..^ Cornell University Library MT 50.C43 1897 Harmonyra course of study /by G.W. Chadw 3 1924 021 756 469 Strata. STeui ^atfc tnr6.a^.\^5^fc« Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924021756469 HARMONY A COURSE OF STUDY BY G. W. CHADWICK, A.M. DIRECTOR OF THE NEW ENGLAND CONSERVATORY OF MUSIC BOSTON, MASS. THIRTY-FIRST EDITION THE B. F. WOOD MUSIC CO. BOSTON— NEW YORK— LONDON LEIPZIG Copy/ight, 1897, by The B. F. Wood Music Co, for all countries. p. H. alLSON COHPANV, PAINTERS AND BOOKaiNOERS, ■ 08T0N, U. S. A. PREFACE. The object of this book is to give the student a working vocabulary of chords for the harmonizing of melodies in the order of their practical value and harmonic importance. The author has endeavored to encourage the student to use his ever-increasing chord material, — not so much by warnings against what is bad, as by examples of what is good, as musicians understand it, and by maxims deduced from such examples. This book is not intended to deprive the teacher of his occu- pation, but rather to furnish him with useful text and material, systematically arranged, which he is to illustrate and elucidate as much as is necessary. To this end copious references and elab- orate explanations of details have been avoided so far as is consistent with lucid statement. The student is supposed to have already a rudimentary knowl- edge of the intervals, scales and chords given in the introduction. THE AUTHOR. CONTENTS. PAGE IXTRODUCTORY Scales .......... ix Intervals ......... x Chords .......... xiii LESSON I. The Principal Triads of the Major Scale . II. The Principal Triads of the Major Scale. (Continued.) III. The Principal Triads of the Minor Scale IV. The Chord of the Sixth V. The Chord of the Sixth and Fourth VI. The Chord of the Dominant Seventh . VII. The First Inversion of the Chord of the Seventh ..... VIII. The Second Inversion of the Same IX. The Third Inversion of the Same X. The Supertonic Triad . XI. The Submediant Triad XII. The Mediant Triad Dominant I 7 lO II 15 17 22 25 28 32 36 40 (v) VI CONTENTS. LESSON PAGE XIII. The Inversions of the Supertonic, Mediant, and Submediant Triads .... XIV. The Leading-Tone Triad .... XV. Secondary Triads in Minor, with their Inversions XVI. Summary of the Triads of the Scale XVII. Open Position. ( Dispersed Harmony. ) XVIII. Open Position in Minor XIX. Dominant Major Ninth Chord and Inversions . XX. Dominant Ninth in Minor .... XXI. The Leading-Tone Seventh Chord in Major . XXII. The Diminished Seventh Chord XXIII. Inversions of the Diminished Seventh Chord . XXIV. The Same, Continued . XXV. Modulation to the Dominant . XXVI. Modulation to the Subdominant XXVII. The Same, Combined XXVIII. Modulation to the Submediant XXIX. Modulation from a Minor Key to its Parallel Major XXX. The Same, Combined XXXI. Modulation from a Major Key to its Mediant . XXXII. Modulation from a Minor Key to its Submediant XXXIII. The Same, Combined ..... XXXIV. Modulation from a Major Key to its Supertonic XXXV. Modulation to the Dominant of the Parallel Major 44 47 50 52 61 66 68 71 74 77 79 8r 83 89 92 94 97 100 102 105 108 109 "3 CONTENTS. Vll LESSON >. GE XXXVI. The Same, Combined ii6 XXXVII. The Supertonic Seventh Chord in Major and Minor 1 18 XXXVIII. The Same, Inverted XXXIX. The Other Secondary Seventh Chords in Major XL. The Other Secondary Seventh Chords in Minor XLI. Inversions of Secondary Seventh Chords XLII. Chromatic Passing Tones .... XLIII. The Augmented Sixth Chord . . . XLIV. The Augmented Six-five Chord . XLV. The Doubly Augmented Fourth Chord . XLVI. The Augmented Six-four-three Chord . XLVII. The Neapolitan Sixth XLVIII. Altered Chords with a Diminished Third XLIX. Enharmonic Changes ..... L. Irregular Resolution of the Dominant and Dimin- ished Seventh Chords .... LI. Modulation a Minor Second Upward LII. The Same, Continued ..... LIII. Modulation an Augmented Fourth Upward . LIV. General Principles of Modulation . \ . LV. The Suspension ...... LVT. The Suspension, Continued .... LVII. The Retardation, Inverted Suspension . . LVIII. The Appoggiatura and Anticipation Vlll CONTENTS. LESSON PAGE LIX. Free Resolution of Suspensions . . . .192 LX. Passing Tones and Embellishments . . . -195 LXI. Accented and Double Passing Tones . . • ^97 LXII. Obligato Melody 199 LXIII. The Pedal. (Organ Point. ) 201 'lXIV. The Inverted Pedal 205 LXV. Melodic Figuration 207 LXVI. Florid Melodies 210 LXVII. Accompaniments . . . , • • • '213 LXVIII. Harmonizing of the Chromatic Scale . . .217 LXIX. The Figured Chorale. Soprano given . . .220 LXX. The Figured Chorale. Bass given .... 223 LXXI. The Figured Chorale. Alto and Tenor given . . 225 LXXII. Analysis 226 Conclusion . . , 231 INTRODUCTORY. SCALES, INTERVALS. AND CHORDS. MAJOR SCALES. The succession of tones CDEFGABC, called the scale of C major, or natural scale, is the form on which the major scales of all other keys are modeled. The distance between the tones is as follows : CiDlEiFiGlAiBiC. 12 3 4 5 6 7 8 That is to say, between 3 and 4, 7 and 8 are half tones ; between all the others consecutively, whole tones. The names of the degrees or steps of the scale are : tonic, supertonic, mediant, subdominant, dominant, submediant, and lead- ing tone. The major scale is called diatonic, i.e., it progresses fi-om degree to degree throughout its compass. While only the signatures of seven sharp and seven flat keys are used in practice besides the natural one, the major scale may be formed on any given tone by the use of double flats and sharps — twenty-six keys in all. To form a major scale on any given tone write the following formula over the keynote, here indicated by x : 112i3i415i6i7|8 X Bnd then fill in the notes required by the fractions. MINOR SCALES. The harmonic minor scale differs from the major in that its third and sixth degrees are minor instead of major. The diatonic suc- cession is therefore : 112i3i415i6|7i8 (IX) INTRODUCTORY. Tliis scale contains the same tones (with the exception of the leading tone) as the major scale whose keynote is its third, and is therefore nearest to it in point of tonality. Hence the term relative or parallel keys. The minor scales are formed in the same manner as the major, using the above formula. The melodic minor scale is not used, strictly speaking, for harmonic purposes (see Lesson 40), but may be here given as a matter of general knowledge. Its sixth and seventh degrees are major in ascending and minor in descending, the third of course being minor in both cases. i \'S> — g- 10 W- INTERVALS. The difference in pitch which separates two tones, w^hether in combination or succession, is called an interval. This term is applied also to the association of the tones themselves in regard to pitch; i.e., we say that i F to C is a fifth, whereas we mean that the distance between them is a fifth. Intervals are counted upward from the lower to the higher, and are named First, from the number of degrees of the scale which they occupy, and Second, from the number of tones and semitones which they contain. Thus to find the general or numerical name of any interval we put ourselves into the major key of its lower tone. Its higher tone then occupies the degree of the scale from which the interval is named; thus. Fifth. Fifth. Fifth. Fifth. Ell ±S= -4-- -0 Fourth. -ito Seventh. i Second. Third. Fifth. Sixth. i -©'^ -s>- -<9- -&- -s"- -ei- The general or numerical name of intervals applies to their visible difference in size, and is not affected by any number of accidentals before their lower or upper tones. INTRODUCTORY. xi The specific name of intervals refers partly to the chromatic modi- fication of the diatonic (normal) intervals of the scale, and partly to their qualities as concords and discords ; thus, Unisons^ ^,., , r" are called perfect, being consonant. Fifths rib Octaves J Seconds Thirds Sixths Sevenths^ >- are called major or minor, according to size. Any interval may be expanded by accidentals into an augmented, or contracted into a diminished interval. A perfect or major interval is augmented by one chromatic semi- tone. A perfect or minor interval is diminished by one chromatic semi- tone. Note. A second or ninth when diminished is called enharmonic — that is, two different notations of the same pitch. A major interval is diminished by two chromatic semitones. A minor interval is augmented by two chromatic semitones. Perfect. Augmented. Minor. Diminished. Major. Diminished. Minor. Augmented. ■ C — fs H_«, O i f?= All the natural (normal ) intervals of a major scale, counting up- virard from its tonic (keynote), are either major or perfect. Intervals greater than the ninth are named the same as those within the compass of the octave. The concords and discords are classified as follows: Perfect unisons"] Perfect fourths I , . ^ . ^r , r perfect Perfect fifths Perfect octaves. consonances. Maior and mmor thirds ) . , . , , ^ , . . ,, J imperfect consonances. Major and minor sixths ) INTRODUCTORY. Major and minor seconds'" Major and minor sevenths and all augmented and di- minished intervals dissonances. Dissonances must be resolved, i. e., followed by consonances. By inversion Octaves i • Unisons. Sevenths Seconds. Sixths Thirds. Fifths Fourths • become - Fourths. Fifths. Thirds Sixths. Seconds Sevenths Unisons . . Octaves. Major intervals become minor. Minor intervals become major. Augmented intervals become diminished. Diminished intervals become augmented. But Perfect consonances remain perfect consonances. Imperfect consonances remain imperfect consonances. Dissonances remain dissonances. To find the name of any interval : Regard the lower tone as the tonic (keynote) of a major scale. Count the degrees to the upper tone. If the upper tone corresponds to the diatonic degree of the scale, the interval is either major or perfect. If it is greater by a half tone the interval is augmented. If it is smaller by a half tone the interval is diminished or minor. If it is greater by two half tones the interval is doubly augmented. If it is smaller by two half tones the interval is diminished. In case the lower tone is not a possible keynote of a major scale, raise or lower it by an accidental to the nearest practicable tonic. Then raise or lower the upper tone of the interval the same distance ; in both cases without altering the letters. INTRODUCTORY. CHORDS. A chord is a combination of three or more tones, erected in thirds on a given tone, or derived by transposition from such a combination. Thus g m is derived from a combination of thirds by iEiS transposing the upper tone an octave lower " -&- '-§»- A chord with three different tones is called a triad. A chord with four different tones is called a seventh chord (chord of the seventh) . A chord with five different tones is called a ninth chord (chord of the ninth) . The number of parts in a chord may be increased by doubling any of its intei-vals. This does not alter its character as a triad (three fold) , seventh chord (four fold) , or ninth chord (five fold) . Triad with root doubled, n Seventh chord with root doubled. Ninth chord with root doubled. U 1 . . X>n 1 J r3-l I h"^- f(\ % 1 "r^ e> \ V-^ s 1 S ej-. 1 ^ -g-J -^J . Chords are designated by the name of the degree of the scale upon which they are founded (tonic, dominant, subdominant, etc.), and this tone is called the root of the chord. When this tone is in the bass the chord is said to be in the funda- mental position. When any other tone is in the bass the chord is said to be inverted. In vocal music the voices from the highest to the lowest are named as follows : Soprano Alto Tenor Bass and their respective (average) compasses are Soprano. Alto. Tenor. Bass^^ *j z^ --/ ■&- xiv INTRODUCTORY. When the three upper voices of a chord lie so closely together that no other interval of the same chord can be placed between them the chord is said to be in close position.* Otherwise it is in open position. A major third and perfect fifth constitute a major triad. A minor third and perfect fifth constitute a minor triad. A major third and augmented fifth constitute an augmented triad. A minor third and diminished fifth constitute a diminished triad. All major and minor triads are consonant. The first inversion of a triad is called the chord of the sixth. The second inversion of a triad is called the chord of the sixth and fourth. The first inversion of a seventh chord is called the chord of the sixth and fifth. The second inversion of a seventh chord is called the chord of the fourth and third. The third inversion of a seventh chord is called the chord of the second, or sixth, fourth, and second. All tones which form combinations not derived from the system of thirds above mentioned are called nonharmonic or unessential tones, and their function is melodic rather than harmonic. * In this case the three upper voices are within tne compass of one octatre. HARMONY. LESSON I. THE PRINCIPAL TRIADS OF THE MAJOR SCALE. The principal triads of the major scale are those founded upon the tonic, subdominant and dominant. As these triads contain every tone of the scale, it is evident that a melody that is strictly diatonic may be harmonized with these chords alone, provided that they can be made to progress legitimately from one to another. In the study of harmony as an art, two principals are of great importance : First, the proper selection and arrangement of the chords which accompany a given melody ; Second, the proper succession or, as it is called, progression of such chords among themselves. I. SELECTION OF CHORDS. Inasmuch as all chords are ff>vmed originally by adding the third, fifth, seventh, and ninth to a given bass tone (see Introduction), it fol- lows that in harmonizing any given melody the bass tone is first to be found, and the chord built upon it. Thus the tones of the major scale will be harmonized as follows, using only the principal triads : Tonic. Supertonic. Mediant. Subdominant. Dominant. Submediant. Leading Tone. 9 ^m i.< P ^ I -^^^^m ^ EXPLANATION. I. I. Root of the tonic or fifth of the subdominant. 2. Fifth of the dominant. 3. Third of the tonic. 4. Root of the subdominant. 5. Fifth of the tonic, or root of the dominant. 6. Third of the sub- dominant. 7. Third of the dominant. From this it will be seen that only three bass tones are needed to harmonize all the tones of the scale. ill 2 HARMONY. II. The root (bass tone) appears twice in such chords. Note. It is said to be doubled. The root is always to be doubled for the preseiy in the fundamental position of the principal triads. III. The chords appear with either root, third, or fifth at the top. but always with the root at the bottom. Note. The position of the triad will be referred to according to the intervals i* the soprano voice, i.e., root position, third position, fifth position. IV. The first and fifth degrees of the scale have two possible harmonizings, while all the others have but one. This is the begin ning of the principle of selection referred to above. (See Para- graph I.) 2. CHORD PROGRESSION. "First — The progression of two voices in the same direction K called parallel motion. This usually takes place when some othe* voice becomes a connecting link between two chords. {a) (i.) (O 1 : :^— 1- ? (d) l^^6 &v,=^,i=- --3-- 2. ^= ^ =F- i (a.) Connected by (A.) Connected by (^.) Connected by (d.) Connected by alto G-G. alto C-C. soprano C-C. tenor C-C. Second — The progression of two voices in opposite directions in called contrary motion. This takes place when chords having no tone in common succeed one another. {") (*). 3. HARMONY. Third — Oblique motion (which is not really progression at all, ex- cepting in one voice) is the movement of a voice in any direction against a stationary tone. Iie5 ::]=± :± 'mmm It is obvious that oblique motion must always exist whenever a connecting tone is present between two chords. As in all harmonic progressions the smoothness and simplicity of the voices is a desideratum, such progressions as the following are un- desirable : * :g= 10 a^^E :§: ig= ^= i iisEt A chord may be repeated, however, in another of its positions. -4- ^^ 1 — 1 I I -tntizit i On general principles the progression of a given chord is to the nearest position of the next chord, and not to a more distant one, utilizing, if possible, a tone common to both chords, and moving the soprano, alto, and tenor in contrary motion to the bass when such tone is not available. HARMONY. The tones of a melody often suggest, by their progression, the harmony which underlies them. 3 6 8 i ^B 3 S I ^-g--;^ =t=tt s- -i -f-w Sec^ J i For harmonizing the succeeding exercises, the following rules will be sufficient : Rule i . When the melody contains successive tones of the same chord, do not change the harmony {bass tone) , but only the position of the chord. But if a tone of the melody is repeated, change the harmony — especially if the repeated note is on the first beat of the measure. feij ;fe=j=j3 q;E ^ i=q^E a S^-S-- -(S- Not. Not. S Rule 2. Unless one tone is common between ttvo different chords., move the bass in contrary motion to the three upper voices. i ^ ^m -& s- 15^^ 10. ^ r!»- Contrary motion. Not. t -»• Contrary Not. 9^=^:^ Ei This involves also another faulty progression, viz. : the skip of a seventh in the bass. ( See Rule 3.) HARMONY. Rule 3. Avoid two fourths or two fifths in succession in the bass, also the skip of a seventh in either direction. ( See Example 10.) I 1. It -TS<.- 3 -z^- i: i The skip of an octave is, on the contrary, of excellent effect. Note. These progressions are uiimelodious, and are easily avoided ; thus, 12. --r 4; -^ I :t ^ I A triad is indicated by the figures 8-5-3, abbreviated to 8 or B or 3. These figures indicate the intervals which lie in the soprano when the bass is given. The fundamental harmonies of each degree of the major scale are indicated as follows : Tonic. Supertonic. Mediant. Subdominant. Dominant. Submediant. Leading Tone. The chords which are major are indicated by large numerals, and all others by small. The diminished fifth of the leading tone triad is indicated by the sign °, (thus vii°). tor the present we have to do only with I, IV, and V. ' All the given models are to be studied systematically as follows* 1. Study the model carefully, without the pianoforte. 2. Write out the bass, add the upper voices, and compare. 3. Write out the soprano, add the other voices, and compare. 4. Play the model from the bass only. 5. Play the model from the soprano orijy. HARMONY. b. a. c. h. i ^ I fe S3 :S!i ■ig— -«i — a- ■^ — =*- -»- -s- I :S=t: - g^ JCH 4=i^ 3¥^P I V I I IV IV I V V IIVIVI IV I a. Connecting tone. i. Change of position in same chord. c. Change of harmony with repeated note. The musical student is cautioned not to write \hejifth in the bass with the tonic triad. Mark Roman numerals under each chord, as in the example given. Write the three upper voices on the upper staff and carefully avoid incorrect notation. EXERCISES TO LESSON I, 1 . Soprano given. II b. a. b. :=!= I i^q^l :4r *4= i 2. yi 2 P & — f= ^S^^S ■(2- ■Ti- ^ «- 3. E0 5i=:tz=ii?z :± -A=±^ ^^. i^^ p^ i -^- -•-• •-hp i-T r rfft I =F?=P==r ^P=f^ 1 1- HARMONY. LESSON II. THE PRINCIPAL TRIADS OF THE MAJOR SCALE. ( Continued.) In the foregoing lesson the subdominant and dominant triads have been invariably preceded and succeeded by that of the tonic. The succession, subdominant to dominant and dominant to subdominant, has been avoided, not because it is in itself faulty, but because the melody to be harmonized has not rendered it necessary. As we have no tone in common between the subdominant and dominant triads, we must progress according to Rule 2, Lesson I. Progi'ession by parallel motion between chords in the fundamental position, having no tone in coinmon, involves First, consecutive parallel fifths. m 13. i Soprano and tenor, D-G, move in parallel motion to C-F respec« tively. Second, consecutive parallel octaves. 14. "^ Tenor and bass, G-G, move in parallel motion to F-F respectively. Both of these errors are avoided by observing Rule 2. The progres- sion of the subdominant and dominant in such positions is as follows : 15. I IV V IV IV V IV V IV S HARMONY. The last progression of dominant and subdominant (see Example 15) a) is invariably to be avoided. Rule i . T^e dominant triad is always succeeded by that of the tonic ( never by that of the subdominant ) , when its third ( leading tone ) is in the uffer voice. 16. i^^^^ I I V IV Rule 2. In harm.onizing a given bass keep all coniTnon tones ih the same voices. 17. IE Ed= -k -a- :^ -si- zSz. not not 9^ -ei- M Not because these progressions are incorrect, but because with a given bass they are not necessary. EXERCISES TO LESSON II. J- ^: i^i ^ Ef^iE^E "S" IS IV IV V \ *=ar ^■ i -^ 1=1^ P* :t I V IV ' IV V The above model illustrates the progression of V and IV. HARMONY. Note. The root of the first chord is in the upper voice unless otherwise indicated ] . Bass given. ^H£ 12= H 2. mm E§ilE ^ :&=tt =^ 4. — |— 5 3 — 1=— ^^9! ^^ — «>— -^ ...,^ -^ 1 — iS — u -^ -4= a m i i^ ■;$> (5" «"- 5-, Soprano given. U ti-^^ IB ^#^§3 t?: - p— / ^E^^^^E^E^? P :^t=±t p.2^Q =s:=l — ^ ^^£ CADENCES. The succession dominant to tonic triad, with the tonic on the accent, is called the authentic cadence. (Example i8, a). The succession subdominant to tonic, with the tonic on the accent, is called the plagal cadence. (Efccample i8, i.) The succession subdominant, dominant, and tonic, is called the complete cadence. (Example iS, c.) These successions are used, generally speaking, at the end of a piece, and if the tonic triad appears with the root in the soprano and lO HARMONY. bass, the cadences are said to be perfect. If with any other inter- val in eitlier soprano or bass they are said to be imperfect. THE PERFECT CADENCES IN C MAJOR. {a.) Authentic. (*.) Plagal. (<:.) 18. -st- -o- tr — zi- ■si -zt =s= -a- s -zi- complete. -f^- it: V Authentic. I IV I I IV THE IMPERFECT CADENCE. V Plagal. 19. 4 -6>- ir- m =t It Play the perfect cadences in every major key. S I I I LESSON III. THE PRINCIPAL TRIADS OF THE MINOR SCALE. The harmonizing of melodies in the minor mode involves no new principle. In harmonizing the exercises the raised leading tone must not be overlooked. In a figured bass, it is indicated by a Jf or ij over the dominant of the key. 1. 3 EXERCISES TO t t LESSON III. J ff tf^» ^ 1 -^ O „ J Z ej t^ z? 1 2. 6 1 i* 3 i i^j: s)- 3. Unfigured bass given. 