system of the late lamented Carl Priedrich Weitzmann, of which I profess to be a follower. New York, June, 1883. J\ HI. Cornell. COKTEKTS. Page Chapter I. Modulation, in general 1 „. II. Major and Minor Triad. — Key. — Mode. — Questions . . 2 „ III. Major and Minor Triads of the Major and the Minor Mode. — Questions 4 IV. The Major and the Minor Mode in all the keys. — Exercises . 7 r, V. Inter-relationships of the Triads of a key. — Exercises . . 8 „ VI. Connection of related Triads, in general 13 „ VII. Connection of the quint-related Triads of a Major Key. — Exercises , 14 „ VIII. Connection of the quint-related Triads of a Minor Key. — Exercises . ... 19 IX. The Tonic Cadence. — Exercises . ... 20 X. Relationships of Keys. — Summary . . .... 29 „ XI. Relationship of one and the same Triad to several keys. — Exercises 35 XII. Classification of Modulations 39 „ XIII. Chains of quint-related major Triads, as preparatory to the modulations of the first division. — Exercises .... 42 r XIV. Application of the principles laid down in Chapter XI. — Table 1 50 „ XV. Modulation, in the Major Mode, to keys in the l»t grade of quint-relationship 54 „ XVI. Modulation, in the Major Mode, to keys in the 2d and 3d grades of quint-relationship 57 XVII. Modulation, in the *Major Mode, to keys in the 4u», 5th, Qth and 7 th grades of quint-relationship . .60 „ XVIII. Preparation for modulation in the Minor Mode (second di- vision). Chains of quint-related Minor Triads .... 63 , XIX. Modulation, in the Minor Mode, to keys in the 1st, 2d, 3d and 4 th grades of quint-relationship 66 „ XX. Modulation, in the Minor Mode, to keys in the 5"», 6th and 7th grades of quint-relationship 69 XXI. Connection of tierce-related Triads, in either Mode, pre- paratory to abbreviations in modulating. — Exercises . 71 „ XXII. General Principles of abbreviation in modulation .... 73 — VII — Page Chapter XXIII. Short Cut (Mod. 6, 1st Division), through the identity of (IV) with ii of another key 74 „ XXIV. Short Cut (Mods. 7, 8, 10, 12, 14, 1st Division), through the identity of (iv) with vi of another key ... 76 „ XXV. Short Cut (Mods. 9, 11, 13, l*t Division), through the identity of m with (iv) of another key 78 „ XXVI. Modulation, in the Major Mode, to keys in the 8th, 9*, 10th. nth j 13th an d 14th grades of quint-relationship 80 XXVII. Short Cuts in the 2« Division (Mods. 35—40), through the identity of (V) with VI of another key ... 90 „ XXVIII. Modulation, in the Minor Mode, to keys in the 8th, 9th, 10th, nth, 13th and 14th grades of quint-relationship 93 „ XXIX. Modulation in the 3 a Division, up to seven grades of Elevation and Depression 100 „ XXX. Modulation in the 4th Division, up to seven grades of Elevation and Depression 105 „ XXXI. Harmonic Variants and Extra-short Cuts Ill „ XXXII. Addenda to the modulation-formulas of the 1st Division H2 „ XXXIII. Addenda to the modulation-formulas of the 2 d Division 114 „ XXXIV. Conclusion of the modulations of the 3 a Division . . 120 „ XXXV. Conclusion of the modulations of the 4th Division . . 127 Table I. The same Triad in seven different keys (Chapter XIV) ... 50 Table II. Universal Modulation-formulas, for each of the Four Grand Divisions of Modulations . . . 135 Index . . . 153 CHAPTER I. Modulation, in general. 1. To modulate, in the strict sense, is, in general, to pass by means of appropriate harmonies from one key to another. For the sake of terseness we will call the key just left the Old Key, the other one the New Key. By "appropriate harmonies" we mean in general such chords as form, in connection with a short harmonic formula called Cadence (see Chapter IX), a bond of union between the old Key and the new. The harmonies used in this work are exclusively major and minor Triads, as being the easiest chords to manage. 2. We may touch upon the domains of a foreign key without intending to leave the original key, — this would not be to modu- late, in the strict sense. If, however, we not only pass over into the domains of a foreign key, but settle down in the key by means of its characteristic harmonies embodied in a Cadence, this plainly indi- cates the relinquishment of the original key, or. in other words, Di- gressive Modulation. Our means of modulation is, then, the Ca- dence, in conjunction with certain preceding Triads connecting it with the old key. 3. In our method we reach the new key in every case . by means of one and the same simple Cadence-formula, which varies only according as it is to be used for the Major or the Minor Mode. The manner of proceeding is perfectly simple. The determination of the new key to which we are going, of course, also determines the Cadence as being in that key, and our task is to leave the old key by such of its harmonies as can be — as shortly as possible — properly connected (either immediately or mediately) with the Cadence itself. A homely illustration may serve to make this clear. We may imagine a railway-train, carrying with it an arrangement in the form of a movable switch, which can be laid, down and l _ 2 — stretched out as may be required, to enable - the train to leave the main track and connect with auxiliary roads, each leading to a particular station, more or less remote. The Cadence may be likened to one of these auxiliary roads; for, as soon as we have safely made the connection (from the old key to the initial Triad of the Cadence) by means of the movable switch (the intermediate Triads , few or many), we leave the main track (the old key), and our task is as good as done, — the auxiliary road (the Cadence) will unerringly conduct us to the desired new station (the new key). CHAPTER II. Major and Minor Triad. — Key. — Mode. 4. A Triad is a chord or harmony of three tones, viz: a tone called Fundamental, or Root; a tone forming an upper Third (major, or minor, as the case may be); and a tone a Third above this last one, and forming an upper Fifth (always major, for the purposes of this work) to the Boot. (N. B. In the Examples the Root is shown by an open note.) If the Third is major, we have a Major Triad, as at a), Fig. 1; if minor, a Minor Triad, as at b). The Triad with minor Third and minor Fifth is not used in this work. a) b) ■» 5. The Triad has three Forms, one primary, two derivative. For, the root of a Triad does not always appear as at a) and b) above, i. e., as lowest tone. When it does so appear, the Triad is in primary form. But, two other arrangements of the tones are possible: the Triad *, for instance (Fig. 2), appearing in primary form at a), may also have as lowest tone its Third — e, as at b), — first deriv- ative form, or its Fifth — g, as at c), — second derivative form. _ a) b) c) 2. 6. Triads are used in this work in the primary and — less frequently — the second derivative form only. The primary form is — 3 — Indispensable for a full close; the second derivative form is peculiarlj adapted for the Cadence (as we shall see later), in which connection only it will appear in this method. Key. 7. A Key, in the harmonic sense (with which alone we are here concerned), is a family of inter-related Chords or harmonies, comprising Consonances and Dissonances (Primer, Chap. X). Only consonant Chords are suited to the purposes of this work. The only- consonant Chords are major and minor Triads; these will form, therefore, our sole harmonic material. Accordingly, a Key is, for us, a family of inter-related major and minor Triads. Modes. 8. Every key has its two Modes, the one called Major, the other, Minor. It will be most convenient to explain this through the structure of the two so-called model keys, viz: C (major) and a (minor). 9. The principal tone of a key being its Tonic, on the 1 st de- gree of the scale of the key, the principal Triad of the key will be that of the Tonic; the Triads next in importance are that of the Sub- dominant, on the 4 th degree, and that of the Dominant, on the 5 th de- gree of the scale. These three Triads of a key, in conjunction, con- stitute its characteristic harmonies, differentiating it from every other key. By them, moreover, the Mode of the key is determined; for if they are, in their normal condition, collectively major Triads, the Mode is Major; if minor Triads, the Mode is Minor. 10. In the model key whose scale is represented in Fig. 3, the three characteristic Triads are seen to be major; in that whose scale is given in Fig. 4, they are minor. The former key is therefore known as that of C in the Major Mode, or shortly, of C-major; the latter as that of a in the Minor Mode, or, of a-minor. As being natural keys and requiring no chromatic signature, they are the best adapted for serving, respectively, as models or representative keys of the two Modes. ^ C:I IV V Tonic. Subdom. Dom. *■ a^==^ EE^^^= | a: I IV V Tonic. Subdom. Dom. — 4 — NB. (a) A large capital expresses the name of a major key; a small capital, or preferably, a lower case letter, that of a minor key. A ccordingly, C stands for the major key of which that tone is Tonic, and a for the minor key having for Tonic the tone a — and similarly of similar cases. (b) A numeral, in this work, represents not merely a degree of the scale, hut also the Triad seated on it. A large numeral always denotes a major, a small one a minor Triad. Thus, I stands for the Tonic Triad of any one of the 12 major keys; I, for that of any one of the 12 minor keys. To express the Tonic or any other Triad of a particular key, a letter, with colon, is placed before the numeral, as G: I, Tonic Triad of ff-major; a: IV, Subdominant Triad of a-minor; C: V, Dominant Triad of C-major, etc. Questions, What is a Triad? Describe the structure of a major Triad; then that of a minor Triad. — How many Forms has the Triad? What is the peculiarity of the primary form? of the 1 st derivative form? of the 2 d derivative form? Which form is necessary for a full closer For what is the 2 d derivative form specially adapted? What is a Key, in the harmonic sense? Among the harmonies of a key which are the only consonant ones? How many Modes has a key? Which are the three principal or characteristic' harmonies of a key? What bearing have they on the Mode of the key? In which Mode, then, is a key whose charac- teristic harmonies are normally major Triads? In which Mode is a key whose characteristic harmonies are normally minor Triads? What is the symbol of the Tonic Triad of a major key? of that of a minor key? of the Subdominant Triad of a major key? of that of a minor key? of the Dominant Triad of a major key? of that of a minor key? CHAPTER III. Major and Minor Triads of the Major and the Minor Mode. 11. The major Triads in the key of C, representing the Major Mode, being severally seated, as we have seen (Fig. 3), on the Tonic — I, Subdominant — IV, and Dominant — V, we construct the minor Triads by addiDg the Third and Fifth to the other degrees of the scale (except the seventh, not suited to our purpose, as being the seat of a dissonance), viz: the Supertonic— -II, the Mediant — HI, and the Submediant — VI, as in the following example. 5 — 6. 9^ -JT- C: I n III IV V VI Supertonio. Mediant. Submediant. 12. The Minor Mode, as represented by the minor key of a (Fig-. 4), shows minor Triads on the Tonic — I, Subdominant — IV, and Dominant— V, respectively. We obtain its major Triads by ad- ding the Third and Fifth to the other degrees of the scale {omit- ting the second, as seat of a dissonant Triad), viz: the Mediant — III, the Submediant — VI, and the Subtonic — VII. , S 4 6. 9i a: I HI IV V VI VII Median^ Submediant. Subtonic. 13. The two Modes have thus far been considered in their normal form. A key of either mode is normal when its tones, with- out exception, correspond strictly to the key-signature. But, in prac- tice, each mode undergoes certain modifications, whence arise the so- called Milder Major Mode, and Bolder Minor Mode. We are concerned here with only two of these modifications, — one for each Mode. 14. The Milder Major Mode arises from the occasional de- pression — by a chromatic half-step — of the sixth degree of the scale. This depression concerns us only so far as it affects the Sub- dominant Triad. As the tone on the sixth degree is the Third in the Subdominant Triad- (Fig. 3), which is normally a major Triad, it is clear that the depression of this Third will make the Subdomi- nant a minor Triad. Hence the name "Milder Major Mode", for the introduction of the minor Subdominant Triad serves to temper the somewhat aggressive character of the normal Major Mode, with its characteristic Triads collectively major. The minor Subdominant Triad* is represented (see Fig. 7) by a small numeral— (IV), the parentheses denoting, here and wherever else they occur in this work, a deviation from the normal mode of the key. 15. According to the following illustration of all the consonant * It is hardly necessary to observe, that though the Subdominant Triad in the Minor Mode is minor, yet the expression "Minor Subdominant Triad", and its symbol— (IV), refer exclusively to the (milder) Major Mode, in which the minor Subdominant is not normal. 6 — Triads of the model major key of C (both normal and in the Milder Major Mode), 9i c: I ii in IV (iv) V vi we may sum up the major and minor Triads of the Major Mode in general in a scheme or formula which will apply to any key of that mode, as follows: three major Triads: I, IV, V. four minor Triads: n, ill, (IV), VI. 16. In the Bolder Minor Mode, on the other hand, the gentle character of the normal mode, with its characteristic harmonies col- lectively minor, is tempered by the occasional chromatic raising of the seventh degree of the scale. This raising (not to speak of other effects, with which we are here not concerned) changes the {normally minor) Dominant Triad into a major Triad, in as much as the tone on the seventh degree is the Third in the Dominant Triad. The major Dominant Triad* is represented (see Fig. 8) by a large numeral, V, in parentheses, thus: (V). 17. From the following illustration of all the consonant Triads of the model minor key of a (both normal and in the Bolder Minor Mode), 8 . 3L-. \ fc zft rifr-3^ *: - a: I HI IV V (V) VI VII we derive the subjoined scheme of the major and minor Triads of the Minor Mode in general, applicable to any key of that mode: three minor Triads: I, IV, V. four major Triads: III, (V), VI, VIJ. Questions. In the Major Mode, on which degrees of the scale are the minor Triads seated? Give the name of each of tbese Triads, and the symbol (numeral) by which it is expressed. — In the Minor Mode, on which degrees of the scale are the major Triads? Give the name of each, then its symbol. -*- In which form has each Mode been represented, * Just as the expression "Minor Subdominant" applies to the (milder) Major Mode only (see Note, p. 5); so the term "Major Dominant", and its symbol — ("V)i refer exclusively to the (bolder) Minor Mode, in which the major Dominant Triad is not normal. — 7 — thus far? When the tones of a key correspond exactly with the key- signature, what is to be said of the key, as to its form? Name the two modifications of the normal Modes. — What is the peculiarity of the so-called Milder Major Mode? Why is this modification so called? What is the symbol of the Minor Subdominant Triad? Since the Subdominant Triad in the Minor Mode is minor, what is to be said of the application of the term "Minor Subdominant" to the Major Mode [Note, p. 5)? Sum up the major and the minor Triads of the Major Mode in general. — What is the peculiarity of the Bolder Minor Mode? Why is this form of the mode so called? What is the symbol of the Major Dominant Triad? Why is this expression used, seeing that the Dominant Triad of the Major Mode is major {Note, p. 6)? Sum up the minor and the major Triads of the Minor Mode in general. CHAPTER IY. The Major and the Minor Mode in all the keys. 18. It being assumed that, the previous chapter has been thoroughly studied and digested, and also that the student is per- fectly familiar with the Diatonic Scales — major and minor — of all the keys, it is the aim of the present chapter to render him equally familiar with the harmonies (meaning here the major and the minor Triads) of each and every key, in both its modes. Be- sides knowing what tone is on such and such a degree of the scale of a given key — major or minor, so as to be able to name the tone or sound it on the piano without the least hesitation, the student should be able also to name or play with equal promptness the particular Triad seated on the degree indicated. To this end the present chapter is devoted to the following practical exercises, to be worked out partly in writing, partly at the piano. Exercises. I. (In writing). — a. Write down in notes* the Triads — grouped according to the two species (as in the scheme given in paragraph 15) * As to the pitch, all that is required is that the Triads be severally seated on the right degrees of the Scale of the key, keeping them, however; as much as possible within the compass of the star? without added lines. For — 8 — — of each major key with sharp signature (G, D, A, etc.). The sig- nature is, however, to be omitted in every case, and the sharps or flats are to be added to each Triad, when necessary to its proper notation. — h. Write in the same way the Triads of each major key with flat signature (F, B\>, E\>, etc.). — c. Write in the same way the Triads — grouped as in the scheme in paragraph 17 — of each minor key with sharp signature {e, b, f%, etc.). — d. Write in the same way the Triads of each minor key with flat signature (d, g, c, etc.). II. (At the piano). The teacher dictates a key — alternating major with minor keys, and requests the pupil to play its various major and minor Triads, grouping together those of the same species, as in the schemes respectively given for the Major and the Minor Mode (paragraphs 15, 17). For accustoming himself to the symbols used to express harmonies, the pupil should at each Triad give its name in relation to the key (Tonic, Dominant, Mediant, etc.) and its sym- bolical expression, as for instance: Tonic Triad, large I, or small I, as the case may be; Dominant, large V, or, large (V) in parentheses; Mediant, small in, or, large HE, etc. CHAPTER V. Inter-relationships of the Triads of a key. 19. The Triads of a key, considered in pairs, are inter-related, either directly or indirectly. The treatment of Triads which are so paired as to show indirect relationship involves certain difficulties; hence we shall have nothing to do with them in this work, which professedly follows the easiest method. 20. Two Triads of a key are directly related when they have a tone, or two tones, in common. This will always be the case when their respective roots form a Third or a Sixth, a Fourth or a Fifth. When, however, the roots of two Triads form a Second or a Seventh, the Triads are indirectly related, as having no tone in common. this purpose the Subdominant Triad — for instance — may be written on either the Fifth below or the Fourth above the Tonic Triad, according to convenience, — and so on of other cases. The Triads should be written .alter- nately on the staff with G-clef and that with .F-clef, for the sake of acquiring equal familiarity with both. — 9 — Remark. With reference to the expressions "a Third or a Sixth", and "a Fourth or a Fifth", used above, be it said here, once for all, to prevent any possible misunderstanding, that for harmonic purposes any two tones of a key may be, with equal correctness, so written as to form either an ascending interval, or its inversion* — a descending one, according to convenience**. Thus, in the key of C-major, for instance, c and d may form, as in the following figure at a), either an ascending Second or a descending Seventh; c and e, as at b), either an ascending Third or a descending Sixth; c and f, as at c), a Fourth or a Fifth, etc. a) b) c) d) e) 9. m zef: =s?