5 g^ -4-d- m HARMONY. II ep W^^m :S5= Jl -fv-=t-4 i±jt -K i^i — F--F— r-i- =^4=^ ^^^mi H ;^=p:: Play the following cadences in every minor key. Authentic. l^E^^ 20. If^ -•— © 2SI- lisi is> e — *tei (»■ 21. ^^ I V Plagal. i i :i= El^ i^^ s =5- -<9- -«- -z:^ I I i LESSON IV. THE CHORD OF THE SIXTH. If the third or fifth of a triad, instead of the root, appears in the bass, the triad is said to be inverted : ii^ I 22. -p- Root. Third. HARMONY. Although a chord 'when inverted consists of the same tones as in its fundamental position, yet its intervals from the bass are radically changed, and from these intervals the chord receives its name : thus, 8 3 6 m 23, it 35: i From the bass tone the tenor is an octave, the alto a third, the soprano a sixth, hence, 8-3-6, or chord of the sixth. The removal of the bass from the root to the third of the chord results in the ■doubling of the third, which we may obviate by removing the tenor from the third to the root, thus leaving us w^ith two roots, as in the fundamental position. The triad, however, when inverted, doubles the fifth or root with equal frequency, and under certain circumstances the third also. 24. i Root. Fifth. Root. 6 3 6 3 6 3 P: s i?^ § The first inversion of a triad is figured 6 or | or 6. These figures applied to the Roman numerals indicate both the fundamental harmony and its inversions, i.e. i -is- 16 IV6 V6 I In harmonizing the exercises of this lesson the same rules apply as in the previous one. Only the roots and fifths of the chord of the sixth will be doubled, and in harmonizing basses the connecting tones are to be strictly observed. HARMONY. 13 25. < J i^^^^ T—r -(=- -gj- yi Note. In case o£ two possible connecting tones on the first or fifth degree of the scale, the lower tone is usually the better to retain. Some available progressions of the chord of the sixth in this lesson are as follows : ;g^|E^ ^F^=^,g=^M=4 :i=g= 26. -61- -6>- I I -f^- =t l3^ m ■s*- -■ei- 16 IV 16 IV 16 V IV6 I (S2 f2_ -^ % gj— Z^ ^ -^—■^—zi- Ll=^=^=^: :^ i =F= -1=2- -<»- ■fil- V6 I 16 IV6 V6 16 V6 16 Possible combinations for harmonizing the different degrees of the "tal'i are a? now follows : ^7.< % P= I i--t^-t^ iHig H • — • — •■ ;^ ^Wf^ f?^ I I 16 IV IV6 V V6 I IV IV6 V V6 I 16 IV V HARMONY. From this it will be seen that we now have three more bass tones at our disposal in the harmonizing of melodies. EXERCISES TO LESSON IV. rmmm^sm \ -0 — ^- =t :pq—^- • — 1*~ z^ i •^-PT -r-^r I-* ' I I I 16 16 16 IV6 16 16 The triads are very often succeeded and preceded by their own inversions. 1 . Bass given. ^- ^FS ^s eH ^i m --?^=x^ m ^=t=^ m i t=^ 3tt -jj-e- i 3. Soprano given. 1 ^ e=#3E -si- s sS ^f^iS3 =S:3i ^=g ^=g=f •-^ -#-: — •- ■ S- ^ d I— V— I I V I V ifa i 1^ H fsi ■ si — g*- Incorrect notation is to be carefully avoided. Play these complete cadences in every major and minor key. 28. iE -^--&. -gH -&- J=±S: ■s- i=i=±|^zg=:^=:g=:g: -ti—^G'- I P^ :f=:==t I HARMONY. »s LESSON V. THE CHORD OF THE SIXTH AND FOURTH. The second inversion of a triad is called the chord of the sixth and fourth. It is formed by placing the fifth of the triad in the bass. 29. Root. Fifth, The bass tone of the chord of the sixth and fourth is usually doubled. Chords of the sixth and fourth of the tonic, subdominant, and dominant, may be introduced as follows : Rule i . When preceded and succeeded by a triad in the funda- mental position on the same bass tone, or its octave. 30. i X- W=E 6 i :^i I ^ i I Rule 2. As the second chord oj a grouf -with two or three as- cending or descending bass tones. I J— 31. i ii=i: -s)- z^ I 9t IS For the present the chord of the sixth and fourth of the dominant only is available for this progression. In this case the tones of the | chord are hardly to be considered as harmonic, but rather as melodic passing tones. l6 HARMONY. Rule 3. As a refetiiion of the same harmony, forming a melodic bass. J. i 32. tr — z)' -s^- '^E^ =t--i -s<- ^ -25l- is: -g^- a 3^ S=p= 6 4 6 X =?±E£ It II Note. These three rules may perhaps be summed up as follows : Avoid skips to or from the bass of a | chord. The bass is either stationary or diatonic when the harmony changes. Bad. Bad. 33. ± Bad. e=iz i i 34. 91: :^ f= I 35. ^i: The principal function of the tonic | chord is to precede the domi- nant chord in the authentic cadence. In fact, the formula I|-V-I, stereotyped as it is, may be perhaps considered the most useful, as well as the most decided, of all authentic forms of cadence. It is used in all the following exercises of this lesson. All other | chords are practically melodic rather than harmonic. It is hardly necessary to state that a piece never begins nor ends with a 2 chord except in very free composition. EXERCISES TO LESSON V. n" ^^s±2= -77 — w — m — 41 — •— |— • ^ -rd— •!— J- aa;H^33s:s ^ ¥=i^- I ivl a. Rule I. b. Rule 3. 1 . Bass c;iven. 4 iS 112: r2: -li- ^ ll ;(=- -St- Hi HARMONY. 17 P5iiag= 6 6 4 6 4 8 ^^^a -i=- -3— t^f -2=1— h*'- 3. Unfigured bass given. 6 ^gl I i±^=Ef= S -ZS- 1*^ 4-. Soprano g-iven. it :^ 2i :^^ ^3^ •z=t -»Ht-«— ^- 5. EEE 63: -»*— '^ =fc Ttlt •~f-^ ■r^^- ~^^ X^^^ » :^=p: ::p=P=tt S^EdEt :F=fi.»- =U=: =t? ■=i-n- Play these cadences in every major and minor key. Same in c minor. 36. -Tri— sj- --§: -tn z^ , -«- %■ -s- 5=*— ^z:^ -a -g- -s>—!i- :^J^5pEi5i^ ^ i;i 9^ «--(=- s>—-f=^ Ejl^I (S'-'lSi- :t=: 1 I IV 16 VI 4 LESSON VI. THE CHORD OF THE DOMINANT SEVENTH. By adding a seventh to the triad on the fifth degree of the scale, a chord is formed called the dominant seventh. 37. < i^ 9fc H Fifth, Third. Root. 38. ^ tJ-i — O- 9 i Seventh. Fifth. Third. Root. i8 HARMONY. Inasmuch as the seventh is a dissonance (see Introduction) , it must be followed by a certain tone demanded by the ear, which is called its resolution. As this tone is the third of the tonic triad, and as the fifth degree of the scale on which the dominant seventh chord is founded is common to both tonic and dominant chords, the third and fifth of the dominant seventh chord have only to progress to the root of the tonic to form a complete tonic chord. Note. The dominant seventh chord consists of a major third, perfect fifth, and minor seventh. This combination of intervals is to be found only on the fifth degree of the scale, and is the same in both major and minor keys. RULES FOR THE USE OF THE DOMINANT SEVENTH. Rule. The dominant seventh chord resolves to the tonic triad, to the third of which its seventh must descend. Note. These rules are for the regular resolution of the dominant seventh chord j later on its irregular resolution will be considered. (See Lesson 54.) The resolutions of the intervals in detail are as follows : The seventh descends to the third of the tonic triad. 39. i =S^=3b: ^ The fifth descends parallel with the seventh, to the root of the tonic triad. 4-0. i Or it may ascend to the third of the tonic triad, in which case the third is doubled, as the seventh is obliged to descend to the same tone. 41. it™ s I HARMONY. The third, being the leading tone of the scale, may always ascend to the root of the tonic triad. (Example 42, a.) It may also descend in the alto or tenor voices to the fifth of the tonic triad, provided that the root ascends. (Example 42, d.) W («.) (f.) (^0 42. 8=1? «= ^~ i I r ^^^^^ The root of a dominant seventh chord ascends or descends to the root of the tonic triad, and is often retained as the bass tone of a I chord. (See Example 43, i.) When the fifth or third of the dominant seventh chord is omitted, or when the chord appears in any of its inversions, the root is retained in the upper voices. (Example 43, c.) In other words, it is often treated as in Lesson I, when it is the bass tone of the dominant triad without the seventh. The dominant seventh chord is used in three different forms demanded by the exigencies of progression, and while the root and the seventh are always present, the third or fifth may be omitted, and another root substituted in their places. (See Example 46. ) (a.) _ {6.) (c.) 43. i Ih 8=r : — *— ljzigiiEigrS=l:^g ^ ' s • * * m The dominant seventh chord may be repeated in different positions before resolving; this is also true of the principal triads and their inversions. It may be approached in parallel motion from a tonic | triad, for the reason that the upper voices move only one degree. 44. i < \ j I " n 20 HARMONY. Note. Parallel motion from any interval to a fifth or octave is to be avoided in outer voices unless : A connecting tone is present in one of the voices, or The other parts move by contrary motion, or All the upper voices (as in Example 44) move one degree. Such progressions are called concealed or hidden fifths and octaves. 7 The full figuring of the chord of the dominant seventh is 5, abbre- viated to \ or 7. In the minor mode the raised leading tone is indi- cated by a jl or if under the 7, (J). The authentic cadence is much strengthened by the addition of the seventh to the dominant chord, and as the forms of the chord with the root doubled all resolve to the complete tonic chord, i.e., without omitted fifth, this chord is perhaps more useful than the complete 7 chord |. On the other hand the positions with third omitted are of 1 comparatively infrequent use, excepting as passing chords, which are melodic rather than harmonic in character. This chord now gives us the use of another bass tone in harmo- nizing the fourth degree of the scale. Whereas thus far this degree has only been possible to harmonize as the root of the subdominant, wfth the root or third in the bass. 45. < i w i 6 s- we can now use it as the seventh of the dominant chord — a decided addition to our harmonic vocabulary ^ ^ IY-. 46. THE DOMINANT SEVENTH CHORD IN C MAJOR. Complete. With 5th omitted and root doubled. With 3d omitted and root doubled. i:^=g±g^Fp-f3:^z:^±ag:rh^-*=lzgg^g.-gfgzz*zH P I ^ HARMONY. EXERCISES TO LESSON VI. 21 fi i Ei I -iSI- -st #-hg gJ-hz^ -gi- J- 7 a — (S>- I :^=-t -«'- -»- -19- -(2- -SI- g s=s? -i ^T '^ I =^ ^t =t f-= r r a. Fifth omitted, i. Complete chord. 1 . Bass ^iven. 8 7 6 8 6 4 7 6 7 6 4 6 ^::t d^ 6 4 7 7 Ste* i 1^ ■^zjtM: :^iii3t -^-* 6 7 6 14, 7 6 6 7 3 t i t 6 663«4$ -=^^ — A .>j — *■ -P=P=pc -W—^^0- :p=?c^c 4:^ 4=}=^ 3. Unfigured bass given iSi ^^^? ]= ^5=^ ^^3 ■^—\- »-^m-m-\ - m-^»-0- 3z±3^ V-E5: I i 4-, Soprano given. S 1# P^ si -c*— — g'-i sJ :^ -Zi-—iS- m ^=i :1==t I S aj s^ — -iS' 1 s) — - S- -6t- -^ -&■ €=g= f=f^ :si 16 IV 16 V 4 7 ^ S I LESSON VII. THE FIRST INVERSION OF THE CHORD OF THE DOMI- NANT SEVENTH. THE SIX-FIVE DOMINANT CHORD. In Lesson IV we have used the leading tone in the bass in the chord of the sixth, or first inversion of the dominant triad. 48. i W ""-1: — ^ 6 I I If we now add the original seventh of the dominant chord to this triad, we have the corresponding first inversion of the chord of the dominant seventh which is called the chord of the sixth and fifth or the six-five chord. 6 6 3 3 6 6 49. ^ » — e — g g a tr— ^ — :g: — s^ — --ST -9- i HARMONY: 23 The full figuring of this chord is 5, abbreviated to *. In this, as in other inversions of the chord of the dominant seventh, the original root is not doubled, nor the fifth omitted. The resolution is the same as in the fundamental position, excepting that the root becomes a connecting tone with the fifth of the tonic triad, and the third (bass tone) always ascends to the root of the same. 1—4- so. P^F :=»- -is>- m t^^jt The fifth descends to the root of the tonic triad, or ascends, doubling the third of the same. 51. tB=^ i i The seventh descends to the third of the tonic triad 62. I ■^^S- ga 3i i The root connects with the fifth of the tonic triad as before stated. 53. i w~t^-t af^>—f- i I By means of these chords we gain the leading tone in the bass in harmonizing the fourth degree of the scale, and we may also utilize it for the fifth and second degrees as before. H HARMONY. EXERCISES TO LESSON VII. ~As\-si—A ^E^ 5zz^:^zr4:=j=:::=l -4-J-J— J =— s^ -*^ -^ -z5(- ;g: -=«';§: -fe- -s^ "tt # c c R I n I a— sj- s p^=g^fg— ^ I 4 7 6 4 8 7 i ^v— »- ZSI The exercises with given bass should end with a complete cadence when possible. 1 . Figured bass given. 3 S i'-2z :t=^ 8 7 i= !»- i :s=t: e 6 8 5 4 6 6 5 6 8 4 I EfclEE^i -z^- 3, Unfigured bass. 3 Pgfi^ElELS e i -;2_ -f-.»-*^j-4-zj-*-« -¥-4 - ■i=Srzl $ 4. Soprano given. m i=t 3 -1© — -"p= nfi — >i- 3t 21 Harmonize the same in minor. rf ^ w ^r f ^ N.B. I EEii S • i^ *l N.B. Plagal cadence. ^i IS ^^^F %H HARMONY. Play in every major and minor key. 25 64. W^ -l^f^ -S^T& I f--s — • — \-i-» — ' — ie>- r ff r ■•- r i I V6 I 16 16 6 4 ^^m ^ i LESSON VIII. THE SECOND INVERSION OF THE CHORD OF THE DOMI- NANT SEVENTH. THE FOUR-THREE DOMINANT CHORD. Thus far we have considered no combination which has allowed us the use of the second degree in the bass, except as a passing J chord. (See Lesson V.) 65. < i w ^ p^ i By placing the fifth of the dominant seventh chord in the bass, we find a convenient harmony for the fourth, fifth, and seventh degrees of the scale. 66. i Fourth. Fifth. Seventh. IE 9i: ifer -s^- i i 26 HARMONY. This chord consists of a third, fourth, and sixth, added to the bass tone, and its three positions in close harmony are as follows : 57. i Hzz:^ 9- '»~or- 3^« ®^ IlTorz .«_*- I Its full figuring is 4, abbreviated to |. In the dominant seventh chords of minor keys, the sixth interval is made sharp or natural, to indicate the raised leading tone of the scale. 58. ii^i ^^■■. ^ 69 4 3 The resolution is the same as in the fundamental position (.Exam- ple 59, a), with the following notable exception, viz : If the bass tone of the I chord ascends, and the original seventh descends, it forms a tonic chord of the sixth with the third doubled (Example 59, d). In order to obviate this the seventh may ascend to the fifth of the tonic chord (Example 59, c), the parallel fifths which result being excusable from the fact that one of them is not perfect. (Example 59. N.B.) (a.) (i.) {c.) N.B. 59. N. B This particular form of consecutive fifths is to be found in the works of every great composer from Handel to Brahms. The real reason for its legitimacy in this connection lies in the fact that the dominant seventh under these circum- stances is a passing chord, and therefore melodic. HARMONY. 27 As in the case of the other positions and inversions of the dominant seventh chord, this may also be exchanged for any inversion of the same chord, or for its fundamental position. Since this chord enables us to utilize the harmonic progression IV- V|, we can harmonize the ascending major scale as follows : 60. i h ^ ^ a ''- -s>-^ 3 -a—oB- 5^=g= Sl3 g rj "'ST^ ofs I §a 4 3 -g -e o o o o tj ~ I This, however, is not possible in the minor mode, on account of the faulty progression of the augmented second between the sixth and seventh degrees. Note. On general principles all progressions of augmented and diminished intervals are to be avoided, especially in the inner voices. The very frequent and effective use of augmented seconds and fourths in instrumental compositions in no wise contradicts this principle. EXERCISES TO LESSON VIII. ■ Unison. ^^ is s s SI P^^ •=?= t=trf t#* Play in every major and minor key, 61. I VJI6I IVIfVI 3 4 LESSON IX. THE THIRD INVERSION OF THE CHORD OF THE DOMI- NANT SEVENTH. THE FOUR-TWO DOMINANT CHORD. The third inversion of the dominant seventh chord consists of a sec- ond, fourth, and sixth, erected on the fourth degree of the scale. It 6 IS called the chord of the 4, often abbreviated to * or 2. Its three 2 2 positions are as follows : 62. ' i -zr- I HARMONY. 29 The resolution is to the first inversion of the tonic triad. Its bass tone, being the original seventh of the dominant chord, must descend to the third of the tonic triad, the leading tone progresses as in the fundamental position, and the root forms a connecting tone with the tonic triad. 63. The fifth, however, has more possible resolutions than in any other position of the dominant seventh chord. First — It may descend to the root of the tonic triad as above. Second — It may ascend to the third of the tonic triad, forming a first inversion with the third doubled. 64. Third. Third — It may ascend or descend to the fifth of the tonic triad. (See Example 65) . The last-named resolution occurs often in dis- persed harmony, and will be discussed later. The ascent to the fifth of the tonic triad in the soprano, however, is a very convenient mode of harmonizing the melodic progression, supertonic to dominant, and is found in numberless compositions of the best masters. 65. • IE i ^ i As in the other inversions of the dominant seventh, all the inter- vals must be present, the root not being doubled. Like these, also, the * chords may be exchanged for any other inversion of the dominant seventh, but must be eventually resolved. In all cases where the inversions of the dominant seventh are found in an incomplete 3° HARMONY. form, the chords may be considered as passing chords, or as forms of melodic progression. The melodic progression, submediant, leading tone, tonic, is conve- niently harmonized by the use of the * chord, allowing the seventh to become a connecting tone in the bass with the root of the subdomi- nant chord. 6 7 8 66. it ~zr- i i There is one more exception to the general rule for the descent of the seventh. It occurs when a fundamental position of the dominant seventh chord, with the root not doubled, and with the third or fifth (never with the seventh) in the soprano voice, is followed by the first inversion of the tonic triad. In this case — a somewhat rare one — the seventh ascends to the fifth of the tonic triad, instead of descending to the third. 7th. ascends. 67. 6 I The regular resolution of the seventh downward would leave the tonic triad in a very weak position, and produce concealed octaves, nearly as objectionable as consecutive' octaves. (See Lesson VI.) 68. 69. ^ i Singular exceptions to the resolution of the * chord are found in the works of Bach. (Toccata in d minor.) HARMONY. 3> When the dominant * chord occurs in the minor mode the fourth requires a q or J^, on account of the raised leading tone. EXERCISES TO LESSON IX. i iff? -^r4^ I I , J I , -i- u fp^^^m^ 4z5 T 5F ^ T * -•- T -•- -«- -•- -•- -•- 4 4" 2 6 3 3 6 3 6 3 6 4 7 gfM Stfte:t -?5^ -*-Ui-.- 1: ^ ^= F I 1 . Figured bass given. 6 4 6 6 2 6, 6 3 6 6 4 6 7 6 P=¥: sm ^^B E2E r^ ^i«— •- :^ ^=F 6" « 4 4ti 2. 8 3 2 6 6 D 6 I" 3 6 J EES *^ 7~P 40 S 2 40 2 6 60 4 3 ^B :S^ t=f f i^- -#-• 3. Unfigured bass. ihgu] ^ #— •- 8 2 ?S=i= i aSsS? -^-li- -^ — a u+ o S ■* ci. & - 4. Soprano given. ite?^* ■'-\-&-\ -s>- 3 chords. (9 — ^•-*i|9-U 32 harmony: ^ ^^H *n '^ s^s :^*v# ^ Play in all major keys. 70. n Also this. t I ■■ / s^ " \ ■ I f{ ■?- /^ *' 11 v-\/ /v S S o » 1 5; c £.' i> ^ II J ^ # -f - -6>- -«i- tS"- -g- -g- -€©--«.- ^- ^ -- ^^ IS'- -37- ^- -»- a- &! « a """S r^ -3 " II y. fi> «? a; f3 ^ a ■' O \l « » eJ a a/ ■ 1 1 S> 1 » IJ ^\ I V4 18 IV 16 V I LESSON X. THE SECONDARY TRIADS IN MAJOR. THE SUPERTONIC TRIAD. The secondary triads of the major scale are the ininor triads formed upon its second, third, and sixth degrees. They stand in the relation of relative minor keys to the principal triads : tonic, subdominant, and dominant. Their inferiority has caused them to be regarded by many theorists as entirely subordinate to the principal triads, not as inde- pendent harmonies, and to be used only as substitutes for, and in con- nection with, them. If we analyze almost any of the simpler pieces by Mozart, Beethoven, or Haydn, we shall find that the proportion of secondary triads is small, and that the fundamental harmonies consist largely of the tonic, dominant, and subdominant, with modulations to the adjacent keys. However, the further we go back towards the contrapuntal era of musical art, the less we find this to be the case, and even in the works of Bach and Handel they are used with great independence and freedom. The most important of these triads is that founded upon the super- tonic of the scale, and, being the parallel minor of the subdominant, HARMONY. 33 its third is often doubled, especially after a tonic chord. Its best usages are as follows : First — To precede the dominant seventh chord, or tonic | triad in a complete cadence, as a substitute for the subdominant. A skip in the bass from II to ]| is allowed. 71. Possible. Never. sj — ai- ^ E± -a- I I ■ LSI — i- a I g ^^rgj _i_-Z-i3 ^^.-t:' §^E =^ PS II V7 II 16 4 Second — As a passing chord between the tonic triad and its first inversion. (Substitute for the dominant |.) 72. - d r' ^ Besides these most useful progressions, the following may also be used : 73. i s=^ ^^ , Poor. I Rare. Never. t5! Z5t- ^zrg: a II a a- F=f= -g^? : ^ 3r- 5 ^ — »■ ^1 ^^ St—si- i tr— si- ll IV --Sr- II vf II ivs 4 4 .L—J- ^ n IV8 II V6 34 HARMONY. i=i I -^ — ^V^ ^^=g^p- :^±t:S -w- =1= 19 iS" (S> «>■ II Vg 5 II v« ^ i If we employ the supertonic of the scale as a connecting tone in progressing from the second to the fifth degree we form a succession of two diatonic major thirds. 74. ^^iB ^^ This progression involves the false relation of the tritone, which has been regarded as objectionable in the strictest writing. This is easily avoided by observing the following Rule. When the dominant triad (without the seventh^ is pre- ceded or succeeded by that of the supertonic, the connecting tone {supertonic') is to be disregarded, and the upper voices are to move in contrary motion to the bass. 75. I ffi^ i rpi- -^r not =i ^r* -(=- -a- T=^ not m -^ -iS"- HARMONY. 35 The tritone is the interval of an augmented fourth which exists be- tween the fourth and seventh degrees of the scale. Progressions which involve these two tones in succession, like 76. § «^^- [= g ^^-J . I are avoided in strict counterpoint. Example 75» page 34, covers alj necessary progressions of this kind in harmonizing the exercises given. EXERCISES TO LESSON X. rcr g.. -••-$- V -g- T»:f:; :^ T -»- ^ h» -0- -fr J ^ II 4 2 6 252:3 r=: : =t=t=: :t: :q= i ■i*— g SE ES itzi -• — i< -•—«!- ,6 a, Third doubled, i. ii followed by I4. 1 . Figured bass given. 6 3 0, 4. 2 i^i^ X ^ m -zi- -is- 2: ■ - i*S 4 4 3 3 2 6 3 6 4 6 6 3 7 3^^^ I «t >g • fg 3. Uniigured bass. e^^^=i^sai aa* 4. Soprano given. w- II II IV6Y6 4 o II IV6 L^?= ^^ 4^i- ^ 4 *^^:it^^3^^^tJ=3=j 1^ 36 HARMONY: ^^ -P^ :P=P: ^^ *-^^^»-l-T^^ la^ §1 p:ip: -F-M — * !d:^=ad:t::t:i Play in every major key. 4— J- 77. -zs! — gi,- =: g— gisf# -dz-«- Szz^ i^ — F-ng — g-rs^^ — -^- ^i=isaEiia: -si- I 16 IV II 16 Vt I 1^ LESSON XI. THE SUBMEDIANT TRIAD. The secondary triad next in importance is that of the submediant, the parallel of the tonic. It is often used to succeed the dominant or dominant seventh chord instead of the tonic triad, forming what is called the deceptive cadence. (a.) Better. {i.) 78. f J- ^=i^=^ 3 4- 3= -g— fiH Efc& -'=> a -g-- not -s*- i^ ^ V^ VI I V, VI I V^ VI In this case the leading tone, if not below the seventh, or in the soprano voice, may descend to the root of the submediant triad. (Example 78, 3) . Otherwise the leading tone generally ascends, in which case the third of the submediant triad is doubled. HARMONY. 37 This progression (the deceptive cadence) can never take place at the close of a piece. This chord is also often used as a connecting link between tonic and subdominant triads, ! 