=f C: I II I HI IIV ^^ etc. IV I vi 21. Two Triads whose roots form a Third are tierce-related, and have two tones in common; the same is the case when the roots form a Sixth (inverted Third), — as here illustrated: common tones, e, jr. P — * »; common tones c, e. b) * C: I III c,T ^ VI 22. It is seen from the example at a) that whether the root of the 1 st Triad rises a Third or falls a Sixth is indifferent, — the 2 d tone — e, and the Triad on it are the same in either case. Exactly the same may be said of the example at b). Instead, therefore, of relating Triads both by the Third above and the Sixth below, the Third below and the Sixth above, it is evidently simpler to reduce these relationships to one kind, in two species, viz: Tierce-rela- TIONSHIP, upper and lower. Thus, relating two Triads by the Third above (identical with the Sixth below), the 2 d one will be in upper tierce-relationship to the 1 st (as at a), Fig. 10); by the Third below (identical with the Sixth above), the 2 d one is in lower tierce-re- lationship to the 1 st (as at b). When, therefore, among the Triads of a key the root of one falls a Sixth — to the root of a 2 d Triad, we will for our purpose regard the progression as an ascending * See Primer, Chapter IX. ** We shall see later, however, that what is not incorrect harmonically may be had melodically ; so that under certain circumstances c e forming a Third may be better melody than c e forming a Sixth, etc. 10 Third, not as a Sixth, except perhaps for memorizing by the con- trariety involved, — falling Sixth, upper tierce-relationship of 2 d Triad to 1 st , and vice- versa. 23. Two Triads whose roots form a Fifth are Quint-related, and have one tone in common; the same is the case when their roots form a Fourth (inverted Fifth), — as in the following examples: common tone, g, common tone, c. b) C:I V C:I IV 24. Here, too, it is harmonically all the same whether the root of the 1 st Triad rises a Fifth or falls a Fourth, as at a); or whether it falls a Fifth or rises a Fourth, as at b). Hence, instead of relating Triads paired as in the above figure by the Fifth above and the Fourth below (as at a), or by the Fifth below and the Fourth above (as at b), we simplify matters by recognizing only Quint-relationship, upper and lower. Accordingly, at a) the 2 d Triad is in upper quint-relationship to the 1 st (related to it by the Fifth above, identical with the Fourth below); at b), the 2 d Triad is in lower quint-relationship to the 1 st (related to it by the Fifth below, same as the Fourth above). Whenever, therefore, in our method the root of a Triad falls a Fourth, we will regard it as, practically, an ascending FIFTH; whenever it rises a Fourth, as a descending Fifth. 25. Accordingly, in a key of either Mode the Tonic Triad — for instance — is tierce-related to the Mediant Triad, on the Third above (Sixth below), and to the Submediant Triad, on the Third below (Sixth above); quint-related to the Dominant Triad, on the Fifth above (Fourth below), and to the Subdominant Triad, on the Fifth below (Fourth above). To put it differently: The Tonic Triad is in lower tierce-relationship to the Mediant ; Mediant „ „ „ upper „ „ „ „ Tonic; Tonic „ „ „ upper „ „ „ „ Submediant; Submediant „ „ „ lower „ „ „ „ Tonic; • Tonic „ „ „ lower quint-jelationship „ „ Dominant; Dominant „ „ „ upper „ „ „ „ Tonic; Tonic „ „ „ upper „ „ „ „ Subdominant; Subdominant „ „ „ lower „ „ „ „ Tonic. 26. After this exemplification of tierce-relationship and quint- - 11 relationship between the Tonic and four other Triads, it will be easy to understand the following tables, showing the relationships of all the consonant Triads of a key, one to another, as illustrated in our two model keys — C-major and a-minor. Tierce-relationships of Triads in C-major. 12. « m * . * • L~\ • 9 • 3 73 • 4 #• • <> 2 * ?J <3 /? 3 ■s rj rj o C: I III I VI II IV III V III w IV vi IV II III VI VI Quint-relationships of Triads in C-major,. 13. ZciZ C:I V M si W (IV) II VI ii V in VI cp-fl^ J IV I V ii VI vi in vi ii l (IV) Tierce-relationships of Triads in a-minor. u. 5 73 )• * • * S » 5 # o s ■ n > * 73 •4 5 rt r» • IV a: I III VI III v III i iv VI m ~SZ v VII III VI VI iv VII v Quint-relationships of Triads in a~minor. 15. Eit jSl. Ill VII III VI a: I IT \(Y) IV IV I m =?- ESE iv VII i(V) VI III VII iv VII III — 12 — Exercises (at the Piano). I. In the Major Mode, in keys severally dictated by the teacher. 1. Strike the Tonic Triad. Then strike, giving its name and symbol, 2. the Triad in upper tierce-relationship to the Tonic; 3. „ „ „ lower „ „ „ „ „ 4. „ „ „ upper quint-relationship „ „ „ 5- >, ;) n lower „ „ „ „ „ 6. Strike the Super tonic Triad. Then strike, giving name and symbol, 7. the Triad in upper tierce-relationship to the Supertonic. 8. „ „ „ „ quint-relationship „ „ „ y. „ „ „ lower „ „ „ „ „ 10. Strike the Mediant Triad. Then strike, giving name and symbol, 11. the Triad in upper tierce-relationship to the Mediant; 12. „ „ „ lower „ „ „ „ 13. „ „ „ lower quint-relationship „ „ „ 14. Strike the Subdominant Triad. Then strike, giving name and symbol, 15. the Triad in upper tierce-relationship to the Subdominant; 16. „ „ „ lower „ „ „ „ „ 17. „ „ „ upper quint-relationship „ „ „ 18. Strike the Dominant Triad. Then strike, giving name and symbol, 19. the Triad in lower tierce-relationship to the Dominant; 20. „ „ „ upper quint-relationship „ „ „ 21. „ „ „ lower „ „ „ „ 22. Strike the SufanediantTriai. Then strike, giving name and symbol, 23. the Triad in upper tierce-relationship to the Submediant; 24. „ „ „ lower „ „ „ „ 25. „ „ „ upper quint-relationship „ „ „ 26. „ „ „ lower „ „ „ „ II. In the Minor Mode, in keys dictated by the teacher. 27. Strike the Tonic Triad. Then strike, giving name and symbol, 28. the Triad in upper tierce-relationship to the Tonic; 29. „ „ „ lower „ „ „ „ 30. „ „ upper quint-relationship n „ „ 31. „ „ „ lower „ „ „ „ 32. Strike the Mediant Triad. Then strike, giving name and symbol, 33. the Triad in upper tierce-relationship to the Mediant; 34. „ „ „ lower „ „ „ „ 35. „ „ „ upper quint- „ „ „ 36. „ „ „ lower „ „ ,. „ „ — 13 — 37. Strike the Subdominant Triad. Then strike, giving name and symbol, 38. the Triad in upper tierce-relationship to the Subdominant; 39. „ „ „ upper quint- „ „ „ 40. „ „ „ lower „ „ „ „ 41. Strike the (normal) Dominant Triad. Then strike, giving name and symbol, 42. the Triad in upper tierce-relationship to the Dominant; 43. „ „ „ lower „ „ „ „ 44. „ „ „ lower quint- „ „ „ „ 45. Strike the SubmediantTmA. Then strike, giving name and symbol, 46. the Triad in upper tierce-relationship to the Submediant; 47. „ „ „ lower „ „ „ „ 48. „ „ „ upper quint- „ „ „ 49. Strike the Subtonic Triad. Then strike, giving name and symbol, 50. the Triad in lower tierce-relationship to the Subtonic; 51. „ „ „ upper quint- „ „ „ oi. „ „ ,, lower „ „ „ „ „ CHAPTER VI. Connection of related Triads, in general. 27. In our illustrations thus far no two Triads have been con- nected, in the proper sense of the word. It is the connection of Triads, specifically of those that are directly related, that is now to be studied, preparatorily to the construction of the Cadence, our instrument of modulation. 28. In a pair of tierce-related Triads there are, as we have seen, two tones in common; in a pair of quint-related Triads there is one tone in common. These common tones are used for smoothly connecting Triads, and called accordingly, connecting-tones. 29. The process by which one chord is connected with another involves what is called Voice-leading. Each tone of a chord is re- garded as a voice, and in chord-connection each voice of the 1 st chord is to be properly led to its place in the 2 d chord. If the voices are led to their several places with the least possible motion, i. e., using the common tones as actual connecting-tones, and avoiding unnecessary skips (moving as much as possible from degree to degree), we have — 14 — the so-called strict voice-leading, as distinguished from a freer kind, in which skips are of frequent occurrence and the connecting-tones are not rigidly observed. The former is the kind which, as reducing the difficulties of chord-connection to a minimum, our method follows, with but one exception, to be noticed later. 30. In the strict method of voice-leading a pair of related Triads are connected by observing the following Rule. (1) Keep the CONNECTING-TONES each in the SAME VOICE in both Triads, i. e., do not move them: (2) lead the other voice — or voices — by a degree (up or down, as the case may be) to another tone — or other tones — in the second Triad. Although the connection of tierce-related Triads is easier, we begin (for reasons which will appear later) with that of quint-rela- ted Triads — those Of a key in the Major Mode, as represented by our model key, C-major. CHAPTER VII. Connection of the quint-related Triads of a Major Key. Upper quint-relationship. 31. In the following example of a pair of quint-related Triads (written in two different ways, a) and b), both expressing the same tWn g). a) b) i 16. =*= ■» C:I V if there is no Chord-connection: the common tone — g, instead of re- maining, according to the Rule, in the same voice (upper voice) in both Triads, thereby serving as connecting-tone between them, moves to another tone — d* thus g appears in the 2 a Triad in another * Thereby making, with c, moving to g, (fa «'*^^ - tn e faulty pro- gressdon called parallel Fifths (less properly, consecutive Fifths), the Ute noire of beginners in Harmony. Fortunately, the modulations of this method are all so arranged that we can never by any possibility encounter so dreadful an apparition. - 15 — voice, viz: the lowest. Again: the two other voices move, to be sure, to tones of the 2 d Triad, but do not move by degree, so as to go to the nearest tones; for, e makes a skip to b, whereas d is on the degree below, and c skips to g, whereas b is on the degree below. We correct all this by following the Eule: 1. The connecting- tone — g — does not move (to signify this we will tie the two g's) % — 2. The tone e moves by a degree — below, in this case, to the nearest tone of the 2 a Triad, viz: d; the tone c fefi The two Triads are moves similarly, and goes to b: y— - ggi C:I V now properly connected. 32. For the sake of practice we will continue from this point and form a small chain of four Triads in upper quint-relationship. As Fig. 13 shows, just as V is in upper quint-relationship to I, so is II to V, and VI to II. The whole chain may accordingly be re- presented as follows (without chord-connection). i7 -i ■§*■ ^ C: I V II VI Having already connected I with V, we must next connect V — in the shape in which we left it at the end of paragraph 31 — with II, the 3 d Triad of the chain. We first look for the con- necting-tone between the two Triads, and find it to be d, root of II. We tie the two ^'s, (fo g^—^^:: then lead g and b respec- tively to other tones of Triad n, — g to the degree below, viz: f*, -i^l and b to the degree below, viz: a. . yy j£^h Finally, we C:V II connect this last Triad — II — to the closing one — VI. The con necting-tone is a— root of IV, and we tie the two a's : fe ^=E I then lead d and /, respectively, to the nearest tones of VI, viz: / * True a, above, is as near tog&sf is. But b — lowest voice — is bound to go to a — degree below, as nearest tone of Triad II. If now g also were to go to a — degree above, the Triad concerned would lack ita> Third — /. Therefore g goes to f, in the same direction with the lowest voice — b. — 16 — to e, and d to c: S These four Triads, which we C: if-fa have thus connected according to our rule, form the following har- monic chain: €: I V II ""VI 33. Here two points deserve notice. We observe, in the first place, that in the above chain, Fig. 18, the two voices which are not tied, i. e, which move, go invariably to the next degree, together (here downward). In connecting Triads (whether tierce-related or quint-related) in our method of Modulation, whenever a voice has to move, in passing from one Triad to another, it will be always (except occasionally in the Cadence, of which later) either to the de- gree next above, or to that next below. Specifically, in connect- ing a pair of Triads in upper quint-relationship, the two voices which move will always move together, each one degree lower, as- in Fig. 18. For the exception, in the case of the Cadence, see par. 52. 34. We observe, in the second place, that two of the Triads of our Chain, viz: V and II, are not in primary form, V being in the first derivative and II in the second derivative form (see para- graph 5). This is perfectly correct, as resulting from correct voice- leading; yet a Triad in either of these forms has no strength or independence. Hence our chain, though its Triads are properly connected, lacks, as a whole, a solid harmonic foundation. This defect we remedy by adding a fourth voice, as lowest voice, or Bass, for carrying the root of each Triad, thereby affording the solid basis required, by placing the Triad in its strongest and most inde- pendent form, viz: its primary form, as in the following example: b) 19. i «iS3Sf C:I v - ,x ^ NB. 1. The addition of a fourth voice to a Triad, as above, is not the addition of a new tone; the Triad remains a three-voiced — 17 — chord (in the sense of having but three different tones), and the additional voice is merely a doubling of one of the tones, viz: of the' Boot. Just as above, so in every Triad throughout this method (with a single exception, to be noticed later) the Boot will be doubled, appearing in one of the three upper voices and at the same time in the Bass, thus insuring the primary form of the Triad, in ac- cordance with paragraph 5. NB. 2. The statement that the voices move, in our method, invariably by degree, must now be explained as applying to the three upper voices only. The Bass will — as has just been said — carry the roots of the Triads; and as these Triads are either tierce- related or quint-related, the Bass must needs move by skip, namely in tierce-relationship, to the Third above or Sixth below, or to the Third below or Sixth above; in quint-relationship, to the Fifth above or Fourth below, or to the Fifth below or Fourth above. In our chain, Fig. 19, representing upper quint-relationship, the Bass, theo- retically, skips by ascending Fifths, for instance: 20. C: I V II VI But such a Bass is objectionable — to say nothing of the exten- sive compass involved — from a melodic point of view, as compris- ing three consecutive Fifths*. A law of good melody requires the correction of such a progression by alternating descending Fourths with ascending Fifths, — as we have done in Fig. 19, a and b. We have already seen (paragraph 24) that harmonically a descend- ing Fourth and an ascending Fifth are equivalent, similarly an ascending Fourth and a descending Fifth. Lower Quint-relationship. 35. We will now form a chain of four Triads in lower quint- relationship, — for instance, vi n V I, — which may be repre- sented, without chord-connection, as follows:' 21 % C: VI II V I * Not parallel Fifths, which are formed by two voices moving together in a certain manner (see Note, p. 14), and which are often miscalled "consecu- tive" Fifths. 2 with II — 18 — We proceed as before. We tie the two a's which connect VI -IP ^s , then lead the two other voices respectively to the tones of Triad n which are a degree distant, viz: c to d, e to f — ill C: VI II Next we tie the two d"s which connect II withV- -HP ■0- then lead / to g , and a to b which connect V with I — ttT)ig'~< f C: II V . Lastly, we tie the two ^'s then lead d to e, and b to c — i£ *3f . The whole chain will appear thus: C: vi ii V I 36. Here too we observe, first, that in passing from one Triad to the next, the two voices which move invariably move each to the degree above, reversing the. order of upper quint-relationship. This will be the ride, in connecting a pair of Triads in lower quint-re- lationship. An occasional exception will be noticed in paragraph 51. 37. We observe, secondly, that Triad II appears in its second, Triad V in its first, derivative form. We therefore add a fourth voice — Bass — as before, to each Triad, for carrying a doubling of the root. This Bass, which theoretically would be this, for instance — 23. 9 : ^ = ls ' "~~- zE C: VI II V I must be so arranged as to appear as at either a or b, in the follow- ing figure: a) . 24.' i ^g^ C: VI ItT - 19 — Exercises. These may be worked out, either in writing or at the piano*, in each major key, first in C, then in G, and so on**; or at the teacher's discretion, in a few major keys dictated by him. I. Form the following four harmonic chains, of four Triads each, by properly connecting the Triads of each chain, and adding the Bass-tones, according to the method illustrated in this chapter. 1 st Chain (upper quint-relationship) IV I V n. 2 d Chain (lower quint-relationship) n VI IV. 3 d Chain (upper quint-relationship) V n VI III. 4 th Chain (lower quint-relationship) m vi II V. II. Form the following chain — which combines both upper and lower quint-relationship, by correctly connecting the Triads and adding the Bass-tones. Play the Subdominant Triad either normal (major) or minor, at option. u pper uppe r I ( IV I V I CHAPTER VIII. Connection of the quint-related Triads of a minor key. 38. The table, Fig. 15, shows how the Triads of «-minor (re- presentative key of the Minor Mode in general) are paired according to quint-relationship. The connection of the Triads — thus paired — * If it is practicable, they should be done off-hand at the piano, the Bass being added at once to each Triad. For some pupils, however, the slower process of renting will be necessary. In either case the following details are implied: 1) Determine the key. 2) Strike, or write down, in a reasonable pitch, neither too high nor too low, the tone on the degree indicated by the first numeral of the chain. 3) Build up, on this tone, by adding Third and Fifth, the proper Triad (major or minor, as may be required). This gives the initial Triad of the Chain. 4) Add a doubling of the Boot, in the Bass. 5) Connect this Triad with the next one indicated, afterwards adding a Bass to the latter, as before, — and so on, to the end of the chain. ** See, as to the order of keys, paragraph 61. 2* — 20 — of a key in the Minor Mode, ought, after the explanations of the connection of quint-related Triads in a major key, in the preceding chapter, not to present any new difficulty. Dispensing, therefore, with special illustrations for this chapter, we will merely give a few exercises, to be worked out in each minor key, or, at the teacher's discretion, in certain minor keys dictated by him. Exercises (see Note, p. 19). I. Form, in various minor keys, the following two chains of three Triads each, by properly connecting the Triads of each chain and adding the Bass-tones, following the method illustrated (for the Major Mode) in Chapter VII. 1" chain (upper quint-relationship) rv 1 V. 2 d chain (lower quint-relationship) (V) I iv. II. Form the following chain of mixed upper and lower quint- relationship, by properly connecting the Triads, and adding the Bass- tones, as above. upper upper i iv i (V) i CHAPTER IX. The Tonic Cadence. 39. In Exercise II, of chapter VII, also of chapter VIII, we have accomplished in a crude form our first harmonic structure, the Tonic Cadence. This is a short formula composed of the character' istic harmonies of a key (see chapter II, paragraph 9), properly con- nected, the final harmony being the Tonic Triad. This Cadence serves for closing a piece of music or one of its smaller or larger divisions, also for modulating into a new key. 40. The characteristic harmonies of a key are, as we have seen, the Tonic Triad, the Subdominant and Dominant Triads, in con- junction. To make this harmonic succession a Tonic Cadence, the Tonic Triad must of course reappear at the close, when the formula for the Cadence would be, for the Major Mode — I IV V I; for the Minor Mode — I iv (V) i. — 21 — Remark. In the Cadence in the Minor Mode, the "Major Domi- nant" Triad — indicated by (V) — is used immediately before the Tonic Triad in preference to the Dominant Triad in its normal con- dition (minor), indicated by V. 41. In the above cadence-formula the Subdominant Triad appears in immediate connection with the Dominant, whereas these two Triads are not directly related, as having no tone in common. But as our plan of modulation implies direct relationship between every two Triads in immediate connection, without any exception, we insert the Tonic Triad between the Subdominant and the Dominant, when the formula will exactly correspond with the two exercises alluded to at the beginning of this chapter, thus:* in major: I f IV I V I 1(IV) in minor: I IV I (V) I We are supposed to have already (in Chapters VII and VIII) constructed the above two chains, by properly connecting the Triads, and adding tbe root -bass to each, thus giving a crude form of the Tonic Cadence, — for instance, in C-major, for the Major Mode: 25. % -foZ 3C m C:I jIV 1 V '(IV) and in «-minor, for the Minor Mode: i ^===£1= ^ a: I IV I (V) I 42. We will now give to our Cadence a good rhythmical form. We will choose for this purpose the 2 / s meter**, and give tbe Ca- * Other cadence-forms are possible; the simplest form has been deemed best adapted to the purposes of this method, which professedly follows the easiest way. ** The rhythmical arrangement of a piece of music, or of a musical thought, 22 — dence an even number of measures — say, four. To this end we extend the duration of the final Tonic Triad by an additional measure, in which, however, that Triad will take up the first kalfovhj, while the second half (the unaccented part) is represented by a half-note rest, as in the following examples. 27. i E^gf: $■ St ~r gr^Efg £ £ ~<2Z =5=^ C: I fIV t(IV) 28. i 3 sT 0* ^fc=3= * ^ -^ a: I IV (V) We proceed to criticize* our work, with a view to making im- provements. In the first place, we find the Bass stiff and monoto- nous; the Tonic Triad is represented in this voice each time by its root, whereas some other tone of the Triad — its Third, or its Fifth — might be introduced once, for giving the .Bass a better melody. For greater facility of management, and for intrinsic reasons, the Fifth is preferable to the Third, for taking the place of the root. The best place for making this change is on the second oc- currence of the Tonic Triad (which Triad must be, on its first and last occurrence, in primary form, as in Figs. 27, 28). Between the Sub-dominant and Dominant Triads, therefore, we give to the Bass implies that its tones are divided up into certain time-portions, called Measures, of equal duration, and marked by regularly recurring accents. An aggregation of measures constitutes a Meter (commonly, but incorrectly, called Time, though this expression in music refers properly to the quickness or slowness of the movement, with .which Meter, as such, has nothing whatever to do). Thus, in a piece in which each measure contains, e. g. two half-notes, the Meter is called Ttvo-ttvo ( 2 / 2 ); in another piece the Meter is Three-four ( 3 / 4 ), or Six- eight (%), etc. * To encourage the pupil to think for himself, the teacher should let him make his own criticisms, asking him what he thinks of the Bass of the Cadence, as a melody; of the effect of the final Triad in its position as above (Fig. 27, 28), compared with the two other positions in which it might appear, etc. 23 the Fifth of the Tonic Triad, instead of the root, and the amended Cadence reads, in major: 29. < 3f5* -r m * fee and in minor: C:l (IV l(IV) ^5 -g>- 30. I^T^ ST flit 3 ^^ 3 a: I IV f (V) I Eemark. — The Bass in the Dominant Triad may be either in unison with the Bass of the preceding Triad or an Octave below it, at pleasure. 44. The above change forms an exception already alluded to (see NB. 1, p. 16), i. e., the only instance in this method in which the Bass is not a doubling of the root of a Triad. The Tonic Triad in this instance appears in its second derivative form (see paragraphs 5 and 34); and its very weakness in this form peculiarly fits it to be — as it is here — immediately followed by and as it were ab- sorbed into the Dominant harmony, only to assert itself the more powerfully the moment after, by appearing in its primary form, as the final chord of the series. 45. In this solitary instance in which- the Tonic Triad — or any other harmony — appears in this work in a derivative form, the tones above the Bass severally count — not as Third and Fifth (as in the primary form), but as Fourth and Sixth (see the 3 d Triad in Fig. 29, or 30, above), whence a Tiiad in its second derivative form is called a Chord of the Fourth and Sixth, or, shortly, of Four-six. In our Cadence the second appearance of the Tonic Triad will invariably be as a Chord of Four-six on the accented part (1 st half-note) of the measure, and indicated by \ over the numeral standing for the Tonic Triad, thus: I, or I, — meaning - 24 — that the tones are those of the Tonic Triad, the lowest voice (Bass) being in this case a doubling of the Fifth of the Triad, not of its root, as in all other cases. It should be borne in mind that the 6 6 4 4 Bass of 1 or I is always the same tone as the Bass (root) of the immediately following Dominant Triad (see Remark to Par. 43). 46. We have applied the word Form to a Triad, in reference to its lowest voice; thus , a Triad is in primary form when its root is in the lowest voice; in first derivative form, when its Third is in the lowest voice; in second derivative form, when its Fifth is in the lowest voice. Now, in reference to the highest voice of a Triad we may apply the word Position. Thus, a Triad is in Root-po- sition, when its highest voice has the root; in Tierce-position, when that voice has the Third; in Quint-position , when it has the Fifth. The position of a Triad is entirely independent of the form of the Triad. To give examples from Figs. 29 and 30: the Tonic Triad is throughout in quint-position; the Subdominant Triad, in tierce-position; the Dominant Triad, in root-position. 47. In the second place, then, we may criticise the closing of the Cadence with the Tonic Triad in quint-position. For, this final Triad should be in every respect in its strongest condition possible. It should therefore be not only in primary form (its root in the lowest voice), but also in root-position (its root in the highest voice also), since either the tierce- position or the quint-position would form a relatively weak ending. 48. In point of fact, every one of the modulation-formulas used in this method (see Table II) is specially arranged to meet this requirement. That is, the initial Triad of each modulation is directed, by a particular sign in the formula, to be in that one of the three positions, which, for the particular modulation concerned, insures the right position of the final Triad of the Cadence. However, as it would be too difficult to memorize*, in so many different modula- tions, the indications for the initial Triad, we must adopt some ex- pedient whereby, with that Triad in any one of the three po- sitions, we may still secure the desired root-position of the final Triad of the Cadence. This expedient is, to introduce, when ne- * The ambitious student will not rest satisfied with being able to play the modulations from the formulas "before him, but will aim at mastering the principles on which the formulas are constructed, so as to be able to play any modulation extemporaneously, as it were. — 25 cessary, in the Cadence itself (and the nearer to its close, the better) such change of position in one of its Triads as will lead to the desired result. This change will be necessary in two cases only, as we shall at once see. 49. We premise that the initial Triad of the Cadence, besides being in- quint-position, as hitherto, may also be either in root-posi- tion, in which case the Cadence will appear thus: i 31. 35t :(to!= ~&- ± fctEfeEgEJ^ Eg C: I I IV J (dV) I V £ rst :t -a- -&t tfs 9t ^ -&- a: I IV (V) or in tierce-position, when the Cadance will appear thus: 33. < % 3=t -ifcoJ- --2 — -&- 1 "St ■6- C:I |lV I (IV) n 1 \ | - ! W A /L- *i ^ iu-j — -*i fm *1 *< ^ «£§ % d, VA) a „> H5S -J %) "■ i» ..I* i> ^* ^: .1 r a -4- ! 1 — a 34. a: I IV 1 (V) I 50. Let us now concentrate our attention upon the second oc- S 6 currence of the Tonic Triad, viz: the I, or I. This Triad is, m — 26 — Figs. "31, 32, in root-position, and so is the final Triad also. When- 4 4 ever, then, the 1 or i is in root-position, no change is to be made in the Cadence, as the final Triad will necessarily be in the proper position. The two cases when a change is necessary are, 1) when I e 4 I or I is in tierce-position, involving that position in the final Triad, as in Figs. 33, 34; and 2) when the I or I is in quint-position, in- volving that position in the final Triad also, as in Figs. 29, 30. In either case the change will occur in the Dominant Triad — V, or (V). 51. In the first case, — when the I or I is in tierce-position; the Soprano of the V or (V), instead of rising to the degree above {Third of the final Triad) — as in Figs. 33, 34, falls to the degree below {root of the final Triad); the Alto is not changed, i. e., it rises one degree, thus doubling with the Soprano; the Tenor will not form, as before, the connecting-tone with the final Triad, but instead will, on the second quarter-note of the V or (V) , fall to the degree be- low, thereby forming a Seventh to the Bass, which Seventh also falls to the degree below to form the Third in the final Triad. With this change of the Dominant Triad, the Cadence, with the I or I in tierce- position, will appear thus: 4 6 4 4 52. In the second case, — when the I or I is in quint-position; the Soprano of that Triad, instead of remaining, to form, in the V 27 — or the (V), the connecting-tone with the final Triad — as in Figs. 29, 30, rises to the Third above, so as to lead up, by one degree, \ e 4 4 to the root of the final Triad: the Alto of the I or i follows the So- prano, rising by a Third to the doubling of the Bass of V or the (V), thus forming the connecting-tone with the final Triad: the Tenor of the 1 or the I, instead of falling, rises to the degree above — Fifth of the V or (V), to lead up by one degree to the Third of the final Triad. With this change of the Dominant Triad the Cadence, with the 1 or I in quint-position, will appear thus: 37. ^toi i z«z 9± =?L a=t c : i/rv HIV) 38. { Or, the Alto of the I or i, instead of following the Soprano upward, as above, may fall one degree, to lead up again by one I e 4 degree to the Third of the final Triad; and the Tenor of the I or i instead of rising, as above, may fall to the Fourth below, which is a doubling of the Bass of the V or (V), and forms the connecting- tone with the final Triad. Thus: 39. % -M& t. d=E F^E -&- ^ m C:I £ jIV <(iv) V 28 40. P ^ "5fc J- fi^H -"o- 1 _*»_ a: IV (V) 4 4 Eemark 1. The variations in the progression 1— V, or I — (V), in the Cadence, as exemplified in Figs. 37, 38, 39, 40, form the single exception in our method to the general rule that the pro- gression of a voice is to the- degree next above or below (see para- graph 33). Eemark 2. When the three upper voices of a Triad lie so close together that no tone of the Triad can be introduced between any two of the voices, we have close harmony, so called, — as in all our examples previous to Figs. 39, 40, 4 th and 5 th Triads: otherwise, the harmony is called dispersed. In Figs. 39, 40, the Dominant and the following Tonic Triad are in dispersed harmony. The cadence- form as in these two figures is specially recommended — for the sake of the greater sonority of the Triads concerned — when a modulation lies in a high register of the instrument, in which case the effect of the two closing Triads in close Jiarmony would be weak and unsatisfactory. 53. It is absolutely necessary that the student should be per- fectly at home in the Cadence in every key, major or minor. There may be some difficulty in connecting the old key with the initial Triad of the Cadence; but, this Triad once reached, the rest of the Ca- dence, implying the consummation of the modulation, should follow without the least hesitancy. This facility may be attained by a diligent practice of the following Exercises. NB. Play in the middle of the instrument, avoiding too high a register, for the sake of sonority. I. Take the Cadence-formula exemplified in Fig. 31 through all the major keys. II. Take the Cadence-formula exemplified in Fig. 35 through all the major keys. III. Take the Cadence-formula exemplified in Fig. 37 through all the major keys. — 29 — IV. Take the Cadence-formula exemplified in Fig. 39 through all the major keys. V. Take the Cadence-formula exemplified in Fig. 32 through all the minor keys. VI. Take the Cadence-formula exemplified in Fig. 36 through all the minor keys. VII. Take the Cadence-formula exemplified in Fig. 38 through all the minor keys. VIII. Take the Cadence-formula exemplified in Fig. 40 through all the minor keys. CHAPTER X. Relationships of Keys. 54 Just as the various Triads of one and the same key are inter-related, some directly, others indirectly, so too there is direct and indirect relationship of keys. And as two Triads are directly related by having a lone or two tones in common, so two keys are directly related by having — in their normal form — at least four Triads in common. This relationship of keys underlies the practice of modulation, and must therefore be carefully studied. 55. Two keys are related, directly or remotely; directly, when their Tonic Triads are directly related, i. e., have a tone, or two tones, in common (implying that the two keys have many common Triads); otherwise, more or less remotely. If the roots of the two Tonic Triads of two keys are on contiguous degrees of the scale, thus forming a Second; or if they form a Seventh — which amounts to the same thing (see Remark, p. 9) — as in the following examples; "■$=g^Ipl§il I D: I C:T B[?:I F: I G: I G: I F: I the two Triads will not be directly related, neither will the two keys of which they are Tonics. There is therefore no direct relation- ship of keys by the Second. But if the Tonic Triads of the two keys are tierce-related Triads (see paragraph 21), the two keys will be directly related — in this case by tierce-relationship; if the Tonic Triads of two keys are — 30 — 42. 43. ;& C:I e: I rv " 1" ** -J n <3 rt C:l 44. 9^ ^P C.I 45. C: I G:I qmnt-r elated Triads, the two keys will be directly related — in this case by quint-relationship. Here follow examples. The two Triads are tierce-related; the two keys, C- major and *-minor, are in direct tierce-relationship. The two Triads are tierce- related; the two keys, C-major and a-minor, are in direct tierce-relationship . The two Triads are quint-related; the two keys, C-major and .F-major, are in direct quint-relationship. The two Triads are quint-related; the two keys, C-major and G-major, are in direct quint-relationship. Kemark. Direct relationship between two keys implies, there- fore, that they are either tierce-related or quint -related. Conversely, tierce- relationship between two keys implies that they are directly related: but this is not implied in every case of ^aw^-relationship, which, as we shall presently see, has its different grades. 56. If the Tonic of the second one of two keys is a Third above (Sixth below), or a major Fifth above (minor Fourth below), the Tonic of the first key, the second key is, in the first case, in upper feVra'-relationship to the first key (see Fig. 42); in the second case, in upper ^'^-relationship to it (Fig. 45). If the Tonic of the second key is a Third below (Sixth above), or a major Fifth below (minor Fourth above), the Tonic of the first key, the second key is, in the first case, in lower ^'m^-relationship to the first key (Fig. 43); in the second case, in lower ^^-relationship to it (Fig. 44). 57. Two keys, then, are tierce-related, when their Tonics form a Third (or a Sixth). This is to be understood in a twofold sense. In the first place, tierce-relationship exists between a major key and the minor key whose Tonic is a minor Third below the Tonic of the major key, and is thus identical with the Submediant of the major key*. Two keys thus related are called Parallel Keys; C-major — for instance — is the parallel of «-minor, and conversely. Two parallel keys always have one and the same signature in com- * The Triad on the Submediant of a major key being minor, the key of which this Triad is the Tonic, is of course a minor key. — 31 — mon. They have — supposing each key to be in its normal form — also the same Triads in common, though these are of course differently inter-related in the two keys, on account of the difference of Tonic. Thus, as in the following example, ^ =5= =£ t- _€L J- I 46. C: I II III IV V VI VII rf= -§- J_ a: I IF III IV V VI VII I II" the Triad which is. n, related to C: I, is IV, related to a: i; Triad IV, related to C: I, is VI, related to a: I, and so on of Triads on other degrees. 58. In the second place; a major key has also above it a tierce^ related minor key, whose Tonic lies a major Third above the Tonic of the major key, and is identical with the tone on the 3 d degree (Mediant) of the latter key*. Above C-major, for instance, is the key of *?-minor, called the upper relative minor of C-major, which latter, conversely, is the lower relative major of ^-minor. The follow- ing illustration shows that such a pair of keys has — normally — four Triads in common. 47. I % o.T 3E ii m fi a II" III * -0- & %z IV -§- =15 IV in IV V VI VII 59. Tierce-relationship, thus understood, implies therefore, that every major key has its upper relative minor (as, C-major, ^-minor), and conversely, that every minor key has its lower relative major (as, tf-minor, C-major); that every major key has its parallel minor (as, f-major, «-minor)„ and conversely, that every minor key has its parallel major (as, a-minor, C-major); that thus the two keys concerned differ in mode; and that the two keys which are parallel have one and^ the same signature and the same Triads in common. 60. The other kind of relationship between two keys — quint- * The Mediant Triad of a major key is minor; hence the key of which this Triad is Tonic, is of course a minor key. — 32 relationship — implies two different SIGNATURES and one and the same mode. Every key has an upper quint-related key of the same mode, whose Tonic is a major Fifth above, and a lower quint-re- lated key of the same mode, whose Tonie lies a major Fifth below. Thus, above the key of C is that of G, whose Tonic is a Fifth higher; above the key of G is that of D, again a Fifth higher, etc. Below the key of C is that of F, a Fifth lower; and below the key of F is that of B\>, also a Fifth lower, etc. In this way the various keys of one and the same mode form a chain of quint-relationship, — upper, when the Tonic is each time a Fifth higher, and lower, when the Tonic is each time a Fifth lower. Each kind of quint- relationship has its several grades; two keys of the same mode, whose Tonics are one major Fifth distant, are related in the first grade, i. e., directly, — as, C and G, C and F; two keys of the same mode whose Tonics are two major Fifths distant are related (in- directly) in the second grade, — as, Cand D, C&ndiB\>, — and so on. Accordingly, in each of the following figures 48 and 49, any key is related in the first grade to the key immediately before, in the second grade to the second key before, in the third grade to the third before, — and so on, the relationship becoming more and more remote. — NB. Read either of the two following figures in the numerical order- of the keys. If we begin with flats, we have a chain of keys in upper quint-relationship; if with sharps, a chain of keys in lower quint-relationship. Chain of major Keys in quint-relationship. •J IS. 14. 13. 12. 11. 10. sz 15. 14. 13. 12. 11. 10. 9. 8. ' q> g> Df ty % fy F c 9. 10. 11. 12. 13. 14. 15. *^ TT^ 6. 5. 4. 3. 2. 1. G D A E B Ftt Cti DA E B Ffl "Cj| Chain of minor Keys in quint-relationship. £ feteflatefefei m 49. t 4^ #* 15. 