1 so. 1 BE^ I VI IV but less often between subdominant and tonic. -J J -l 81. ^EiE^lE \m IV VI I Other possible progressions are as follows : S^SSiE^is 82. VI V, (See Lesson XII.) Rare, Poor, Possible. Harsh. ^ ^- -t- FF= -^= =ld^- f^= — u ^T=F\ -f -;i-| -*- -4- ^ r rpS— Tf— - fF^ 3^ h-M * etc. 1 -^ r : r ^ 1 3— «i- u_ =^J VI Vf VI I VI V6 VI V2 3 6 All the above progressions must retain the connecting tones in the same voices, or move in contrary motion to the bass. The descend- 38 HARMONY. ing major scales may now be harmonized by means of the submediant chord, in the following manner : 83. i -nf.- ^S^ f^ - VI IV '\ Or much better : 84. i ZlZi i i I i the melodic progression submediant to leading tone having been thus far impossible on account of the succession dominant — subdominant. This may now be combined with the ascending major scale (Lesson IX) , and must be play6d in every key from the following model : 12 3 4 5 6 I V I IV 16 IV 78765 4321 V* I V VI V6 Vg I Vv I or I IV I| EXERCISES TO LESSON XI. i ^i^ i^ ^ ^f=P §S T—rs-p>- -z)^ IV 16 i *=i= I i ;a^ i^J3 g=g-=, 6 4 e£ I XJ±l a. Deceptive cadence. ^. I vi IV. HARMONY. 39 S I , Figured bass given. 8 7 a. ^ -z^ E^ Deceptive cadences are indicated at a. 2. 7 3 wm 6 7 a. i 4^ e t^^Ei 3. UnfiE^ured bass. 6 i*i^ =?=^ 5=P= r — r^ -• — r- ^t i M=±±:±jt ,-±^^=Lt-:dt^ i 4-. Soprano given. ^ g fg . p^ -<& — sJ- i I -S> .1=- 3^^=± -!° s^- 5 — ,ig — g'- lE - o sJ — s- It will be found a useful exercise to set this melody to the words of a hymn. i ^ &^^^^*-^ S^iS 11=^=^: 3^^: PD »-* 4 4\ l -4-^e- -6 IfeESESEsfegta^J i=± ga ;h li 4—* I g i- Play in every key. 85. Ei=E5t 3Z i -f=- V7 VI IV I| 40 HARMONY. LESSON XII. THE MEDIANT TRIAD. The triad of the mediant, the parallel minor of the dominant, is the least useful of the secondary harmonies of the scale. Its princi- pal use is to harmonize the leading tone, in the descending melodic succession, tonic, leading tone, submediant. This, like the super- tonic and submediant triads, is indicated by a small numeral, from the fact that the triad is minor. 86. i i^ S ft I :t -Ti- =P =t I I III IV Like the submediant, its third is often doubled. It should only be preceded by the tonic, dominant or submediant, the progression supertonic to mediant, or subdominant to mediant, being harsh in any position. 87. < ■i — r =t =^ ^ :t I It may best be succeeded by the subdominant or tonic triad, the succession mediant to submediant, mediant to dominant, or mediant to supertonic being either harsh or weak. 88. < i d= -^ ^M ■5 :tl=S: -s>- not better -s>- -& s etc. ?=^ F=F On general principles the progression of a secondary triad to its relative major, supertonic to subdominant, submediant to tonic, or HARMONY. 41 mediant to dominant, although possible, is not to be recommended, although the reverse progression is always of good effect. In each position. Good. Possible. 89. | 9^=f=f= |-f=fe g^ ;g^ pEg^Eg I II IV III V I VI I IV II V III The progressions from secondary triads to those a fourth above, although freely used by all modern composers, are avoided in the strictest vocal writing, or else are used in contrary motion, disre- garding the connecting tone. not not not m^& 4 -SI- m^mw^mm 90. T ^ I Note. The apparently contradictory relation which exists in these successions is due to the fact that these chords appear to be tonics, preceded by minor dominants. As all dominant chords are major, these progressions must be false. Our general rules of progression may now be re-stated, as follows : First — Move to the nearest position of the succeeding chord. Second — Use connecting tones with parallel motion. Note. Parallel motion by perfect fifths or octaves is wrong. Parallel motion by thirds or sixths is right. Third — Use contrary motion when no connecting tones are present. Note. Also in the succession supertonic to dominant, or its reverse. Fourth The third of the supertonic, mediant, and submediant is usually to be doubled. Fifth Avoid progressions of augmented and diminished intervals in all voices ; also progressions of two fourths or fifths in the same direction in the bass. With our present chord material the various degrees of the major scale may now be harmonized as follows : 42 HARMONY. Tonic. TABLE. Rare. Supertonic. Rare. B m =1A U: W^^ ff rwm "• r t-\ E£l I 16 IV If IV6 VI IV6 Vl6 V V„ V6 II V2 Il6 Vf Ilf 4 4 7 5 4 4 Mediant. Rare, Subdominant. ^^ u ?^^^^ 5^ -B- -^ vpv m =t=t: 16 HI 16 VI 4 vr IV V^ IV6 V6 IV6 V4 i Dogiinant. Rare. m m ■ — [ IE i^^ -r^- ^m i^= =t=t: \ ^^ ^6 ^ V4 l6 V2 IIl6 Iil6 V6 4 4 i Submediant. Rare. Leading tone. Rare. Rare. a p r ii^ t=s f^^TTrt iStej^i If VI Vl6 II n6 IV IVg IV6 n6 V V7 V6 V4 V4 III in6 V6 V6 4 4 '32 4 From this table it will be seen that the dominant and the sub- dominant have a much more varied relationship to the triads of the scale than the other degrees, being possible to harmonize with every degree of the scale in the bass except one. The tonic may be har- monized with every degree of the scale except two. HARMONY. EXERCISES TO LESSON XII. 43 a* =iT'=- Ttt i f- V III IV 1 . Figured bass given. 8 6 V6V4 6 3 2 6 III V6 I6VV I 4 7 ifc ?^ S 1=- 111 &J f hS- 2. » B 6 8 7 3=t ^ ^F I I52E - gi — >s- 3. Unfigured bass given. 3 ^i» -©■—(2 •«'— I =^:t ^ 4 15>- 4". Melody given. tf 3=P?f ti: -(2- ■ z^ »i- fzztci: -5)- ^1-*- 5. 3^ -La > J — a — I.I l-"6fe^ E^ X =7i=* -t=tfcit •-«-#-• Play this cadence in every major key. (a.) (*.) 91. :^=^ Zj '51 Si- -a — si- :g— g^g §^=3 it g g^ (^.) I III IV II I| v^ I i S=fe^^^ IB f^t -f2- I 44 HARMONY. LESSON XIII. INVERSIONS OF THE SECONDARY TRIADS. The first inversions (chords of the sixth) of the supertonic, mediant, and submediant, are used with almost as much frequency as the fundamental positions, and like them often double the third. The first inversion of the mediant is of somewhat less value, but has one peculiar usage which is of singularly interesting effect. It consists of the doubling of the third, which, with the root, progresses to the seventh and fifth of the dominant seventh. The effect is that of a double suspension on the dominant seventh, and later on will be so regarded. (Example 92.) i 92. ^ -gl- T 1 I 1116 V„ -is- I 1116 V. 7 7 Note. The second inversion of the mediant is often used in the same way. 93. 1 =t= '-■^ The second inversion of the secondary triads is of very slight value, and its usage is governed by the strictest rules of Lesson V. All I chords sound like tonic chords, and it is only when their re- lationship as passing chords is made plainly obvious to the ear, that they can be used in any other way. ■ ' I. 94. l¥z 9^ i£: „e =g= -s^g — S- 3^ir :2i:^i =1=4= eI^^Se n| V16 W- s HARMONY. 45 As the possible successions of the first inversions of triads require some special treatment they may be summed up as follows : Rule i. A single chord of the sixth doubles the root or fifth (see Example 95, a), and may double the third in the inversion of the chord of the supertonic, mediant, or submediant. (See Ex- ample 95, b.) («•) 95. m (b.) Third doubled. ^ =& -IS> B' &■ f= ^= -s^- i :gz=:^ I sr 6 etc. iS ■S f2- -t=P -zS- RuLE 2. In two successive chords of the sixth, first double the root^ or fifth, then the third, or vice versa. 4- 96. I^ =d= 6 6 Si- i -3 <2Z Note. An exception to this is found in the connection of the tonic with the sub- dominant, if two connecting tones are present. 97. i W- E3E 9^=^ 16 IV6 Rule 3. In three successive chords of the sixth, tvith ascending diatonic bass, double the fifth, third, and root in succession. Fifth. Third. Root. 4S- 98. m -ri. — g'- l^SI IS g - PUfe 3 6 7 3 6 3 6 ^647 a =P=iC :t^:pc:^ -HS--(^ ±±s g 3. Unfigured bass. 3 ■ ■• — ^ tes^= ^^= 47 I EE3 4j Melody given. ^^^^i =35 ^iPi tt i •m 33g la S^-4- m-1 r3:^=g:--g: "Si—ii-*- -si—'^' ^2?- Play in every major key. ioi.< J-^-J- '^ *—•- X^ =^ I IV6 ni6 Il6 16 V_ 4 gpgJB LESSON XIV. THE LEADING TONE TRIAD. There remains but one more triad of the major scale to be consid- ered, i. e., that of the seventh or leading tone. As the fifth of this chord is a dissonance, being imperfect, it follows that a resolution is required. This we find to be identical with that of the dominant seventh ; in fact the triad in itself is a dominant seventh with the root 48 HARMONY. omitted, and is subject to the same treatment as that chord. Its root being a leading tone, is almost never to be doubled (See Example I02, a), nor used in the fundamental position, except in sequences. (a.) Poor. , Never. {i.) I 02. N.B. This resolution to the tonic triad with doubled third is very common in strict part writing. Generally it appears as a chord of the sixth, -with the third or fifth doubled, and resolves to the tonic triad. In this form it is frequently found in strict writing, in which it forms some cadences of the church modes. (See Example io3, i>). In modern writing it has been practically superseded by the dominant * chord in the authentic cadence. Note. The fundamental position with doubled root appears often in sequences. (See Example 103, a, and also Lesson XXXIX.) Its use in the second inversion is also very infrequent, but it is occasionally used in the place of a dominant » chord, in the same manner as any other | chord. (See Example 103, 6.) 103: r ^^g=g=^: d=i mz i :^=a±^ etc. * 3»-l-iS- 9- ■si g- -zi- -& — I — a. trrf^ V1106 4 VIlO viio This triad is employed as follows in harmonizing the major scale : 104-. i — I ' I I — J — J— «l — ■ -g m J i ll ,' XX X It -d- is)- IB M-^£-^ff - «! b! -I — r— r I V8 I IV 18 IV VII°« I VIIO6 II V6 V6 I V I 8 7 HARMONY EXERCISES TO LESSON XIV 49 I -m- -0- s^ -0- '^ -m- ' ^-J- IB -•-r^ t=± I :pjE^^t^ 5:*:^^ VII06 1 , Figured bass given. VIlOi a±*^ m ■^z ^ Sequence. 6 7 ^m=^ 3 6 6 4, 7 l^g:l fct ^^ 3. Unfigured bass. 8 I ^ifeg 5^: -s< — •- 4. Melody given. i^ Ti- • t- i fe^ t=F* g^ [0 e^d -i° — f — ^- =j ^ ^ — y — ^^F- ^^ It X I :^ -*— ^ Play in every major key. Also all the major scales with the harmony given in Example 104. 105. ^^^^^^^gB i^ T=W- t^^ m =S=P- I VII°6I6 IV 18 V^ I i 5° HARMONY. LESSON XV. THE SECONDARY TRIADS IN MINOR, WITH THEIR INVERSIONS. The secondary triads in the minor mode are treated similarly to those of the major. Being formed from the tones of the minor scale, *hey all contain discords except the submediant, which is a major triad. The supertonic is a diminished triad, corresponding to the leading tone triad of the relative major. io6. ^rt>— g^ c : 11° equivalent to E\): vil° The mediant is an augmented triad, consisting of a major third and augmented fifth. The augmentation of the fifth is indicated by the sign + (III + ). 107. - This chord is not used for the present for harmonizing purposes, but will be considered later on under chromatic passing tones. The submediant is a major triad, the parallel of the subdominant. Note. The root of the submediant in major is frequently doubled, but in minor never if preceded or succeeded by the dominant. 108. The leading tone triad is also diminished, and coincides with the leading tone of the tonic major triad, being in fact the third, fifth, and seventh of a dominant seventh chord. Both diminished triads are more frequently used in their first inversions, usually with the third or fifth doubled, than in the fundamental position. HARMONY. 5» 109./ If^g^^^ etc. Ett f=F eI:I The root of the supertonic triad is more frequently doubled, how- ever, than that of the leading tone. The rules for the use of these chords are the same as in Lessons X, XI, XII, and XIII. EXERCISES TO LESSON XV. ■^ 71TJ T ^ F ^^ j— 1— ^ ' a tr" m=^ g-'-»- f Si^ d=i: 1^ -0 IF' • ^ S I II°8 VI IlP8 vn06 n s -a^ y^ i ^t -• •- ^^ 1=?^ i »z± Ei 1K 1 . Figured bass given, 6 6 6d 8 7 4 II ^ :ifife^ ^^¥^=^ -ri- -s?- D 6 6p IB agfc -z^ f 2. S6 6 6 7 4 7 6 4 S ^*^^^:^^ '^ 66 6 7 43 6 4 « -•— F — i^^^g^B I ffl U p=i 52 HARMONY. 3. Unfigured bass. i^ afefta^ P =g^-8=^ -si- |«^^3Ett 4-. Melody given. =1= r§Ei=r=P^^SgS a=:^^^=[:FHjp^:^=F=^^irpz^ •i- -■ — d — I — ^— I rpz=ir !S -^—=5- ?^=:f ili P=?= I :p=^»- :^=n: ^?=:ii=* -•-^- =t=t t^:=F :Ui.^ j— t- LESSON XVI. SUMMARY. We have now considered all the triads that it is possible to erect upon the various degrees of the major and minor scales, with their first and second inversions. The following table gives a summarized view of these triads in the key of F major and minor. The term rare is of course to be understood in a comparative sense. TABLE I. TABLE OF TRIADS OF THE MAJOR AND MINOR SCALES, WITH THEIR FIRST AND SECOND INVERSIONS. In F major, HARMONY. 53 i i4^ j=^— g i^»=^ Sii ^^1 ft^s^^=«^J=^^ ^^ I 1 I J— J— J- -F=¥=F ^ J 1^ — »^- ;F=f=F m6 iii§ 4 IV IV6 E^^^ -•• — • — ■ — # e^^ 3 — •- ^^- j=^i W=W=^ ^»=P=P= -^ ^ K- t— I u :^ t=t=t IV6 V V6 V| Rare. Rare. ^M=^ ^=i=^ -« — s — •- -^ — Jt — I — ^»- .._^_ ^« — ^ — ^ # — P- — ft- I -• — • — •- =P=fi=P= - i — r : i — r— r -x ^\ V Vl6 Vif 4 VII°6 VII°5 4 In /minor. Rare. 1 — r- 9iu5=| 1— t EE =t:=:t l6 1 IlO Rare. , ( Melodic forms.) i fi^^^^- n06 1,06 4 III+ IIIJ IIIJ 4 54 HARMONY. iii^i -^ ^ p j:t)»— I* — I*- S! ;^F^ liz^zut -l-i— U t=t=t IV IV6 IV? ft V6 V6 fii-H- Rare. Rare. ^^^^^^^^ -#-H«— I*- P P P - I aisa i P=F=?E^ i=i: ^ I I L VI VI6 VIS 4 VIlO VII06 VII06 4 It will be observed that all the above | chords are to be used in strict accordance with the rules of Lesson V. This table is to be played at the instrument in every major and mi- nor key, from the Roman numerals. The next table presents the available progression of each triad to the other triads of the scale. TABLE II. Prom I. Poor. » — h* -d p — I-* % I — • » — Vf S-F w P — I m =#=^ • J- -J -L 1^1 3^3^-i 18 — *^ — ^"*" l l^g r^ i=i=^i=^='=i=^=t|--! n-1— J- ^^ \ — i- s t—t -A-\ "N I V* I VIlO« s I II I 16 I III I VI6 HARMONY. 55 etc. I I I I leic. iCtC, ^^^^p^^^i »-5^.»Hi-l Never. etc j= =t=^= i^S^ I i=t :5^E^s I IV Va 1 ii6 1^1^ I VIl6 I V7 1 II16 Rare. I 16 4 m — F— I — f — f^-f — * — * — r * — p—t * — -^1 i 4_J — I. Ei=ei=t^^; J J-J^4 Only i=iE^=^a=l ^=;= p^ «— i«^N; • — ^ Never. .0. -0. .0. .0. .4. .0. r .0.9 .0. •• j^- .0. •-«-_(. ^ I VI I IV6 I 116 I V6 I V6 4 6 From II. Poor. I VII I 1116 4 Rare. msm -\ As passing- .cnord. J— I . 1 ^^^or« ^- l—j—t • — J— !-• — ih-F« — • — • — h* i l^fL^^I * i=p II 16 pSa? II III II V16 II IV II 11* ^ II va VljOJ 56 HARMONY. i g ^i ^ ^jg^ g^Eg^ gEgj T— r-1 r^ fe^ESEtt =F* :^= P iEi.^Eti^3:i l^^t^EEj II 9t li 1= It =t ^ ^11 V II Vt II IIl6 II 16 II VI II IV6 II In sequence. Rare. Il6 4 1 5^ 1^; /fr: s PNiif^f^ i -i—j- j— .1- -a — y— p«i=J|- ^^^^E ;i^S i=^^i=i ^- H^i^ 1^ -ii- |EEiEt|=S=^|=| --T- P^^ Si^i^_g • II V6 From III. Poor. II V6 II VII° II Illf 5 4 II I II Vl6 IV6 4 Rare. i T-T— r r^T-^-^ i ^^: S — J— '— S — J- .s=^g=2EE£ s 133 IPiUgS is E--^± rr4= V * r^ -r^r"i =?«±s -#- -# TT V ii g^ HI IV III ii6 III vii°;iii V2 III V III yi III xii6 III 18 4 m HARMONY. Never. 57 Rare. F?=?r None of these progressions z\=t- m m d= -t- i^ z^ ^i^ :irpi:| -. -a- are desirable. H- i P^ ^8=2: -•- T(- -•- V rr 9i=p= It ^ III VI III IV6 in ii6 in HfegfeHg ^ V6 III V6 III VIlO III III 6 6 4 m-H-r-i :tF/f Never. r^ — r -« — «- s?:;* m Rare. r— r -• F • — u- r— r-^r-r i=i- S^i lEtgE^ p — • — I m '■T~ i-l-i:^St=|EFt=zg± •- T^- -#- v • V "^ ^= li=ill p^ t vt= t:=F=tt=F III I III Vl6 III IV6 III V4 III II III VII°6 III V6 4 3 4 From IV. Rare. i %=^^i r — r Avoid. ^^i^^M ra W^L SI^IeI^ coriijc:] m ilE^^g^ t=:t:=t ^r^ ^ IV V IV V7 IV 1116 IV 16 IV IV8 IV VI IV K« 58 HARMONY. From V. In passing. Rare. iO-J U-rJ— » , I J- - j— , \^J 1 dzz^lidi a^-=d=i 3^S:i=^ i=S^i=|F^=- g: — ir^^ — «i»- 9t ^^?^ ^ i i V VI V IV6 V iiS V V6 Bad. V vii° V V6 V iilf 4 m ^—r zi= ■-± ^ ilE m ^— •^ F^fH^ fe2=f^ n =t =F VI V vi6 V IV? Bad. V4 V II V VII°6 V V5 1=^ 1^^1^iS^@ £ ^^= i=«=3: -^ J ^ — 1-* — a •^ — 5 — 1-« — • P ^_H_ i 1— ii^ — >-J S=|^^g^ i— 1- J— ^^^ir^ Jr^^ ^r^ m Never. 16 V III V vif 4 V V2 V IV V ii6 V viios * HARMONY. 59 From VI. Poor. In sequence squen nly. 1^ only, Poor. Better., 9 -- | I [-\-^ P^PPP^^^ 3=F ^J=F^4 -B m. ;i=H:or:ifti^^^: 4— J-^ •- •■ -#- -J- -•--•- -•- iS; "•■ "*■ • ^ • — 1«- ;t=t nt=t VI V6 VI Vg VI vii° VI 11x5 VI I VI vi6 4 VI IV6 4 ^^i^^^^g^^^g^^^ ^r^^ r- I I I ^1 J 1 i^& ^=|^^:| -• — j^ J ^ -M IS P^fg-J^^I^^^J ^TI^ ^ :4=*: 4— r ±=p: ^ lizzit: ^ VI II VI VIl06 VI V* VI V? VI 16 VI III VI VI? 3 4 4 Rare. ^1 ■F=^i — F-^ — F=L| — iM iSEEi P==S: iEI ■0 ^ m. '^^i tr^^i—^' -^ ^ — h i < l-S — • I ! J_L ^-i iF^^ 1= 4: VI IV VI Il6 VI vii°5vi V2 VI V VI V_vi ni6 VI I? 4 7 4 6o HARMONY. From vii°( only -« a r« an fc -i w » in sequences) up Never. r! • — rt f- r^f s^g^ J UJ. l^^gS=f=l3=t s=s^ ^=il p^' t=|z^=i^:J= =*=^='f ti=g=tj. S=^ i VUO I vhO VI ^1 VII<^IVf vn° II VII° VII°6 VI1° V6 VII°V4 4 4 3 Poor. i ^ Rare. ^H^^j^^i^^^ i i II J^E ltZ3£ I d=ri: t I §fcj=^f=Jd.j=|=tS=^JEEES3fE^;^=EjE5 s -N vil° 16 VII° III VII° Vlf VII° IV VII° V^ VII° Il6 VII° VI1°6 4 4 Poor. Bad. /5=r:3=q ^S¥^ Very poor. ^1 See 3C :t=fc i 1^ "S=*=±t i=jF ggE^ I Vt: 's vii** V vii° Vt vii° 11X8 VIlO i; VIlO VI VIl" IV6 VIjO Il6 4 4 HARMONY. 61 It is indispensable that the student should have both of these tables at his finger ends as well as in his head, in every major and minor key. They form in themselves alone a fundamental system of dia- tonic harmony which, if thoroughly mastered, will be of the greatest assistance in all future studies. Although these tables do not give the possible progression between inversions of chords, these are suffr ciently controlled by the principles heretofore given. This second table remains the same in the minor mode, omitting all progressions to and from the mediant triad (HI"*"). Like the ma- jor, it should be played in every key. After a careful stud}' of these tables it may be well to omit the progressions to all the | chords, excepting those of the tonic. The examples marked "rare," "poor," and "never" (always comparative terms) are improved by the use of other chords in harmonizing the soprano. LESSON XVII. DISPERSED HARMONY. (Open Position.) If the three upper voices of a chord, soprano, alto, and tenor, lie outside the compass of one octave, the harmony is usually said to be dispersed, or in open position.* (See Introduction.) This does not alter the original doubling of the intervals, however ; that is to say, when the root, third, or fifth is doubled in close position, it is also doubled in the corresponding position in open harmony. In short the difference between the two positions consists only in the inversion of the alto and tenor voices. Close. Open. Close. Open. Close. Open. Close. Open. -a es.— I 10. i W- 3^^ — ^J^V SI- 9t i From this it will be seen that the tenor of the close, becomes the alto of the open position. The alto being inverted into the lower octave forms the tenor, and vice versa. • There are some exceptions to this definition. It is given as the most practical general rule 62 HARMON y. The following table shows triad and seventh chords in every position and inversion, in close and open harmony. TABLE. Triad, 6th 4 7th i-t Open. ^J PP=l SEE!EE==| J=Sz i J — * — ^»— 1^ — » — g=|- * s=\ W^ Close. ^ » * ' m i 7th without 5th. IB p Open. I I i Bi-?= ^^^ ^:*^d ^ I •m m»- Close. I i M The open positions of the above chords are to be played in every key. The preference between close and open harmony in vocal music is decided by the compass of the voices. For instance : -s>- i i HARMONY. 63 This is equally favorable and practicable in both positions of the chord, as all the voices lie well within their own compass. I 12. f) Tenor too high. Mto and tenor too low. Tenor too high. Tenor too low. Tenor too high. Tenor too low. u ^ 1 J CJ t l(\ '^ n 1 \^) n n 1 «J ^ -&- V 'S ^ But this leads the tenor too high in close position, and, on the other hand, too low in open position. Well written four part harmony contains both open and close positions, according to the compass and effect of the voices, but it is safe to harmonize any soprano above i in open, and below that tone in close position. Exceptions may be made for special effects, but these may be safely deferred to a later period. It is by no means to be under- stood, however, that the inner voices are to skip from open to close position and vice versa whenever the soprano voice moves above or :^= The change from one to the other is always below i lYz combined with a smooth progression of the alto and tenor. Not. Thus. 1 13. It often happens that, in the case of a wide leap in the soprano or bass, the thirds and fifths of the principal triads are doubled temporarily. Third. Fi'";- =11=2= ■rf- i H HARMONY. The tenor may, if necessary, lie more than an octave above the bass, but the soprano and alto, or alto and tenor, should not move more than an octave apart. 1 15. Not. -^ S>- I The following position, however, is often found, especially for instruments, and is of very good effect. 1 16. ^ 1=1 The major scale may now be harmonized in open and close position "as follows : Cadence. r i 1 17 i 3 s? — 7i — gi — gy -&=t ^=^ -'gj— SJ- ■^- ei cJ g^ — ?^ — g'pg^ ■(S(- -19- I- -(gj- ei - A -^ -'g- a. -a- ^ -A o. =t Close. I I M f Open. ■ s^'g— s*- S Close. IV I IV 16 IV V4 I in IV 16 IV If V I 3 4 7 I The major scale in the bass may be harmonized as follows : Sequence. I 18. rt ■g- li^l -e- ^ W i^ngz - g i ai -«— fZ- -zsf- -«>— ^- -PT- 3E ^ ^ -g>- FfTFp a HARMONY. EXERCISES TO LESSON XVII i^^4^^M^H=i^^^^^=J^^^=^=J"^hT~H _ ,-•- -(2- -•-•- ^- .^ ^ — *-lJ _ RVf -f- I! LVhk-g-f4- k^^sH F ^hi-l-l ^^-W — ^ ^b-r-H+p ipl-^ lf^-f+~^ — ^^ ^— "" I . Figured bass given. 4 8 6 i I ^9igE :^=t -zsi- 2. 3 6 3 4 6 B 4 2 6 3 4 3 2 6, 6 3 3 6 668 6 377 S m ^1 i±± ^=i^ ^ ^ 1=F 3. Unfigiired bass given. 3 6 6 6 3 I !gi q=p=^ :^ ■< g ■ ' a; •- 4. Melody given. a^B?: ->«—•- ^^ i ;^Et? All the major scales are now to be harmonized at the pianoforte, in the soprano and bass voices, according to the models given above. These cadences are also to be transposed into every major key. 1 19. ■ cj g |— ;s | - -fi > aT ^- 'i9 C i g - -g^ ydE^E^di _a^_g- ii -^ S=^ 12. I g j g= i - ij -S3~S>—SI- .(2_ I 66 HARMONY. LESSON XVIII. DISPERSED HARMONY (Open Position) IN MINOR. The exercises for harmonizing in the open position in minor keys present no new difficulties. The interval of the augmented second between the sixth and seventh degrees, although much used in all forms of instrumental composition, is much better avoided, especially in the alto and tenor voices. m 120. \ I ^ ^J 0^^ ^^ +8 -fi>- ■»- I As a purely melodic interval it often has rare expressiveness. ^ 121. :^— t ■^ 'r^ i §g^JiE^^g g=^=^ Harmonize the harmonic minor scale in every key, ascending and descending, according to the following model : =d— J 4 - fe 122. i^ r * X ^ i±fe& rf V6 -^ IV6 :^ V11O6 I I ?^^ =P # -g- ■ -g- g - yi;'H^-f=| g ^ -^ ^ i i Is V7 I V| I I VIIO8 IV 16 IV HARMONY. 67 Review all previous work. Play all the exercises from the given soprano and bass, and transpose them, up and down, a half and •whole step. It will also be found useful to play the exercises of the first fifteen lessons in dispersed harmony. The student may also compose such exercises for himself, according to the given models. EXERCISE TO LESSON XVI 4i! i^ r I I H 1 . Figured bass given. 