10. 14. 13. 12. 11. 10. 9. 8. f c s d a 13. 14. .. 15. P i^^fe^gfeft^^^fe i ft >* e# '■# d# a| — 33 S9 N! f 3 ^3 n u o ¥ o o u o I ■*-> B o 3 *= s .a s at « l It J* ', starts from natural key, - nt-cirele to the (S ed the "Qui and returns ti r- *: ft J- -J J keys, call with flats, r ^ ? .e.-f .£ mt-related ind those s F J£s.t -H »- in of qu sharps a T Hi 4 ged chf sys with T Hf 5 5 -a) fferently arra hrough the k mples: <> ^«= «) 35 61. A di key, passes 1 following exa 5 w ^ w >> o fl «f-( o I— < o ■1-1 o I +-> ■ r-( a ► "?£: *■«■ % © is ft -A ■** 3' ,5 "b -2 « +s k ,y T- It- 's P « .p t> -a | P "!« ° ° fix 2- ° a p © p o*,P =C S ~ 60 s o "S Sis ■J-l J_ .1-1 g> .. fl ^ CD •"■" P 00 "^ p O 61) Q, a .-a - I .a § fed £" 6D p cd a S .p •« i O QQ 1 ££ o ^ S tW^ '1^ S-a^ — 34 - relationship, each key after the first having one flat less in its signa- ture: reading from Z to X, lower quint-relationship, each key after the first having one additional flat in its signature. — To make the complete circle of upper quint-relationship , start at U ; arriving at the last key on that line, change the name of its Tonic for that of the Tonic of the key with flats, immediately below; then continue to the right, each new key having one flat less, till the natural key is again reached. To make the circle of lower quint-relationship, start at Z; arriving at the last key on the line, change the name of its Tonic for that of the Tonic of the key with sharps, immediately above; then continue to the left, each new key having one sharp less, up to the natural key. NB. The change of the name and notation of a tone (the sound remaining the same) is called an Enharmonic Change. As the above Figs. 50 and 51 show, the following tones, as Tonics of keys, may be changed thus: B into C\>, F% into Gp, C\ into Dp, g$ into dp, d\ into ep, a\ into b?*. For making the quint-circle, the enhar- monic change of Tonic, instead of being deferred till the last key on the line (as above suggested), may be made at any one of the three places where a key with sharps coincides with one with flats. Summary of this Chapter. 62. The basis of the direct relationship of two keys is their common possession of a certain number ot Triads. 63. Two keys are directly related, when, and in the same way as, their Tonic Triads are directly related, i. e., by either tierce-re- lationship or quint-relationship. 64. Two tierce-related keys are directly related; but two quint- related keys may be either directly or more or less remotely related. 65. If, in the case of two keys, the Tonic of the second one is a Third or a major Fifth above that of the first, the second key is in upper tierce-relationship or quint-relationship to the first; if a Third or a major Fifth below, the second key is in lower tier-ce-re- lationship or quint-relationship to the first. 66. Tierce-relationship exists in the following two cases: 1. be- tween a major key and the minor key whose Tonic is a minor Third below the Tonic of the major key. Such two keys are called * Hence, B and Cp may be called interchangeable keys; similarly, FJJL and Gp, Cjj and Dp, g& and dp, etc. — 35 - parallels; they have one signature and the same Triads in common. 2. Between a major key and the minor key whose Tonic is a major Third above the Tonic of the major key. This minor key is the tipper relative mifior of the major key, and conversely, the latter is the lower relative major of the minor key. Two keys thus related have four Triads in common. 67. Tierce-relationship implies therefore that the two keys differ in mode; and that parallel keys have the same signature and Triads in common. 68. The quint-relationship of two keys implies the same mode and different signatures. 69. Quint-relationship has its grades: two keys whose Tonics are otie major Fifth apart are related in the first grade, that is, directly; when their Tonics are two major Fifths apart, in the second grade, etc. 70. A chain of quint-related keys is formed 1) in upper quint- relationship, by starting from the key with seven flats, up to the natural key, then continuing up to the key with seven sharps: 2) in lower quint-relationship, by starting from the key with seven sharps, up to the natural key, then continuing up to the key with seven flats. See Figs. 48, 49. 71. The Quint- circle implies a return to the starting-key, which is effected by means of an Enharmonic change in the case of the interchangeable keys, of which there are three pairs for the Major and three for the Minor Mode. See Figs. 50, 51. CHAPTER XL Relationship of one and the same Triad to several keys. 72. As we shall presently see, the modulation to a directly re- lated key involves no special difficulty, for it is immediate; where- as to modulate to any other key we must pass through a key (or keys) having an intermediate relationship. In going, for instance, to a key distant by two grades of quint-relationship, we introduce an intermediate third key, directly related to each of the two others ; the same principle is applied in going to a more remote key, — there will be a chain of intermediate keys, each two in succession being directly related. We have seen that the direct relationship 3* — 36 — of two tierce-related or quint-related keys implies that they have eertain Triads in common. This, then, is what enables a key to mediate, or bridge over the chasm, between two different keys: by means of one of the Triads which it has in common with the first key it connects that key with itself, tben reaches out to the other key by another one of its Triads (not possessed by the first key) which it has in common with this other key, — and so on, up to the Cadence. As has been already observed, this linking the old key to the Cadence by intermediate Triads — getting "switched off", to employ our old figure — is (supposing the student's perfect familiarity with the practice of the Cadence in any key whatsoever) the only real difficulty in modulation; hence the importance of the present chapter, which treats of the relationship of one and the same Triad to various keys. 73. In Fig. 46 we have seen how two parallel keys have all their Triads in common, and in Fig. 47, how a major key and its upper relative minor have four Triads in common. The following figure shows how C-major, for instance, has four Triads in common with its upper ^«>rf-relative, C-major, and its lower quint-relative, /'"-major; also, how «-minor has four Triads in common with its upper quint-relative, e- minor, and its lower quint-relative, d- minor. 9£ G.IV =& VI f II zs^ C:I 52 F:V W e=VI a: HI a 3f d:VII II VI IV III IV llT =8= III m VII VI in IV — 37 74. It will be seen that the first and the last Triads in the above figure are common to all the six keys there represented. Every other Triad In the figure may also be shown to belong to six different keys. Specifically: every major Triad is severally I, IV, V, in three different major keys; and III, VI, VII, in three different minor keys. Every minor Triad is severally I, iv, V, in three different minor keys; and II, in, VI, in three different major keys. - 75. Moreover, every major Triad is (V) — major Dominant — of a key in the Bolder Minor Mode (see paragraph 16); and every minor Triad is (iv) — minor Subdominant — of a key in the Milder Major Mode (paragraph 14), — as in the following examples: -#- 53. m^^^m f=(V) m C:l m f: I C:(IV) 76. We may therefore 'lay down the following general principle as to the relationship of one and the same Triad to various keys Every major and every minor Triad has each SEVEN places in as ■many different keys, major and minor. Kemark. The above statement holds good as a general rule. Its application must be limited by the consideration that there are some Triads which could not each have all the seven relation ships without implying keys that are not in use, as they would require more than seven sharps or flats in the signature. The for instance, as I, would imply the major key of G%, with 8 sharps; as IV, that of D%, with 9 sharps; as III, the minor key of e%, with 8 sharps, — etc., — and similarly of certain other Triads. Triad Exercises. I. On the following 40 major and minor Triads, comprising all of these two species that are in use. 38 2. 4. 8. 54. $ &z z%z 9. D: I c: I A:IVf; I/ G: V f - V eVIl At each of the above Triads the pupil should state its various rela- tionships. At Nos. 1 and 2, examples are given of the proper method of doing this. Thus: No. 1, major Triad, is I in Z>-major, IV in A- major, V in G-major; III in £-minor, VI in /jf -minor, major Do- minant (V) in ^--minor, and VII in .?-minor. — No. 2, minor Triad, is I in c-minor, IV in ^-minor, V in /-minor; II in B|?-major, in in ^l?-major, VI in .Eb-major, minor Subdominant (iv) in C-major. It might be well to have the relationships written down by the pupil, in the manner above exemplified under Triads 1 and 2. NB. The Remark immediately before this exercise should not be forgotten. II. The pupil will strike on the piano, or write down in no- tation, the Triads expressed by the following formulas: F-.TV. — Z?:*V. 8 F:Y. 14 3 *:HI. /ft: IV. — Blpim. - £>M- G: IV. 13 a: IV. 19 20 21 23 24 g: (V). - c: IV. - E: (iv). - B\>: IV. - % I. - /: III. - 25 ,,26 27 28 29 in G,x IV. - Fl IV. _ 4: v. - d: (V). - CfrV. - C%: IV. - 5>: II. - /ft: I. - rf: HI. - A\>: (IV). c%: (V). - Zty: III. - B: V. — £|?: IV. — £: (V) — 39 - 31 32 33 34 35 36 37 e\>; I. - *: V. - i4: IV. — 4?: III. - ^ft: (V).- «: V. - 5: IV. — 38 39 ■ 40 41 42 43 ^2: II. - ^:VI. - e\>: (V). - Gt>: II. - ,5": III. - 3:111. — 44 45 46 ■ 47 48 49 c: (V). — e\>: IV. - £>\>: (IV). - .fft: IV. - %VI. - E\> VI. - 50 51 52 53 54 55 /ft: VI. - F: VI. — -4: V. — A\>: VI. - Zty: II. - B\>: (IV). — 56 57 58 59 60 61 D: VI. — *: VI. — F%: (iv). - C: II. — £": IV. — «{>: (V). - ,. 62 63 64 65 „ 66 67 C%. (IV). — b\>: IV. - *: IV. - Op: III. - £#: V. - rf#: (V). - 68 69 70 71 72 73 G>: VI. - f||: IV — £>: (IV). — % (V). — i?|>: V. — F: III. — 74 75 76 77 78 79 a\>: [V. - Z>: II. — £-ft: I. — F: (IV). — &\>: V. — /: (V). — 80 ,,81 82 83 ,, 84 85 E\>: V. - g%: VI. - Of. I. - G: III. - /ft: (V). - g: III. - 86 87 88 89 90. 91 C>: IV. - G: (IV). - a: VII. — G: n. - D\>: VI. — a: VI. — 92 93 94 95 , 96 97 e\f: VI. - E: VI. - B: VI. — Ep: (IV). - /ft: III. — : III. — 104 105 106 107 108 109 C: III. - c: III. — G\>: (iv). — d: VI. - a: (V). — eft: I. — 110 111 112 113 114 ,115 116 a ? : I. - F%: V - G: V. - c:VI. - /: 1. - /ft: V. - A\>: IV. 117 ,118 119 120 121 122 — D V : IV. - F%: I. — E: V. — £f: V. - A\>:V. - a\>: V. - 123 124 125 126 ,127 128 d: IV. - /: IV. - G>: V. - C%: II. — 4'- IV. - B ? : VI. — 129 130 131 132 133 134 a\>: VI. - D: IV. - /: VI. - b: I. - 4: III. — C>: VI. — 135 136 137 138 139 140 g: V. - /ft: VII. - A\>: II. - 5: (IV). - G: VI. - Cp: V. — 141 142 ,, 143 144 145 146 c: VII. — e: I. — ^ft: VII. - t^: (IV). — *|>: VII. - a%: I. - CHAPTER XII. Classification of Modulations. 77. In modulating we shall follow a certain systematic order. In the first place, we classify the modulations according to the mode or modes involved, and get four grand divisions: 1. Modu- lations from major keys to others of the same mode (Major to — 40 — Major) ; 2. Modulations from minor keys to others of the same mode (Minor to Minor); 3. Modulations from major to minor keys (Major to Minor); 4. Modulations from minor to major keys (Minor to Major). 78. A modulation belonging to the 1 st or 2 d division we will call homo-modal, as involving one and the same mode (major or minor, as the case may be) in both keys: a modulation of the 3 d or the 4 th division we will call hetero- modal, as implying a difference of mode, a change of the mode in going to the new key. 79. In the second place, each of the four grand divisions falls into two subdivisions, with reference to the signatures of the keys concerned: 1. the transition to a key with more signs of raising ($ or fc|), or fewer of depression (|? or %; and this process we will call Elevation: 2. the transition to a key with more signs of depression, or fewer of raising, — which we will call Depression. Here follow examples of each: to 1. 55. C: : III f0 a: |g _ a: g|| toB[, : |jjg_A:||g|| ±_ 5. G. C:^I^-G.9^ to stt:5Jj§g-*8toG.-D:|||E toEta§|g_eb: g^^ to a: §£| - C>: g^5 10. to F: §fj=E| — F to C>. - AJ>: 9^^ to A: §f In the above figure, Modulation 1 — C to G — implies one grade of elevation, the latter key having one more sign of raising (jf). — Mod. 2 — a to B\> — implies two grades of depression, the latter key having two more signs of depression ([?). — Mod. 3 — A to C — implies three grades of depression, as there are three sharps to be cancelled (the cancelling of sharps is depression, just as the cancelling of flats is elevation). — Mod. 4 — G to g% — is elevation by four grades, the latter key having four sharps more. — Mod. 5 — g% to G — is depression by four grades, the latter key having four sharps fewer. — Mod. 6 — D to E\> — is depression by five grades, the cancelling of the two sharps counting as two de- - 41 — pressions, and the three flats of the 2 d key making the sum five. — Mod. 7 — e\> to a — is elevation by six grades, there being six flats to be cancelled. — Mod. 8 — Cp to F — is also, for the same reason, elevation by six grades. — Mod. 9 — F to C\> — is de- pression by six grades, as six flats are to be added. — Finally, Mod. 10 — A\> to A — is elevation by seven grades, the cancelling of the four flats counting as four raisings, to be added to the three raisings (by sharps) in the second key. 80. The processes of modulation — with the single exception of the transition from a key to its parallel (see Par. 57) — can therefore be stated thus, in general terms: a) From either natural key (C-major or «-minor) to a key of either mode with sharp signature, is Elevation, by as many grades as there are sharps in that signature. b) From either natural key to a key of either mode with flat signature, is Depression, by as many grades as there are flats in that signature. c) From a key of either mode with sharp signature to either natural key, is Depression, by as many grades as there are sharps to be cancelled. d) From a key of either mode with sharp signature to another with sharp signature, is either Elevation, — by as many grades as the 2 d key has more sharps ; or Depression, — by as many grades as the 2 d key has fewer sharps. e) From a key of either mode with sharp signature to another with flat signature, is Depression, by as many grades as are indicated by the sum of the sharps (to be cancelled) and the flats (to be added). f) From a key of either mode with flat signature to either natural key, is Elevation, by as many grades as there are flats to be cancelled. g) From a key of either mode with flat signature to another with flat signature, is either Elevation, — by as many grades as the 2 d key has fewer flats; or Depression, — by as many grades as the 2 d key has more flats. h) Lastly, from a key of either mode with flat signature to another with sharp signature, is Elevation, by as many grades as are indicated by the sum of the flats (to be can- celled) and the sharps (to be added). — 42 — 81. In the third and last place, we shall classify the modulations according to the relationships of keys. This applies, of course, only to the 1 st and 2 d grand divisions, in which alone such relationship exists (excepting the two cases of tierce-relationship — mentioned in paragraphs 57, 58 — which belong to the 3 d and 4 th divisions). The modulations, therefore, of the 1 st and 2 d divisions will follow each other in the order of the successive grades of quint-relationship of the keys, — first, upper quint-relationship, implying elevation, then lower, implying depression, — and so on, alternately. In the 3 d and 4 th divisions there is no quint- relationship of keys, hence the modulations here are ranged according to the successive grades of Elevation and Depression, exclusively, the latter alternating with the former, as before. CHAPTER XIII. Chains of quint-related major Triads, as pre- paratory to the modulations of the first division. 82. In forming chains of quint-related Triads, in Chapters VII and VIII, we confined ourselves to harmonies of one key — either C, or a, implying major and minor Triads mingled. Now, how- ever, that we are preparing to modulate, our chain of harmonies must be such as can lead us away from our starting-key, and for this purpose, as we propose to begin with quint-related major keys, our best method will be to use (at least, for the present) chains of quint-related tnajor Triads. The reason of this will appear immediately. 83. To modulate from C — for instance — to its nearest quint- relative, is to go either to G (Fifth above), by one grade of Ele- vation, or to F (Fifth below), by one grade of Depression. Now, in the key of C, G is V, and F is IV; hence the modulation here is either from a key to that of its Dominant— C to G; or, from a key to that of its Subdominant — C to F. In the former case, G, which in the key of Cwas V, now becomes Tonic, or I: in the latter case, F, which in C was IV, now becomes I. This change of V into I, or IV — 44 — 86. It should, then, be well understood, that in modulating among quint-related keys, all Elevation is in the direction of the Dominant, all Depression is in the direction of the Subdominant; and that in the former case the first link in the chain of major Triads will be the progression from I to V, in the old key; while in the latter case the first link will be the progression from I to IV, in the old key. 87. It will be indispensable to practise, as a preparatory exer- cise, the connection of links of I V, and of I IV, forming the harmonic chains about to follow, which, however, are not to be mistaken for modulations, properly so called, since the Cadence is wanting to them. NB. 1. All the Exercises which follow are supposed to be in 2 / 2 meter, and every single numeral — as, I, V, (IV), I, etc, — or every pair of numerals grouped thus: V: I, I: IV, etc. — or thus: l/jyy represents rithmically the value of one half -note, invari- ably. A numeral filling the measure, as indicated by a dash after it, — for instance, 1 1 — | , of course represents rithmically a whole note. Sometimes an exercise ends with but one numeral, immediately followed by the double-bar; this means of course a half-note, the other part of the measure being the half-note with which the exer- cise begins (see examples in Figs. 58, 60, etc.). 2. A certain order of keys should, generally, be observed in the Exercises. In Elevation, begin with the key with most flats; these are cancelled one by one, up to the natural key, from which, pass to the keys with sharp signature, ending in the key with most sharps. In Depression, begin with the key with most sharps; these are cancelled one by one, up to the natural key, thence pass to the keys with flat signature, closing in the key with most flats (see Figs. 48, 49, where reading from left to right is the order of Ele- vation; contrariwise, the order of Depression). Exercises. I. Chain of I V,V constantly changing to I. {Tonic to Dominant). l \TJ: YjJ | "Y^i va I "v^i va i v 11 The following is an illustration of the above chain, starting from the key of C\>. The initial Triad is here in root-position, but any other position will be equally good. 45 58. I gSES E0: ~B»- ::*§= ^ tefeFEt^E £# E^PI is f M q,. .Gfc» — m» — ib^Jb. -B?_ I V:I V:I VI V:I V:IV:I V The same formula should be taken through the other major keys, G\>, D\>, and so on, as already indicated. Each time vary the position of the initial Triad. NB. In working out the above chain in keys with sharp sig- nature, from D on, an enharmonic change will be necessary. We know that the major key of G%, as such, is not used, as* also that of D%, — A\> being substituted for the former, E\> for the latter. Now, in beginning the chain of I V with D:\, or with A:l, E:\, etc., the major Triad on G% would appear as I, implying the major key of G%. This Triad must therefore be enharmonically changed, on its second quarter-note, into the major Triad on A\>, as I, to be followed by that on £\> as V, and continuing, when necessary, in keys with flat signature, as for example: 69. I PSgj£f^§ m at- Ei ^ T £=^= n V:I C# G#AJ> V:I V:I II. Chain 0/ V I, I constantly changing toV. (Dominant to Tonic). V|I:V I:V|I:V I:V|I.V I:V I I II The above inversion of the first chain should be taken through all the major keys, beginning with most sharps. The pupil is cautioned against the mistake of beginning the chain with the Tonic (instead of the Dominant) Triad of a key. An illustration of this chain here follows; next in order would be the chain beginning with F$:V — and so on. 46 60. I:V I:V I:V IV IV NB. In taking the above chain through the keys with flat sig- nature, beginning with the key of i?>, the major Triad on C\> would appear as V, implying the key of F\>, which is not used, as such, the key of E being used instead. In this case the major Triad on C\f must be enharmonically changed, on its second quarter- note, into that dn B, as V, to be followed by that on E, as I, and con- tinuing, when necessary, in keys with sharps. Thus: 61. ^ ■frjgdrag g P^¥^=fej Gb Cb_ J>E I:V I:V I:V III. Chain of I IV, IV constantly changing to I. (Tonic to Subdominant.) I I IV;I IV:I I IV:I IV:I I IV:I IV:I I IV The above Chain should be taken through the different major keys, beginning — as in the case of the previous Chain — with most sharps. Thus: 62. P tife Bsi ;feB±i=$S§i!! -zt- ^E| £ i ^ Cjt FJJ B E A D ft F IV.FlV:! IVIIVI IV:IIV:I IV 47 — NB. In taking the above chain through the keys with flat sig- nature, beginning with Z?b, the major Triad on F\> would appear as I, which, as we have seen, is contrary to usage. In. this case, we enharmonically change this Triad into the major Triad on E, as I, continuing, when necessary, in keys with sharp signature. For example: 63. m * tnr ^1 fe# i r IS ^ W—-Qp Ffc> E i£ nm ivj rva iv IV. Chain of IV I, I constantly changing to IV. (Sub dominant to Tonic.) iv | my my | my my|my ijiy 1 1 jj The above chain should be taken through the different major keys, beginning with most flats. Thus: 64. m fr-a-gg i^E f^^i=3=#=#s^=fi^, : Istes ^^ 3 =3= q> Gb_Dt; ty—fy B[,_F. iv my my my my my my i NB. In taking the above chain through keys with sharp signature, beginning with D, the major Triad on C\ would appear as IV, im- plying the major key of G\, not in use. This Triad is therefore enharmonically changed into the major Triad on £>\>, as IV, fol- lowed by that on Ab, as I, and continuing, when necessary, in keys with flats. For example: 65. i ■#= -f S St fp fefr i ftr ^j S IV I IV IIV 48 V. The last one of the preceding four chains — IV I (Sub- dominant to Tonic) — is the same, as to its mode of progression, as the first — IV (Tonic to Dominant), the root-bass in each case moving to the major Fifth above or the minor Fourth below. The principle involved is this: the progression IV to I in one key is absolutely the same, as to the Triads concerned, as the progression I to V in the quint-related key one grade lower, as for instance: 66. I T- P ^ ^^ C:IV ^ a F:I The student should write out — starting each time from a new key (following the order of Elevation) — several chains of the above kind, combining double relationship of each one of its Triads — as in the following example: g^Si IV IIVIIV I:IVI:IV I:IVI:IV VI. On the principle (the converse of the preceding one), that the progression V to I (Dominant to Tonic) in one key is the same, as to the Triads concerned, as the progression I to IV (Tonic to Subdominant) in the quint-related key one grade higher — for instance : 68. ji In T- £ -fir I PS C:V I G:I IV 49 the second chain — V I — is identical, as to its mode of progres- sion, with the third — I IV, the root-bass in each case moving to the major Fifth below or the minor Fourth above. The student should write out — starting each time from a new key (following the order of De- pression) — various chains like that just described, combining double key^relationship of each of its Triads, after the following example: ft 69. si mis * & X i q=s^= 1p T* =fcl^= 3^*3 Y 9* B_ _E. J>. G_ I:V IV I:V I:V I:V I:V ?«^? _E_ D G i iva -ivi ivi iv i ivi IV: i iv NB. The various cases in which the enharmonic change will be necessary are here summarized, for the sake of facility of mem- orizing or of reference. a) The Triad H j a PP eai ' in g as G h^ changes to A\> : I, ) as in Fig. 70 ,Jf' )appearingasi?Jf:IV,change8to-£t?:IV,l at a. b) The Triad e\> , appearing as Fv : V, changes to E : V, as at b. c\, c) The Triad $, j a PP™S a * f ^ Ranges to ^:I,j as c _ ^ | appearing asi?Pp:V, changes to ^:V,j , appearing as Cft:lV, changes to y2[?:IV, as at d. d) The Triad e c a) 70. b) c) d) 8 H&-J55 S*RW sw * ^a=M =* g I_I B# % IV IV V V F|? E I_I Bft A' V V «tf A> IV IV — 50 CHAPTER XIV. Application of the principles laid down in Chapter XL 88. Before proceeding to the practice of modulation , we give in the present chapter a comprehensive Table showing the identity of major and minor Triads' in different keys. This chapter is in fact a specific application of Chapter XI, and is inserted here, im- mediately before setting about modulation, as a summary. — for the sake of reference — of the general principles on which the modu- lations are severally based. The enunciation of each of these prin- ciples is accompanied by its symbolical expression in the peculiar formula of identify adopted for this work, viz: two numerals (re- presenting two relationships of one and the same Triad) connected with a horizontal brace. The vertical brace connecting two keys indicates that these keys are parallels. TABLE I. The same Triad in seven different keys. 1. I of a major key coincides with IV of the ( major key 1 grade above. I 1 !)..?}, .!» VI „ , v., m„ (V)„ VII „ B. J5' V »J ( minor major „ „ „ below. parallel minor key . . minor key 4 grades below. „ „ 1 grade „ IV VI V m (V) vn 1. II of a major key coincides with III of the (major key 2 grades below. 2 - »"»_>' » » ■■•■> r,"-_ V » ., (minor „ ,, „ „ 3- „ „ „ ,', ., „ „ (IV) „ „ major „ 3 „ above. 4- „ ,. „ ,, „ „ VI ■„ „ i major „ 1 grade below. 5 - ;» » >• » ■> >. ■, I » ■, (minor „ „ IV parallel minor key n : III n : v IL(IV) II: VI 11:1 ILIV — 51 — C. 1. Ill of a major key coincides with II of the ( major key 2 grades above, m.-ir 2. „ „ 3. ., „ 4. „ ,. 5. „ „ 6. „ „ „ VI „ „ | major „ 1 grade „ I „ „ (minor „ „ „ ., (IV) „ „ major „ 5 grades „ „ V „ „ parallel minor. key . . D. IILIV III: VI IILI III:(1V) . III:V 1. IV of a major key coincides with I of the ( major key 1 grade below. IV:I 2. 11 11 » 3. 5J 511 4. •>•> 11 J 5. 11 11 1 6. 11 11 11 1 H 11 ., ni „ v ,. (V) „ „ vi „ „vii „ E. (minor „ „ „ „ IVHI major „ 2 grades „ IV: V minor „ 5 „ „ IV. (V) parallel minor key . . IV: VI minorkey2gradesbelow.IV:VII 1. (IV) of a major key coincides with II of the major key 3 grades below. (IV):II 2. 3. 4. 5. 6. 1 11 11 11 7 ') 11 11 1 11 11 11 i 91 11 11 in „ „ j „ „ 5 „ „ (iv>ni V „ „ (minor „ „ „ „ (TV>V , VI „ „ (major „ 4 „ „ (IV>VI I „ „ ( minor „ „ „ „ (iv):I , IV „ „ samekeyintheMinorMode.(iV) : iv 1. V of a major key coincides with I of the ( major key 1 grade above. VI 2. 11 11 11 71 71 3. :i 11 11 11 '1 4. ii ii iy -ii ii 5. ii 11 J) 11 17 6. )» Ii ?j if »J „ III,, „ (minor „ „ „ „ VIII „ IV „ „ (major „ 2 grades „ VIV „ VI „ „ (minor „ „ „ „ VVI „ (V) „ „ samekeyintheMinorMode.V(V) „V1I„ „ parallel minor key . . . V.VEL 4* 52 — a. 1. VI of a major key coincides with II of the (major key 1 grade above. VLII > i) )» 5' >i ?» a »j 2. , 3 4. „ „ „ v 6. ,. „ „ 6 JJ it "•> )) J) IV [ minor ») jj jj VI:IV „ (IV) „ „ major „ 4 grades „ VL(IV) » m ., .. ( ., .,1 grade below. VLin 55 JJ 9> I minor parallel minor key VLV Vl:l II. 1. I of a minor key coincides with IV of the (minor key 1 grade above. LIV 2. '» t i jj it 3. a J i ?i >* 4. " 1 J >' Jj 5. Ji 5 1 )J JJ 6. ,, , J J) »> II 1 major LII „ V„ „ (minor „ 1 grade below. LV „ III „ „ (major „ „ „ „ tffl .. ( IV ) » » major „ 4 grades above. L(IV) „ VI „ „ parallel major key . . . I: VI I. 1. Ill of a minorkey coincides with (V) of the minor key 4grades below. IH:(V) •. VI „ „ ( „ „ 1 grade above. Ill: VI „ IV „ „ ( major „ „ „ „ IMV „ VII „ „ minor „ „ „ below. IILVII 2. J' JJ 7i 3. it it it 4. JJ Jj " 5. j» jj j; 6. a i) ?; 5» >) )> )> •> .. ,1 I » „ parallel major key . . TTT:T V„„ major key 1 grade below. HLV J. 1. IV of a minor key coincides with I of the ( minor key 1 grade below. iv : I 2- „ „ „.. „ „ „ „ VI „ „ (major „ „ „ „ IV: vi 3- ,. >. » ,. „ „ „ V„ „ (minor „ 2 grades ^** •' a a )) j> jj jj J-J-J- 5. 6. ■)•> .J) )) ») IV.-V ma J 01 ' » » „ „ IV:III >» ll » ». parallel major key . . iv : ll „ (IV) „ „ samekeyintheMajorMode.iV:(iy) — 53 — K. 1. V of a minor key coincides with I of the ( minor key 1 grade above. V:I 2. -, „„ „ „ „ „ VI „ „ (major „ „ „ „ VVV1 3. „ „ „ „ „ „ „ IV „ „ (minor „ 2 grades „ yjy 4. „ „„ ., „ „ „ II „ „ (major „ „ „ „ VJI 5. „ „ „ „ „ „ „ III „ „ parallel major key . . . VHII 6. ., „ „ „ „ „ „ (IV) „ „ major key 5 grades above. vj(IV) Ii. 1. (V) of a minor key coincides with III of the I minor key 4 grades above (V):IH 2. „ „ „ „ „ „ „ I„ „ (major „ „ „ „ (V>I 3. „ „„ „ „ „ „ VI „„ minor key 5 „ „ (V>VI 4=- )i ,; j. ,1 ,, ;> ;, V IX ,, ,, ,, „ O ,, ,, ^VJ. Vll 5. „ „ „ „ „ „ „ IV „ „ major key 5 „ „ (V)jy 6. ,. „ „ „ „ „ „ V „ „ samekeyintheMajorMode(V):V M. 1 . VI of a minor key coincides with III of the ( minor key 1 grade below. VLIII *■ tt tt tt tt a n ji X ;; tt (major „ „ ,, , t Vl.l 3. „ „ „ „ „ „ „ (V) „ „ minor „ 5 grades ,, VL(V) 4 VII 2 VIVII ^* tt tt tt I) tt tt tt ' XM - It J) tt tt "it tt T A- T J-l 5. „ „ „ „ „ „ „ IV „ „ parallel major key . . VLIV 6. „ „„ „ „ „ „ V„„ majorkey 2 grades below. VLV N. 1. VII ofaminorkey coincides withlll ofthe minor key 1 grade above. VILIII „ „ (V)„ „ „ „ 3 grades below. VII:(V) „ VI „ „ „ „ 2 „ above. VII: VI „ „ I„ „ major „ 1 grade above.VII:I „ IV „ „ „ „ 2 grades „ TOTV „ „ V„ „ parallel major key . .VII:V 2. }} ji j) u }j 3. }} jj j) }) )i 4. fl jj it 17 jj 5. J7 )} >; >i jj 6. ;> J) j) jt >} — 54 CHAPTER XY. Modulation, in the Major mode, to keys in the 1st grade of quint-relationship. 89. We. will now attempt modulation, confining ourselves at first to major keys (the modulations of the first grand division), and in this particular chapter to major keys distant one grade ot quint-relationship. 90. The .first step is, to settle from which key to which key we are going: this of course determines the Cadence as being in the new, key. Now, our modulation, once chosen, will be in the direc- tion either of the Dominant — in which case the first link of the chain is I V, or of the Subdominant — when the first link is I IV. If the 2 d Triad — V, or IV — of this first link happens to be the very same Triad which is I, or IV, of the Cadence, we are al- ready "switched off", and have only to finish the Cadence, leading into the new key. But in most cases we shall need more than one link to reach the Cadence: at' any rate, whenever a second or a third link brings us to a Triad identical with I or IV of the Cadence, we must not add another link, but follow the Cadence- formula into the new key. 91. Our first modulation is to a key one grade higher (Ele- vation, direction of the Dominant), — as, from C to G. The Cadence for the new key — G — will begin thus: 71. I gftfe-3 EEgE ft:I IV -» 3 2 etc. The first link in the modulation will consist of the Triads I (Tonic) and V (Dominant) of the old key, here C. — 55 72. i -r E^£ etc. C:I But this second Triad — V — is identical (see Table I,* F, 1) with the initial Triad of the Cadence for the new key (Fig. 71); so being now "switched off" from the old key we follow the Cadence into the new key. This modulation will read thus, the Tonic Triad in the link being made a whole-note, and the close of the Cadence being so arranged that the whole forms a rhythm of four measures. 73. < V:I IV Or, in this modulation, we may apply the general principle (Table I, A, 1) that the Tonic Triad of a key is identical with the Subdominant Triad of the key one grade higher. Omitting therefore — for this particular case — the link IV, we regard the Tonic of the old key as identical with the Subdominant Triad of the new key, and initiate the Cadence with this Subdominant Triad (as we shall often have occasion to do, omitting the initial Tonic Triad). In the following example of this modulation the duration of the final Tonic Triad is extended, so as to form a rhythm of four measures. 74. < I 2=i- 3= -& m 2 ° ^: X -$- c a I:IV_ -&- ^ (See, with reference to the 1st Triad .of this modulation, Remark to Par. 94.) 92. Our second modulation is to a key one grade lower (De- pression, direction of the Subdominant), — as, from C to F. * This Table, to which frequent reference will be made throughout the modulations, is found, it will be remembered, in Chapter XIV. 56 The Cadence for F begins thus : 75. < The first link in the modulation is I IV of the old key, thus: 76. ^ C:I IV and since the second Triad — IV — is identical (Table I, D, 1) with the initial Tiiad of the Cadence (Fig. 75), we regard it now as I of the new key, initiating the Cadence, which latter we finish as usual, making the first Triad — I — of the link a whole-note. Thus : 77. < ^ si—ZTsct^ zzn On r^ -Bt- c. I IV: If IV ^- t(IV) V Remark. The principle of the above modulation, viz: that IV of a key coincides with I of the key one grade lower, is, it will be noticed, merely the converse of the principle underlying Modu- lation 1, — i. e. that I of a key coincides with IV of the key one grade higher. 93. The exercise now in order is the practice of these first two modulations in the other major keys, following the order of keys given in Fig. 48, reading from left to right for Elevation, and from right to left for Depression. Thus, in Modulation 1, from C\i to Gp, then Gp to Dp, — and so on: in Modulation 2, from C% to 7*$," then F% to B, — and so on. Be it said here, once for all, that every one of the modulations to follow is to be taken through the various keys in the same order of Elevation or of Depression, — 57 — as the case may be. For the purpose of this practice our modulation- formulas, each one numbered and accurately described, are grouped and classified at the end of this work (Table II). The particular formulas to be practised in connection with this chapter are those of Modulation 1 and Modulation 2. In every modulation having more than one formula, use, for the present, only the first one given, marked A. CHAPTER XVI. Modulations, in the Major Mode, to keys in the 2 d and 3 d grades of quint-relationship. 94. Our third modulation is to a key two grades above, — as, from C to D. It is based on the principle (Table I, F, 3), that the Dominant Triad of a key coincides with the Subdominant Triad of the key two grades higher. Accordingly, the first link being I V, with the 2 d Triad — V — of the link we at once touch the Sub- dominant Triad of the new key, and have only to go on and finish the Cadence. Thus: 78. i ^>f-2- 2 -2~-£- C. I ** 38t §m D_ V:IV f & _ Eemark. As we have already seen, the explicit sign that the Subdominant Triad in the Cadence may be either major or minor, at pleasure, is this: iiy, — as, for example, in Mod. 2, Fig. 77. (i In general, the Subdominant Triad immediately preceding the |, even when expressed only by IV, may, after entering as major Triad, be changed at pleasure into minor by lowering its Third on the note equal to half the value of the Triad. Thus, for instance, in Mod. 1, Fig. 74, — 58 — 79. < and in Mod. 3, Fig. 78. 80. 95. Our fourth modulation is to a key two grades below, — as, from C to Bfc. Principle, as in Mod. 2. The first link is I IV; this IV, not reaching far enough, is in its turn regarded as I and followed by its IV, which coincides with the Tonic of the new key and initiates the Cadence. Thus : 81. I IV:I IV:I|IV — ■—- I (IV) 96. Our fifth modulation is to a key three grades higher, — as , from C to A. The first link is I V, the 2 d Triad of which not being identical with either the Tonic or the Subdominant Triad of the new key, we must go on till we reach a Triad that is. We therefore regard this 2 d Triad as Tonic — G:\, followed by its Dominant, which latter, as coinciding with the Subdominant Triad of the new key — A, initiates the Cadence. Thus : _, Or thus: ~ac ^ G. I V:I V;IV V I — 59 — 97. Our sixth modulation is to a key three grades lower, — • as, from C to E\i. First link, I IV, the 2 d Triad of which does not coincide with either the Tonic or the Subdominant Triad of the new key. We therefore regard this 2 d Triad as Tonic — F:\, followed by its Subdominant. This latter Triad, not yet reaching far enough for our purpose, is in its turn regarded as Tonic — Bfr-.l, followed by its Subdominant, which latter, at length, we find identical with the Tonic Triad of the new key, wherewith the Cadence is initiated. Thus: 83.- 9i=s; P '-=^$i r ■ B[7 _— — Eb IV:I IV:I IV:I IIV — T— •— I(1V) 98. The examples thus far given sufficiently illustrate the meth- od to be followed (for the present) in the remaining modulations of the 1 st Division. The process may be summed up in a few words. — 1. In Elevation, the first link is I V, and the 2 d Triad of this link is followed by one upper quint-related major Triad after another, till the Subdominant Triad of the new key is reached: this Triad initiates the Cadence, which should then be finished. — 2. In Depres- sion, the first link is I IV, and its 2 d Triad is followed by one lower quint-related major Triad after another, till the Tonic Triad of the new key — and with it the beginning of the Cadence — is reached. The Cadence is then to be finished. Eemark. From the fact that in the regular* formulas for De- pression the Cadence is always initiated with the Tonic Triad, and in those for Elevation with the Subdominant, it follows that Depres- sion is a longer process than Elevation. In fact, every one of our regular formulas for Depression shows two Triads more than the formula for Elevation by the same number of grades. This Remark applies equally to the regular modulations of the 2 d Division. 99. As a test-exercise, the pupil should now endeavor to apply the principles of modulation, as far .as explained, and work out, at * The regular formulas are those marked A, in the Table of Formulas, in distinction to Variants, Short cuts, etc. — 60 — the piano, the first six modulations without looking at the formulas. For, the aim of this work is, not nferely to give a table of universal formulas for modulation (thereby dispensing with the immense number of examples which musical notation would involve) — the purely mechanical rendering of which formulas would constitute the first stage of advancement, — but, chiefly, to educate the student in the theory and principles of modulation, up to the point that he may be able to work out by himself a given modulation without looking at the formula ■ — in a word, to extemporize a modulation. 100. The preliminary steps in this exercise are two: 1. De- termination of keys. From what key to what key is the modulation? (At first, it might be necessary to write down in notes the Cadence for the new key, keeping it in sight, for reference.) — 2. Determi- nation of the direction of the modulation. Is .it towards the Domi- nant, or the Subdominant ? What, accordingly, will be the first link of the Chain of Triads ? This point having been settled, the student should follow out the process of modulation , as indicated in para- graph 98. As to a precise rhythmical form, this can hardly be ob- served in extemporizing a modulation: it will suffice that the Tonic Triad in \ form (in the Cadence, immediately preceding the Domi- nant Triad) be somewhat strongly accentuated, so as to appear as the first half-note of the measure. CHAPTER XY1I. Modulation, in the Major Mode, to keys in the 4 th , 5 th , 6 th and 7 th grades of quint-relationship. 101. Our seventh modulation is to a key four grades higher, — as, from C to E. ^ Or, 2 d measure, thus 61 — 102. Our eighth modulation is to a key four grades lower, — as, from C to A^. 