8 S 6S 4 S t=X s^ i s 3a: 1^ eg 4 6 3 eg 4 3 ^^^ ^=t 6 3 8 6 _ g^ __ Idz^ -■5^^- d • <^ ^- -sl — # 3. Unfigured bass. 5 eS ;b I?: ES -z^ ■glHg- -JK i 4-. Melody given. a=^ :P=?e: m^ I N« ^ I ViitT P^-lS" — •--^ t±^4=:^^ P -•+al- ^ = j=i£ r 1 F- t^ -^r^ d=rt: s^ * 68 HARMONY. Play in all minor keys. 123. ^^g-j^j^ Bufz^ -•--»- -^ i^^ ^^ V=Ft^ ■^- - ♦ ^ 1 — r =^ T— 1- I I I IV Il06 16 V7 I 4 f ^ -» •- i eJ i 5>" i.^. I i -g- LESSON XIX. THE DOMINANT NINTH IN MAJOR, AND ITS INVERSIONS. The chord of the dominant ninth is formed by adding a third to the seventh in its highest voice. In four-voiced harmony the original fifth is often omitted, the third at times, but the seventh never. The ninth of this chord (the third above the seventh) must not aopear less than nine degrees above the root. Such positions as 124. i -a- wm ^ =§s= ~e> :^r i etc. are therefore inadmissible. This chord being a dominant harmony like the dominant seventh, its resolution is to the tonic chord, the ninth descending parallel to the seventh. In this case the bass is HARMONY. 69 sometimes connected with the fifth of the tonic, forming a tonic | chord. 125. i 1^1 I II or 6 4 or 6 4 or 6 4 ^^^ :(^): ^(; P If the fifth is present in the chord, and lies belowthe ninth, it must ascend to the third of the tonic, to avoid the consecutive fifths which would result from parallel motion with the ninth. Not. Thus. 126. i -^"^rsr m •^ I Sometimes the ninth resolves before the rest of the chord, forming a dominant seventh, which is then resolved. J 27. zSi±=Sz ^ i I I This, however, can hardly be considered as a true chord of the ninth, being simply a dislocation of the root of the dominant seventh. Thisw^ill be considered later under the subject of Suspensions. (See Lesson LV.) The first and third inversions of this chord are the most practicable in four-part harmony, although the second may be used if the ninth lies in the upper voice. The fourth inversion is not used tor the reasons before stated. (Example 134.) Harsh. 128.< i Harsh. -e> r =S^ I 70 HARMONY. In the exercises of this lesson the ninth appears only in the soprano and tenor voices. The following resolution has been used by some modern composers : 129.< i 9^ ■g 1 I The major ninth is useful in harmonizing the major scale descend- ing, as it facilitates the progression from the seventh to the sixth degree. ISO. i w r- ^-4— s<- 4. Chant. mi m -lo- ^^=^ =t=t=p 4 -zg-f'g- -^ -s^- I EM! " S>~ g ' g^ i p=i«=F ^ M^=^ ^J=^ H 1^ -• — ^ It t:=^F=t * Seventh in the Bass. Play in all minor keys. i^^. 135. J g iS- -*- -(S- 3 S=g=X^ P -g. ij : ^ -1 -r I :g=t 74 HARMONY. , LESSON XXI. TH-E CHORD OF THE SEVENTH ON THE LEADING TONE. If we erect a chord of the seventh on the seventh degree of a major key, we shall find that its tones are identical with those of the dominant ninth chord considered in the preceding lessons, in fact such chords, from the fact that their tones find a common generator in the dominant, are practically dominant ninth chords with their roots omitted. They are in every respect so introduced and resolved. The positions and inversions of the leading tone seventh in major are as follows : 136. \ r t 1 1 & 9 1 / • • ((■ T" <3 r? ° a " eA> _ M^\j s n s o f J -Z7- 6 6 -8- f> • 9 • ^ m ^ • ^' « s * ■c i Harsh. ^— %^rA 10 w -ggr- -z^g- -g- ^•- ^ 5- 9- It will be seen that in all positions in which the third, originally the fifth of the dominant ninth, appears below the seventh, it ascends, doubling the third of the tonic triad. Parallel fifths would result from the fifth descending to the tonic, (see Example 137 ) therefore Rule i. The third, when below the seventh, must ascend to the third of the tonic, doubling its third. Otherwise it may ascend or descend. Note. This is of course the same rule which applies to the fifth of the dominant ninth chord in the preceding lesson. IS7. HARMONY: 75 :SS2 m ^i Rule 2. JVb intervals of this chord are doubled or omitted, I Not. 1SS. 1 Note. The third of this chord is sometimes resolved upward or downward to the fifth of the tonic, in order to avoid doubling its third (see Lessons XIX and XX), but this usage is not to be recommended for the present. At best It exchanges the doubled third for very audible concealed octaves. isg. I W i ^ Otherwise the treatment of this chord and its inversions is identical with that of the dominant major ninth. EXERCISES TO LESSON XXI. f^A- f a a zSz:^ f ^1-^ -zr -ei — ei ^ Z) ei W^ -fit -z;^ -s^ p^ •<=z — &- iM IB VII^ VII°« 76 HARMONY. 1 . Melody given. 7 ( g I l g; ^ ;B -?iTTT • ^ I g VIlO VII°4 VII°6 VII°8 VII°4 VII°8 VII°4 6 2 •-' VII°4' Vlios' 3 6 4., Ik. ?» H«-»- i;- j-^-^— r^ ^r zB±iz d^e:t i =t r-^W :Uit^ i i |-:i 5^4 =?c^ -»-^- :n m ^^T^- =?t? a •- ~«-^ Play in all major keys. 4 I40. i ^3^i-|^S^ ^i= Erf tz^: :a±g — ^ ^ *^^ (S^ IS :(= — T^f i=& 6 -S- -Gi- -^ -Sf I 1 I :^=z^f=p -fZ- gg^l I viio I I IV V I ^ii i s -si--c'- ^ g=t^r=:g± g— I) : "-r 4 3 9^ -«>- ^ -es- -*— :'5'7 -J L.^ .fi, — I 4 2 liD HARMONY. 77 LESSON XXII. THE CHORD OF THE DIMINISHED SEVENTH. The cliord of the diminished seventh so-called, or the dominant minor ninth, -without its root (vii'^q) is one of the most useful and valuable combinations with which we have to deal. Although having no existence, fundamentally speaking, in the major mode, it is yet freely introduced in that mode by flatting the minor seventh melodi- cally. 141. i P= ^ —a— sr ■!=? r^rr I It consists of a combination of three minor thirds, and contains no perfect consonance between the intervals of any position or inversion. It may therefore be approached by parallel motion with impunity. Correct 142. ^m. w ^^ I etc. i Its three positions with resolution in the key of c minor are as follows : Close. a. Open. a. 143. i W ^ -^'—^ i i_^ &-t?- ^ ' 1 ^ %t ^^^B At a the third ascends to the third of the tonic, as in the corre- , spending chord in the major mode. The descent to the fifth of the tonic is also occasionally used. The augmented second between the 78 HARMONY sixth and seventh degrees is to be avoided in the innet voices. The rules for the introduction and resolution of the diminished seventh are the same as given in the preceding lesson. EXERCISES TO LESSON XXII. i ^ — ^o J. A id. EES :^i=S-:^ ■m-A-. :^ S ^±s!--g=b ^— •-^ j^b^=g: Jwa L a^i^i -(ffi- g= EfcttiS :^=»: I^SEt it=F: vn°o 1 , Melody given. VII°o n=^=^r^-:¥- i^ M-4^^l^\ -s—^ S5 ::±3t m 2. 1^ ^ 5^*^ -\ — H~+-i :^^ll i5z^3 I -^ ^ J $!e&™e^e?e -•— ^- ff^ :S=± ^t i 1 - g— g=p^ I p^^ .■g — gi- 42_< i5_ -ffl. Play in every minor key. 144. i^^^i^^^^ s^ -.jzrppr ^ I vn§o I IV I* Vt I HARMONY. 79 i k=i=^ i ^-i—k \ t~r i -0- -^ ^H EEE LESSON XXIII. THE INVERSIONS OF THE DIMINISHED SEVENTH. The three inversions of the diniinished seventh are used with as much freedom as the fundamental position. The peculiarities of the resolution of each inversion are as follows : First — Since the third, in the first inversion, is in the bass, and therefore necessarily below the seventh, it must ascend to the third of the tonic triad, forming a chord of the sixth with the third doubled. (Example 145, a.) 145. IE Efc|gi=2=|siz E | El^=:r^g-:-|: ^^^— g zj ] §±^ EE i ^ I Sometimes, however, the fifth also ascends, forming a chord of the sixth with the fifth doubled. (Example 145, 3.) This happens only when it is below the root ; otherwise consecutive fifths [result. (Example 145, c.) Second — The fifth of the chord, when in the bass, may descend, forming a chord of the sixth with the third or root doubled (Example 146, a), or ascend to the fifth of the tonic, forming a | chord. (Ex- ample 146, 3.) 146. 8o HARMONY. The first resolution given at b is bad on account of the progres- sion of the two fourths in the outer voices. An exceptional pro- gression of the vii°* is seen at Example 146, c, in which the first fifth is imperfect. It is not recommended, however. The second is much better. Third — Since the seventh descends to the fifth of the tonic triad, when in the bass it must form a 5 chord with the root or third doubled. 147.^ EE EE zszz'^- Fossible. etc. -♦- -^- . / ^ I As in all preceding lessons, the different positions of all these inversions may be exchanged for one another before resolving, avoid- ing, however, all progressions of augmented seconds, augmented fourths, diminished thirds, etc. Play all the inversions of the diminished seventh chord in eveiy minor key. EXERCISES TO LESSON XXIII. EE mx %- 6 r 6D 6 4l] 3 3 ia •m :J=i ^ Efc ■#-ife T fe 4d 6 2 4 t" i. 7| HARMONY. 8l ^^ 1 . Figuted bass given. •8 41 3 6 3 J (^ — r«'- « i« -^- :^3 6 7 &2^ I I I E^ ^s 4S 63 3 3 3 6 7 m- 6 6 4 6 7 6 4 6 4t7386 2$4| i»- "fg 1 --I 1- S I S F -f^ --H 3. Unfi^rured Bass. 3 i ^mr=f ^ Efet iS"-^ i i 4. Melody given. I & 1=^ S^ 1^ ^^ ^ M zSzM: (2- i^ 1^ =F The I| chord in the cadence may be approached by the vii°4 or * instead of by the subdominant chord, viz : Or better. Another position. 148. I dt SE -^ -g- -s^ y-|^ | d =H= l lil t :a:-2!»- ^^-^^ :^ -S>--f2- I P ^^ ^ ^ -s--^^ ^ fe£S 3'isH-)=- Play in every key. i LESSON XXIV. THE CHORD OF THE DIMINISHED SEVENTH. (Continued.) As before stated the diminished seventh chord is freely used in the major mode as a melodic alteration, and since it only differs from the dominant seventh by one semitone, as a matter of scale degrees. _ J — s> G> f % hgc? it follows that the second, fourth, and seventh, vhi which are common to both, may be harmonized as well by one as • Changes of position are indicated by the order of the figures. 82 HARMONY, the other. In case of change of position by repetition, the figures indicate the intervals in the upper voice. EXERCISES TO LESSON XXIV. ^S -SA ^ -4—tr -zi- m=^ ^ 1=F 170 VIlO i 1 j=it ij^ p -zi- m zJz ib i i^^ jfi. 4 I f— V VII^o VIIOq 1 . Figured bass given. 6 4 8 7b 4 3 6 3 6 ■be 6 7b 6 7 -ft— g- S I ^4= 1 2. Open position, g 4 4 8 7b5b6682 63 •(= — s>- 7b 8 6 7 izfc:2: ■<= — «■- ^=^ li2E -p — #— ^- :t=^ -gj — g- 3. gife 7U -,_^_ 6 51] 7tl 3 6 4 6 4 sq 6 H K=fc 4. 4 4 3 a*? 6 6 3 1 4^ 8 6 3b i=^ 3b 3b 3 S 4 6 66 S? # I * I * -,— P F^£S 7 4 7b HARMONY, 83 5. Open position. 6 3i> SB 6 7b 6l> 4 6 ^±^^ i »^t5z:^ trt:^ 6. Melody given. ^^ ■3=^f=PTP=i: ^gpa 3t:^ =p=r=^ 7. i ^ :^=* i Sf »^4 I I r ! i & =?2=if t^ y-r IB =t:t -»-■ =f=i=^ ■(=—«'- :t=f LESSON XXV. MODULATION. The connection of chords belonging to different keys by means of harmonies common to both is called modulation. As different keys are said to be related, in proportion to the number of tones in com- mon between them, it follows that those of the dominant and sub- dominant, with their parallels, are the nearest in tonality to any given tonic. For instance : Scale ofC— CDEFGABC. Scale ofG— GABCDEF#G. Scale of F — F G A Bl? C D E F. Every tone of G major exists in C major except Fj}. Every tone of F major exists in C major except Bt>. 84 HARMONY. These chords as tonics coincide with the triads erected on the II, III, IV, V, and VI degrees of the scale, and when a simple melody forsakes its own key temporarily, it usually does not stray further away than to one of these related keys. Note. Some theorists define related keys as those whose signatures differ by not more than one accidental. The mere succession of chords of different keys does not necessarily constitute a modulation. In fact, it is entirely possible for any two major tonic chords to succeed one another in such a manner as to be mutually related to the same key. For example : (a.) {i.) 149. SE * i ^i C:I D^:l C:I B:l I /:V VI e-yi V The tonic triad of C major and the tonic triad of Di? major are also the dominant and submediant of the key of f minor. (Example 149, a.) The tonic triad of C major and the tonic triad of B major are also the submediant and dominant of the key of e minor. (Example 149, b.) All the other possible successions of major tonic triads will be found demonstrated in Lesson LII. A change of tonality convincing to the ear must be accomplished b)' means of a connecting harmony which contains a tone or tones foreign to the original key and belonging to the new one. The chords most commonly used for this purpose are the dominant and diminished seventh, and for this reason : The major and minor triads and seventh chords on every degi ee of the scale excepting the fifth and seventh (seventh of minor only) are ambiguous in tonality, being related to more than one key. u\ ISO. i -IS>- g=s=tg^ =i; ^ I etc. C:ii %=i^^^^^ \, " |e d:i B^: F: VI C:V c:viiO„ always always I HARMONY. 85 Such triads are supertonics, mediants, dominants, subdominants, and submediants, according to their relation to a given tonic. The combination of major triad and minor seventh (V_) , or diminished triad with diminished seventh (vii°7q), however, is never found except on the dominant of the major and minor, and on the seventh of the minor, respectively. (See Example 150, a.) Aa.) {6.} 151. m X I § C:l V C: I <7:V I 7 If now the progression at a is given the ear recognizes the G triad as the dominant of the key of C. But if we interpose the chord x in Example 6, the G triad becomes a new tonic preceded by its dominant. If, then, the change of key is to be made permanent, the modula- tion must be followed by a complete cadence in the new key, in any of the forms which reiterate its fundamental harmonies. A formula for a permanent change of key might therefore be as follows : Let X represent the key to be modulated into. Let a represent the key to be modulated from. Original key. Modulating cliord and resolution. Cadence in new key. IV If V I 4 a: 1 AT : VII 70 or V_ I Since any given tonic may be succeeded by the dominant or diminished seventh of any other key (in one way or another), it follows that the above formula stands for a complete system of modulation from any kej- to all the others. For this lesson, however, we will confine cur work to the modu- lation from tonic to dominant, using both the dominant and diminished sevenths as modulating chords. (a.)- {i.) (c.) (d.) 152. -2^- i-r— 1^ - — 7^ — »- -gR i --k Ei t^ ^ ztin i 11 C: \GNi I 86 HARMONY. At «, b, c, and d the root of the tonic chord of C major becomes the seventh of the dominant qhord of G. Any inversion of the dominant seventh may be used. {e.) {/.) (^.) (h.) Bad. 153.< iE=^R -^-61 ^—s. ^ppSS^^P rife si- i 3»- ^m C: I C: vil§o At e,f, and ^ the root of the tonic chord of C major becomes the fifth of the chord on the seventh degree (diminished seventh chord) in the key of G. The position at h is not possible on account of the false relation E-Eb between the baas and tenor. Note. The false relation, or unhartnonic cross relation, is a contradictory suc- cession of harmony. It consists in progressing to any tone which is chromatically altered, when that tone is preceded, in some other voice, by its octave without the chromatic alteration. The following tables give the modulation from the tonic to the dominant of C major, with every position and inversion of the V_ and vii'70 chords. TABLE I. MODULATIONS TO THE DOMINANT, WITH THE V'. ^r i^S With 8 6 I^P^^ \^. '■¥ i ^Sr^^-griS-- =*=^: -25*- f=r^^-^-#^ -• — n — J — « * — ^ s^- =5^:p < £> \ ' s ^ \ s < ^ I ^ " ^ 1 P W~P' r^ Best. rti i =t=T:j i^S^ ■i^5=^ ^i: atat itzt i9E F? 88 HARMONY. Poor. i^pi^^st^i^B ^ 2t -(=- : pfar:# I ^iz^fep=^a ip= ^F=F ;F3 feN= Note. As before explained, the diminished seventh belongs strictly to the minor mode only, but its use as a melodic form in major is so common and familiar, even in the most unpretentious compositions, and its usefulness so great as a medium for modulation, that the student may well become familiar with it at this point. 1 . Melody given. 2. i Sg^agagjgi >peE»&H ^ W^ 3:JttE«-zt#t G:V7 (7:vil9o i :]=ld=l4 m ^^ ■f2-l«- tlti^tt iz±t 5=^3^ :2^ m •id F:Y7 e-.vii^o ^ h =ts: m=^^ i— P—f—r ■.^z=M=^=Jt itat f-b^3^- B-.y Transpose to every key. 154. -J 4- ^l=SS ■^ — tfg- m :g=n^i S — g— -'a— tt a— l- g- ■^# 9i -5i- -K ;i i^r ^= ii a=BgJ i=^ I =4= ^^•^ ^_ All the other positions of this modulation should also be thoroughly- learned in every key at the pianoforte. HARMONY. 89 LESSON XXVI. THE MODULATION TO THE SUBDOMINANT. The subdominant (or dominant below the tonic) is as nearly re- lated (as a key) to the tonic as is the dominant above the tonic. In modulating to this key through its dominant seventh, the root, third, and fifth of the tonic become the root, third, and fifth of the domi- nant seventh in the new key, the minor seventh being merely added to the chord. 155. i j- I'iE^ ^^i ^-<9— S-- J- 5£ -5— ua i 1 — r- st- -^=^t O.l FN-, I In the modulation through the diminished seventh the position at a is bad on account of the augmented second in the tenor, (a.) (*.) ic.) (rf.) 156. ^i =17=^- l^ -d: -J- ^— > — 12 — u :^=5i i ■zi- C:\F:v\\%o ^i This may be avoided by approaching the seventh from the root of the tonic as at b, c, and d, whereby the third of C: I becomes the root of J^: V1I70, and the fifth of C: I becomes the third of I*': V1170. Transpose into all keys. ^^^^ :r ^=i: IF-.Vi I 9° HARMONY. i ^=d? S^EEiS^SEEfe^feife %=^ ^a: f^p^^i-^^^!^S^ w- 5:=rt -^ -•- =^=:t :^=|i- I/-:vil?o I II im tr— •■ ±=:d=::^= I :ii= :gi= The following tables give the modulation from the tonic to the subdominant of C major, with every position and inversion of the V_ and VII70 chords. TABLE I. MODULATION TO THE IV, WITH V, CHORD. ^H :t3|=^ It 'f=^ ^ i-i— 1- ^z P^ rg: rSzzftn^ .izSz^ I 7b pu^^r "--"^r^i^W^: -»i— »i- ^ 53^ :t3t -•- -1©- :(i=t :«= int -J-J-J I ^ ^^^ ;f53E5^3:??^Ea -! ! I :fiSt^:^Ej^ ^^=f=gE^£ :^E ^ 3=F= V^-, ^ * Undesirable position. HARMONY. ^p=ge fe^ W^ i^ z^i ^^^i>^r 4 3b -*- -^- -»■ i=s=±j I I * g - :t= H«-^- =P^-=F^ 1^=^^ :^ :*=* :^=t: -5»- E ^ i -l—\-,-L-4 'M=^ ■^—M — g^ 3= f=f=^«=^JF^^^S-S-^-^-^ "P==pc :t^3 * rj I p b » L I * b J=g W a J ^^-^^^p TABLE II. WITH THE VII 5o CHORD. ^^^^^P^ l5*5 7b Bb ^ ^=T^ 2|IL_s.- m '■-^ ^ 5F F?^ ilti^ 3^ :^- :&: ^"^^ -s!-r| a- -_gri?*» W- t± It 6 Bb 3b ^Si. -#-!/» -fi.. -V -b*-:;?!- i^ ;=E I I i ^ ^-t^j- j=i-^H=j^ p^ ^ If I h w yi:^:=r^^ :g^*=^:^: -T^- 6 4 3b *I2F 1 I -«>- :^=^ S W J l .^ -S^ i=& '»— s<- 53= -gi- ■«t5i=E I Undesirable position. 92 HARMONY. i^^^^i t=t IgB :^& ^^-^^^^^'^^ 5-^:1 1^ 2 ^ . l/ l I . f -t ^f ^ I I . ^^3 p^ :^3:tfti^ EXERCISES TO LESSON XXVI. is 1 ■ Melody given. Modulating tone in soprano. :p=5i :5i= IB 'f — ?^ ^ tt^ ^f^- ^^^ 3. Modulating tone in alto. 4-. In tenor. -i&— i 4=t 5. In bass. I I^ 3t=3t i i3: (t)- -?^ -(S^-Lgy LESSON XXVII. FORM OF MODULATION BETWEEN THE TONIC, DOMINANT, AND SUBDOMINANT CHORDS. The two modulations, from tonic to dominant and from dominant to tonic, may now be combined, forming a temporary modulation from the tonic to dominant and return, or from the tonic to the sub- dominant and return, since the relation of tonic to subdominant is the same as that of dominant to tonic, and that of tonic to dominant the * Undesirable position. HARMONY. 93 same as that of subdominant to tonic. Such a combination would be as follows : Cadence. 157. Play in every major and minor key, using this formula. EXERCISES TO LESSON XXVII. 1. Soprano given. 2. fc ^ ifcME^^ J-tu-« 1 3 :^ES da SE^JgS^ G:V7 I C:V7 -£:I':V7 3. |g^%j.f^S5sa^^Hm^Bip .5b:V7 .ffl':V7 £'':V7 ^^ -fghH-g ^^ ^^-4^ Hi »^- -•-Sci- . Chant. # . ij J r^ :|t =^=r iF =^=^ -« =^- 94 HARMONY. LESSON XXVIII. MODULATION FROM A MAJOR KEY TO ITS PARALLEL MINOR. In modulating from a given major key to its parallel minor through the dominant seventh, the third of the original tonic triad becomes the root of the new dominant seventh chord. 158 ^^ Third. Root. I This connecting tone is not utilized in case the complete dominant seventh is used in the fundamental position. 159. i E^ s i The progression from the original tonic to the diminished seventh of the new key may be by either contrary or parallel motion. (See Lesson XXII.) 160. ^^^i etc. IS * Efe * i=E^ ^ r The bass progression to the root of the diminished seventh at a is not to be recommended, as it involves either an augmented fifth or a HARMONY. 95 diminished fourth. This is true also when the first inversion of the dominant seventh (V|) is used. 1 161., I gJ P Both of these progressions may be disguised by using the modu- lating tone as a chromatic passing tone from the root of the original dominant. i 162. -^ 1 P^ ^ I This properly belongs to the subject of chromatic alterations, howrever, which will be considered later. The available positions of this modulation are here given : TABLE. With V7 Poor. Better. With second inversion. i ^ ^ f Usm^m^ *m w*^ '•^ s :i--^ .J-J — u ^ '• «> j J ii=t=;^ t=t *—^ 96 HARMONY. With third inversion. 7=1. — ^-•-?5l.r.r d-itd- ^ ■iftSr^57-^=t5=§i:^or: :tl8=il.or^dlJ=J_5U*:^ ;^l 4S 2. ^^^, ;^3- z=t=*z3t ititi^z - ^-* ^ :^d |g > ^ With viio _ il^^iPi^MS^ 6 -ritf -.--^ip P^ Ei^ -g- :pE=fe=^ J: --tai:--'^ -•--J--S'- -•--r-^ r r 3 *=t s ?% 9- P^ s=t •"-s>- ^Wi i I ^=^J^^E±J^.^S*p^ 2Jt ■J= ^t^=i= S=^ I 9^ ^=E EEE rt: ^g^ • Weak position. This table is to be transposed into every key at the pianoforte. HARMONY. EXERCISES TO LESSON XXVIII. 97 Either the dominant or diminished seventh may be used as a modulating chord. 1. 2. i : 2_r ? f\ ^r { ^r=^ i :p=l=iT: I ^ #2= *EEf^=^ )=^ 4. ^S 1^^ -*-^ -'S'-z^ s 5. i I ^ f^^^sE^ ^E5 =#t LESSON XXIX. MODULATION FROM A MINOR KEY TO ITS PARALLEL MAJOR. The modulation from a given minor key to its parallel major by means of the dominant seventh involves progression by contrai-y motion. The different positions are as follows : TABLE I. (a.) With the V7. I. 2. ? j^^ J_l ' • , 6- -s^ -0- -ft- 1^ ^b^^ i^ *-g- =t=S- It^ ^1^ ^r-r 98 HARMONY. W. With the |. „ Bad. Bad. i i ■*■ -#-3- ' -p- i!i^ -•- -d-^ f-Tf- EE (c.) With the 1 I. 2 If the third of the dominant seventh chord in the fundamental position ascends while the fifth descends, a tonic chord with three roots is formed, which is not a strong position. (See Table I, a, i, 4, 5.) This is avoided (see Table I, Oj 2, 3, 6 ) by allowing the leading tone to descend — not being in the upper voice. The consecutive fifths at a, 2, 3, and b, 1,4, 5, and 6, etc., are correct on account of the second one being imperfect, and of the contrary motion in the bass, and above all from the fact that* they are in the inner voices. Inversely the consecutive fifths at b, 2 and 3 are to be avoided for the same reason, even though the second one is imperfect. The third inversion of the dominant seventh (V| ) is not useful in this modulation on account of the rough skip to the seventh in the bass. 163. \f EE 1=« -^ f^^^ ;n Bi-t?= ^ -t^^- ■x=t t HARMONY. 99 In all the other positions the seventh enters diatonically. The positions of the diminished seventh chord in this modulation are as follows : W I. TABLE II. 3- m in-4- -J , I I I 1,1 Efc -W---S- ^ ^ rr avoid. ^ :0 -^- ^t^^ 3^ f^ ^ r=f=t= tt <=- («■) ^fcit :P 3t f^it "zSf^Jf^"^^ • 4 < ^ -it—^ T^ -^ -g- 6 bs -*^-^ ^P £: ^ i^ S (/.) i e ^ i^; $y-lg:>=^"i:^ S ori:^< j=Jorzt • ^ g-g-lS — '-t-J-^ ■*t:^ -7^ -zi- -z?- 4 173 ?=•= »b»-. 