85. i h-r 3 ■¥c s^^^^s^y wm^^^g^ c. F. ^s:: Je 3 .Bb % AJ, V I. I IVd ivjivj IVJJIV 103. Our ninth modulation is to a key five grades higher, as, from C to 2?. | i , Or, 3d meas. thus 86. % «=3 3- ty_B[? G> I I IV:I IV:I IV:I IVJ1V:I IV:I)IV I V I (IV) 107. Our thirteenth modulation is to a key seven grades above, as, from C to C%. Or, meas. 3. _G_ _B i ^ A I V:I V:I V \ E_B-Cjt I IV:IV:IV:IVI -r V I 108. Our fourteenth modulation is to a key seven grades lower* as, from C to Op. I Ps2 91.< *p ■f p^ 3iS -M p^N^g * ss ^ 4k i?E te C F-Bj 7 -Eb-Ai 7 -»t'-Gb ty | i ^^^rjjiiV|iiva: rvvi jiv i v i NB. The remaining modulations of this division — Nos. 15 to 26^ inclusively — are for special reasons passed over for the present. The eight modulations illustrated in this chapter should now be worked out at the piano in the manner recommended in paragraphs .99, 100. They ought to be thoroughly mastered before proceeding to the next chapter. At this point, when the principles of modu- lation according to this system may be assumed to be pretty' well — 63 — understood, it would be an excellent exercise for the student to construct, the formulas of the first fourteen modulations for himself, i. e., not writing them down from memory, but, as it were, inventing them, in accordance with the principles ou which the several modu- lations are carried out. In this the chief thing would be the proper succession of Triads for a given modulation and its proper expression by our method of numerals, etc: the metrification, i. e., the arrange- ment according to Meter, might be done, afterwards, as a secondary matter. CHAPTER XVIII. Preparation for modulation in the Minor Mode •■• (2 a Division). Chains of quint-related minor Triads. 109. We' now turn our attention to the modulations of the 2 d grand division, in the Minor Mode, devoting the present Chapter to certain preparatory exercises in forming chains of quint-related Triads — sueh chains, that is, as shall be fitted to leading from one key to another. For this purpose we use here chains of minor Triads (with a modification to be noticed presently), just as, in the Major Mode, the chains consisted of major Triads. . 110. We begin with the chain of I V (v constantly changing into i), as a preparation for modulation in the direction of the Do- minant. Its formula is this : IIV:T V:I|V:I V:1|V:I V: 1 1 V 1 1 to be worked out as in the following example, starting — as usually in Elevation - — from the (minor) key with most flats : 92. < I ifeto=g *P irifc^E T= ai m #=t ifefefed; ^ i — e\, — b[?_f- c_ V:I V:L V:I V:I P mm V:I V:I 64 The above chain should be taken through the other minor keys in the usual order, varying the position of the initial Triad. In the keys with sharp signature, from b on, an Enharmonic Change will be necessary, viz: the Triad A — which can appear as V only — must, when the formula would make it appear as I, be changed into «b, in which shape it may be i, or V, for subsequent minor keys with flat signature. Thus: etc. instead of; etc. g^^Hg^g ajf e| f c. #^ feEf S I V:I V:I ail eij \& I* V:rV:I ff V 111. The next chain is in the direction of the Subdominant, thus: iiiva ivriiivj; ivniiva ivjiivii which formula is worked out as in the following example, starting — as usually in Depression — from the (minor) key with most sharps. 94. < pn mm. mm s=s= £S mi 1 w ^ s f t~ 44#- M~ =** IV:f IV: I IV: F IV^I IV^I IVd IV 112. The chain in the direction of Tonic to Subdominant may — as we have already seen (Fig. 68, 69) — be also interpreted as iu the direction of Dominant to Tonic — V I:V I:V I, etc.; hence the above example may take the following additional reading: 95. fife 3 ISSsgg =te ^n i^d=# * -^ V - g# — 4- I:V I:V A— b- 1F ^ I:V I:V I:V I:V — 65 Now, generally speaking, it is smoother and more natural that in the progression of Dominant to Tonic — V to I — the Dominant Triad should be major, for at least part of its value — say, for its second half, immediately preceding the Tonic Triad. This applies, accordingly, to the chain represented in Fig. 94, as being identical with that in Fig. 95. Therefore, in the chain in the direction of the Subdominant — which we shall treat as being in the direction of Dominant to Tonic — , we will introduce in each Triad (except the last) a so-called leading-tone*}, by raising the Third a chromatic half step on the 2 d ' quarter-note. In this way each Triad of the chain will represent, on its first quarter-note, a Minor Tonic harmony, on its second, a major Dominant harmony. This latter progresses to its minor Tonic harmony, which is treated as before, and so on, to the end of the chain, as in the following example: I ftt 3 *^m^pp^«w 96. < Wr P^fepp^ m m sr f=t r ajfdfl I:V(V) I:V(V)I:V(V)I:V(V) I:V(V)I:V(V)l:V(V) I 4^-4^-4- i 113. In accordance with the above principle, our formula for the chain in the direction of the Subdominant (Depression) will be this — I:V|I^V I:V|I:V I:V|I:V I r V J 1 11 with the understanding that it is to be rendered as in the example (Fig. 96), the introduction of the leading-tcme — implying the change from v to (V), as expressed in the example — being omitted in the formula, for the sake of greater simplicity. The foregoing chain should be practised in all the other minor keys in the usual order, varying the position of the initial Triad. * Technical term for a tone which shows a tendency to lead to a tone on the degree above or below. Thus, e. g., the seventh degree of a major scale naturally leads, under certain circumstances, to the Ionic (Octave of the first degree), rather than to the sixth degree, or the fifth, etc. The progression of the leading-tone is generally by a diatonic half-step \ hence a, step is often changed, by chromatic alteration, into a half-step, to form a leading-tone, — as, for instance, in the scale of a-minor, the progression g — a, into ffjj. — a, etc. 5 — 66 — In the keys with flat signature, from d on, whenever the formula «b would make the Triad c\> appear as V (implying the minor key 4> of d\>, not in use), this Triad must be enharmonically changed into * , as V of c\, the key which replaces d\> minor. The chain will then continue in keys with sharp signature, thus: etc. instead ^yW j^Zr^f &Em I:V I:V I CHAPTER XIX. Modulation, in the Minor Mode, to keys in the 1 st , 2 d , 3 d and 4 th grades of quint -relationship. 114. The principles already laid down for modulation in the 1 st Division will apply — mutatis mutandis — for the 2 d Division also, and the formulas for both modes will have a general corre- spondence, though this is to be said specifically of the formulas for Elevation, on account of our representing Depression, in the Minor Mode exclusively, by the formula VI, rather than iiv, as already explained. 115. In our twenty-seventh modulation — to a minor key one grade above, — as, from a to e, the connection is immediate, on the principle (Table I, H, 1), that I of a key coincides with iv of the key one grade above. Thus: 98. Is -^ -0- T a e_ I:IV. (V) — 67 — 116. Our twenty - eighth modulation is to a key one grade below, — as: 99. fta 1 /T* S^i fli'» « r>. <> — »- 1-^ =P^— fl a «> | S>— L_ <2 a (1 . I:V. IV t (V) I Eemark. As in the above example, the change of a minor Triad into major by raising the Third for a leading-tone, is to stop as soon as the I of the Cadence is reached. 117. Our twenty -ninth modulation is to a key two grades above, — as: 100. i s =3 r gEfE ^Fff^B g f a. I V:IV ¥ (V) 118. Our thirtieth modulation is to a key two grades below, — as: 101. i 1 ^^ rr -&=n % §^ T ad_ I:V ^ H- ^ JM —s- I:V IV "f (V) 119. Our thirty -first modulation is to a key three grades above, — as: 102.^ $ zsz gr w^l$rW^ « fe=*= =s* — e fJM V:I V:IV I (V) 5* 68 — 120. Our thirty-second modulation is to a key three grades below, as: 103. <^ Jrt>j ^^ife i ~f*f =3 r ^U-hJt^U a (1 . I:V P J-J^J I:V I:V IV ^ (V) 121. Our thirty-third modulation is to a key four grades above, — as: e_ V:I V:I V:IV I (V) I 122. Our thirty-fourth modulation is to a key four grades below, — as : 105. < ^PBiPliP T 1 ^" T- M v ■»•»■» -gg^Ks ? < g a; i pfi > ■ €> ad I:V I:V I:V I:V I IV *-f (V) I. The preceding eight modulations should be practised in the various minor keys without looking at the formulas, as suggested in paragraphs 99, 100. 69 CHAPTER XX. Modulation, in the Minor Mode, to keys in the 5 th , 6 th and 7 th grades of quint-relationship, 123. Our thirty -fifth modulation is to a key five grades above, — as: 4- 106. 3T ^ 1* - — i i j=j£ ^-g-4 2-^^ ^E&&E E$ *- ■*»-«# 2 V:I V:I V:I V:IV I (V) 124. Our thirty -sixth modulation is to a key five grades below, — as: M WE£ ^jj^^^ mm l&il&r 3St » X \ fi~%. \ % ±® 107.< M -laO* ^|= r a d . t~ ^ F I:V I:V I:V I:V IjV I IV 1 (V) I 125. Our thirty - seventh modulation is to a key six grades above, — as: 108. < % 2=g i ^ -=9" 3 * # eP ^ a e b f| cty djj f I V:I V:I V:I V:I V:IV I (V) I ^ P a d_ -tyl d? ajz i3F* IWj- u >g?1> I:V I:V I:V I:V I:V I:V I:V IV (V) i- 129. The six modulations of this chapter should be worked out, without looking at the formulas, in the various minor keys; moreover, the student might be called on to construct for himself (as recommended for the Major Mode, at the close of Chapter XVII) all the formulas for modulation in the Minor Mode which have been thus far exemplified. — 71 — CHAPTER XXL Connection of tierce-related Triads in either Mode, preparatory to abbreviations in modulating. 130. As a preparation for the abbreviation of the longer modu- lations by means of "short cuts" (see Chap. XXII), we must now study the connection of tierce-related Triads. Here there will be, for eveiy pair of Triads, two connecting-tones , and only one of the upper voices has to move. In this chapter we include both the Major and the Minor Mode, and a single Triad-connection in upper tierce-relationship, and another in lower, will be a sufficient illustration for the Exercises. The tierce-relationships of Triads are exemplified in the key of C-major, for the Major Mode, in Fig. 12; in the key of «-minor, for the Minor Mode, in Fig. 14. Upper Tierce-relationship. 131. In connecting — for instance — C:I feE upper tierce-relative, III IF with its fc£ , we first, according to the Rule & — ' (paragraph 30), keep the connecting-tones — here, e and g — in their respective voices: /ry - then we move the other tone, c, to the next degree — Below, as invariably in Upper tierce-relation ship, — and the two Triads being thus properly connected ^ eZ^g, - we have only to add the root-basses, thus: a) b) J 112. c=I ni 72 NB. The progression of the Bass as at a is generally better, melodically , than as at b (a Third being a smoother progression than a Sixth), though the latter may under certain circumstances not be conveniently avoidable. 132. In the above example (Fig. 112) the initial Triad is in quint-position (see paragraph 46), involving the tierce-position of the 2 a Triad. The initial Triad may also appear either in ratf-position, with the 2 d Triad in ^VzAposition , as at a, in Fig. 113; or, in /zVrci-position, with the 2 d Triad in root-position, as at b: a) b) 113. P fee •* •*■ ^ ^ C:I III ^ Lower tierce-relationship. 133. For illustrating the connection of a pair of Triads in lower tierce-relationship, we will connect a: I jfc)~~^~ and VI ^ . The connecting-tones are a and c, 1= the other tone, e, moves to the next degree — Above, as invariably in lower tierce-relationship: fa _^.J ^ The root-basses, being added, form ■&ZZ.-+- a progression of either a descending Third or an ascending Sixth (here — mutatis mutandis — the remark at NB., paragraph 131, will apply): 1st Triad in Do. in Root- Do. in Tierce- Quint-position. position. position. 114. -$=£ 9t sal STZ2 — VI =2:^ "*-+*&- — 73 — Exercises. NB. These Exercises should he done at the piano. Take each one through all the keys of the mode indicated, as it is just as ne- cessary, for our purpose, to he perfectly familiar with the connection of tierce - related as with that of quint - related Triads. (For the order of keys see Figs. 48, 49 .) Each exercise is to he done (in the same key) three times, according to the following Scheme: a. Strike the Eoot-hass of the 1 st Triad indicated. 1 st time. { b. Add the three upper voices in root-position. c. Connect with 2 d Triad indicated, the Bass moving simultaneously with the upper voice which moves. a. As at a, 1 st time. 2 a time. ^ b. Add the three upper voices, in tierce-position. ?. As at c, l et time. !a. As at a, 1 st time. b. Add the three upper voices, in quint-position. c. As at c, 1 st time. Then take the same exercise, in the same three ways, through each of the other keys ot the mode indicated. Then take the next exercise, going through it in the same different ways, — and so on of the others. I. (In Major ^ Connect Triads paired thus: 1. I in. — 2. I VI. - 3. II IV. — 4. Ill V. - 5. Ill I. — 6. IV VI. — 7. IV II. — 8. V III. — 9. VI I. — 10. VI IV. II. (In Minor.) Connect Triads paired thus: 11. I III. — 12. i vi. — 13. m v. — 14. in i. — 15. iv vi. - i6. v vn. 17. V HI. - 18. VI I. — 19. VI IV. - 20. VII V. CHAPTER XXII. General Principles of Abbreviation in Modulation, 134. It is desirable to shorten the modulations to come (they being very long), and even some of the preceding ones. In the present chapter therefore we, interrupt the course of our modulations to consider the general principles on which we may effect abbreviation by means of "short cuts." — 74 — 135. In the formulas hitherto followed we have had, in modu- lating to the more remote keys, to pass through many intermediary keys, making the modulation both tedious and — as our Triads are all quint-related — harmonically monotonous. What we need, then, is a means of eliminating some of these Triads, and this in such a manner as also to afford the desired harmonic variety. The insertion of a pair of tierce -related Triads in our modulations will accomplish this double end, as we shall presently see. 136. Our means of effecting the desired abbreviations in modu- lating will be: (1) for the Major Mode, the introduction — outside of the Cadence — of the Minor Subdominant Triad — (IV) — of some key, in connection with some major Triad to which it is tierce- related, — as will presently be explained: and (2) for the Minor Mode , the introduction — also outside of the Cadence — of the Major Dominant Triad of some key, in connection with some minor Triad to which it is tierce-related. The short cuts for the Major Mode (first grand division of modulations) will come first under consideration. 137. The minor Subdominant Triad, which now becomes so important, has thus far in our modulations appeared only in the Cadence, and then, either ad libitum — instead of the normal (major) Subdominant Triad, or following this Triad, for the sake of more rhythmical motion. In our short cuts, however, when the minor Subdominant Triad — which we will henceforth designate , for brevity's sake, by its symbol, (IV) — is introduced outside of the Cadence*, as an express means of abbreviation, it is not preceded by the normal Subdominant Triad, IV. 138. The basis of our short cuts, in the Major Mode, is the identity of (iv) of a key — introduced as just explained — either (1) with II, or (2) with VI, or (3) with III, of some other major key. CHAPTER XXIII. Short Cut for Depression (Mod. 6), through the identity of (iv) with n of another key. 139. We begin our short cuts with Modulation 6, to a key three grades below. In its regular form (Fig. 83) three major Triads come between the first one and the Subdominant Triad in the Ca- * Except in Mod. 7, formula B, when it begins the Cadence. — 75 — dence. The short cut is effected by eliminating two of these Triads*, on the following principle (Table I, E, 1): In Depression, (IV) of the old key coincides with II of the hey THREE GRADES BELOW. Now, Modulation 6 being precisely to a key three grades below, it follows that the (IV) of the old key coincides with n of the new; and since II and IV of a key are tierce-related Triads (see Fig. 12), we have, on reaching n of the new key, only to let it be followed by IV of the same key, thus initiating the Cadence. The following illustration shows six Triads as against eight in the same modu- lation in Fig. 83. 115. I Abbreviation of Mod. 6. % ^ H i r2 «- ^ 9. ac 4^= -Kt~ _Et». I (IV): II IV I Eemark. In the above short cut the Cadence, it will be seen, does not begin — as it has hitherto begun in Depression — with I of the new key (see Par. 98). This will be the case, in our short cuts. generally, so that the Cadence will be alike for both Elevation and Depression. 140. The above short cut — and (be it said here, once for all) each of the following ones — should be practised, according to the proper formula — marked B — in Table II, in the various keys. * Only in this sense can it be called a short cut, since it does, in fact, not lessen the number of measures (compare Figs. 115 and 83). In general, the necessity for abbreviation is of course greater in the longer modulations (to come) than in those which are already short, in which the short cuts are introduced chiefly for the sake of harmonic variety. — 76 — CHAPTER XXIY. Short cut (Mods. 7, 8, 10, 12, 14) through the identity of (iv) with vi of another key. Elevation. 141. In the regular form of Mod. 7 , to a key four degrees above (see Fig. 84), two major Triads come between the first one and the IV beginning the Cadence. The short cut is eflected by eliminating these two Triads (thereby making an immediate connec- tion with the Cadence), on the following principle (Table I, G, 3): In Elevation, VI of the old key coincides with (IV) of the key FOUR GRADES ABOVE. In the present modulation, which is precisely to a key four grades above, we connect the first Triad — I — with its lower tierce-relative — VI, which, as coinciding with (IV) of the new key, initiates the Cadence in that key. Thus: 116. Abbreviation of Mod. 7. _E \ I VI: (IV) 1 V Kemark. In the above short cut exception might be taken to the abrupt change from I of a key to I of a key four grades higher, involving the objectionable melody illustrated in the tenor {g — a — g§) in Fig. 116 (which would be still worse in the soprano voice). It would therefore be as well to use the principle of this short cut — applied, of course, differently — for Depression only, especially as there is no great necessity for a short cut in Mod. 7. — 77 Depression. 142, In Mod. 8, to a key four grades below, there are in the regular form (Fig. 85) four Triads between the first one and the IV of the Cadence. We eliminate three of these Triads on the principle (Table I, E, 4) — the converse of that which we have just used for Elevation — that (IV) of a key coincides with vi ot the key four grades below; or, differently stated for our particular purpose : In DEPRESSION, VI of the NEW KEY coincides with (IV) of the key FOUR GRADES ABOVE. It follows from the above, that whenever, in Depression, we introduce a minor Triad which is at once (IV) of a key and VI of another, we enter the sphere of this latter key, which is four grades below the former. In the present modulation, therefore, which is to a key four grades below, the (IV) of the old key connects immediately with the VI of the new, and whereas, in Elevation (Fig. 116), the major Triad — I — is followed by the tierce-related minor Triad — VI; here, in Depression, the minor Triad — VI — of the new key is followed by a tierce -related major Triad — IV, of the same key, thus initiating the Cadence. For example: Abbreviation of Mod. 8. 117. < % BiH2- 2 sS^S 2 n £ I — (IV): VI IV 143. In Modulations 10, 11, 12 and 14, the Cadence begins, as in Mod. 8, with IV, preceded by its upper tierce -relative, VI. The modulation sets out in the regular form — I IV: I, etc., — continuing until a Triad is reached whose lower quint-relative — minor Triad, so that the two are correlated as I (IV) — coincides with VI of the new key. The Cadence then follows, as above stated. — 78 — Abbreviation of Mod. 10. 118. $ * -2^ f= * ^ff %f ZJE ±* 4E 35 -I I IV1 (IV): VI IV 1 Abbreviation of Mod. 12 V I ^ §@5 £ 119. 9t =^= =fe E^ S P-1T-J£. f C F Bb Gb_ I IV: I IV: I (IV):VlIV I Abbreviation of Mod. 14. PTl'^^s^ EUfcrrj- 120. |>g=qFfe: J^fe » I l O fc^- ^ J£= 2 '1° r F ^^ _F__BU E> CU I IV:I IV:I IV I (IV). VI IV I V I_ CHAPTER XXV. Short Cut for Elevation (Mods. 9, 11,13), through the identity of in with (iv> of another key. 144. In Mod. 9, to a key five grades above (Fig. 86), we have, before reaching the IV of the Cadence, three intermediary Triads; in Mod. 11, to a key six grades above (Fig. 88), four; and in Mod. 13, to a key seven grades above (Fig. 90), five. In each of — 79 — these modulations three of these intermediary Triads may be elimi- nated, on the following principle (Table I, C, 5): In Elevation, hi of the old key coincides with (IV) of the key FIVE GRADES ABOVE. This short cut is the shortest of all the ordinary ones*, and is used in Elevation only.** Its application is illustrated in the follow- ing examples: Abbreviation of Mod. 9. 145. In modulating to a key more than five grades higher, the best application of this short cut is as follows: Begin the modu- lation in the usual way, IV: I, etc., and continue until a Triad — I — is reached whose upper tierce -relative — m — coincides with (IV) of the new key; let this latter Triad — in : (iV) — im- mediately follow, initiating the Cadence. * Farther on, we shall become acquainted with formulas called extra- short cuts, also with some involving an Enharmonic Change. ** In Depression, this short cut would be based on the principle, that HI of the new hey coincides with (IV) of the key five grades above. Now, although hereby a depression of five grades is at once effected, yet, not only is this short cut relatively inharmonious, but — which is the chief point — the modu- lation is not made one whit shorter than by the short cut treated of in Chap. XXIV, which effects a depression of four grades. This is seen by com- paring the following two formulas — Mod. 10. I ( iv ): ni | I IV | I V | I - Mod. 12. I | IVI (iy^ni | I IV | I V 1 1 1| with those of Figs. 118, 119, respectively. The simple reason is, that in the above two formulas the Cadence begins with I, which is omitted in Figs. 118, 119. 80 122. % m=^ Abbreviation of Mod. 11. &=t I PEEE3 r n V:I III: (IV) I 123. 1 Abbreviation of Mod. 13. V:I V:I III:(IV) 1 CHAPTER XXVI. Modulation, in the Major Mode, to keys in the 8 th up to the 14 th grade of quint-relationship. 146. In the present chapter we conclude, for the present, the modulations of the 1 st Division, the course of which was interrupted at Mod. 14 (to a key seven grades below), in Chapter XVII. Here no key will have the natural signature, but in each modulation one of the keys will have such a sharp signature, the other, such &flat signature, that the sum of the sharps and flats (see Par. 80, e, h) will be eight, nine, ten, eleven, thirteen or fourteen, according to the grades of quint- relationship involved. Thus, eight gr imply that the 1 st key has 7t? >> >> >> u - op >l 1) )} it "r >> o ;; >) 4)7 or, of Elevation or Depression — of Elevation, for instance, will the 2 A key, 1# : 7 + 1 = 8.' „ „ ',, 2|:6 + 2 = 8. „ „ „ 3fl : 5 + 3 = 8. „ „ „ 4J( : 4 + 4 = 8. 81 the 1 st key has 3[? )! II !! >! 2|? >) >> JJ >> ■*■!? the 2 d key, 5# !) >! J! 6jJ 7# 3 + 5 = 8. 2 + 6 = 8. 1 + 7 = 8. 147. That the modulations of this chapter may not be tedious, we shall, instead of working them out in the regular way, apply to them two methods of abbreviation, viz: (1) the short cuts already introduced in Chapters XXIV and XXV; and (2) a new short cut, involving an Enharmonic Change. The modulations will follow each other in regular order, Depression alternating with Elevation. First method of Abbreviation. 148. In the first method of abbreviation we use, for Elevation, the short cut expressed thus: ili:(iV), effecting an elevation of. five grades, as explained in Chapter XXV; for Depression, the short cut (IV): VI, effecting a depression of four grades, as in Chapter XXIV. 149. Our fifteenth modulation is to a key eight grades higher, — as : 124. I V:I V:I V:I III:(IV) I 150. Our sixteenth modulation is to a key eight grades lower. — as: 125. < ^ri-^m m d . , s CpsJ bj l ire^fS J W-fy M *fe£ ^ T ~^= *t 3=fcifc gs itet fc a* ^^ S£* a M D G CJ F Bb_Eb Cb | i ivj iyjiyi rvjiiyj(iv):viiv i v i 153. Our nineteenth modulation is to a key ten grades higher , — as : 128. < r S^ % rapp 3i« ^s= m T< EJ7_ ^ k -\> as equivalent to C%, A appears only four grades below. The formula is therefore that for Mod. 8, Table II; and we shall, in this and every subsequent case of the kind, always choose the short cut, when there is one. The short cut — B — in the present case is: (IV):VI IV | I V | I - which is applied as follows, the Enharmonic Change being made at the outset. Abbreviation (with Enharmonic Change) of Mod. 15. 132. < ¥^4m^ ^^^^M -p^r* 3% I r^J^ ^f >- t <*- (IV):VI IV 158. We will now take the case of the interchangeable key being the new key, in which the Enharmonic Change of course cannot be made at the outset. In modulating — for instance — from F to Cjf, likewise eight grades of Elevation, we enharmoni- cally change C% into D\>, implying four grades below _F. The for- mula will therefore be the same as above, applied as follows: Abbreviation (with Enharmonic Change) of Mod. 15. 133. < n =fc ^= F_ I_ w^^m %y® I ^^hf*^=r J>t,C#_ .(IV): VI IV 1 ^ 3C 159. As an example of a modulation by many grades of De- pression, shortened into one by few grades of Elevation, we will take — for instance — Modulation 16, — say, from F% to B\>, eight 85 grades lower. Considering F\ as G\>, the modulation to Zfy will be by four grades si Elevation, — formula B for Mod. 7, Table II: 6 i vi:(iv) i i v 1 1 I -. Abbreviation (with Enharmonic Change) of Mod. 16. ita? g4=ffi|B fe| -3r- j! w y®—y ^ 134. I ^fett j-r^ ^ ^^ -H -^g -f F S T %■ VI: (IV) 1 Or, should formula A be preferred to B (see Remark to Para- graph 141), it would appear thus: 135. Bfeifellii! I VI Dtz A> BJ, f V:I V:IV I 160. In the following example of the above modulation with formula A the interchangeable key is the new key, not the old. Abbreviation (with Enharmonic Change) of Mod. 16. 136. I w 3^ Pfe 3f£ r A_ I ^B 1" I _BJ r. * ^ V:I VI V : IV V >- ¥=& =>&£ I 161. It is in Modulations 15 and 16 that we find the exceptions referred to in Par. 157, viz: the only two eases — in the 1 st Division 86 — — of two keys more than seven grades distant, yet neither one an interchangeable key. These two cases are: AV to E, Elevation by eight grades (Mod. 15) ; and E to AV, Depression Jy eight grades (Mod. 16). Here the Enharmonic Change is practicable only at (IV) and at VI, supposing the use of formula B (Mods. 8 and 7). That is: in Elevation the (IV) in A\l — /\> — is changed into the notation d\> of VI in E — e ; in Depression the process is the contrary one, 4 VI in E being changed into the notation of (I v ) in Afy. If, for De- pression (E to A\i), the formula A (Mod. 7) is used, the V in B\ — is changed into the notation of IV in AV •b / for initiating d\, the Cadence in this key. 162. In Mod. 17, to a key nine grades kigfier, — as, from DV to E — we mentally change D\/ into C\. C$ to E, Depression by three grades; formula (B) for Mod. 6, exemplified as follows: Abbreviation (with Enharmonic Change) of Mod. 17. 137. ffi'U di^ i ^ Pl m& m : J— ^ ^ E^ i^J p-LjJ t p— bi mr^i f* — |J— i-fr t <*- (IV): II IV V 3* I Example of Mod. 17, the new key being the interchangeable one: 138. 163. In Mod. 18, to a key nine grades below, — as, from F% to Ep. — we identify F§ with G\t. G\> to Ep, Elevation by three grades; formula for Mod. 5. — 87 — Abbreviation (with Enharmonic Change) of Mod. 18. te i^Si; =te W % 139. i K Sfcl * ^ =fc*= ?^Gb_ I _V:I V:IV I 164. In Mod. 19*, to a key ten grades above, — as, from Afy to Fft, we identify F§ with GV. A\l to £b, Depression by too grades; formula for Mod. 4. Abbreviation (with Enharmonic Change) of Mod. 19. 140. n pfc& 3=S8^P^I nfe m^=f=^^^^^ f F IV: I IVI J- -R _D_ % *jf- IV -1 165. In Mod. 20, to a key ten grades below, — as, from B to G[i, we identify G\i with F§. E to F%, Elevation by two grades; formula for Mod. 3. Abbreviation (with Enharmonic Change) of Mod. 20. 141. % fSS £1 nti he ^H i: ^^t^ ^l pljfe:- -b<2^ ^ E_ I ■G>_ V:IV * The examples, in this modulation and the three following, are of cases in which the interchangeable key is the new one. 88 — 166. In Mod. 21, to a key eleven grades above , — as, from AV to C%, we identify C% with DV. A\f to DV, Depression by one grade; formula for Mod. 2. Abbreviation (with Enharmonic Change) of Mod. 21. 142. 1 ^ «£ ^T 3 ^ =HiJS * fc IVI »t> eft f— ir-^ ^e» IV I 167. In Mod. 22, to a key eleven grades below, — as, from E to C\f, we identify Cb with B. E to B, Elevation by one grade; formula for Mod. 1. Abbreviation (with Enharmonic Change) of Mod. 22. M 143. i s §£ t* gfejEp; iz?— — — ; J*£ $#= f$ E I IV. #E Keys twelve grades distant. 5[z + 7Jf. 168. The modulation to a key distant by more thati eleven grades implies that both the keys concerned are interchangeable ones. Thus: (Elev. D\f to C%. — bV to a\. \Depr. CfltoZty. — «# to % fiU 4- fitt i Elev - G ^ t0 ^ - ^ t0 ^#- ba + • \Defir. FttoG\> - d$toe],. a\i to^fl. ^fl to afe. Keys thirteen grades distant. 6^ + % |f^ ^ t0 ?" ~~ e \ toa ^ \Depr. Cjf to Gjz. — a% to ^ [j>. 7^ 4- fitt I £/w * ^ t0 ^- _ °b t0 ^S- '" + bff- |^ /r _ ^ tQ ^ _ ^ t() flJ|> Keys fourteen grades distant. 7b + 7J| i^" 5 t0 ^ ~~ ^ t0 " # " \Depr. cfl to C|z. — «{f to «|j. 7 k + 5ti \ BeV - °* t0R 89 — 169. In every case of a modulation to a key twelve grades distant, the new key is, as is seen in the above table, the enharmonic equivalent of the old. Consequently, an actual modulation may in this ease be dispensed with, it being sufficient to mentally change the signature of the old key for that of the new. 170. After the foregoing explanations and examples, the follow- ing illustrations of the application of the Enharmonic Change for abbreviation, in the remaining four modulations of this Division, will be easily understood. It will be noticed that here the principle of contrariety, hitherto applied, no longer obtains, — in other words, that for Elevation we now use formulas of Elevation (not Depression), and for Depression, formulas of Depression (not Elevation). a) Mod. 23. Elevation, 13 grades. 144. < b) Mod. 24. Depression, 13 grades. Fit Gb I_ IV: I IV c) Mod. 25. Elevation, 14 grades. m^^M I VIV I V — 90 — d) Mod. 26, Depression, 14 grades. CHAPTER XXVII. Short Cuts in the 2 d Division (Mods. 35—40) through the identity of (V) with VI of another key. 171. The short cuts in the 2 a Division — Minor to Minor — are based on the following principle (Table I, L, 3): In Elevation, (V) of a key coincides with VI of the key five grades higher \ and its converse: In Depression, VI of a key coincides with (V) of the key five grades lower. 172. In applying this short cut in Elevation, the starting- triad of the modulation is immediately followed by its (V), which coincides with VI of the key five grades higher. In Mod. 35, to a key five grades higher, this VI belongs to the new key, and is im- mediately followed by its lower tierce-relative — rv, which initiates 6 4 the Cadence, as illustrated in Fig. 145. The sequence: I (V):VI rv I is, however, objectionable, as involving an inharmonious progression* — for instance (in Fig. 145) — c b c\ b. For this reason this particular application of the principle: (V):VI must, at least, not be extended beyond Mod. 35. In modulations to keys more than five grades higher (Mods. 37, 39, 41, e,tc), the VI is followed by its UPPER tierce-relative , I. If this I coincides with IV of the new key (as in Mod. 37), let this IV initiate the Cadence (as in Fig. 147); * We have already had something similar to this. Remark immediately following. See Fig. 116, and — 91 — otherwise, the I following the VI is to be succeeded by one upper quint-relative after another — V:I V:l, etc., — until a Triad is reached which, coinciding- with iv of the new key, initiates the Cadence (as in Figs. 149, 151, etc.). 173. The application of this short cut in Depression is the following. The Cadence begins with i (of the new key), preceded by its (V), which latter coincides with VI of the preceding key, five grades higher. This VI, as such, is preceded by its upper tierce-relative — i. Accordingly, the Cadence, with these prefixes, will appear thus: 6 | I VL(V) | I IV | I (V) | etc. The procedure is, after Mod. 36, to form the usual chain — lv I (see Par. 113), etc. — until that particular I is reached, whose lower tierce -relative — VI — coincides with (V) of the new key, whereupon the complete Cadence begins, as above represented. Abbreviation of Mod. 35. 145. » 4 3®z -■0- _(V):VI IV (V) w ^ Abbreviation of Mod. 36. 146. M a=g- i VL( #7 W IV (V) — 92 — Abbreviation of Mod. 37. I B 147. H=2- M M ■t- ^ eP &#— *S 1 (V):VIl:IV I K (V) i_ m i r Abbreviation of Mod. 38. i 148. < "m s^ i v». =± 3 i a d_ F fe E 0= =H -«b- *S J: I^V I VI:(V) I IV I (V) I Abbreviation of Mod. 39. I i « 149. =# ^ # f* F #* HzzSz tig- -»> K i -*»— i It =*= I (V):VI I VIV I (V) I Abbreviation of Mod. 40. 04tU^ 3= JfeN; ^ 4isf ^s * -*g *- 4¥*= W I. _ b i s 4^ 5^ r iv i (V) 93 — CHAPTER XXVIII. Modulation in the Minor Mode, to keys in the 8 th up to the 14 th grade of quint-relationship. 174. The present Chapter concludes, for the present, the modu- lations of the 2 d Division. Here, as in the corresponding modulations of the l Bt Division (Chapter XXVI), no key has the natural signature, hut in each modulation one of the keys will have such a sharp signature, the other, such a flat signature, that the sum of the sharps and flats will be eight, nine, ten, eleven, thirteen or fourteen, accor- ding to the grades of quint-relationship involved. Here, too, the modulation to a key twelve grades distant is omitted, the two keys being in this case enharmonically equivalent (see Par. 169). 175. In these modulations, as in the corresponding ones of the 1 st Division, two methods of abbreviation will be followed. The first one consists of the short cuts employed in the preceding Chapter; the second one, of the very short cut by means of an Enharmonic Change, exactly corresponding to that employed for the concluding- modulations of the 1 st Division (see Par. 157). First Method of Abbreviation. 176. Our forty -first modulation is to a key eight grades higher, — as: 151. pm ri^^ =fe a=§ Pfff = Tf = gf^B^^ 4- I (V):VI I * * =t » =l f5 i V : I V : IV I ( V) ^ 177. Our forty-second modulation is to a key eight grades lower, — as: — 94 — ^faAfr^fgld** * 152. PH W rcr: I0__a_ ^ fc r i i _a^_ I I:V I:V I:V I VL(V) I IV I (V) I_ 178. Ouv forty -third modulation is to a key »?>z a ) i^P^* Formula, M. 32. I j -_ v 1 ■ v : : v : Iv I OO : Mod. 43. As above, with interchangeable new key. i 163. faT2" I - -2=g= Mr ¥• =3= lS>- 'I ^P^wwr^ ^ r=a= r ±*fc S^ i ran & c_ -ty »fr P-- fe=i Formula, Mod. 32. J_^v i_^y i_rv i iv i (V) i Mod. 44. Depression. 9 grades (Elevation, 3 grades). 164, - S z$*z H&Z Formula, Mod. 31. I ftf stf ai, 2 V:I V:IV I (V) — 98 — Mod. 45. Elevation, 10 grades (Depression, 2 grades). ffef Ji j J | , H^^hLj f^^p^^H^t ^ 165. < ±# ^ ^Bl=^ Pi ^V-g-p 1* **= c f_ -*b- Formula, Moil. 30. I •• v i : v IV 1 (V) i Mod. 46. Depression, 10 grades (Elevation, 2 grades). 166. < gEg ^1= ^P g ^ * ^fe i^=iM ^ HE: 4e Formula, Mod. 29. i V:iv "i (V) i_ Mod. 47. Elevation, 11 grades (Depression, 1 grade). $#j j *i i a ^S -fe f^K - V Ht^l* 167. < ^ * m^ w «s t£f£=» ^ *^ f ty- rormuia, Mod. 28. i : v - a 8- K IV i (V) i Mod. 48. Depression, 11 grades (Elevation, 1 grade). 168.< § iiHrg— r a=ai ^b^^M i * afet^|p^^ c# g# a> J rormuia, Mod. 27. i : iv i (V) i_ 99 Mod. 49. Elevation, 13 grades (Elevation, 1 grade). %^mm=3m^m J ^WW**^ i £=^ If /y i — ja Formula, Mod. 27. I : iv i (V) i_ Mod. 50. Depression, 13 grades (Depression, ] grade). m^^m i^£ Wm^m^m 170.< Sfaa ffip =ffig B 1 E 3=tfei Pormula, Mod. 28. I : v_ -a^_ iv i (V) i Mod. 51. Elevation, 14 grades (Elevation, 2 grades). 171. i te^^ &§S ^^^1^^^^^^ * Hi- Ss ffife=£ ^ Ml *■= g=== -i» a^ b^ a| J rormula, Mod. 29. I v : iv I ( V) i_ Mod. 52. Depression, 14 grades (Depression, 2 grades.) rWcrr* fst 172. <^ II I i=at *j* gw ^ sste ^E* a^E ESSf * iltiSE: rormula, Mod. 30. i = v i : v i _a^_ iv i (V) i — 100 CHAPTER XXIX. Modulation in the 3 d Division, up to seven grades of Elevation and Depression. 185. If the modulations of the preceding two divisions have been well understood and diligently practised, those of the following two divisions will be found comparatively easy. Here the grades are — not those of quint- relationship, but — of Elevation or De-- pression according to the key-signatures involved (see Par. 79). 186. The modulations of this Division may be worked out in regular forms on two general principles, viz: (1) for Elevation, V of a key coincides with VI of the minor key two grades higher (Table I, F, 4). Procedure (after Mod. 54): Let the first Triad — I of the old key — be followed by one upper quint-related major Triad after another (I V:I, etc.), until a V is reached which coin- cides with VI of the new (minor) key. This VI is succeeded by its lower tierce-related minor Triad — iv, which initiates the Cadence. (2) For Depression , IV of a key coincides with VI of the parallel minor key (Table I, D, 5). Procedure: Let the first Triad be followed by one lower quint-related major Triad after another (I IV: I, etc.), until a IV is reached which coincides with VI of the new key; then form the Cadence, as in Elevation. In Depression, the new key is always the parallel minor of the last major key of the modu- lation, i. e., that key whose IV coincided with VI of the new key. These regular forms become, after a while, undeniably tedious. At any rate, assuming the student's ability to work out the modu- lations of this Division in these regular forms, we dispense with illustrations of the greater part of them, and give abbreviated forms instead. For economizing space we merely give an example of each modulation, omitting, as unnecessary, all explanation other than the indication (in brackets, and referring of course to Table I, Chapter XIV) of the principle on which the modulation is based. 101 Mod. 53. To the parallel minor key. [D, 5.} 173. < $ 2r~ - Z2C JVVI iv (V) Mod. 54, Elevation, 1 grade. [A, 2.] 174. i -■6- C e_ r =s * d^ si- I VI iv "i (V) i_ 'f Mod. 55. Depression, 1 grade. [D, 5.] 175. { i ^=* wm m. ^2" 9£ ^# =» £ 176. ^ I IV: I IV VI iv I (V) Mod. 56. Elevation, 2 grades. [F, 4.] si I 3E *t m c_ I- V:VI IV (V) 187. The principle: IIL1V (C, 2), analogous to lll:(IV) (C, 5), for the 1 st Division, affords a formula shorter than the above by one Triad, as in Fig. 177. The principle; V:VI (F, 4), as in Fig. 176, 102 may be extended to following modulations in Elevation, by prefixing to tbe formula one additional lower quint-related Triad for eacb additional grade. Mod. 56. Elevation, 2 grades. [C, 2.] 177. ^rt^M r* a^M^ c_ I III: IV I (V) ^>— = r Mod. 57. Depression, 2 grades. [D, 5.] 178. < $ B &z * t- 'z&i d= % * H-2-^ : s=;a =t m teso -%h ^ TS 1 ^ u Bet T^- j-L I IV: I IV :T IV: VI IV I (V) Mod. 58. Elevation, 3 grades. [C, 2.] 179. < Mod. 59. Depression, 3 grades. [D, 5.] $ & # i 5fc WS, f ^ -w- ?«S»T 180.< ?.«- J^ ^ 3t P Ifci: M &T- ^ r ^ Jfy ^ -1 I IVI IV: I IV:I IV: VI IV I (V) I — 103 — Mod. 60. Elevation, 4 grades. [C, 2.] 181. % m % ~ri- =3= \^Ei wm i X ~ef- ~$r^' & rsz t !_G T) ' 4 J I V:I V:I IILIV I (V) ■ff* I !od. 61. Depression, 4 grades. [A, 5.] 182. i s * af m ^r— a ~ J ± 1 j—T-b ~&z C f_ ICV) i iv i (V) i Mod. 62. Elevation, 5 grades. [C, 2.] 183. { t i % m~ m * ^S 3^ % &. C lG D A gi J I V:I V:I V:I IIHV I (V) Mod. 63. Depression, 5 grades. [D, 4.] 184. pFtW ^—4^=^ 3=3=§ >* V- : 1 g J F ^fe _b^_ r =*t i I IV:(V) I IV I (V) I (Additional illustrations of the formula thus far used for Ele- vation are deemed unnecessary. And as this formula is becoming tedious, its place will be supplied by shorter ones.) 104 — Mod. 64. Elevation, 6 grades. [G, 3; C, 2.] 