9i.fc=fe -!ir ^-- Efc^ ^J r ^ ^M " ^ ^ I !LjLJ I p-»'-^ ±p|cp I^ I -± — ^ — <9 — ■-•■ (§1 — g — j— g: 3E^E^ :i-^«^ r-r-T ♦ ^^ r-r I ^ ' n I n I 1 ^^ tfetr^ ■ Weak position. lOO HARMONY. The consecutive fifths at d, 2 and 3 are to be avoided for the same reason as at 3, 2 and 3. The third inversion of the diminished seventh (vii°|) resolves to the tonic | chord with the third doubled (see Example g, i, 3, 6), or with the root doubled {g, 2, 4* 5). These two tables are to be transposed into all keys. EXERCISES TO LESSON XXIX. gig^i^iH^^p^^g^ g^^i^gi^^^^ 4. =6d?.^ — l^jzrti-^ L_| g -r /t_, I |_L 4 W- -s^-^ m s I -»l — •- LESSON XXX. PRACTICAL MODULATIONS RESULTING FROM THE PRECEDING LESSONS. The modulations mentioned in Lessons XXVIII and XXIX may now be combined, forming a modulating phrase in major, thus : I : modulation to vi : modulation to I and cadence. Or in minor, thus : I : modulation to III : modulation to i and cadence. In other words, from a major tonic to its parallel minor and return, or from a HARMONY. lOI minor tonic to its parallel major and return. Such modulating phrases might be as follows : 164. 1 ^ — fS — 1 -1 M 1 — 1 l-i r-J 1 1 y^^^ \r — ^ ^^=s=^ — 1 -J — -^ — J hH m «> ^ — 1 -» — = -tI — ^- a , g-,. :g> ^ S -^ -f- -^ -^ ^ -^ -^ I i ^ l^3E 4 f • J— r r~t f--^ -• — ^ i js ai^ig^ Ttiz^ *Z±3t I'-'-id- -•-4- i 5. ^-fr^- =l?=t -(=2— • Itti -(«—•- I02 HARMONY. LESSON XXXI. MODULATION FROM A MAJOR KEY TO THE PARALLEL MINOR OF ITS DOMINANT. This modulation presents a slight difficulty in the progression from the original tonic to the dominant seventh chord, i.e., if the ba.ss progresses from the root of the one to that of the other, it compels either an augmented second or consecutive octave in some other voice ; +2, "^ tr — - m M 165.< '^- therefore Rule. When the root of the tonic triad in the bass progresses to the root of the modulating dominant seventh, double the third of the tonic (see Table I, i, 2, 3, 4, 5, 6), or in other words, ap- proach the third of the dominant seventh from, the third of the tonic, never from the root. Consecutive fifths also result from parallel motion of the root and fifth of the tonic to the root and fifth of the dominant seventh, when the fifth is above the root. 166. -i^g 2 9t I i For this reason, as well as on account of the abrupt skip to the dissonance of the seventh (as in Lesson XXIX), it is better not to use the third inversion in this modulation. HARMONY. 103 The available positions of this modulation through the dominant seventh are as follows : TABLE I. la.) I. 2. ^^^^p^^^:^l ^S^i 1^ Tph -iih"-»- rri ppB^S stut; 3t=it 7 j=z^i| iU ^ _|_^_^ zg=|=i: ^,^__|- q=q;!::!1 rq=:F J=zj^ P ^—C- ■•- ?tt^-p «=FE=t=t=t 9^ :^=^: t=^= eeS^^^P^^ :£Ft i E^ 3^ (..)!• W=#i: ■«" — iSh ]=4=T -^5^^- -^- -s^ "t^r -i^ >^ or ^ ^E=f« -l«— ,l=- 9t =t3 -g- =ir^ ffs( I' 3 ^y 6. -•- -& St ^Pi^fe(^)=-^^^^^W^ .-f-J^ At c an inversion of the intervals in the bass, while not forbidden, would be much less melodious, ^ I04 HARMONY. The modulation through the diminished seventh chord is somewhat smoother than that through the dominant seventh, because the root of the original tonic becomes the seventh of the modulating chord. The free upward progression, of the third at Table II, _/", 3, 4, 6, is particularly strong. The positions are as follows : (d.) TABLE -M»^=- & ^ -^ — \-'* — P—^ — js' — hi — jt ^=^=^#i **^ V^ IS « 0- -25»- ^ i -kk ^^ S=P- :t#=t ^ fel -i— J- («.) L tita - IZZlt - #— ^- fc^3fcSi^5^i^|EE* s^ •- -s- w^m^^^ 13 ^^^ -r (/•)i. ^^S^^^ ri: ^ d=:4 :a-Jr -iH i^ :^; f»^ i=^ -z*- :^ P^^ f=^ ^^^ •#- -IK P ^^^^P^^^ t=i=± s^ •- -^ g^ ^F=f 4* 2t HARMONY. 105 Poor. ^ I ^t=^- #j^#H=^M=-^i=# =p=^ ^ 1^ al- Si=^: ftf3§=3t=t Transpose into every key. EXERCISES TO LESSON XXXI. 1. -r-r-f i I ^: -l=2~ ■f2- ^ ^2 12- :t t=f-^^=^ nt= «— t^ i .« 't fij-f • i\f-^ I - ti »-; Pi-I 1- I P E EEia :t3tzt:tt 3. ;^^^^^gE^ S :S^f tii -*-»-=^ ^Q: i=t :tzt ^^^l^^gg^^^ rfe?^ I W o rg =g: ^J^g:z:g=^gHJ ; LESSON XXXII. MODULATION FROM A MINOR KEY TO THE SUBDOMINANT OF ITS PARALLEL MAJOR. This smooth change of key (the reverse of the preceding one) presents no difficulty, as two tones are common between the tonic triad and the dominant seventh, and one between the tonic triad and the diminished seventh. io6 HARMONY. The third and fifth of the original tonic triad become the root and third of the new dominant seventh, and the fifth of the original tonic triad becomes the root of the diminished seventh. The positions and inversions, all of which are entirely practicable, are as follows : TABLE I. 7b ^-V :s^= i^ :fEE^ =P=^: i^E t=^=h^t=r^ i (*.) ^=Fd=5! ^Z^t^ZII^t^ feliidzH^^ =fc=?=fe= S=^= S=^= -i=^->- "-5=:t 6 -12^ :t=P 4-J — I i 9iife :g=^ :^: 3 EE -N=4 zJ-h ' J ;:>■ rr^-* rr -J- -j^ -«- I 1^ I ?E J-J— J- gs ts - sJ- I^ - •—si- f- id.) Bt -J^ ^f^Eg j -z^ •— »— p^— -•— •— ^— -•— •— £q — d— •■ [iteSSE ^^ st - »H^*^- ?^ p^ i HARMONY. 107 The positions and inversions with the diminished seventh are as follows : TABLE II. ("•) I. 2. The weak positions of the tonic triad in Table II, a, i and 2, b, I, c, I, etc, are caused by the necessary ascent of the third of the chord of the diminished seventh which lies below the seventh. This is avoided by the free leading at c, 5 and 6. The third inversion of the chord of the diminished seventh is not practicable in this modulation, as it involves the very unmelodious bass progression of a diminished fourth upward, or an augmented fifth downward. No interval is perhaps more difficult for the singer. p^g^^n 167. is^S VX- =5i= eB io8 1 . Chant. HARMONY. EXERCISES TO LESSON XXXII. lEE j^^|g=|: 3=t:^ i=t I \s> Mz- < => eJ ^ fc I -z^- .^ — -g-^ 3. .Sb: I IV IV6 I II IB^p T I M f^^f ^gl^EfcS I ra^fei^^^i^^^^a G:I|V7l LESSON XXXIII, MODULATION FROM A MINOR KEY TO THE SUBDOMI- NANT OF ITS PARALLEL MAJOR. (Continued.) The combination of the two modulations given in Lessons XXXI and XXXII gives us the successions : with major tonic, I : modula- tion to III : modulation to I ; with minor tonic, i : modulation to VI : modulation to i. Such a modulating phrase with a plagal cadence might be as follows : lea ^^ E3: ;a i "0^=^ I C: I z^-ip -f^- =B?i E^ e: Vt C:V« ^ l@ HARMONY. And with an authentic cadence ^^U-J-l lhEi 109 S fe!?r=biS: ^^ - i^ ■27" Ifl -<2- C:V7 I e■N^ Harmonize the following exercises in like manner, using either dominant or diminished sevenths with their inversions. EXERCISES TO LESSON XXXIII. 1. i •:5=?= ^^^^^^m^\ 4 I r' ^si— t 2. e? pltyTZtJJ -(2-rS'- 1^1 ^ ^SSl % -(=- -^=^ EiEEE :^ ^faj^^i pgfeg ^=rs: -^—d-^ LESSON XXXIV. MODULATION FROM A MAJOR KEY TO THE PARALLEL MINOR OF ITS SUBDOMINANT. When the dominant seventh is used to effect this change of key the third of the tonic triad becomes the fifth of the dominant seventh chord, the fifth of the tonic triad becomes the seventh of the no HARMONY. dominant seventh chord, and the root is raised a semitone to become the third of the dominant seventh chord, and in order to avoid a false relation this must also take place in whichever voice it originally exists. It is for this reason that b, 5 and 6, is to be avoided. (See Table I.) Note, To be sure, this example is not strictly a false relation, but it has the effect of one on account of the skip in the soprano voice. When the diminished seventh chord is used, the third and fifth of the tonic triad remain the third and fifth of the diminished seventh chord. One root (when doubled) is raised to become the root of the diminished seventh chord, while the other descends a whole tone, becoming the diminished seventh of the chord : or the seventh may enter freely if all the parts move in parallel motion, as stated in Lesson XXIL The consecutive fifths in Example 169, a, axe quite correct. 169. (i.) t^ -s- I etc. zj^ ^t=4 -zji- ¥ The third is doubled in the resolution of the seventh chord, and sometimes the root. (See Table II.) Transpose these tables into all keys at the pianoforte. TABLE 3- 6. ^1 U) I. HARMONY. 3'Poor. 4- Better. Ill -M JE^ I i=r- tti ^ ^I -«1 *U gi - ni=^ fcz^ -J-tf-»- -iS>- 6f 4 ■•=s«= -•-s-#- i.-2^ -fl-^* a .j *lf^ s m 9i=^ :?c^= w.) I- 2. i ^CI^ d=i: J^J-4 M P i^r:^: -J-tf-«- ■ 6 4S 2 f= ^^ ^iSLt-a'z "-*f =«^^^ ^i^ ilriEiS =st • * gj 'If-^ f-^r-t -J-i- I -* — I — h- > -l— u *=\=r -•— zJ- rr ■•— s^ -•— s*- zi-^f— •-=*- a, I. Dominant seventh chord, either complete or incomplete. 6, 5 and 6. See note, page no. c, 3. The only way to raise the root in the same voice. d, I. The second progression is the better. The skip to the seventh in the bass is excused by the lack of motion in other parts. (.1.) TABLE II. 3- 4- 6. E^ lorj: :p=P= ^^^ E^^ -M-r^ S=S=^ -•l?-*^: ■*^ -gg^ ^ ^^S ^^i^^t^f^^^^ ^^#^^^p^ 6S 617 IS .-fit: ^Iz >- 112 i 6. tl^^^^rT^W^n^ -.-i-.- -^ :p±Cz;t: ^^^^mm ■-td- 9?=^ It::^ *-&t=t »=^r r a, I and 2. Doubles either root or third. a, 4-6. Doubles the third. 6, 6. The fifth, being below the root, may ascend. c, 2. The second progression is useful. c, 5. The consecutive fifths are allowed in the inner voices. d, 4. The free ascent of the third avoids a poor position of the new tonic. All the other positions at d are weak, but unavoidable. The exercises to be harmonized contain no other positions than those given. Either the V7 or VII7Q may be used. EXERCISES TO LESSON XXXIV. i m i m ^ PE?^3S t=^ i^^^^^^^ WEES: iS^ M HARMONY. I'3 i^^ m^ =P=?c: =t: it^it eB The last exercise may be written freely for the pianoforte with parts doubled. LESSON XXXV. MODULATION FROM A MINOR KEY TO THE DOMINANT OF ITS PARALLEL MAJOR. This modulation is the reverse of the preceding one. In connect- ing the tonic triad with the new dominant seventh, the root and third of the tonic triad become the fifth and seventh of the dominant seventh chord. (See Table I.) In connecting the tonic triad with the new^ diminished seventh chord, the root and third of the tonic triad become the third and fifth of the diminished seventh chord, and the fifth is chromatically lowered, becoming the diminished seventh of the modulating chord. As in the preceding lesson, this must take place in the same voice in which the fifth is already present. (See Table II.) (a.) 2. TABLE I. 3- 4- 6. i rc_C-^-H-4r^;^n"T ^ j-J^J-M =^^ i i^ f^^-i- ^ ^2fj*=^r^' ^s -st- -(=- 4-^ t g±ife ^ S W5= Efc (*•) i '^ (f a i g gfe ^zna p5S^33^53a» • •* ^ *=^ 6 6 ^ ^^=^"^i"^r^^ =id= i^ 5^3= gs= y-^Mfi ^^^="'^f^J=^=p^ifea^ 114 HARMONY. ^^^^#1 *s-n&^ rr m '■F^ ^ *=ti 6. -TShsr-*- -*' -zi- -•-zir -#--3!- j-f-;g- '- t=^f^f- ^Ff p:^S^- =t=l= ^ =t The inversions b and c are somewhat smoother than the funda- mental position at a. At d, 3 and 5, the connecting tone is disregarded in order to avoid 170. ^^B i^ ^ which is less audible when distributed as at d, 4. TABLE II. fc :fC=P= S i 1 — ^-til-isl-J-H-H =^= =53 -si- =gs^ ,/'j_ a - ?^g±?:i=3=' r:r^ -•- -•- h!2- pb^ §±^ td 7b I I BtiS: ^^^1 (b.) ^ ^^P^gp^Sigigg i^t^ii:f g-5-b-^ ]==Ji eb - #— # ?#3 HARMONY "5 Poor. The skip to the diminished seventh in the bass at d, 2, is excused by the smoothness of the other voices. At d, 2, the free ascent of the third completes the chord. At d, 4, the crossing of the tenor by the bass compels the change of position. EXERCISES TO LESSON XXXV. a I g^ I ^E2= e i < ^ ,, -f- # ".bit I I i eJ i^ ~(2 (s -^ — r ^=^ /:i i -i — r IB ^ 1^ -•-; • •- =P=P= -*-s^- 1 — r jr:i ^ I * • I=t I Jz^ijtl? I ^ -• — #- ^ =P=P= 1 — r -•-L#- i ri6 HARMONY. LESSON XXXVI. COMBINATION OF THE PRECEDING MODULATIONS. The exercises of Lessons XXXIV and XXXV may be combined, forming complete periods containing a modulation from a major key to its supertonic and return, or from a minor key to its minor seventh degree and return. 171. i l# r i^ -251- :t -^— g— -g— §f ^ ^%-i=i- % ^ ■» — O — IS — 1©- -i?Ts- -^^ -#- 2^- i T i 1=^ I ::^= ->^- -g- :d= J J- -i g-zyg- I :«^ t=t — r-f- The others may be combined in the same way. Our modulating phrases may be made more artistic by the use of the deceptive cadence (the triad of the submediant following the dominant seventh chord), since the chord of the subdominant in the final cadence may succeed the submediant as well as the tonic. a : I xV ^ VI or vi IV I| V I X stands for the new key. This formula, applied to some of the given modulations, gives ' I to V. I to IV. 4 I to VI. G:wi F-.Vq HARMONY. 117 I to III. I to 11. I «:V7 VI d:\l II to I. Ill to I. i ^ -sj- 11 1^ • ^ 1 — a « — ^ d — 1 — I •- H»=^^c: ^ t=t C:V7 C:V7 By combining all the modulations already given, as temporary transitions from key to key, an interesting harmonic fabric is formed, which may be illustrated thus : 172. i ::^=2g: P^¥ :£=et:S=S J ii, ~si- - gHi^ 4=i 'g>! gj - g- i G* a =t=4 ^-'-si-j|5i-L,9^=t>£ a Write such passages through the following keys, ending with a perfect cadence. 1. D-b-G-e-D-G-D. 2. a-d-F-BP-g-d-a. 3. Bb - F - g - D - g - F - Bb. 4. e — C — a — G — C — e. 5. F - g - Bl7 - Et> - c - F - Bb - F. 6. f# - A - CS - E - g8 - B - c# - ft. ii8 HARMONY. LESSON XXXVII. THE SUPERTONIC SEVENTH IN MAJOR AND MINOR. The chords of the seventh formed by adding a third to the upper interval of the secondary triads of the scale are much more restricted in their use than the primary discords, the dominant seventh and ninth. Of these the seventh chord upon the supertonic is the most impor- tant in both major and minor, being often used to precede the domi- nant or tonic triad in the complete cadence ; in fact, it is utilized in practically the same maimer as the supertonic triad (see Lesson X), and, being a combination of the tones of the subdominant and super- tonic, is of great effect in strengthening the cadence. The seventh of the chord, which may appear in any voice, should be introduced : First, as a connecting tone with the tonic : 173. tr— «• ^^ -(2 S- I Second, as a descending passing tone between the second and seventh degrees of the scale : 174. i I i^ This principle applies to the other secondary seventh chords as well. Rule. Sevenths of secondary chords should be' introduced as connecting" or descending passing tones. Note. This of course does not deprive the seventh of its right to enter from an- other interval of the same chord. HARMONY. 119 The supertonic seventh chord has two resolutions. First, to the dominant chord, with or without the seventh ( see Table I), in which the seventh ( of the supertonic chord ) de- scends one degree. This resolution is perhaps the more frequent when the seventh enters as a passing tone. Second, to the tonic chord or one of its inversions, in which case the seventh of the chord remains stationary. (See Table II.) The resolutions are here given in every position. TABLE I. RESOLUTION TO THE DOMINANT. {«.) 116 V V2 B I j I -^ — 0- ^ li=W^ It— » ^'^ t=t m #' l |g n4 V Vt I s I20 HARMONY. Il4 V6 V6 a e a. The fifth of the dominant seventh chord is omitted; or it might be connected in the bass, forming a V* chord. b. Perhaps the strongest of the three inversions. The resolution to V* is much the better* TABLE II. (a.) a RESOLUTION TO THE TONIC. ! I rare. 3P IIE3. in^- ^ --^i -•-«»-•- :2^=stt= -#- -S^ -0-t-0- ^ 1 — s> — ^— I— =fc^ 1 rf=^^ ;F=F 17 16 16 4 (*.) ^ «i^ 5- -5- I9 ^r f7T ^=?= i -'5'- "•--#-•- II i ^^ jcfc 116 16 16 6 4 ^ i Tcrpi ;l t=d= :^=^=^: :^ rr^- r r -zst— •- 9 — ^ -.sS^S-— 5: geg II n4 16 16 3 4 HARMONY. 121 (rf.) i^-r £ ■s^ — £ I 3= lizzt^ • l a 9 \ s) - tm 9- s ^ -"T :t S S^ ■szz ^ ^ - ^ * F r Il2 I I a, b. The resolution to the I^ with the third doubled is not to be recommended. c. Here the resolution to the I* is still weaker on account of the skip in the bass. d. The use of this position may well be deferred to a later period. These positions and resolutions are the same in the minor mode. 175. < i W i EXERCISES TO LESSON XXXVll. ^1— gi I a g spgi— .6^ - s i I t ^g £1=^ ^ -■g— ^ p U. -f s^-i 1— :t=t IV II I (7. Seventh prepared from IV. b. Seventh enters as a passing tone, e. Seventh enters from another interval of the same chord. ^ g'— ^F-F^— ^+ .^ • F hlg r I ^t=t ;£ •-*-# 117 117 117' ^^ii a i tf J3 ^ ^ o^ : a; - -;si- I j EEgE :iii— «■- ^ ^a-^z^ ^F ^-z=»- Play Tables I and II, a, in every major and minor key. LESSON XXXVIII. INVERSIONS OF THE SUPERTONIC SEVENTH CHORD. The inversions of the siipertonic seventh chord present no new difficulties. The introduction and resolution of the intervals are the same as in the fundamental position, and are alike in both major and minor. EXERCISES TO LESSON XXXVIII. m ^^=a ^- =r ^^ :d=iti=J :i=iES^^ m ^ ^^=0- -a) g- ^ - g - al "I II* "t I "i 1. B^Et ii=*= 116116 =i=p: ^ 1?— •- i "i "I 2. ?!:2^^p^^ ^ -•— s<- I HARMONY. 123 ^ *=±:t ^ m - s i * •- -g^ ±JC3t. ^ at=«r3t^f 6. Chorale. ^ g -Tb- -(=- B^E :^=t: i 3 I ■ ^- .ftf i #^ li-* -f-z^ Play Tables I and II, 6, c, d, in every major and minor key. LESSON XXXIX. SECONDARY SEVENTH CHORDS OF THE TONIC, MEDIANT, SUBDOMINANT, AND SUBMEDIANT, IN MAJOR. The chords of the seventh on the tonic, mediant, subdominant, and submediant are practically dislocations ( suspensions and passing chords ) of the other chords of the scale, and are treated as such. The sevenths of the tonic and subdominant, being major, often resolve upward when prepared by a connection with the preceding chord. On the other hand the sevenths of the mediant and submediant, which are minor, usually descend one degree. (See Table. ) The fifth of these chords is often omitted, especially when they succeed one an- other in sequences, which is not seldom the case. When the bass 124 HARMONY. progresses by fourths and fifths, the third of one chord becomes the seventh of the next, and the fifth is omitted in alternate chords. 176. i 1= -25»- j? tf§==a£t:a a^r ^^ ■z^- -z^ r^ •«- ^3^ T- i vi_ ii_ V_ 7 7 7 7 7 The following Table shows the common resolutions of these chords m-G ta ajor. - TABLE. («.) The I. 7 J-.LU ii^iigil^iS j=^ ^^r ^ 3 1 5^ i==lq=^=?c:»or: ^=tt= ^ f= — «• ^^=-V- Il6 The III i d= 5=1^ -^ — :ir ■-g-- 9-^=^ (*.) The IV_ Uj_J- i lie t ■^ :r-^' -si — 1^ : .eJ <=-F^= It 7 7 I7 117 2. i :t=t: F I 1^ i g -f g- ^a=^ i <^ < ^ i^SEEt 8 7 87 87 8 7 77 I » 15=^ 3tzi: • 4 - "t? Fg) 8 LL :?^i: J=7s,, 3^ T=P= -^^-4- *ztfc-|t :^g -fl- ^H EPES ±»t»±±itii3t:p: 126 HARMONY. I 6. Choral. I i^ -(=Z- ^E:2^3t g — ei ' \ --■^ y=f-^^ a^ a i ^i I b— (? f^ P -f^ s^^^ o ci -ei- o < g I g^ Play in every key, also the given Table. ^s I it=^ LESSON XL. SECONDARY SEVENTH CHORDS IN MINOR. The remaining dissonant chords of our harmonic system, founded upon the tonic, mediant, subdominant, and submediant of the minor mode, require but little explanation. The tonic seventh chord consists of a minor triad with a major seventh. The mediant seventh chord consists of an augmented triad with a major seventh. The subdominant seventh chord consists of a minor triad with a minor seventh. The submediant seventh chord consists of a major triad with a major seventh. HARMONY. 127 The major seventh of the tonic and the augmented fifth of the mediant chord, being the leading tone of the scale, must ascend to the tonic. (See Example 177, a.) Note. In this connection it may be remarked that the seventh of the descending melodic minor scale is often used in these chords instead of the leading tone, both chords being indicated by it. and IIlT. ( See Example 177, *, c.) 177 {^M ( ^^ (*.) ( 1=2 1= «>■ ^^ :t=t: r\ L /r\ J . \l •:j .^ 1 r^ 1 / fr K 1 r-7 1 ^ -^ /p T^ ff 1 " " 11 .i a LiL .* U 1 1 1 U ^ — M 1 i s ^ w^^- «! ^^ — gr- w -isM— g ;^ SP^5P^ I 3=5 eS - si— s*- ^^E^g !&— S*t r I ^^(g) ^^jj=^.B fe| G: I 116 6 I The chord at a is of course the first inversion of the supertonic seventh of G major. At d the third is chromatically raised, forming a chord equivalent to the dominant seventh of the key of D. At c the root is also raised, forming the equivalent of the diminished seventh in the key of B minor, yet the identity of the chord remains the same as before the chromatic alterations took place. This principle remains the same in the analysis of all chords formed by chromatic alteration. All augmented and diminished intervals are discords requiring a resolution, the general principle of which is as follows : The augmented tones of an interval ascend one diatonic semitone. The diminished tones of an interval descend one diatonic semitone. Augmented Fifth. Augmented Octave. Diminished Seventh. Augmented Sixth. 179. ^^^ -U zg- -4 i , gH>i i ^^^ |e^ 1^ I Intervals chromatically altered may not be doubled, as the resolution of both voices at once would involve consecutive octaves. Chromatic alterations are often written enharmonically, partly on account of greater simplicity of notation, and partly in order to con- form to the notation of the chroinatic scale, in which the seventh de- gree remains lowered and the fourth degree remains raised both in as- cending and descending, the other degrees being raised in ascending and lowered in descending. Thus : Instead of Instead of ISO. z^ St 1^ -sr =F ^ * I It may also be observed that, in the harmonies formed by chromatic alteration, there are many instances of incorrect notation in the works of the greatest masters. 132 HARMONY. The following example illustrates the chromatic alteration of chords without change of key. If the chromatic passing tones are omitted, the progression from chord to chord remains the same. I. 2. 3. 4. 5. 6. 7. 8. 9- 181 P -j^m. ^^ T ^U ±^ -s>- -&■ -'9\l9- X^ i^J^X II. 13- i I - ^ sJ r -s H^asl- W -fr I I, 2, 4, 5. Fifth raised. 3. Chromatic scale written enharmonically. 6, 10. Seventh lowered. 7. Third raised. 8, II, 12. Third lowered. 9. Fifth lowered. 13. Root raised. EXERCISES TO LESSON XLII. 1 . Figured bass given. 6 6 6 6 6 88.56 4J 5 5 3 3tt 6 6b 3t 3 8 7 6 6l7 6 3 3 3$ B l=5=n % :9tzfc:a=— =:^ -^ 63: la S -fi; Its' L^. =»^"-a- jSL- 8- 2. 3 Sfl 3 3fl 1 io ^ 6 — •S' — 1 3 sir -a. 6 6 6 5^ 3l7 8 7 ^^-n :^^#-S-^— '=' -« — ^ tf= ..L ,. . eJ LfS" — 1 -0 L^J i 3. Melody given. iiS ^ - p — •- -J=a- ^ =1= HARMONY- 133 ^ ==fer: -mm ;F t^F I -ig Sai ;;! b^- ^-^=¥=-'i?~- iE [tte^ ^fc ^EE^E i ^' — »^ i-»i- =M! x=-^ j=t= LESSON XLIII. MIXED CHORDS. THE CHORDS OF THE AUGMENTED SIXTH. The principle of chromatic alteration applied to certain intervals results in the combination of tones not found in any one key. (Such intervals as diminished and augmented thirds and sixths, doubly augmented and diminished fourths and fifths, are thus formed.) Such combinations are called mixed or altered chords, and to this class belong the useful and interesting harmonies known as the augmented sixth chords (in some text-books styled French, German, and Italian sixths). Their distinguishing feature is the melodic con- traction of the minor third into a diminished third, which being inverted becomes an augmented sixth. This is effected either by raising the lower tone or by lowering the upper tone of the original third. When formed from a major third the alteration takes pli.ce in both parts at once. It is from this interval of an augmented sixth that these chords derive their name. Although in the conventional usage of these chords the lower tone of the augmented sixth lies in the bass, their '34 HARMONY. use in the fundamental position, and in the inversions which bring the augmented sixth between the inner voices, is by no means rare. In fact, the works of modern masters contain numberless examples of all these combinations. It may be remarked, however, that the fundamental positions and inversions involving the diminished third are more frequent in the form of tenths, by which their harshness is somewhat neutralized. These chords are formed, according to the principle of chromatic alteration explained in the preceding lesson, from any chords which contain a minor third or major sixth, provided that the alteration of the interval consists of a whole step divided into two half steps. (See Lesson XLII.) Certain chords, however, are more frequently used than others for this purpose, and they may be classified as follows : The minor triad with the root raised, (a.) , . 182. S^ S I The minor triad and minor seventh with the root raised, ^^ 9-\? ^ mi Diminished triad with the third lowered. ^^ T Diminished seventh chord with the third lowered, (rf.) :9i^te=^ ^ Diminished triad and minor seventh with the third raised, ■^ ^ ^ Dominant seventh chord with the fifth lowered, M± -f-r HARMONY. 135 Dominant seventh chord with the fifth raised, ^ia i± -gig- I These chords are here given in their most familiar form (that of the augmented sixth between the bass tone and the otlier voices), and although they are formed in various keys in order to show their derivation, it will be seen that their augmented interval in this instance always lies between Al? in the bass and F| in the soprano. The tendency of the interval of the augmented sixth is to resolve to a perfect octave, both parts moving a semitone. ' In fact, when the interval is formed by the alteration of both its tones, both tones must of course resolve. If we omit all the accidentals we shall find that these chords (excepting the last one) are formed from the chords of the IV, IV7, and 117, in C major, to which key they may all resolve ; and inasmuch as any or all of the intervals may be altered, either singly or in combination, it follows that the relation of the chord to a given key must depend upon the alteration. This is clearly shown at e, f, the first chord of which is formed by raising the third of the chord of the 117 in C minor ; the second by lowering the fifth of the chord of the V7 in G major. These chords will be here indicated by the Roman numerals, as in the preceding lessons, with the signs + and o added to the figures for augmented and diminished intervals. I I u. i etc. i rs^ =1^ -s>^ i 183. w 2 \2^Z V7 f:ll§ f:IV7 (l») (Ob) • (3«) THE AUGMENTED SIXTH CHORD. The first inversion of triads with a diminished third, which invert into augmented sixths, is formed in two ways, as seen in Example 182, a and c. First, by raising the root of a minor triad in a major key 184. i EE M M from either the chord of the supertonic or the submediant, but not of the mediant, because the raised root of the latter would resolve out of the key. In minor it may be formed from the chord of either the subdominant or the tonic. 136 HARMONY. Second, by lowering the third of a diminished triad 185. i IE -Tfz3z r-lS^- fe ^^ i from either the chord of the leading tone of a major or minor key, or from that of the supertonic in minor. The latter form, however, is not usual. The fifth of the chord is doubled, either in unison or octaves, and one of the fifths must be below the root. For this reason the chord of the augmented sixth cannot well be used in the fundamental position. In harmonizing melodies the position formed by lowering 6 the second degree of the scale is often used, instead of the dominant 4 chord. (See exercises.) It is by no means to be understood that an altered chord is necessarily preceded by its unaltered form, i. e. by the chord from which it is derived. An altered chord may be preceded by any chord from which a legitimate progression can be made, its relationship to a key being established by its resolution. 186. r m I i i EXERCISES TO LESSON XLIIl d=F^: w- --^ -st- st- ^ VII (39) yd VI (8S) i f- F^^^r ^r~T -^ ^ 9 1- ' ^— b- Il8« HARMONY. 137 1 . Figured bass. ^^^- — «> - — \ — — zi -'5' ^ 6« ^'^ i i=:^ =l^f zit: At 6 6t 6t 2 '6 7 6 « PI Be^i =?c=Cpz =a= :?=^ 1 L— J 6 6t 3 6 6 8 I ^tt— «- -2^- 8 6 6$ 6 3 6 6 7 4 3. Melody. IS ip=p= I ^i^ W F=t=t ^^=^ :^ H— ^ -'S- jb- i I I I 53 i^fea^t^ 3t=lt ^r=F=-fc^ -z^ feEPEEp-E^^ fE^^E^ E^^ -Pf 0- ^ ai ^* — r EiE ^^ :^ ^~:^^4- j^j^j::.jj ^ The sign + indicates where the augmented chord is to be introduced. 138 HARMONY. ^^^^^^^^ ^^. *-^t-^ I -s<- ?^ These cadences are to be played in every key. LESSON XLIV. MIXED CHORDS. (Continued.) THE AUGMENTED SIX-FIVE CHORD. If to the minor triad with raised root, or to the diminished triad on the leading tone in minor with lowered third, explained in Lesson XLIII, we now add another third, there results the altered seventh chord, whose first inversion is known as the chord of the augmented sixth and fifth. The first-mentioned form is found, of course, on the intervals of the supertonic, mediant, and submediant in major, and on the subdominant in minor. Of these chords the mediant is not avail- able, from the fact that the raised root resolves out of the key. (See Lesson XLIII.) The chords on the subdominant in minor (see Example 187, a ) and of the super- tonic in major (see Example 187, b ) are the most useful, on account of their direct resolution, with a stationary seventh, to the tonic triad. ^^^^^^^m 187. -^ ^ c:rf_ (1» (*.) I -ggg— ; '•§■ '-^ iS^orzSI ^ggg- P C.w. (itt The second-mentioned form is simply a chord of the diminished seventh with the third lowered. i^^^ HARMONY. '39 It will be remembered that the third of the vii? is not allowed to descend when below the seventh, on account of the consecutive fifths involved. If, then, this third be lowered, it must descend ; therefore, this chord cannot be used in the first inversion (|+ ), as the third would descend in parallel fifths with the seventh (Example 187, c). It may be noted, however, that these parallel fifths are often to be found in mastei-works, especially in those of Mozart. They are still more often avoided when the third is below the seventh, by first resolving the seventh a semitone downward, 188. ^i ^ e: I then the rest of the chord. These same consecutive fifths are formed also when the augmented g chord, derived from the iv^ in minor, resolves to the dominant triad, 189. ^i^ SE ^^B and these are likewise avoided by first resolving the seventh. 190.^t± m -^^ r^fct P H 6S 4 3 This forms temporarily another form of augmented sixth chord ( the chord of the augmented six-four-three or *+. ), which will be con- 3 sidered later. The following summary shows the augmented six-five chord, with its possible derivations and resolutions in major and minor. The form at 4 is less usual than the others. SUMMARY. I. i i -jSZ. m fe^ £E & |S|^^^^3ia:p^ fe I t -e>~ ii fip c: iv_ O.ii ^ 140 HARMONY. ^ IE * «i=-(as'-Si^ — » kJ^ p Ij* * "p^^j^ ^ I C:vi_ i [i ^^ yai^p^teg^ j-^ ^ 1^3:^— z^j EEE '70 The fundamental positions, as well as the second and third inversions, of this chord are entirely practicable and are analyzed in Lesson XLVIII. EXERCISE TO LESSON XLIV. Exercise including modulation. m ^=d: ^ Jz J rA- i =^ £ -i- A i: 1x2 ^^ C: I "i „6+ o VI6+ 6 j=^ s F h=z;3: gg — jg- « »— ^f. — *25*s'-^-g: -r~f- I 1^ ^ si— <^- -(2- a: II ij V_ I 4 7 IV6+ o HARMONY. 141 \ zSzzzzs^ s W^^ ^ — F^ j 3= -&- -ie>- -IS- i II ^ p ivf+ 6 C: V. The augmented | chord is indicated by a + in the following exer- cises. 1. i ^ :S=F^ IS =#^ ■fS'- ii 3=^^ -f2 S"- -(2- ^ 3 3. S^W ^SS + ^^ + L#? -^- ?e=±:j=1^ 15^^ ^ 3t?|! tZ^jt 4. i -^ — s- EfcB — ^^4- I — I r— ^ -1 1 F- -F F-Bl iE^ F=^ 5# 15^23 fS: si — g^ zf- ■»— '-s p^^ifee -z;^- :^=t: 6)—\ ,■&■ ■z<-^ ^ i S ' ' z^ Play in every major key. i SE ^^^PM :^ ^B 1*- -•- -rS'- =»=i:J i C I n«+ 6 142 HARMONY. LESSON XLV. MIXED CHORDS. (Continued.) THE CHORD OF THE DOUBLY AUGMENTED FOURTH. The enharmonic equivalent (or one of them) of the augmented | chord is formed by adding a major third, doubly augmented fourth, and augmented sixth to a given bass tone. 191. ■ ^^m Thus the interval Ab-Djf corresponds enharmonically to the perfect fifth Ab-Ei? of the augmented sixth chord explained in the previous lesson. This chord is formed : First, by lowering the fifth of a diminished seventh chord (see Example 192, a) ; Second, by lowering the fifth of a minor triad with minor seventh, and raising the root and third at the same time. (Example 192, 6.) In this form (the most useful one) it resolves to the | tonic or domi- nant chord, according to its derivation. Thus : (o.) 192. ^ #:«= ^ES r ^ ir: vil^o ("!') T- =i= ^- W- \'?SZ lE^ r (*.), i %. m3 ■g^=^^ C:ii7 ^V-^% ^ s- S35E =p: -f -TSi- "^ % a. The second inversion of the diminished seventh in e minor, with the fifth chromatically lowered. b. The second inversion of the supertonic seventh of C major, with the fifth lowered, root and third raised. c. The same on the submediant of F major. HARMONY. H3 The other minor seventh chords are not available for forming this chord, from the fact that the altered intervals resolve out of the key. In the following exercises the altered chord is often preceded, as well as succeeded, by the | tonic chord. te s EXERCISES TO LESSON XLV. MODEL. J=i zsz §se [i ^ES 'is m "t fe^Efc 6D I«SI "t 'X rfe 4$ 3 le -X '% 'X=X ■¥^ Vt I i^ Jbl g * J2pr: Eg Sp-y -j-rt ^ ^^#* ^ ^ ^^^ 1 — \ : j_J_^J^iL-| fc t- fclJ ^ i I 144 HARMONY. i ■*k iSE; M-^=if3ti^l F^^^lS^ T=5: m^^^m ^E^^^^^t 3=± -»i-*^ I + Augmented triad. i B^^ ?»=i= -J — I — I— (- i^ip gJEgp^ •=ti±t±±:bt^ Add the inner parts. 5. &^^e -y-'— #- S^ ■*K a^^g^e EHEi^mE^ :^3=r :M?-h- J=ii :^:i^:n -H- *fc?=; ^^^1 rfcr -l — ^ -^ ^^-r-t ^^^ ^rS I :^33Pt 9^iH&p^r^ir=^.^=i^-^^^i «-=-»=• I -*-•-* — — -I — f — I — -^ I I Ep -^- i — ^ — f — r- h^ — •- -«-^— r =g=:C=g- i =j I 1 — I i j I F^i — I — ^=P * ^-^ H HARMONY. H5 LESSON XLVI. MIXED CHORDS. (Continued.) THE CHORD OF THE AUGMENTED SIX-FOUR-THREE. The augmented six-four-three chord proper is erected upon any given tone by adding a major third, augmented fourth, and aug- mented sixth, and is derived from the second inversion of several seventh chords, viz. : I. From a dominant seventh chord by lowering the fifth (Example 193, a). g -^S'- 193. m -sag- r^ -te II. From a diminished triad with minor seventh by raising the third (Example 193, 6). i 1^ # S ^ 1^ -^zr >i2 III. From a minor triad with a minor seventh by lowering the fifth and raising the third at the same time (Example 193, c). "Sis'- ^^^^^^ I ^ :t r The first form (a) belongs to both the major and minor modes. The second form (6) is founded on the 1I7 in minor, resolving to the dominant triiad or tonic | chord, and on the VII7 in major, resolv- ing to the tonic or tonic | chord. The third form (c) is founded on either the 117, 1117, or VI7 in ma- jor, and resolves the seventh downward, while the root remains sta- tionary. The following model illustrates the use of this chord in its various derivations. Every accidental in this example may be omitted with- 14.6 HARMONY. out affecting any of the harmonic progressions, therefore the logical conclusion is that these tones foreign to the original key are not modu- lating tones, and do not in any way affect the original tonality, except- ing as melodic passing tones. ig4. 2EM^ i --¥ J V6 (8b) -^^f=f-- -(2- "It 3 h. S _L V„6J 3 i 4. d. I =^=tt^ ^ s) »- M -•-Tt- f- -J- J-J-^- J ^ 9^ £ f^^^f^rrr 3 mix 8+ Note. The altered chord is indicated by 4+ Although this apparently contradicts the principle stated at the be ginning of our lessons in modulation (see Lesson XXV) , that the domi- nant seventh and diminished seventh chords are related to one tonality only, the fact of their being formed by melodic alteration causes them to lose something of their harmonic significance ; otherwise we must recognize the chords at Example 194, 6, c, and . h -I 1 1 1 * a 1 1 }•, 1' '• • a kJ J , L r r » v> s< 1 r • ^ ^ fell 1 1 £. • * S S 1 " [JM- 1 1 L ?• 1 - 1 1- 1 =f m-^-^ 4 S I -J— »-^- ^=* n=^=S=i-- =S=^ j= ^ i i ^ zg 4-. With modulation. \^^^. :l=t F ^- • »i- =P=t ^ H?-^g- Change the positions in the melody freely. 5, Melody given. ^-4- 3^ s ^ i i^ hs^ ^ ^^E P^^^=f= ^ ^^^^^^^^^ m^ ^1 ^ ]= -^ j = ^^ -si < i- :#t :i^ i feE ifczzqf. i^3 -» — ^- 8. i ) A4^?-^ ! -* S j— J— J^^ g=:£=g ^ i i 1:1 W ¥ 8, With modulation. ^ -• J f=t: « ! ^ - y- ^ -g^- j=t -i^ — •- -' '> eJ 148 HARMONY. LESSON XLVII. MIXED CHORDS. (Continued.) THE NEAPOLITAN SIXTH. When the root of the supertonic triad in minor is chromatically lowered in progressing toward the tonic, a major triad is formed, popularly known as the Neapolitan sixth (indicated by N®). 195.< i W m ^a^^^a ^ £E N6 I fg 1=t N6 It is most commonly introduced as a chord of the sixth with the third doubled, resolving to the tonic | or dominant chord, with or without the seventh. Sometimes, also, the resolution is to a domi- nant minor ninth chord, especially if the fifth of the chord lies in the soprano voice, thus preparing the ninth. 196. i w EE ^- ^ ^- N6 ©1^1 Both the fundamental position with the root doubled (see Example 197, a) and the second inversion with the third doubled (see Example 197, 6) are occasionally used, and all the positions of the chord may be used in the major mode by the simultaneous lowering of the root ■and fifth of the supertonic triad. (See Example 197, c.) (a.) 197. I p. 4 IbJ: (*,) EE g ^g^d ^^3^te -fS'- m i NoT£. Such positions as a, i, are not really sixth chords. N6 HARMONY. 149 m («.), S' tz-^- IB :tSi P=P S I Sf N6 M6 Like all the altered chords previously explained, it is not in the least indispensable that the Neapolitan sixth should be preceded by its unaltered form. It may be pieceded by any chord which pro- gresses legitimately to the supertonic triad, giving the preference, possibly, to those of the subdominant and tonic. Numberless exam- ples of this chord in all its forms are found in the works of the mas- ters since the time of Bach. Though soinewhat stereotyped in effect, it is expressive and often very useful in harmonizing certain chromatic progressions, as for example : 198. Jfe -r- ^ *=±f i etc. ^^ l± I N6 When it resolves directly to the dominant chord, the otherwise awkward interval of a diminished third — 199. is often written, if it does not occur in the inner voices, lustrated at 6 in the following model : ^ This is il- P 200. ^^±g; ^5:^ S*- -si-s<- ^ip ia St N6 ^g= ^i yri- N6 N6 a. First inversion, third doubled. d. Fundamental position, root doubled. c. Second inversion, third doubled. ISO HARMONY. EXERCISES TO LESSON XLVII. i i I m ^:a- -jsi- r> i ■ f3 g - iiEE^ 2. m W- :1=± Ep -g s>- 6 6 4D i <= — «■ tts* ^"S^- (3 chords.) 5 ^-Ff=^ .=rj--faHFg Efe -r"^ ^^ ■ it (3 chords.) ^^^ ^^ r-f - |^=^ te^^ I I --s- -(= (=■ =E^ i^^ :fep= ■|!2-^ ■•~-3l- (3 chords.) •J AT S - a; • — P^ y- i ~s ^ 1 V- Nlb fA^ P=:^^^^3i=^ LESSON XLVIII. ALTERED CHORDS WITH A DIMINISHED THIRD. The chords of the seventh containing a diminished third, from which the augmented sixth chords are derived by inversion, are freely HARMONY. 151 used in the fundamental position, as well as in the inversions not al- ready explained. As stated in Lesson 43, the triad with a dimin- ished third and fifth is not practicable in the fundamental position, as its fifth must be below the root. (See Lesson XLIIL) In the second inversion, however, (|) it is occasionally utilized en passant. 201. i m m^^ "P-" m ^E ^ ir-- The chord with diminished third, fifth, and seventh is the enhar- monic equivalent of the third inversion of the dominant seventh, and is therefore of special value as a modulating chord, as will be more fully demonstrated hereafter. 202. {a.) ^ Kb. w >J- F B'^-Nn m a: IV 6+ a is V7 of, and resolves to, Bb : I. h is iv|+ of, and resolves to, a : i. Since the seventh of this chord is a perfect fifth above the third, consecutive fifths result from a resolution to the dominant. (See Table I, -5, 3.) Note. These particular fifths are not especiaHy bad, being in the inner voices. Such fifths are avoided by placing the third of the chord above the seventh (see Table I, a, 2, 3, 6, <5, 4, 5, 6, etc.), in which case it may resolve to either the tonic or dominant triad. Observe that the interval of the diminished third or tenth invariably resolves to a perfect unison or octave. I. TABLE I. 3- 4- 5- — • g ^ ^^m- '^h ^^ 7?g * -& — •- HARMONY, The chord with major third, diminished fifth, and minor seventh, 6+ wliose second inversion is a chord of the 4+, resolves in all positions and inversions to a tonic or dominant chord. The positions in which the third lies below the fifth are very harsh, but may be improved by transposing the third into a tenth (see Table II, 3, all positions, also 3, i, 5, c, 3, 4, 5). All the other positions are entirely practicable and in common use by all modern composers. TABLE II. I. 2. 3. 4. 5. 6. b: fc ^ K=E3irz^=i ^sTp^T -^^y^ l! % m :2=??r?tr±te:t2f; n- ¥ ~»~y - *— y gS e^ I. 6. ^ ^53E--^"|^3?^^.i^ -• — •- In the harmonizing of melodies involving the tones of the melodic minor scale, the raised sixth degree in ascending and lowered seventh degree in descending are to be considered as melodic alterations. The enharmonic form of the chord in Table I, page 151, formed Vkfith a minor third, doubly diminished fifth, and diminished seventh, is also useful in modulating, as its resolution is invariably to the tonic triad of a major key. As before, the diminished third resolves to the perfect unison. The positions and inversions are as follows : i TABLE III. ^ w jt zg *- -^fe ^ ■h^ Tjfisr- isi i^ :?*= I 3t 7 h>5 a =■ r- "fi *■ * f, Su^s — S p * 64 6 3b i ^^s=§ M ^ sfesz P^^5^z=-^f Pg w- -^g' r I rt> p — r zsz 2« We have treated the subject of chromatic alterations and the chords '54 HARMONY. derived from them thus specifically and exhaustively, because the principle involved is one of the most important with which we have to do. Innumerable other combinations may be formed in the same way, in which the dominant seventh and ninth with the fifth raised is of frequent occurrence. J The student is recommended to study out such combinations for himself. A knowledge of the preceding tables is indispensable for the analysis of modern compositions. 204. ± &*=?: M^. El ^ ■-¥^ 9ifei Fbg- fepE^Ei ^.JL ^^- ^- -f^ 1>5 — W- jQ. — Sue t i EE ?a=:g: I ■ ^ s)- ■e' tes! ■ « -(S- 9iife --^ ^=Z2Z- ^ .. J_J_ -SI- E £E <= ^ P2- -4- _!2_ -H EXERCISES TO LESSON XLVIII. 1 , Soprano and bass given. ^- 8= I fctjft g -S-T- 4f' F^H^-^=^^3^tl^^ ^ 5ilS 3iS ^^^Ts" ^ =#^ -& — (=- i HARMONY. 155 i #3^ =^5^ •-^ TX^^z^ri I ^"^ 9^ ^^a 4- W — ' — *^ ^' 1 ^ * » i5 15 ft 4-^ ^ EES^^ t=i?i S ^ ?= .J ^ I as ^ -|2_ EE S: 5. Melody alone given. m s •2 ^ r qg :t=t: -gj' si ■gl— 'g ^— Zj-hsr =P=i"- a ^T4^4f-R ^ -•— ih LESSON XLIX. ENHARMONIC CHANGES. It has been already shown (in Lesson XLV) that the dominant seventh chord coincides enharmonically with the mixed chords 4++ and I"*". Both of these altered chords, in their most natural relation (ivj in 7 minor and ^'i in major) resolve to the tonic chord of the minor '56 HARMONY. or major key one half step below that of the corresponding domi- nant seventh. For example =^i ^Ell :12: St m Q y= ^ IB 205.< :b^ ±^ -Mz D\):N'l I <:: IV8+ o ^ 6+ C:ii44+ 3 If now, all the tonic triads are connected with these altered chords, as they have already been connected with the corresponding dominant sevenths, a new and complete series of modulations is formed. (See Table) . Tlius with one chord and its enharmonic equivalent, modu- lations are formed to any given key and to that of a semitone below. TABLE. i^^^^^=ii^^ ^^-=^ :j uu ,_ I- 3t=gS ct or C £>b C d or i^^^^^^^^^ i ~r— I — T F ^=J: is: -g' U— gi^ i=bt5i D C ^bor C E\) \ gg=^j=|^^^i^=t^=^^^EE;^ « or C / " HARMONY. 157 i 1 — r '^^ ESlfe^^^l P ^ffPfl 3ffZZ± -^^-5-1^ ^B St^it /iorC i^« i^i^EiEi^^^ iz^ d zj q^ 2±=3t: i?c *=^ g or C g» or i^ -I — I H |=|i s^^^^^^Pi §3 -: -/ bj -^:=t:s; ife: -r 5=p^.-*i;5=^^r II :^ -P-=P= ^b a or C m^^^^^^i^ ^^^^^^ -» • St- -^^ 1^ I b\> or C Bb C b or ^M^ iEl^Ete C" " n^ B Ctoc or Transpose this table into all keys. 158 HARMONY. The diminished seventh chord, on account of its peculiar construc- tion, is more subject to enharmonic treatment than any other chord. If the twelve semitones of the chromatic scale be divided at any point into four equal parts, the dividing tones form the intervals of a dimin- ished seventh chord : 206. i w= -Az t t :#= -4-- t SH^^^^ and since there are but three points where this can begin (without reiteration) , it follows that there are but three fundamental dimin- ished seventh chords in sound. If we include^Jf minor, each one of these has five enharmonic forms. resolves to ^g=ttf^gi4M^si^ 207. f' dm°a i w =?5Sfc?^ '5izzE:|?£2r^ resolves to =5i= /••vn?o % -|725»-^ ii I S ■•-^Sg— ^htes"- ^ 3s resolves to^ is^ddzs^ fe 9 <*:VIl5o resolves to =& ^ sr-r gX^Sr siflxs^ -?«.vnOo I *i I i resolves to IE ^ ■pr^^- ■ ^..^ ' g - i -^^^ffg' iS»!Z^ x:«s; *: VIiOq The other two diminished seventh chords may be analyzed in the same way. i Thus it is seen that every inversion of a diminished seventh chord HARMONY. 159 is equivalent to another inversion or fundamental position of some Other diminished seventh chord. As all of these positions and inversions sound alike, the identity of the chord can be established only by finding the leading tone upon which the chord is founded (see Lesson XXII) and upon which its legitimate resolution depends. The diminished seventh chord in all its forms is much used as an altered chord, and many irregular resolutions are thereby accounted for. 208. ^^^m e: viiOq Another peculiarity is its capacity for progressing chromatically or diatonically by parallel motion in all parts at once. 200. By lowering any one of its intervals a chord is formed enharmon- ically equivalent to the V7, ^^, or ^, iiJ . ^^^M^^jte ^ E s#! -^ I 210. ±^ ^ r This of course is simply the resolution of a dominant minor ninth to the root of a dominant seventh chord. 21 1. J ! ,A S-i 1 I b ^4 ^^ -a ■^- i ^ i w 98 98 98 98: Note. The entire chromatic scale harmonized in diminished sevehth chords is so written that every fourth chord is in the same key, viz.: d a g d a g d 2JZ. ^ ^-- d $ g d * :etc. r6o HARMONY. This is in order to conform to the notation of the chromatic scale, in which the fourth degree is raised and the seventh degree lowered, both in ascending and de- scending. ^ ds^z I 213. * =^^ LESSON L. IRREGULAR RESOLUTIONS OF THE DOMINANT SEVENTH CHORD. Besides the regular resolution of the dominant seventh chord to the tonic triad there are many irregular ones, which, however, are gov- erned by one general principle of progression. The dominant sev- enth may progress to any other chord : First, with which it has tones in common ; or, Second, if the parts move only one whole or half step. Consecutive fifths and octaves, augmented intervals and false rela- tions are of course to be avoided, as always. The seventh may remain stationary or become enharmonically changed, 214. ^^^^ r ^ i-p r~pf= I It may also descend as in the regular resolution. Thus we see that the dominant seventh chord may progress : First, to any triad of its own key ; 217. Possible, but not desirable. i Vt ^^ -J5i- i =t V7 III etc. ^H=^=H : i i =^«= IIE^ :^ etc. etc. E^^^P i V7 IV V7 V7 viio Second, to any seventh chord of its own key ; 218. 