186. i &: W&; 4. ~^S "T5»- ^ ^ c_ I E^_ f=3=$* ^ 4 VI:(IV) I III: IV I (V) Remark. The necessity for a short cut in Mod. 64 may induce us to tolerate the harmonic progression in the first three Triads in the above example. See Remark to Par. 141. Mod. 65. Depression, 6 grades. [D, 4.] % E i EE ^ =3^=^ W^W &r 186. ^S V» K- \k i k *fc i* E^E *f=T I ■<^- IVI IV:(V) I IV I (V) I Mod. 66. Elevation, 7 grades. [C, 5; F, 4.] i -k X s E3 187. < ^B ^ JgEEEEf^ ES PS ^ c_ I JB_ #- III:(IV) I V:VI IV (V) 105 — Mod. 67. Depression, 7 grades. [D, 4.] i SS3 t- 3=^5 SOT txs&i rp- l ^3^& 188.< •M jf^^^m -1=S= ^ -4e _B^ a^. I IVI IVI IV:(V) I IV I (V) I. (The remaining modulations of this Division are passed over for the present.) CHAPTER XXX. Modulation in the 4 th Division, up to seven grades of Elevation and Depression. 188. Whereas, in the regular modulation - formulas of the 3 d Division the IV of the Cadence was preceded by its (major) upper tierce-relative — VI, which VI coincided with I, IV or V of the key immediately before; here, in the regular formulas of the 4 th Division, the IV of the Cadence coincides with VI of the key immediately before (always the parallel minor key) , which VI is preceded by its (minor) upper tierce -relative — I. The Cadence, accordingly, will begin with IV, with the prefix, for Elevation, V:I VI; for Depression, I VI, — thus : ~~ 6 4 In Elevation: V:l VI:IV j I V I etc. In Depression: I VLIV | I V | etc. The procedure will be (after Mod. 82), to begin with the usual formula— I V:I, etc., or, LTI, etc., continuing in the Minor Mode till a I is reached whose lower tierce-relative — VI — coincides with the IV of the new (major) key, which IV initiates the Cadence. — 106 — Mod. 82. To the parallel major key. [M, 5.] 189. < p I \ — J fe 3 J^l ^E VI nV-{ 190. Mod. 83. Elevation, 1 grade. [M, 5.] i zzr. ~& X -^rir -0- _V:I VI IV t V I 191. Mod. 83. Elevation, 1 grade. [H, 2.] t 3 % -&: ^ 3 =3c ^ a 6 1:11 IV 192. Mod. 84. Depression, 1 grade. [M, 5.] i gr 3 K =r ' ): 2 O-i* -**- -&- a d_ I:V_ i VI aH 107 Mod. 85. Elevation, 2 grades. [M, 5.] i 193. -8r -- V- I VI.(V) i VIIV I -•b- Mod. 95. Elevation, 7 grades. [K, 6.] 204. <^ I J ~W %i$$ d ft 3£ f^- taN ? l =Sa= I V:I V:I V:(IV) 1 V I Mod. 96. Depression, 7 grades. [M, 3; M, 5.] 205. 3 !=Tt=ra ■»*» A : fe TsM** & «J"~>-I r^ bfe J \\ >*t A *# 5£ m I 1T|- frg^frg! 'b^ 3E Bx-g- S a d_ I i& * 'fr o' a ^ -gf V- M m ^ \- (IV): III I (IV): VI IV Mod. 19. D. Elevation, 10 grades (see Pig. 128). [C, 5, twice.] 207. , $ fr 2 4 - 4= f 2 ei - 4 &3=H te— > '^ ^ H hP— M E|? D C# 1 I III: (IV) I III:(IV) I Mod. 20. D. Depression, 10 grades (see Fig. 129). [E, 2; E, 4] 208. , I* 1 =2= ^=i^ J^~grnB s el b ^ ' ^=W ^> ' b < 5 Rig±fr JB 5^ pfe ¥^2 r * fe— tf-fg:^ ^ r ^ k^ ^ V?»\jp - A D Eb — Cb i I IV I (IV):III I (IV):VI IV I V I. — 113 — Mod. 21. D. Elevation, 11 grades (see Fig. 130). [C, 5, twice. £ rar s: 1 * ■M 1$ I 209. sa zal: rJ ^*=He s± ^=2= A[, E> D C|- I V:I IIL(IV) I IIL(IV)' W = ^E $* V I Mod. 22. D. Depression, 11 grades (see Fig. 131). [E, 2; E, 4.] 210. ) fgjfi sa^sa* -si — r— ^ r^n- fefei i § a^=?^ ag ^ ^±^=c^=t^±; ^fcjt^gd-gz=te^ SSI J3f3^ SBS &—rrr& — bfc&~ ma & I T-ga-J^a * ** F ?S G : A[?_ -Cb- I (1V):III I (IV):III I (IV):II IV V I- — 114 — Mod. 25. D. Elevation, 14 grades. [G, 3; C, 5, twice.] ^^ m -m « ^tplmWi 213. i t i & — rB a 1 g* ife^a ^jLL^i^ §±HF g ^p^p^ T ^ng^ I VI:(IV) I ITLflV) I III:(lV) I (IV) V I Mod. 26. D. Depression, 14 grades. [E, 2, twice; E, 4.] 214. J v " " — " 3 ?< g j, ^ ■ pg §Sta j ^w^ffe^ ^ ^^q iH | |Tfnr^i #^;* * C# D Ep Cp — I (IV):III I (IV):III I (IV): VI IV 1 V I CHAPTER XXXIII. Addenda to the Modulation-formulas of the Second Division. Mod. 33. AA. Elevation, 4 grades (see Fig. 104). [L, 1.] 215. i _2_ a a afy a\f ^ -W -&- A *== ^ ±* Wrong. Eight. a a J? >— A A 4ec Wrong. Eight. & -O. V -&- za: $BZ -& a 6» *sr^A *-\> etc. 'A> - " " A^ It is true that in Fig. 217 the a and a\> (measures 1 and 2) are both in the same voice (Tenor) ; but the Bass also begins with A, against which the a\f clashing, two chords later, produces a harsh effect, which comes under the category of Cross Eelation. 8* — 116 — fore insert between 1 and III:(V) the normal Dominant Triad — V — of the old key, to be followed by its lower tierce-relative — III, which coincides with (V) of the new key. We thus obtain the Harmonic Variant as in Fig. 216. The exceptional syncopation of the V here is for the express purpose of weakening the impression of the start- ing-triad, so soon to be followed by the I of the new key, with its Third (afy) a chromatic half-step below the root (a) of the starting-triad. Mod. 38. BB. Depression, 6 grades (see Fig. 148). [M, 1; M, 3.] 218. , i j=? fefe r ■«b u~ I VI:IIIVI:(V) 1 IV I (V) I Mod. 40. BB. Depression, 7 grades (see Fig. 150). [M, 4; M, 3.] 219. , I VI: VII III VL(V) i iv I (V) i Mod. 42. BB. Depression, 8 grades (see Fig. 152). [M, 4; M, 3.] 220. I:V I VI: VII III VI:(V) I IV I (V) I - — 117 Mod. 44. BB. Depression, 9 grades (see Fig. 154). [M, 4; M, 3.J 221. ) fc $ ¥ t^ bse 3 »— ?§h &*» #s 1>m^ *# ffl w I J-J lt» e a_ I:V I:V T fe^ 'f IJH^ • a|?- iVI:VIIIIIVI:(V)i iv I (V) i Mod. 45. D. Elevation, 10 grades (see Fig. 155). [L, 3, twice.]; P i EB i g | i==J2^S tdatzgz t* «rb^»- &~^r & fr» ■&■ 1 C-U, T I ^ £ f 5^ ^ I VI:(V)I VI:(V) I IV I (V) I. -aj? . — 118 Mod. 47. D. Elevation, 11 grades (see Fig. 157). [L, 3, twice; H, 1.] 224. aN ^^s a^u j^^^ ag w ie>^—^&- w a 3= PPP r ^ ^f=W^^^^^^ dg_atf _ 2 I (V):VI I (V):VI I: IV I (V) I «p Mod. 48. D. Depression, 11 grades (see Fig. 158). [M, 3, twice; M, 1.] I VI:(V)I VIIIIVI(V) I IV I (V) I- Mod. 49. D. Elevation, 13 grades. [L, 3, twice.] i^S^&mW^ 226. P^^ ^^ m 3£^I aa: > I (V):VI I (V):Vt I V:IV:IV I q^ (V) I- 119 — — 120 CHAPTER XXXIV. Conclusion of the Modulations of the Third Division. Mod. 68. CC. Elevation, 8 grades. [C, 5; C, 2.] 230. io^^cfW f -« ^^f^ft^g C B. 3jp^ ,aa_s I V:I III:(IV) I III: IV I (V) I Mod. 68. E. (Formula B, Depression, 4 grades. Mod. 61.) 231. , £= l ^=fr r^p^ ^^g #^ *= F_a#_ % I : (V) I IV I (V) I Mod. 69. B. Depression, 8 grades [D, 4.] I h nz i i ^ ^m^m^m » *■ n ^ -» pg^kg 232. , Pg g< | ^- I IV:I IV:I IV.I IV:(V) I IV I (V) I - — 121 — Mod. 69. E. (Formula B, Elevation, 4 grades. Mod, 60.) 233. , I b 3=£ ^H -d ^^tfej 5fs KK V r^ 5* -2 — P* -&- tes i T 223; G D X ab 2 I V:I V:I III:IV I (V) I Mod. 70. CC. Elevation, 9 grades. [C, 5; C, 2.] I 1 $m ^ us g % 3 iES^e^PlpFfS 234. z &tFTtt-^ £=fag=3d?=*g: BSSI J^-.. ■■_ B iE: ^ *5f Y B|?__F C B 4 — i I V:I V:I III:(IV) I III: IV I (V) I - Mod. 70. E. (Formula, Depression, 3 grades. Mod. 59.) 235. , ft ^ «b fc§3^ •r 3e # S 9^ ^ 5S? g fcsfc=t £ 1^ fc I IV.I IV:IIV:IIV:VIlV I (V) I $ Mod. 71. D. Depression, 9 grades. [A, 5; M, 3.] w= t^rn r: ^ 1 i^^HdQ Iff 9*f3 W=J N^^F pac S Ei 1®- Bg I ■ *\>- I:(V) I VI:(V) I IV I (V) I — 122 Mod. 71.. E. (Formula B, Elevation, 3 grades. Mod. 58.) 237. £ -gf 1—W$ v fr£ t* g S^ *=£ D. I f [frgjg" ittf I to ■ ■> — V:I III: IV I (V) I Mod. 72. CC. Elevation, 10 grades. [C, 5; C, 2.] ^ffl=^#f^^ 238. , ^rrrr^T^ ai UN S--*a- #ff El? D A E B ajj % I III:(IV)I V:IV:I V:IlII:IV I (V) I - Mod. 72. E. (Formula, Depression, 2 grades. Mod. 57.) 239. 4LJirr»Utf^^iKUi j ^^^^^^ 9i ^rtfHN^r4^ ^ ^ ^ EJ, _ AJ,_D(, _ ty ' ^ ajf I I IV:I IV:I IV: VI IV I (V) Mod. 73. D. Depression, 10 grades. [D, 4; M, 3.] i 32=S P=£ P| ^ ^ J 7 H&^frg ^g 240. y^ J-«l ^ ^ A g. 5£ 2a: ■ »!?. I IV:(V) I VI:(V) I IV I (V) I — 123 — Mod. 73. E. (Formula B, Elevation, 2 grades. Mod. 56.) 4J- 241. % iH7J — h- 4 : 3=» I W* y wm Sa§E* -JX&&1 -Xtaz 32= »b I III : IV i (V) i [od. 74. D. Elevation, 11 grades. [C, 5, twice.] v 2 ,y~~% . 3 d4&*&tE&&=& m T5C fc^l^KVSF g y^ s ris ag g FTPT^ ^^ w^ A^ G F# a#. I III: (IV) I III:(IV) I: VI IV I (V) I. Mod. 74. E. (Formula, Depression, 1 grade. Mod. 55.) 243. | Sa^^ a 3= -B»- 33 *F # m-M=^^ & si «t= I IV:I IV:VI IV I (V) I Mod. 75. D. Depression, 11 grades. [D, 4; M, 3.] I IV! IV:(V) I VI:(V) I IV I (V) I — 124 — Mod. 75. E. (Formula, Elevation, 1 grade. Mod. 54.) 245. / ■ v ^~—^ iSfei p=* m 5^ \fo I J I : VI IV I (V) i. Mod. 76. D. Elevation, 12 grades. [C, 5, twice.] P^^^M^mmi 246. , w m i f H ^t# ^ ^535 »b — c B. -•» 1 ■H I III: (IV) I III:(IV) I V:VI IV I (V) 1 Mod. 76. E, (Formula, Mod. 53, ,to the Parallel Minor key.) 247. , ^ fr=i= ¥= =t s s m i m~w ^p ipt^ SS ^ C — r~fr~ ^ft # Bb bl>=: I IV:VI IV I (V) I Mod. 77. D. Depression, 12 grades. [D, 4; M, 3.] i fi# g^fe fe^fe ^fe 248. , ^8 ^-2=g^^ jze! £ rt^r p j 4**= $ 2-.«g- y- T ' 'n B E A g:. ■ «b- -2 I IV:I IV:I IV:(V) I VI:(V) I IV I (V) I 125 — Mod. 77. E. (Formula, Mod. 53, to the Parallel Minor key.) 249. w^ an^i ^m Vbz B I IV:VI IV S%^^*\>- I (V) I Mod. 78. D. Elevation, 13 grades. [C, 5, twice, etc.] a ftfr-3 P T^ ^^#^^4 l ^ 250. ^-4-i \ Q\>. F E I IIL(IV)I III:(IV)I III: I V:IV I (V) I [od. 78. E. (Formula, Elevation, 1 grade. Mod. 54.) W^^^^^M ^ hvJz 3 =8? 251. to^^ ^f^ ^ * G^-F#a# I : VI IV I (V) i. [od. 79. D. Depression, 13 grades. [D, 4; M, 3.] i ite ^s l^pp^piiGH P* 252. , fW^fTp ifei-agEOl K »TPffia ^ *z m pt ^^ Fjf B E A gr. -»b- IIVIIVlIVIIV(V)lVI:(V) I IV I (V) I 126 Mod. 79. E. (Formula, Depression, 1 grade. Mod. 55.) 253. i it w ^=1=% js^fe jj ■Jtiu K g^ H= ^^ ^ E# 5s: t*E Fti_B I IV:I IV : VI IV I (V) I :c\> alp. Mod. 80. D. Elevation, 14 grades. [C, 5, twice, etc.) i to 'g^H^ -^Ui^ ix— r- s * "F|,2 et—g I frgj V ^=fcfe: : ^hlSS : =» a: * T5<- 251. »s 2=F=ft U HE;^=bi3 fc&ag » Esse ^W #= _C B. I V:I I1I:(IV) I V:I IIL(IV) I I1LIV I (V) I -»#— i Mod. 80. E. (Formula, Elevation, 2 grades. Mod. 56.) f^^ fe^jj^ ri~k7J Tl^HJ— _ m fe* g~C~1fr-Hg =tt* ^SMbp=§pl g=^ SanrqsE I 111:1V I (V) I Mod. 81. D. Depression, 14 grades. [E, 5; M, 3, twice.] ^s fij^^^J^^ Sgl 256. SrtS S n«>- tJ^gfe^JLllJ. I ;:*tffll *E 3H * Stt^ zjascfe b#_f«— -»- -a|?- £ I (IV):I VI:(V) I VI:(V) I IV I (V) I 127 — Mod. 81. E. (Formula, Depression, 2 grades. Mod. 57.) m m *:^^=^d=fe£S * f^Pp^^pK| 267. ate ^ 4fe yggN^ 4bc P^f? 3f I IV:IIV:I IV : VI IV I (V) I CHAPTER XXXV (Final). Conclusion of the Modulations of the Fourth Division. Mod. 97. B. Elevation, 8 grades. 258. $m m * H - a ) ^¥^^^ =s^ 1 ^ * : ^jtT^f :*»°lf- I VI:(V) • I : II IV IV — 128 — Mod. 98. B. Depression, 8 grades. g^ ^fe k- 1 I ff I T." I .£IK £ =t# s=^ £ £*==£ ^ ^ £C e F_ -B[7 a[>_ ■C>_J I V1:V I IV:I IV.(V) I VIIV I V L Mod. 98. E. (Formula, Elevation, 4 grades. Mod. 89.) 261. , p^ f § | £ *2 £ mac 91 g P ik jSS? 4^ .^ *= -0- 1 : (IV) Mod. 99. D. Elevation, 9 grades. [H, 5; C, 5.] 262. , £=2= » m 1==^ #£<#£ \ Z7 %~ r=* %e? &D. ^E^ ng- I:(IV) I III: (IV) V IK I_ Mod. 99. E. (Formula, Depression, 3 grades. Mod. 88.) p^m j i bj-dzjWsJ bttJ-H tj ^=^=^^r SBP^ -B>- ^ I ^ 3£ s= -& &=t ^ fr ifE a)?. r ^s- g^ ^^^ ^^j g*f#3s -Cb- I VI: V I IV:I IV:I IV:(V) I VI:IV I V I Mod. 100. E. (Formula, Elevation, 3 grades. Mod. 87.) 265. ) J L= jj^ 2— « ^ g= pi — ^- ^5 ^=J=teh^ ffpH-fc fc^ 7=>- zsc *s>- ^^^¥^af¥F^ 5 f#J_B I V:I V:II IV -CI,. Mod. 101. D. Elevation, 10 grades. [K, 6; C, 5.] i fc=E3E v=w- s ^J fc^-m j^g^l S> 2 ^rN :flV) I D- I V:(IV) I III: (IV) V I Mod. 101. E. (Formula, Depression, 2 grades. Mod. 86.) 267. i k— 2 J ;J -j-^y -^p^ ^ TV^J P=jfer#=y=py c f . .bt>: -^ £= ZSti =T I:V I:V =atf_CJJ_i I VI:IV I V 130 Mod. 102. D. Depression, 10 grades. [M, 3, twice.] i ± 4"= ^ - }t> > ! 3 ^ fc f Hi * $BL g_q j»ng^ t& -et~ 268. t— n i pt. $ g^^g I VL(V)i VI(V) i VI IV I V I. Mod. 102. E. (Formula B, Elevation, 2 grades. Mod. 85.) 269. i tf r * a-TiH- ^ ^^-LtJ *S -fr*r i^=i % ^ Jf— 2— ,%- 4> p' p -» %l p£ '»- f =t T5H V:II IV I * *z =& Mod. 103., D. Elevation, 11 grades. [K, 6; C, 5J i *=e ^ SF itf 270. J P; 'S- *=►• y -i— g pM m& is m ^ -(9- ¥~*r ip- 1 c D. C#- I Y:I V-(IV) I HI: (XV) I 8f V I Mod. 103. E. (Formula, Depression, 1 grade. Mod. .84.) 271. P !£=£: :a= S P ^Ef E$3 PS I . rft JFf * ~^> I *> I:V VLIV I V *■= 131 272. Mod. 104. D. Depression, 11 grades. [M, 3, twice.] fcfe I ^Sl 3h^^3-^ k^ ate^^^^^^ ^^aa Off f|f *_ -«t>- -C[>_« ^_-fi^ I : V I VI:(V) 1 VI:(V) I VI:IV I V I- Mod. 104. E. (Formula B, Elevation, 1 grade. Mod. 83.) 273. 3=SE5^:ES^^ ^5— t^S'-t^ 9* ' F-"=2=F =**P=f i Baa < *. c 8 B C ^ — rtF I : II IV * ^ -l«- ^ i v i_ Mod. 105. D. Elevation, 12 grades. [K, 6; C, 5.] fr$F=g m^g ^Tft tfl^wf^^ 274. , ©• «■ f .c D «#- I V:I V:I V:(1V) I III:(IV) I V __ I 8* Mod. 105. E. (Formula, Mod. 82, to the Parallel major key.) 275. ) i%3t£^ j— Mtit 2&5 «^^ fj^jj^fe^ j^p^ PPg^? ffa^p b> ajf C||_J I VI:IV I V 9* — 132 276. , Mod. 106. D. Depression, 12 grades. [M, 3, twice) m m 2o «,- f'f l^gi=> -<6>- 9* -*» — taS" fe §£ ip J- is: ^Pfc^ P^ a^. -Cb-2 I .g# cjj! — ffl iW I:V I VI:(V) I VI:(V) I VIIV I V I Mod. 106. E. (Formula, Mod. 82, to the Parallel major key.) 277. i tf rgSrii^H H fejk^t b^U -l^ p-^lfcMffs s s4 ^ s# a i>- VI:IV I Mod. 107. D. Elevation, 13 grades. [K, 6; C, 5.] P»Pii "a" 9" W&'~-~' ^^SmW% 278. §te £^fc s* ek b^_f e B C#_f ™^ I V:1V:I V:IV:(TV) I IILffV) I V I— (IV) Mod. 107. E. (Formula B, Elevation, 1 grade. Mod. 83.) ' fftFF 279. r* K£fc=^ ? ££ ^ t# n j. ± ek dji Clf I : II IV f gfaf^=tM V I_ 133 Mod. 108. D. Depression, 13 grades. [M, 3, twice; M, 6.] 280. m^mm$. I VI:(V) I VI:(V) I VLV I IV:I IV I V I Mod. 108. E. (Formula, Depression, 1 grade. Mod. 84.) 281. , Mod. 109. D. Elevation, 14 grades. [H, 5; C, 5, twice.] I ss feffiFg 282. # 2fe ■ & — i H ^ ~s? =diaia==g=rtfflai aF^i^T isS m=?=f$^ =^ i ab Eb D 1 Lcjt 2 C»- tff I: (IV) I III: (IV) I III:(IV) I V I Mod. 109. E. (Formula B, Elevation, 2 grades. Mod. 85.) ffi w 4^y-^s t$sfJi|8# |£ % 3 ^b - b _ i-^ i fe ^^^'-^ g ^ ^ q-tf ^%=^^p^^% ab g# C# I V : II IV v i_ — 134 — t<9 •ae QO T& l— I BJ — H TO Cl- L I IV I* V I I -T I II Fig - 21 °- E*. 1 t:IV - |i V| 1^1 - || Fig. 143. Mod. 23. Elevation, thirteen grades: new Tonic, a diminished Fifth below (augmented Fourth above). d. 1 1 1 V:i Yji|Yji ijw^l 1 ^Pl V I I 'H^ I II Fig - 21L E*. tlV— 1 1 VIIHI-ll Fig. 144a. — 140 — Mod. 24. Depression, thirteen grades: new Tonic, a dimin- ished Fifth above (augmented Fourth below). D. li|(IV):III I|(lV):III 1|(IV:II) IV|I V|I— | I || Fig. 212. E*. \ i - I IV:I flV II VI I - II Fig. 144b. I l(IV) ' ' II Mod. 25. Elevation, fourteen grades : new Tonie, achromatic step above. 3 £ - — - D. |lvL(IV)|I HI:(rV)|I III:(IV)|I V|I— |I—|| Fig. 213. E*. \\ V:IV|f V|I^fl-|| Fig. 144c. Mod. 26. Depression, fourteen grades: new Tonic, a chro- matic step below. D. ft |(IV):III I|(IV):III I|(IV):VI IV|I V| I^fl || Fig. 214. E*. \\ IV:I| IV:I (IV li VI I — I Fig. 144d. 1 t(iv) ' ' '■ Second Division. Minor to Minor. Mod. 27. Elevation, one grade: new Tonic, a major Fifth above (minor Fourth below): Key of the Dominant. A. 1 try — | 1 (V) | I — | I - || Fig. 98. Mod. 28. Depression, one grade: new Tonic, a major Fifth below (minor Fourth above): Key of the Sub dominant. A. Jf:V- |I IVjf (V)| I - || Fig, 99. M o d. 29. Elevation, two grades : new Tonic, amajor Second above. A. 1 1 V:IV | i (V) | iCji - | Fig. 100. Mod. 30. Depression, two grades: new Tonic, a major Second below. A. Jfry LV|I IV|! (V)|I-|| Fig. 101. — 141 — Mod. 31. Elevation, three grades: new Tonic, a minor Third below. A. !!i-|ya V:IV|I (V)|l-|| Fig. 102. Mod. 32. Depression, three grades: new Tonic, a minor Third above. A. lijiyJLV LV|I IV 1 1 (V)|I|| Fig. 103. Mod. 33. Elevation, four grades: new Tonic, a major Third above. A. if. va|Vjl V:IV|! (V)|l-J| Fig, 104. AA. 1 f (V):m I VH IV |l (V)| I - || Fig. 215. Mod. 34. Depression, four grades: new Tonic, a major Third below. A. aLV LV|LV LV|I IV|f (V)| f^fl- || Fig. 105. AA. Ji^ m(V)|l IV|I (V)|lC|i« || Fig. 216. Mod. 35. Elevation, five grades: new Tonic, a minor Second below. A. If j V:I V|I|VI Vjiyjf (V)|I|| Fig. 106. B. 2 2 i-|(V):VI IV 1 1 (V)|I-[| Fig. 145. See Par. 172. Mod. 36. Depression, five grades: new Tonic, a minor Second above. A. !£v|LV L7|=-|I IV|! (V)|rqi|[ Fig. 107. B. \\ VL(V)|I IV|I (V)|I-|1 Fig. 146. Mod. 37. Elevation, six grades: new Tonic, a major Fourth above (minor Fifth below). A. U vi|-vi vjilyji viy|f (V)|i^7i-|| Fig. 108. C. !f|(V):VI mv|l (V) | l^T|l|| Fig. 147. — 142 — Mod. 38. Depression, six grades: new Tonic, a minor Fifth above (major Fourth below). A. liv LVlljV LV| == |I IV|I (V)|l-|| Fig. 109. B. |tv|i VL(y)|l iv|l (V)|l|| Fig. 148. BB. |I|VLm VI(y)|l 1V|I (V)|I|| Fig. 218. Mod. 39. Elevation, seven grades: new Tonic, a chromatic half-step above. A. ll\va va|--.|ya: vov|! (V)|iCfi|| Fig. no. c. if I YJ(V)|I yjKV)|l IV| l (V)| r^p[|| Fig. 256. E*. |1|IV:I IV:I|IV:VI IV | 1 (V)| I || Fig. 257. Fourth Division. Minor to Major. To the parallel major key a. ll vliv| i vi *Hl- || Fi s- 189 - Mod. 82. To the parallel major key: new Tonic, a minor Third above. — 149 — Mod. 83. Elevation, one grade: new Tonic, a major Second below: Key of the (normal) Subtonic. A. Si- |^I VI:IV|I V|t-|( Fig. 190. E. Sin lV|i V.|_I^fl-|| Fig. i9i. Mod. 84. Depression, one grade: new Tonic, a major Third below: Key of the Submediant. A. HjV-\l VLIV|i V|t- || Fig. 192. Mod. 85. Elevation, two grades: new Tonic, a minor Fourth above (major Fifth below). A. Si v^va VI:IV|I V|t— || Fig. 193. B. 1! ly^ri IV|1 V| I^jl || Fig. 194. Mod. 86. Depression, two grades: new Tonic, a minor Second above. A. SljV IjV|l Viiyjl V|'I-|| Fig.. 195. Mod. 87. Elevation, three grades: new Tonic, the same, change of M o d e only. B. Si v^v^i IV|I V| I- || Fig. 196. Mod. 88. Depression, three grades: new Tonic, a minor Fifth above (major Fourth below). 2 ? i£ l 1 ^! i^t 1 IHII 1 V l I II Fig " 197 ' Mod. 89. Elevation, four grades: new Tonic, a minor Fourth below (major Fifth above). B. % Uw) - \l V| l^fl - || Fig. 198. Mod. 90. Depression, four grades: new Tonic, a chromatic half-step below. B. Si VI:(V)|HI IV 1 1 V|f-|| Fig. 199. — 150 — Mod. 91. Elevation, five grades : new Tonic, a major Second above. B. !IV:(TV)|I V|f^fl-|| Fig. 200. Mod. 92. Depression, five grades: new Tonic, a diminished Fourth above. 2 .3 B. if VI:(V)|I VLIV|I V|I— || Fig. 201. Mod. 93. Elevation, six grades: new Tonic, a minor Third below. B. \\ — (v:I V:(IV)|1 V|l— || Fig. 202. Mod. 94. Depression, six grades: new Tonic, a diminished Seventh above. B. VUv\l VL(V)|i VLIV|! V|I|| Fig. 203. Mod. 95. Elevation, seven grades: new Tonic, a major Third above. B. f V:I|V:I V:(IV)|1 VI T— 11 Fig. 204. Mod. 96. Depression, seven grades: new Tonic, a diminished Third above. B. !i|V IjV|l VI:(V)|I VI:IV|I Vlf^I-H Fig. 205. Mod. 97. Elevation, eight grades: new Tonic, a minor Second below. B. |f I V: I yj|V:I V:(1V)|I V|I|| Fig. 258. E*. if VI:(V)|LII IV|1 V|t- || Fig. 259. Mod. 98. Depression, eight grades: new Tonic, an augmented Third below (diminished Sixth above). B. 1?|VLV I|rVI IV(V)|I VLiy|l V| 1^1 || Fig. 260. E*. I !:(IV) — II V| t^Tl- || Fig. 261. — 151 — Mod. 99. Elevation, nine grades: new Tonic, a major Fourth above (minor Fifth below). D. |t(lV)|I III:(1V)|1 V| t^\l || Fig. 262. E*. llv\ljV LV|I VLIV|I V| I || Fig. 263. Mod. 100. Depression, nine grades: new Tonic, the enharmonic equivalent. B. HI VLV|I tViJI^L 1 IV: ( V )|! VIIV|l-|V-|f^jl-|] E*. If Vjl|V£[ IV|I V|l— || Fig. 265. Mod. 101. Elevation, ten grades: new Tonic, a chromatic half- step above. D. llV:(IV)|I IIL(IV)|I V| I— , Fig. 266. E*. IfsV IrV[l VI:IV|I V|T-|| Fig. 267. Mod. 102. Depression, ten grades: new Tonic, an augmented Fourth below (diminished Fifth above). D. \l VI:(V)|l VI:(V)|I VI:IV|1 V|t^|l-|| Fig. 268. E*. !f|yni IV|i V|]^|l|| Fig. 269. Mod. 103. Elevation, eleven grades : new Tonic, a diminished Fourth below (augmented Fifth above). D. i 1 Jy^I V:(IV)|I I1I:(IV)|I V| I'|| Fig. 270. E* Sf|V-|l VHV|I V|l-|j Fig. 271. Mod. 104. Depression, eleven grades: new Tonic, a chromatic Step below. D. |frv|i VI:(V)|I VLCV)|I yjIV|i V|l^l|| Fig. 272. E*. Stn IVll V|f^Tl-l| Fig. 273. — 152 — Mod. 105. Elevation, twelve grades: new Tonic, an aug- mented Second above. D. aiA^llVd V:(IV)|I III:(lV)|I V| f^fl -. || Fig. 274. E*. If VI:IV|{ V| l^fl- || Fig. 275. Mod. 106. Depression, twelve grades: new Tonic, a doable- diminished Fourth above. d. Si^ i=yji yW|i vi