4— i- $E^^^^m In sequence, etc ! I I -^ ^[—-^—^i=S- i^—fi — ^r-^-i-zsi §■ -(2 ^- V 117 =^ V7 1117 =^=^ i I etc. V7 IV7 ^^ V7 Vl7 1 62 HARMONY. Third, to the tonic triad of any other key ; « 219. ^^^^^^^^^m ^^ ^3glL | X3£^^ l;Mg C: V7 I CyiJD-.l Better. h J^-l-J-i=^'-Fit sii^^a^a^E^ f= r -^s>- ]P -^«- -bs"- -(= (2- C D'^ * Really an enharmonic form of 6+. 6 A^ E^ B'o Fourth, to the dominant seventh chords of any other key ; 220. ^%\ -^ §±g^^a^ES?^^y^ lE2=g| :j^ C:V7 G:V4 Z?:V4 ^:V6 £-:V4 B■N^ 6 ^:V« 3 2 6 2 '56 .^:V4 G'»:V4 Z»'':V4 /il':V6 ^l':V6 £t:V4 F-.Y^ 3 3 2 6 5 2 3 Fifth, and to the diminished seventh chords of any other key ; (a.) I- 2. 3. 221. i^=-^^^gi^g^^; giEg^^^g^g ^ -F'— ^ *- * HARMONY. 163 A) i -■, 2^1 -M^- fe^- ^f^ ¥E -i^ * ■^1;^ etc. I After these modulations have been thoroughly learned in all keys with the cadences, they may be reviewed in sequence form. EXERCISES TO LESSON LI. i t=f^ U^ iEi^ -^ -t>Bl- f ^^pi^S it J- i!E =^= Effi ZZI^_ IZ^_ i»:V4 I 3 Cadence. i 4^ i SS^ES F^i^=3^ -|7g- ^fe §^rfe^^= -1= — -iS- i =F rf:V| Cadence. HARMONY. 167 Omit no necessary accidentals. 1 . Unfigured bass. 8 P^^=^^-f^,f=^ ^^ .b«,_biffi- i^ ■^m b«- I t=h±d2^ £»:Vj 2. 6 =i=ic is i5c^ I 'F=i==i- li=^P^ £t?:V7 S. ^L-_|t f^m^^-^^dm p^ i ^ -[ ^ g < - »-[■ It /ib:V7 VI i 4. Melody given. !£2: ^^^^^E^ (Z-r-6>- ^p=^ |p'Jp:^ j ^-i^g=[. [ P # iTP=^=4 i &^ ^^^EE^ f-= >=F= F= -F=f=J E^ g*-#- 5»-'^i727- i & m *.* V f *^ u t.l-r4-tj— 1 I ife£ ijt^— *— ^=p tt aai J. , ^^JJiS^ gJii^^^a tflE V7 V7 V7 i68 HARMONY. LESSON LII. MODULATION A MINOR SECOND DOWNWARD. The modulation downward a diatonic semitone ( the reverse of the preceding one), is effected through the dominant seventh, preferably in its first or second inversion, as follows : The third of the original tonic becomes the seventh of the modu- lating chord. The fifth descends a semitone to the root of the modulating chord. One root is raised chromatically, becoming the fifth of the modulat- ing chord. The Other root descends a diminished third, to the third of the modulating chord. This awkward interval may be avoided by interposing a passing tone. (See Ex. 227.) J— J- ^^l^^^l: |#^ 227. >iz=g==feiE ^-0l f -:rJ~\-- 6 !p i^ ^ri ^ ^m w t^^^=^^=^^^^^m- zJ 1- IS B ^^ga 9i ^ *=t ^ The second inversion of the dominant seventh chord (| ) may be used with equal advantage. -J— J- 228. fi^ :^S^ ^ ^^=tf^ HARMONY. 169 i :^ -'-r^ Ili«- ,r ttSS fL^t^ ^^=4=-^^^'^ -4^ '^^^ The modulation with the diminished seventh is somewhat smoother than that with the dominant seventh, on account of the two connect- ing tones between the tonic and the modulating chord. The third and fifth of the original tonic become the fifth and sev- enth of the diminished seventh chord. One root is raised chromatic- ally, becoming the third of that chord, while the Other root descends a diminished third, becoming the root of the diminished seventh chord. The passing tone may be interposed as before. 229. i 1^* II! -^ t- =^ ^ _-f_fe_^if: -^f' i =r =#^ m # I s I^H ^^$ i*= M ^ I 4^ * -gi- t=t 4' :^ Consecutive fifths must be guarded against in the resolution of the diminished seventh chord, when the third is below the seventh. (See Lesson XX.) The descent of the third in Example 229, d and c, may now be freely used. This modulation, like the preceding one, may be continued in the form of a sequence. 230. I 170 HARMONY. It may also be combined with the preceding, forming a modulation upward or downward a diatonic semitone, and returning to the orig- inal key. 231. (i d=j= 232. U *: f- iz^ f=^-- -b^- ^ M^ *=* m * ^^=r i i Both the sequences given above, and Examples 231 and 232 are to be transposed at the pianoforte into every major key. EXERCISES TO LESSON Lll. S g ^E :5#; '«22- J mEJE^EEEm £ P=F=^ C:I ±^ ^i^i^i * ^S^K— fei- 4* -j2_ i-^Eg=^ I f- ^ ^:V3 I 1 . Melody given. Cadence. J 1 1 1 1 ■ 1 ■ >' ft ,rj ^ u. ! r U 1- i IL Tf TW ttsJ nc p» a ]^J V- iii_i 1: 1^ — , t #:^ J i ^lE T© fi- i HARMONY. 171 a ^^^^ s^ m^it t i fe==^ 15 =ttt -¥^ -^* ' a i /?: VIiOq i 5^ P^^ iE^a #^^f=^ r^i^feff^= Pr^ ^ I pt=jt Sequence. fe :t==t ^ b ^ — S» Sequence. idi i ^«— i^ r=i^ N6 5. i^^^^^^^s^ i^^^^^^^^^^i p * The rests indicate where the melody is not to be accompanied. LESSON LIII. MODULATIONS OF AN AUGMENTED FOURTH OR A DIMIN- ISHED FIFTH. This, the most abrupt of all modulattons, is easily effected through the third inversion of the dominant seventh chord, with all the uppei voices progressing in contrary motion to the bass. 172 233. ^ V-6 HARMONY. u, not. ^£='=f= =g ^= i?-i^ d2^ -t— gg- (?b:V4 ^ ±!i SI 1— t- r^zite i ^i 3.- not. £7. not. i ^ S= i^ =bS= -t>g= -^s-- -t?p~ i =d — ^j=i^^E^i:i i^^ -S'S'- — r r— Even in this restricted form the fifth of the tonic triad must always be below the root, otherwise parallel fifths result, as at Example 233, a, b, c. The use of any inversion or fundamental position of the dominant seventh chord involves parallel fifths, augmented seconds, or still more undesirable false relations, as shown in Example 234. (See note to Lesson XXV.) 234. This modulation may also be made through the chord of the dimin- ished seventh. Its harshness may be justified by the very slight movement of the three upper voices, none of which progresses more than two half steps. As in the preceding lesson, the diminished sev- enth of the modulating chord is often written enharmonically. 235. ,t?e: f^ L bg b.g - -t>^ m M W=W :ft^ :\Mz -ri- ib^ and Fj major, and of Eb and D| minor, to be identical, which is the case in the tempered scale, we form a circle of keys leading in one direction from the natural key to that of six sharps, and in the other direction from the natural key to that of six flats. 237. HARMONY. 175 The outer circle represents the series of major keys, with their sig- natures ; the inner one that of the minor. Note. Keys which include more than seven sharps and flats cannot be indicated, since there are no signatures for them, but such keys often occur in composition, especially in modulations where enharmonic changes are undesirable. 238. Ui f ^. %J ^B m^. Efe G% -xzsi- ^^: ;pz i Bi Starting from any point in the large circle, the next following key to the right is that of a fifth above : {\. e. it stands in the relation of a dominant) . The next following key to the left is that of the fifth below: (i. e. subdominant). The three keys indicated by small letters on the small circle are the parallel minors of the same, and the six keys included in the dotted lines represent the key of C major with what might be styled its nearest relatives. If the triangle be revolved on the axis x, the keys included in the dotted lines will bear the same mutual relations as those given above. This diagram shows at a glance the mutual relationship of all keys to one another. It is useful in many ways for oral work in modula- tion, of which the following is recommended : 239. To* C Major and Minor.' C I. Modulate from every key to C major and minor. Then into all other major and minor keys in the same way. 176 HARMONY. 240. From C Major and Minor. C II. Modulate from C major to every major and minor key. Then from all other major and minor keys in the same way. III. Combine the preceding, forming a modulation from C major to •dnAfrom every other major and minor key. Then from all other keys in the same way. 241. IV. Modulate by a sequence of fifths through all the major keys (C, G, D, A, etc.). Modulate by a sequence of fourths through all major keys (C, F, Bb, Et7, etc.). Then through all minor keys. V. Modulate by sequences of major seconds in both directions, and in two series : C, D, E, F#, etc., C, Bb, Ab, Gt>, etc., and G, A, B, etc., G, F, Eb, etc. The same through the minor keys. HARMONY. 243, 177 VI. Modulate by sequences of minor thirds in both directions, and in three series : C, Eb, Gb, A, C. \ and in the Dt7, E, G, Bb, Db. [ reverse di- D, F, Ab, B, D. ) rection. The same through the minor keys. VII. Modulate by sequences of major thirds in both directions, and in four series : C, E, Ab, C. Db, F, A, Db. D, Gb, Bb, D. Eb, G, B, Eb. The modulation by sequences of semi tones is given in Lessons LI and LIL MODULATION BY TONIC CHORDS. Although an exhaustive system of modulation by the connection of dominant and diminished seventh chords is here completed, our stud- ies are by no means finished. Practically, any chords may be utilized to effect a change of key. The secondary triads (as well as their seventh chords), being neutral or ambiguous in tonality (i. e. belong- ing to more than one key), are identified as real tonics by the keys into which they lead. For example : « d= s ^- 245. CA Not C:ii 117 d-.ni 1 Vt J 78 HARMONY. (*•) i Jz -7i- 1 5^ % :£ i C: I «: I not C:lll i6 III IV VI V-7 VI VI This shows that tonic triads may succeed one another and form a permanent modulation, if succeeded by a complete cadence. The following table shows how this applies to all successions of major tonics, on account of their mutual relationship to the same key. (See Lesson XXV.) TABLE. ^ fe ^y^i^ :|i= -J= =i=M I ^^ i i S :^ ■^i eI i^f^ bi :?^ :5f: 3- C X* EXPLANATION. I, C major I and V; 2, G major IV and V; 3, Parallel and tonic major of a minor; 4, Parallel major and dominant of a •ninor; 5, e minor VI and V; 6, N^ and V of ^ minor (see note) ; 7, V and VI of f minor; 8, Dominant and parallel major of f minor; 9, Tonic and parallel major of c minor; 10, F major V and IV ; 1 1, F major V and I. HARMONY. 179 Note. The corresponding succession of C and FJ at No. 6 occurs whenever the Neapolitan sixth is followed by a dominant chord. The derivation of the Neapol- tan sixth in this case is as follows : 246. ' I w m b:i "^-i- M^ 11° N6 I and its continuation as a permanent modulation through the major subdominant is as follows : 247. i ^ElE 9- c.i 4= * |-ff -(=- ■: e= &N6 --^ J^save 16 4 V7 I ^i CIRCLES OF KEYS. The entire series of circles of keys shown in diagrams 339 to 244 should now be studied as tonic successions. The circles of the fifth and fourth are shown at Example 241. The circles of major thirds are shown at Example 244. The circles of minor thirds are shown at Example 243. Note. A key may be established by a cadence at any of the points on these circles, forming a permanent modulation. Finally, it is not the case that all modulations must tajte place by the most direct means. In practical composition keys are finally changed by modulations through other keys, thus avoiding both abruptness and monotony. The shortest road between the two keys is not always the best from an artistic point of view ; but one who knows the shortest road will surely know the longest. A general principle for modulating to a given key through other keys is as fol- lows: Choose connecting keys -which tend in the direction of the new key to be established, by adding sharps or Jlats to the signature. l8p HARMONY. In this connection the following table may be found useful. TABLE. I to V adds one sharp to (or removes one flat from) the signature. I to IV adds one flat to (or removes one sharp from) the signature. I up a minor third adds three flats to (or removes three sharps from) the signature. I down a minor third adds three sharps to (or removes three flats from) the signature. I up a major third adds four sharps to (or removes four flats from) the signature. I down a major third adds four flats to (or removes four sharps from) the signature. This is illustrated by diagrams 239 to 244. Play this table, beginning on each major tonic triad successively, and succeed the second triad with a complete cadence. LESSON LV. THE NON-HARMONIC TONES. Since the tones of a chord are called harmonic tones, the non-har- monic tones may be defined as tones combined with a harmony to which they do not belong. A chord being a combination of tones de- rived from thirds ( see Lesson I ) , its non-harmonic tones are those which lie between or adjacent to those of the chord. ( Harmonic tones.) The non-harmonic tones are divided into several classes, according to the relation which they bear to the harmonic tones, viz : the sus- pension, the anticipation, the appoggiatura, the passing tone, the embellishment, the changing tone, and the pedal or organ point. SUSPENSIONS. A suspension is a temporary dislocation of any interval or intervals of a chord. It is caused by delaying the diatonic downward or up- HARMONY. l8l ward progression of any voice during a change of chords, forming a suspended tone foreign to the harmony. 248. I -J J i 2 3 Thus, the tone C ( i ), the root of the triad of C, is delayed in its progression to b ( 3 ), the third of the dominant seventh, by being suspended ( at 2 ) in the tenor voice while the rest of the harmony changes, thus dislocating temporarily the third of the dominant seventh chord. The three tones in question aie named the prepara- tion ( I ) , the suspension ( 2 ) , and the resolution ( 3 ) . Any intei-val may be suspended in any voice, provided that it event- ually progresses diatonically. Strictly speaking the seventh of the dominant chord cannot be sus- pended, for the reason that it simply adds another third to an exist- ing triad. But in combination with other suspended intervals it has the effect of a true suspension. (.a.) , (*.) 24-9. < i p ESE i 4 ts- ms It At a the progression of the soprano adds a seventh to the dominant chord, while At 6 the seventh and fifth of the dominant seventh are practically suspended, even though the mediant triad is temporarily formed. The preparation, in strict writing, is not allowed to be shorter ( in time value ) than the suspension, but as this is purely a question of rhythm, it is not necessary to observe the rule here, excepting when the preparation and suspension are tied. The suspension occurs on the accented beat, although not necessa- rily on the strongest accent. On the other hand, the resolution is al- ways unaccented. l82 HARMONY. Since the suspension of any interval which is doubled must inerita- bly form a dissonant seventh or ninth, and since a ninth must never lie less than nine degrees above a root (see Lesson XIX), the follovvr- ing rules must be observed : Rule i. The resolution of a suspension may be doubled simul- taneously with the suspension itself if it is a ninth below the suspension, but the suspension and resolution must not appear simultaneously in the same octave. Not good. 250. tt r I m i It is even better to avoid the doubling of suspended intervals whenever possible, by the use of other positions and inversions, and for the reason that the leading tone is seldom doubled, — Rule 2. The leading tone is not suspended when already pres- ent in some other voice. Not. Good. 251. 1 X- g | ''-~^?^ ^5^- ^s=t -Ki- - gj — [-z^ 9^= E^E eI^ •gg- I I Rule 3. The resolution of a suspension in the bass must not be doubled in an upper voice. 252. i Not. i^^ lS>- i TP-^ ibrrt: I The suspension may or may not be connected by a tie. HARMONY. EXERCISES TO LESSON LV. 183 n !>- a. i. . I 1^ 1 1^ 1 1-^ t. ■^ 1 / <> 1 '^ iJ J. - ej ! fi fi/i ^ « \ " ,! Vs\J -a '--„ /3 " .•3 _«/ t \" ir> - = g^J ^ g ^ LESSON LVII. THE INVERTED SUSPENSION. (Retardation.) The suspension resolving upward is called retardation or inverted suspension. It is of less frequent occuiTence than the regular suspension, and is most commonly found on the third or seventh degrees of the major scale, and on the second and fifth of the minor. It may, however, be formed on any degree of the scale, provided that the tone to which it progresses is not present at the same time in some other voice and in the same octave. a. b. c. 254-. \ -&s- ^^i r5^ ^- 1 9i=^ :sz I a. Retardation of the third. b. Retardation of the root. c. Retardation of the fifth. HARMONY. 1S7 Here again many combinations are formed, identical with the secondary seventh chords, as in the case of the regular suspensions. (See Lesson LVI and XL.) The upward and downward suspensions are very often combined, forming three parts of a legitimate chord with its regular resolution. Such cases are practically the anticipation in the bass of the har- mony to which the suspension resolves, thus : —si — I— d- 255. w Tzys-t - ^jB) — ,a- ^ i -^- C:I i a:vii§o a. Root retarded, third suspended. b. Root, third, and fifth suspended. Or practically, the roots of the C major and A minor triads are anticipated in the bass. (See Lesson LVIII.) EXERCISES TO LESSON LVII. |-9-t>-n=: J= ^4— *. l_ -. 1 [^ . ( ^^ :=d= d" 1 -gj — SIS' tb^Ts — -=* \-H^. ** 1 I \k1 — S- -si — s — \-~\ — Ji—S 1 '^ %. f <3 \ 1 I *-!., 7^1 ' 1 ■-^ b i Jz /. ^ i ■ a> \ a > ej - -i — 2S^ . a= gj ^ I i±^ r ^ a. Third suspended and retarded, fifth suspended. b. Seventh retarded. c. Third retarded. d. Third retarded, fifth and third suspended. e. Root retarded and suspended. f. Root retarded, third suspended. 1 88 HARMONY. 1. i g=? r-r^-^ S:5rr^- ^=f=fF=^ ^ _^2 f^ E m \ \ \ tf:=t=^ FMt G-y i ? _, 11 J IP s>—\-^- Kl I g/l 'S'- 15*- P 2=rs C:I i M TJ 1 ±=t r=a=q=^ i^^ ^t=t3e^^^ ■•— >-• — i^- ^^^^^Slfe g^ :^ -<-i-*-g^— ^- ^=d^:j=^: ^ #=(»= -*= — •- P=:p^ -» — *■ itt±t i -P — n-X^ — s- -^ »— Hi -^-gi a=^ -* eJ ^^ P ^ ^l 5-fQ=^ ^^ w — ^ — ft # — • — I*- # H 1 1 (— *' — '"V 1 ^ £^S :f=^=F^ ^ 33^^ I LESSON LVIII. THE APPOGGIATURA AND ANTICIPATION. A suspension which enters freely, i. e. without being prepared in the preceding chord, is called an appoggiatura. It is resolved down- HARMONY. 189 ward or upward one degree, like the prepared suspension, but in the latter case commonly progresses only one half step. 256. -z^- gi=Sr m i ^ I ■H :Sz i 9^ ^ The appoggiatura may be approached by a skip (if not unmelodic) from any tone of the preceding chord, but tiot from another non-har- monic tone. The only exception is when it moves upward or down- ward by a third to another appoggiatura of the same harmonic tone. 257. m ^ -Hi- =J=^ =B — f-*— *= The appoggiatura may appear in any voice, but is somewhat more easily assimilated by the ear when it lies in the upper voice. It may also occur in two or more voices at the same time ( double and triple appoggiatura) . 258. i ^^ Pi^ i: rrT" i^ f I ^ etc. Such combinations may often be analyzed, however, as altered chords. 190 HARMONY. ANTICIPATION, An anticipation is an unaccented tone which moves to its position in a chord in advance of the other voices. It is thus the literal re- verse of the suspension, becoming a non-harmonic tone by advancing a progression, whereas the suspension becomes non-harmonic by delaying one. delayed, advanced. 259. u=m-=Ur 7^ f=1 f l^^^pl^^^ Like the appoggiatura the anticipation may appear in two or more voices at once. (See Example 260, a, ^.) In fact, an entire chord may appear in advance of the beat to which it naturally belongs, form- ing anticipations in all the voices. (See Example 260, c.) 4 — IV 260. i ^ -*—* «*E^^^^gi -9- ^ I tilt ^fr ?SB I The anticipation is not always repeated or tied over as a harmonic tone, but may progress to another tone of the same chord. This is called irregular anticipation. 261. These are sometimes called changing tones, or Fux changing tones, being a license in strict counterpoint allowed by that authority. HARMONY. 191 EXERCISES TO LESSON LVIII. $E * i. m. t. i -Ps:^ ■z, — ^«z7^K^5:--4fr §±E ej- \ cr i a. Appoggiatura. b. Anticipation. i p •- (^ • d - 3ttt-iZ tat »^ i 1^ ?o=f= JETf^ ^ P^ i-=P=£ \^ i S FF^ F— • ^IrE 1-f-^ i ^-- ! Litt^zj=3 = g_p_!i j^ ^^^^i ^ -* — • Szt 4^3E^^3^3:^5t^^ i ^^-•- I *= #-f-» W- ^^=13 4. is -f-St:^ r^^^x =P=?=if a..L |.L^ Jt^i:g=*=|i=^fci!LP*n:g ! ^^ iS=l^ is 3^^^ ^ ^ ::Pa: •r— t- :f»+|g- ^ 3t=p3 L#E ^ :^= ^ • p rrrr; ^ \\ xJk -I— w — ^ Note: The anticipation is commonly shorter in time value than the tone whick succeeds it. IQ2 HARMONY. LESSON LIX. THE DELAYED RESOLUTION OF THE SUSPENSION. The resoluticJn of the suspension and appoggiatura, whether down- ward or upward, is often delayed by the interpolation of other tones belonging to the chord, or forming an embellishment of the resolving tone. When the suspension comes on the first beat of the measure, the resolution may be delayed until the third or fourth beat, or even longer, by devices like the following. a. u. b. 262. I ie3 m -s* — ig: d= E^ -i Jz e 321 I i etc. s s i a. Fifth of the chord interposed. b. Ornamental or embellished resolution. c. Ornamental or embellished resolution delayed until the fourth beat. d. Ornamental or embellished resolution delayed until the first beat of the succeeding measure. The suspension or suspended chord is occasionally extended during a change of harmony (see Example 263, a, 3), (a.) , (*.) (<:.) J 1 I I I u U 263. ?L-|-»- ^i^ rr^ M HARMONY. 193 or exchanged in position with some other voice (see Example 263, c ) ; and the harmony may be changed at the resolution of the sus- pension by the chromatic alteration of existing intervals, or by the addition of sevenths to the chord, or by any other legitimate means, which do not interfere with the original resolution of the suspension. (a.) 264. 9-f^^ ^ ;* I y a i g-' -8 1 i Here again the sequences of secondary seventh chords (see Lesson XXXIX) are of frequent occurrence. (See Example 264, 6.) i EXERCISES TO LESSON LIX. ^ ^ ii£ -i-»s l^^T- f- ■i i -. -St gfe 5^ (S'-=- ^ i i= I I ^i^ -^ — pi — ^ p^ '94 HARMONY. In harmonizing basses, add the suspensions after the fundamental harmony has been selected. 1 , Bass given. rs \ s>- u ea Z=SI 13e 2. Suspensions in any part. 3 ^ ^ i g:nz3=er:tti£ -m_ EE^PE^E^SS §8 3. 3 tt^ P^^3 eH EE^2S±^^^t#-=^Eztf2 4. 8 3 * ^ :i^= :?-;:*■ ^gsa^ gpj^FE g^p^E^gy g — — «■ i 6. Soprano given. m fc^I - &» r p' EE&¥^ i [0 *l VM^g=fcft': 8. rllfe: s 3tztJ -^- ^ SEa^EiE -#-»h2- :?i=4: =t,spt=±t=t 3gi ^-r^ 9. E^ :?c=i=P=1=:* ^ ^ it=t -•—si- 0]^^^ St e -I — K- - «-i. Z -^-f-y- Ji^-IZEZl 3=tt 1-0= :^izt3tr^=s?±t i I HARMONY. 195 LESSON LX. THE PASSING TONE AND EMBELLISHMENT. The chromatic alterations referred to in Lesson XLII have been characterized as melodic passing tones, and to this class belong also f the successive tones of the diatonic scale which occur between the in- tervals of a chord. Thus : 265. i ::^ The passing tone, w^hich, instead of proceeding to the next har- monic tone above or below, returns to the same one, is called the embellishment. 266. etc. Many ornamental figures and melodic embellishments such as the turn, the mordent, and the trill, have their origin in such nonhar- monic tones. The passing tone and embellishment may occur in any part and between any intervals of a chord, but like the appoggiatura the ascending embellishment is more often by a semitone. (See Exam- ple 266.) Consecutive perfect fifths, formed by the combination of a passing tone with a harmonic tone, are universally condemned in text-books but nevertheless written with impunity by all masters. Such fifths are often of strikingly beautiful effect. Such examples, however, 196 HARMONY. should be regarded as the exceptions which prove the rule, rather than the rule itself. Thus : Mendelssohn. 267 0=e^=t? i ^m 9^ & =t Intervals of a second between a harmonic and a passing tone should not be followed by a unison in the same voices, viz. : 268. §5^ r f^-r-j p ' f f r~^-r I and this is easily avoided by changing the direction of the voice containing the passing tones. -J_ 269. i m r *TT ^ rr r r i EXERCISES TO LESSON LX. Harmonize the given basses with either diatonic or chromatic pass- ing tones (half notes ) in each voice in turn ( three exercises to each bass ) , thus ^ {a.) In soprano. 1 ■ 1 i * ^ S^ — - 121 etc. P zsz ( *.) In alto. HARMONY. 197 ao— ^ ^^^ -(=ht. Is etc. (^.) In tenor. i -J2- 122: 94 J I^^ t-I lj. ^ - i 1 . Bass given. i i Efc^i Eli In harmonizing basses in triple rhythm, two notes in each measure may be harmonic ; suspensions and retardations, prepared and un- prepared, may be freely introduced ; the third of the chord may be temporarily omitted, and the seventh doubled on an unaccented beat. Harmonize given basses, with three half notes in each measure, ia each voice in turn (three exercises to each bass). 3. —<5>- —SI- <> • \ — g)j — 1 !*'•■ ■a . . . -1 — s-^— ^ ^ LESSON LXI. ACCENTED AND DOUBLE PASSING TONES. The accented passing tone does not differ from unprepared suspen- sion except by being preceded by the nearest adjacent tone above or 198 HARMONY. below. The following example shows the difference in effect of ac- cented and unaccented passing tones with the same harmony. 270. II: -F-i- =t=F t =p=i= isi ■^2- -'S^ i I ^ Unaccented passing tones. o Accented passing tones. * Anticipation. f Embellishment. Accented passing tones may occur in any voice but are somewhat less discordant when in the outer voices. They are often combined with other dislocations of intervals (suspensions, etc.) in same har- mony. 271. IE ^ -i-L^- W- 1 etc. -1®- X ■^ I Both accented and unaccented passing tones may be written in two voices at once. Such double passing tones usually move in parallel sixths and thirds, or by contrary motion. Continue the following exercises, with quarter notes in each voice, in turn utilizing accented and unaccented passingtones and suspensions as before. Write each exercise three times. i J ! Continue soprano (with harmony). I?li i etc. 121 ^z: * Accented passing tone in the bass combined with suspension in the alto. I HARMONY. Continue alto (with liarmony). 199 I rrftTrrT I 3. \f—^ T n;:-! \ \ r— 1 1 II t)—^ r^ h^=R= — 1 1 — \ 1 F -#- J 1 t, 1 ^ Continue tenor (with harmony). ^^ ?-Pi- J* . -s> G>— -a — M zf- s> s> ^ t=S-J h^H i 4*. Soprano given. Continue bass (with harmony). I P£^^ LESSON LXII. OBLIGATO MELODY. In the preceding lesson the notes of the running part derived from the given harmony were all of the same rhythmical value. Such a part can have but little melodic significance in the aesthetic sense, on account of its rhythmic monotony, and is, in fact, a species of count- erpoint formed by embellishing the tones of the fundamental harmony. If, however, we add to a given harmony a melody possessing rhyth- mic variety and interest, as well as contrapuntal accuracy, we are at once outside the proper sphere of harmony in its strictest sense. 200 HARMONY. Gounod (in his well known Ave Maria) has added a striking melody of this kind to a fundamental harmony by Bach : 272. J^. " ^- 1 ~~^ -! ! n 1 ^ -l^ V '^ o 1 ! _' . • J ^ 1 / " # ! J i _ -S 1^ f l' ' ^ • " 1 CI ^ ej £• -*- 'St- ly o «'■ ?•/» . K-" _ J V ( / G a a CJ 1 1 i;^ -'^J III ^-4 Uj^.L4f^^ — » — .Ljl? :_ -^^ g _- p *4-a-H etc. » (9 - ■ » ^r^ hi-H No arbitrary rules can be given for the construction of such melodies. The beautiful in art is not developed by rules, but by reasons, and it is hoped that the musical pupil, with the experience gained in the preceding lessons, will be able to avoid the illogical, common-place, and ugly for himself. The cadences on pages 14, 17, 22, etc., may be used as fundamen- tal harmonies for melodies, which for the present are to be written only in the upper voice. Also to the following harmonies. 1. i m 1 ^ ^- V=^ li -b«- -«.- ^=5=^ =^ i i HARMONY. 20 1 2. ^ W ^~^ l=^n^4— ^- IB ■g-l — g— l- '^fs^ Ir -«- -!&■ g^ ■<= «- I l£ ^fe T :i ttg_J-_i ^ z? h g;! — s ? i^ l^=t S: ^^^^^ 2: +— ^ — <& --(Sj- I i-:^- — r 7^ G^ ca^- n»- i^gg 1 li -» — f^ ■J^i- ::s= LESSON LXIII. THE PEDAL OR ORGAN POINT. A tone which is held or repeated against a series of passing chords is called a pedal point, or organ point. It commonly occurs in the bass voice, but is also to be met with in the alto, tenor, or soprano. In the latter case it is sometimes called an inverted pedal. It is most frequently used on the dominant to postpone an ending: Bach. 273. F =^=^ rJ 5=^1 TrxX'-^rt 19 I T- ■~s — ^ 202 HARMONY. F=f=R^: 3=^5EB^i=^;^Eg^^ U=a^^ 9iiz|l2t3=3t ^fet 1 W-^ — ^ =^^ T r r r ^ ^P^ 3tit:ilut^^3zxS 35*^=^-^*^==^^ V -i<- ^±^^ t:j:^j==:g=j:f^ q J Cl -J: r r Or on the tonic to extend and elaborate a final cadence : 274. In this example the pedal point is in both the upper and lower voices. The sustained tone is used occasionally on other degrees of the scale, and sometimes on both tonic and dominant at once : 275. Numberless examples of such passages are to be found, especially ; in works of a pastoral character. The principle involved is the same as that of combined passing tones or passing chords, the moving voices progressing independently of their connection with the sustained tone. Thus many dissonant combinations are formed, among which the dominant and diminished HARMONY. 203 seventh chords on a tonic pedal (a) and the secondary seventh chords on a dominant pedal (6) are the most important. 276. ^^^^^m ^ ±s ^ 9^ pr itis: =Fs: :t:2= ^ f ^r cj"r CTf^ I i£ -•— #^ i^'.-' -1 Xi 11 -2:*- :t: r No arbitrary limitations can be given governing the use of dissonant hai-monies against the organ point. The sustained tone should be a root, third, or fifth of the first and last chords of the succession, and the intervening chords should not be so persistently dissonant as to be unsatisfactory to the ear. The more intricate the harmony, the less should the voices skip, and diatonic and chromatic scale progression are always of good effect. The following is a stupendous example of an organ point on the dominant, from the Symphony in C minor by Brahms. Brahms. 8va Allegro, j ^ ^J _ ^^J , j 277. ^^^^^ ^mrfT^^ ^ ^rr" 8va.. mr- 5^1 i=^i LU m * p=^T=r=f wr\ ^P 204 HARMONY. i^^^^^^^ I etc. .jM*p»-g_.._^^.^,^^^„T_^.:jZS^-_^^^^l EXERCISES TO LESSON LXIII Add the inner parts. tP £^ g5^r^3=^ gi— |^-|-p^^-t-g'-b»'y-fg p"-^^^ i^^^^T^c ^ + ^^JJ^J J J J_ 1 i ia3 ■^ (S 1 — & 1 — &- -S> — Si~ w ^^ES^t^^f '^=^ "^ 3 :5t ^^^ m §iEE i^ p 5EEC=rf=*^ -^ — s- tatz^ I I HARMONY. 205 Add one inner part. 4. =*it -i^«- -iS^I-S-r B i^fe ^ g± ^ ^ ^S=^g; r^r Harmonize, with a pedal point in any voice. ^^^^^^ m^^m Write original periods on tonic and dominant pedals, using the exercises as models. LESSON LXIV. THE INVERTED PEDAL. The organ point in the upper and inner voices is frequently used, especially in organ and choral works. The following is a beautiful example by Mendelssohn : Mendelssohn. j rn i ^1 .^ ! 278, i m r ^ I ^^^^p^^^^^^^i 5ise ^ '^^pf-»-^ V r f^gT^M S: a sm ; 1 ■'?=gFiJ j^ ^ 3C^ ^ P^L .^ feJ,i^4^j!fe g EE^ ^ _4fl I >f 2o6 HARMONY. i ii ggE^^^^^ ;Ek fc^^^^S^3 1 9iib=I Fl7-^=fj ^ ^ i Another one by Bach : Bach. 279. & ^^5 ^ i^ 3^ s=i^ f-^ ±3t fe^ DTI -w^- 1^ ^M^: ^\r4^ T-^ w ^:^±^ -•-'•H ^iJ-r;J. -•-zt, ;s^^ n i^ t-Mzzti :^ :B LJJ -^ EXERCISES TO LESSON LXIV. Add the inner parts. I T — z? I ]- ! rJ -^;J=iS c i\ rj <^ i =a=^=t^ ^^ ^=p:#t=t^^^ -»H-P*- f i ^LL^'=^ i=i ^-. ^ — ^- r r -fS'-r- ^4-hs !— ^I^s — h- ^^ I tf—^ -fS'-r- -!-«• r' -(SI-. -<&- Add the soprano and alto. 6. -•-iS>- _ _ -s> =t=F ::t ^^B =P=F I ^ ^l »=?= b» — f- : ^H-H-^^ F=^ Write original periods, with pedal notes in upper and middle voi- LESSON LXV. MELODIC FIGURATION. The elaboration and embellishment of a melody by means of non- harmonic tones is called figuration. The nonharmonic tones are often combined with the harmonic, in arpeggio forms and otherwise, and all forms of nonharmonic tones may be utilized. 2o8 HARMONY. The following examples illustrate melodic variation oi a given melody, the fundamental harmony remaining unchanged : (a.) Original melody. 280. =P=^ • — • i i Beethoven. « ^ f±^=t :t: fifa&ff y^ Etea^JlB s a a&g ^laE SBaB ^iB^ ^ ^ (6.) Figuration. f^^,-i=^^to^ ^Sl ^grr^Frf I ■P-^ — — h*lF-#H — — — I — r-a-i — r r 1 i i O — y- t=ft=f^ i^tri 1^ — ^,^^y_^^ (a.) Original melody. Chopin. 281. m ^^^ Efe :?=* m (i.) Figuration. 5 ^^ 14 .HARMONY. (a.) Original melody. 209 Brahms. 282. i a-itr£ te4E^3 It ■ etc. {i.) Figuration. ^^^ ^ ^^^ ^ jzt etc. ^ The following example shows the different species of nonharmonic tones, applied to a single given tone : (D) Accented Accented passing Suspension. Appoggiatura. Anticipation. tone. 283. fe :^^ tsg'- i=qEai r^=T- ^^^^^^ m Retardation. Unaccented passing tone. Embellishment. I J I- i^ ^^^ -iS'-J- r r r f f I^^B TI^ -TS)-. All nonharmonic tones are governed by the same general law as the discords in counterpoint, viz. I. Accented discords must be prepared or enter from a tone oj the previous chord. II. Unaccented discords are introduced and resolved by dia- tonic succession. No arbitrary rules can be given for the selection of nonharmonic tones in a melodic variation. As the object is to evolve a simple and graceful melodic outline, unmelodious successions of intervals are principally to be avoided. The given exercises are to be worked as follows : First. Harmonize the given melody. Second. Make a figuration of the melody, two notes to each beat, and reharmonize, retaining the original bass. 2IO HARMONY. Third. The same with four notes to each beat. Fourth. The same with three notes to each beat. 1. :fe I 13 T— Ig- -g eJ fa^^^7^P ^ |^ #fr^.Vfi^ i m fc I P .S>--i5(- »— i^- -sf «> -S-^fS- 4. i ^i -tz=£: j=[j=d= i« — •- r^zzd* f M f * ^ F=^ itit 15^3^ -p — •' m ^ ^=JB=E fe 3F:=t i I 2±: Dtrt LESSON LXVI. HARMONIZING OF FLORID MELODIES. In harmonizing any embellished melody the harmonic tones must first be mentally separated from the nonharmonic ones. The process is the exact reverse of inventing a "melody for a given harmony ( see Lessons LXI and LXII). On general principles, the harmony will HARMONY. 211 consist principally of the three principal triads and their inversions, with modulations to the adjacent keys. Tones progressing by a skip are usually ( though not invariably ) harmonic in character ; the em- bellishment, on the other hand, being diatonic. A melody may be simplified by omitting the ornamentation, ren- dering the fundamental harmony somewhat more obvious. Thus : a. Original. 284. i a iE 2 ^^^^ip=p=i= -j_jq_itJ :^3q:p ^|^;ir p= :3z±lz±di j^q -*-=-^ -^ ' — I — Vi-j — m — *— « ¥- h. Simplitied. ^'^r •- T* \ 1^-^— -r" — 1 — yj 1 • — 1 — 1 1 9—'^ -1 i I In harmonj-iing the given exercise : I. Reduce the melody to its simplest form, as above. II. Harmonize the same, as simply as possible. III. Write the original melody with the same harmony with what- ever alterations are necessitated by the part writing. IV. Harmonize the melody, off-hand, without reference to the studies, I, II, and HI, already made. I I , I I I I v^ i ^^^^^ '¥- Et^ -¥- P^ ^ i ±ita3t?;t i — I- g X=^ i=i= -«>-=- w ^ ,. I j J J ^a-^^^^E^ I I I [— r-y- 912 HARMONY. i ai ^£B - fg »- m^s^B^i^^^^^^^^^:,^^^ r I i I r i te ii^^=F t T=^ i^ t==l==^-l— t 3. i w E±Ei^^:^ii^^ =^=^ • '-SI — ^ 3^^S£3^ 3=^ ^g :4 i^ r ^^gi^^^^^^^^l 4. # • ■ ^-» fe *^e4 m^ ^^^^^^^ 1— rT=i' I ¥' i ^=f^^^ :t ^^F=^~rd'f ^g fctacF^t ^pg * EfEEfEt =a^ ^ 6. Bass given. gWFfW^^^^ HARMONY. 213 ^^E^^^^^EEiE^ ^ i «h ^ f i g^^ai^g •-f-#- p=ii rm^:3;1= ^^pjj^_=J=j--^-^=^ ^ I I I ^J-J ^ ^"^^^^f: ii ^J^|J.^i^ J3^^^ g^ggg^a LESSON LXVII. ACCOMPANIMENTS. Accompaniments are formed by representing the tones of a funda- mental harmony in succession or by repetition, or as combinations of both, forming figures which are reiterated at each accented beat, or oftener. These conventional figures, of which there are practically an infinite number, trace their rhythmic origin to the broken chords and arpeggio forms in one, two, three, or even more voices. In the more idealized forms, the nonharmonic tones and embellishments are also freely combined with the tones of the chord ; two ( or more ) figures of accompaniment may be used simultaneously; contrapuntal parts are even introduced, or secondary melodies in the form of an obligato added to the accompaniment. The function of the accompaniment is to furnish a harmonic and rhythmic background, which shall enhance the beauty and effect of the melody itself. 214 HARMONY. Some of the more common figures of accompaniment are as follows. Beginning with the broken chord form, in one voice : («.) (*.) ^^^ £ M ::p=t: '-0-^-lf ("-a-^ ^i 285. *S^ ^p=w-o (d.) («.) i ^ p=.-p=«=t=pq=r1=p=t=E=p=4 ^^^1^ I ^^ =P=t=t -i — I — I — I- etc. H 1 1 1 1 In two voices. (a.) (*•) U.) 5Et 2Et^3E 286. W.) ('.) (/■) ^ L — t— .C^^ — t_J— 'h — ■ I III -' — I — I — I '-I — h=^^ — I — ■ 1^ — 'J In three voices. ic.) J=i '■ ^^ ^^m^^m. 287. =P=t^ (rf.) («.) (/.) ^^^i^=||ei^^^ I -J- 8' ^ 3t=t etc. With embellishment, (a.) =P=i=iE 1 288. ±=t=t W (rf.) i^- n^ («.) :t:=t:=p=t:ii=pi-|=qcif=p:|i=iK ^^IJetc. ^i 'EtM -, — '. — r HARMONY. 215 Two simultaneous figfures. (a.) 289. i (*.) W- §a m r r (^.) (or.). ^H^^ (^ a^ By repetition. (a.) 290. M i^ 1-1-1-31 -s- -S- -8- -•- ^ -71^ y (^.) («0 -' -■ — • — • — • — • — ■ — • — "-■ IF 1=^1= -*- v V -^ -#- etc. I J J e ^—t=t=i=^i=iF=t^ I gg^ The following examples are from the " Songs without Words" by Mendelssohn, which present a remarkable variety of accompaniment for lyric melodies. 29..S^^^^g r r j 2l6 HARMONY. No. 6. ^K^^ ri J^ ri -.n -*^ OE ^■s=q: r— "T -n-L#. No. 7. it |te ^TXlt ' zJ Uj 1 — T S^E r r No. 14. k=f- m ^ J^^: it:gjzl:3=^ I S23; p P P p p i ' ^ I I B S « -?-^?- -fL^ -»—»—»—»—»- :fc:4: a »= ^b:H=f^-^ S Form the following chords ( at the pianoforte ) into figures of ac- companiment like the examples at 285—290. -&- s- 'M'- -g- ^z i^i^ri^^^ -^s'- Rewrite the following harmony in the form of an accompaniment, according to above models. -s>- -%' -&- JS- -52. JSL. rs -&- ^ -9- f: ■^'^ -^- -«>- c\- " _^ -o % Ki ,(5^2. $* a ■ s^ (^ in -S u ^ i./i ej CJ s f ^3 M ^ \ 1 XJ "s^ k^fs— 1 -^^ u rs ^1 H ^ The cadences on pages 14, 17, 22, 28, and 36 may be used in the same way. The melodies given in Lesson LXVI may now be accompanied, with- out altering the harmony previously written. Accompanying figures should be chosen, best adapted to the character of the melody, and each piece when completed must be possible to play. The simpler melodies in the preceding lessons may also be used. HARMONY. 217 LESSON LXVIII. THE CHROMATIC SCALE, HARMONIZED. The nature of chromatic progression is such that the greatest possi- ble scope can be given to any harmony which accompanies the chro- matic scale. The simplest method is to consider each sharped or flatted tone as a chromatic alteration of one of the degrees of the dia- tonic scale. Thus : 292. 5=^ ^ ■ffS^ '^^l -sir- r ^F=9f i^ ^^PE s- ^=0^ m pf^^ i =i ^4 J t. . ^ ^^tei^pg^is g r ~r~ p = i7V r^=-rl=2- k j^-^?:- I It ^ :|=t ^ I — 'r— r ^P=r PZ- Or we may modulate in sequences between the tones of the chro- matic scale, regarding them as the roots of a series of tonic chords in a circle of major seconds, major and minor thirds, perfect and aug- mented fourths, etc. For example : ( For the sake of convenience the chromatic scale is here based on F.) \, F G A B Db E\) F 2l8 4. F HARMONY. Bo E\> 3 l^fl ^ =t=t fe I 5 ±=4 i^^^ S :it^^ :#?^ p=^-| -1=2- 5F=t=t I The harmony which connects the tonic chords must progress as smoothly as possible. Each tonic chord will be immediately preceded by a dominant, or diminished seventh, or other chord of which it is the resolution, and any chord may be written enharmonically. Continue each sequence as here indicated. Each exercise should be carefully written out and transposed, at the Pianoforte. The dia- grams in Lesson LIV will be found useful in this connection. 1. te^^^i^^ j?g=t|grzp ± -bs etc. ilEi e: ? Also beginning on E. 2. 1:'^=l:t2 ^= i^^gg^*! ^:33Js: .;s>_i/ia!_C(S :t==l=t n li^^^ i=— rf^-iW-S' ES Also beginning on E and Eb. 3. -I — ^ 1 V- s r#^lff ^=S^j^fe- i^i^ i=i=l .^i --m Also beginning on Gb, G, and Ab. HARMONY. 219 ^gffl -VV^ la r ' -J. ps^^^t^^ i etc. ^|g^=F m ea eS Continue the circle of fourths until complete. 5. :&^^ ^ s-"fi=^ s^s^^a^ ^^ Continue ii J J_ 1^^ J. in open position. ^i fei§ i 51= Also beginning on Gt>, G, Ab, A, and BP. The harmony of the descending chromatic scale may be founded on the following sequences. I J2^ s I rWU-^- m^m^^^ 4: etc. I ififeg IS :^ Also beginning on E. 2. ^g-ftr^ ^^g p=B g>°2r p p i etc. I 9^§^ e= #^ 3^-ttz:^ Also beginning on E and Et^. 3 ^ ^' g fg ! > ; s i ^ #g~^r^^P i^i* P f= 1^ 'isH- z^ i g ^ ' etc. I i_^ eE e) ei ei \ m ■^i Also beginning on E, Eb, and D. 220 HARMONY. fe ^j^ fej;^ I w^ tTT ^^-•-s^ etc. Pg ^Op-r-r- iE^ etc. 11 =p=^ ime Continue the circle of fourths until complete. i*J=i^ 1^1^'^i ^ f^i ji^t^t^j.^^ ^ B etc. ^g^ f=f^r= ^ §iiS s= -#i Also beginning on E, Efe, D, Di?, and C. Sequences may also be formed of tonic chords in the third aiid fifth positions, connected by modulating chords. The student ih recommended to invent such forms for himself and carry them through all the keys. LESSON LXIX. THE FIGURED CHORALE. Passing tones may occur in two or more voices at once, provided that objectionable progressions (parallel fifths, octaves, etc.) are avoided. Thus it happens that parallel thirds and sixths are of very frequent occurrence as passing voices and are sometimes combined with a fourth, forming passing chords of the sixth, and sixth and fourth. ( See Lesson LXII.) HARMONY. 221 294. i w ^ ¥mi^m etc. i By combining the different species of nonharmonic tones, in har- monizing a given melody, the voice parts are made interesting and melodious. In fact, the licenses of progression which are allowed in part writing are entirely due to the independence and individual interest of the separate parts. Thus Bach illuminates the following simple harmony : Original harmony. 295. Bach version. T-r ;g^q|=r -^ ^F=t i=r m T 3 B T^ ^ =t=t i^ I The following chorals are to be first worked by altering the given parts (except the soprano ) with nonharmonic tones, so that a contin- uous rhythm of four quarter notes is present in every measure. 1. ^1 wm ^^M i '-U=^'^- --'g^ -si. E 3= ^=g: -a-'g J. I m& ■p-f -(2- JJ- J- S^S -l=--(2- i & ^r-1 P m 2. ife i ■^ ■i n ^ I ,1 I-' -^ 151 — s- r^^"T^[ ■- ^=t 2±£z=tr== ig^ ?^=S 2. a%^-^LJq:p=ffftf=ff^°=^ ! ^'Trg a - r? .- i I i^ ^- -fS- -s>- 42-.es-. etc '& ^- :fez i II. With nonharmonic tones in the other voices : i ^ i -Sl-r- w ky-"—^- ^- -^ etc. i— rfr §3 1 — i- ^ III. And in triple rhythm : i 1 — ^.1 j «E3 1-« W— I &^-I^ f^ 1^ -— J-s;) g ' a; ^ p^^gE^^i^ 3 - rn — g= q: g^tfg= =#^ 3z6 HARMONY. i 3 m s No. 2. The following is to be worked the same way, in the tenor. ^^^ ^^^m ^=:f=F i iS> |S> 'S> «>■ IE m m =S=P= -g* — ^ P=^F=^ f^ ig— -7i s)- LESSON LXXIL ANALYSIS. In analyzing the given excerpts the follo^ving method is to be observed : I. Copy the piece carefully. II. Analyze the piece harmonically, indicating the derivation of each chord and all the modulations. Chromatic changes may or may not produce modulations. In all cases define the harmony as simply as possible. III. Indicate carefully all the nonharmonic tones ( suspensions, anticipations, organ points, etc.) and account for every tone. The following model shows the method of writing out the anal- ysis. This work may profitably be extended indefinitely. A com- prehensive study of the chorales by Bach is earnestly recommended gs H B HARMONY. & ■ft- -^ -f- * ^^^ i :^= *=J3 « £ t3 .£ 10 £-° * I i! 337 m i •a o Q •2 .i Q 2 i The chorale is continued as follows : 1 I i 1 — r 1^ ^1 -0-^^ ^^ w m 9^ ^t^ ^ ^£ ^P= EP ^^f-j-j =*=?= =i ^EIE F=s=i= ^*;*= =& ^ * — ^ I /^ tj =?t=p= i ^ ^^^fi^ J^^iJL^ ^^Effl :t=t ES t=N= ^ ^^^^ I t;-r-gri^ ^ i^^tef ^ i 228 HARMONY. 2. Andante canlabile. ir. Mozart. M i ^M r^ =^ ^d^j^^ §±i^=P l^^g I3E r 1 -x^- E£i^£ * ^^^J gj-A -■P^ i W H^^-r-f-V t -0 ^ =P=p: rj- m^^ is s^^^^^^m^. i-,-1- IfflS =^^=F cg^ jE r^j^=j^^p^^^ H ^:^ Haydn. r ^^ ^'-l^r^^ ^ I/Cj -r i^^^^i m^. ' ^ rj - s HARMONY. 229 4-. Adagio. Beethoven. S iSz ^^^^^^ 5^ 1 I I ^ ^r-^ ^^^^ 1^ ■^ — ^ -•-■ ! -•-^- ^ -I 1- It=t :}=:1^^ -* << - f-; ' tti^ i^ > =M^ -4^.J- izi 1^ I Mendelsohn. m^ ^= i=^ . -p-i—f^ =^?F^ ip f :it*- i i^S^iEE^E 3t=S= =4^ :^ ^i a S^-^-£S ^=^^ i=^ I a J — L.-5. — - ' j I — ^ — p^i^ — 1 j-i — H^— ^ Tt±Xtt iMzF -- g- r^P , T - -O -! 1- < » . J - — e a 6. -sf- SCHUMANN. ^ il I I :^ ^3#. I I I ^33=^ ii^s 23° HARMONY. ^ ff — ^— 1 — -^ ,-.r rr^_ -I I [— ^-•t-*-^-" iiitt i^ — s- :^— T— ,-^i * ^ : — ^ It: c=t=3=3EEz3i«!ild ^ i Waoneb. ^i-a -^-9 — i— -« — •- ^ Wf ;-:j: :^- 1=* m^^^^^^ agi I peE|E ^:^: ±±^z stzS- rrr :ri *i=FP^^^=^J t« 3 -»t-» — a — i-H;t3 — ^— a-d— ^■^— •-M — !-• — V'i-^ — I — |- li: -1-. -I- -4 ,c-H- -d- -^ W-31- ^-tt=t -Tfq 1— I — I nr-t ^tt"'- -«•<= I*: !^:=5=td 8va., \ 44 ^fiM -ffli ;=2r»S3i ??^ lic*- ■it?= -•— (i- -J- -^ r :^d; ^^§^^a 9tt ip^- r r --^#-^ r :g= :ti#t HARMONY. 231 fefe^ ^^ r#f *f ^ S-T- T- lit :♦• -1*- 15: IB i-rSt r m= i^^m t=i^=^ I CHORAL MELODIES TO BE HARMONIZED. 5i=P= ^ :f=t^ ^ It -•»E itrt ^ t=^ -•— hatizt ■I *^F — h- # -^ ,*- ^m i =F=P= 3*-#-*- i=^ ■tit ±=3t *— T^ i ^^ me ^» — •- ?=p= ^tji^^^^:^ P t=r : ^ I ^ =t=F i^^^ -f-,«-1— it: # ^ =*"-=?»=*: -• — ^ ■# — •- •|* • f g~ il IF :f=* P=F i^ =feiE :^r ^ ¥ 4=t 5=^ S :^ -^snis ♦zz^ -e— =5- t=*: -• — ^ I 332 HARMONY. ^ms^ m J \ I -^-m-^ 7t=3tZ Itt-^— .- 5=p ^^^^ r^ fe^^^B ■#^t^^^^^sEEt? ■F-=-»-» • — F-| ^-= • — i- =£^ e^g ^g ^^^ i^ -fg #— =p — ^ E: 3 H« •- m m f=r=^- S -A^ lis T— jg I ^^y — • — — m — fg (g- -(2 fZ- ^=f^rri 1^ -H*- • — ^-t-VP — ^ W ^ — I I- c^ HARMONY. 233 tt I * I * ^ t=t f 4 •- f F 4 H? :t=t i M p^p^ «= H=^-T~r"gg^f^ ^ * I '^ '-^ Si 1^ trrrtf^ ^p^i ^r ii I ii I u [^ al: ^ a= =p=p= I #— #- ^^^^ f • • s)-. w ttP=t: :^ ^it^ =3=P= «: ^=3 ^ • ^ I* ^ -S #^—3 i F^=P= I 10. f^ f4F^t^=^ ^ rS' •- -iS' •- -f—^ ^^E ^ r? > -fZ- w 1 1. ^^ ft— I — r- -d * si- i m^^ ^ ai •- irS: I 234 HARMONY. 12. g P *=t ^ •- =tiz ^ ESi -^ — » :t=t * al- -•— ^ m. m j-4i_J J 1 B-« J IJ ■^X=^~ m^ itrt • — *- 13. i^s n= H«— I*- i ±E3E ^f=F= -f — ii- -ci- It=t •g — -zi^- i t=^ ^H g= 35--3: -^ — ^- 3t3t f ^- '^=-\ - St ^ 14. i 5^ s^ 1^2 p=4=- ■ici- •sH — « g p^ ^rr=p= ?^^^^ -i i> 4 —