STACK ANNEX F.H.5HEPARD NEW YORK C . PODPIC TWENTY-FIRST EDITION R EVISED AND AUGMENTED HARMONY SIMPLIFIED A SIMPLE AND SYSTEMATIC EXPOSITION OF THE PRINCIPLES OF HARMONY DESIGNED NOT ONLY TO CULTIVATE A THOROUGH KNOWLEDGE OF CHORD- CONSTRUCTION BUT ALSO TO PRACTICALLY APPLY THAT KNOWLEDGE AND TO DEVELOP THE PERCEPTIVE FACULTIES BY F. H. SHEPARD AUTHOR OF "HOW TO MODULATE," "A KEY TO 'HARMONY SIMPLIFIED, 1 " "CHILDREN'S HARMONY," "GRADED LESSONS IN HARMONY," ETC. Price, $1.50, net NEW YORK G. SCHIRMER, INC. 3 E. 43d STREET Copyright, 1896, by G. SCHIRMER, INC. Printed in the U. S. A. PREFACE. This little work offers no apology for its publication. It aims at the following distinct objects: I. To treat tL* subjects of Scales, Keys, Signatures, and Intervals so thoroughly that the pupil will be prepared to understand with ease the principles of chord-construction. II. To present the subject of Chord-Construction in such a man- ner that the pupil will be obliged to form all chords him- self, thus deriving a practical knowledge of the subject. III. To discard all arbitrary rules. Instead of blindly struggling with a mass of contradictory rules, the pupil is made acquainted with the original principles from which the rules are derived, and his judgment cultivated to apply them with discretion. IV. The principles of the natural resolution of dissonances are shown, instead of giving the rules for the resolution of chords of the seventh. The pupil will apply these principles not only to chords of the seventh, but to all fundamental dissonances. V. The chords of the Dominant Seventh, the Diminished Seventh, the Major and Minor Ninth, and the Italian, French and German Sixth, are shown to be but diffeient forms of the same chord, with a perfectly uniform reso- lution, thus enormously i educing the difhcuicy o under- standing these harmonies, and diminishing the complex- ity of the whole Harmonic System. VI. The system of " Attendant " Chords will be found very helpful in under- standing those chords which, though outside the key, evidently are closely related to some of its triads. It is also of much assistance in reducing the art of Modulation 2223840 iv PREFACE. to a condition in which it can be studied step by step. VII. After the regular course in chord-connection is com- pleted, a supplementary course of study is outlined, in order to gain proficiency in practically using all the means of giving variety to a composition or improvisation. This proficiency is indispensable to young composers and organists, but it is usually allowed to develop itself, as nearly all manuals of Harmony stop at this point. To expect a pupil to be able to introduce Suspensions, Pass- ing-notes, Sequences, Anticipations, etc., into his improv- isations, or even into his compositions, after reading the explanation of them, is like explaining to a novice how the Piano is played, and then expecting him to be able to per- form. VIII. A course in the Development of the Percep- tive Faculties is given, training the pupil to listen intelli- gently to music, to distinguish between the various chords, etc., and to write, in musical notation, what he hears. IX. A chapter on Musical Form is added, together with suggestions in regard to the Analysis of standard works. Owing to the pressure of professional duties, as well as to the consciousness of his inability to improve on them, the author has taken the exercises with figured basses chiefly from the "Manual of Harmony" by Jadassohn, and the "Manual of Harmony" by Richter, indicating the exercises of the former by the letter J., and those of the latter by R. These exercises are supplemented by others, designed for special purposes. TABLE OF CONTENTS. PART I. SCALES: KEYS: INTERVALS. SCALES AND KEYS. CHAPTER I, pp. 3-25. The major scale Sharps and flats Double sharps and flats Keys Signatures Circle of keys To distinguish keys having many sharps or flats Relative sharpness of keys and notes Related keys Specific names of scale-notes Rela- tive minor Chromatic and Diatonic Synopsis Historical The perceptive faculties. INTERVALS. CHAPTER II, pp. 25-42. General names Specific names Standard of measurement- Major, minor, augmented and diminished Extended and in- verted Consonant and dissonant Application of terms- Definitions Enharmonic Historical Perceptive faculties Complementary Intervals. PART II. CHORDS. TRIADS. CHAPTER III, pp. 43-66. Foundation of the harmonic system Natural harmonics Triads Marking Specific names Principal and secondary Doubling Position Four-part writing Connection of triads Consecutive fifths and octaves Open and close harmony Connection of triads in minor Harmonizing the scale. INVERSION OF TRIADS. CHAPTER IV, pp. 66- 81. Figuring Figured bass Hidden octaves and fifths Per- ceptive faculties Transposition. CHORD OF THE SEVENTH. CHAPTER V, pp. 81 - 100. Its construction Resolution Inversions On the Preparac vi TABLE OF CONTENTS. tion of dissonant intervals Cadencing resolution Leading of the parts Influences, combined and opposed Directions for part-writing. INVERSIONS OF THE CHORD OF THE SEVENTH. CHAPTER VI, pp. 101 105. Figuring and naming To find the root Resolution. SECONDARY CHORDS OF THE SEVENTH. CHAPTER VII, pp. 105-127. Formation Resolution Preparation of dissonant intervals Succession of chords of the seventh Secondary sevenths in minor Inversions Cadences Closing formula Non-ca- dencing resolutions Analytical and comparative review His- torical. CHORD OF THE DOMINANT SEVENTH AND NINTH. CHAPTER VIII, pp. 127-130. Construction Resolution Inversions. CHORD OF THE DIMINISHED SEVENTH. CHAPTER IX, pp. 130-136. Construction Use in major Similarity of sound Resolu tion Inversions Figuring. CHORDS OF THE AUGMENTED SIXTH. CHAPTER X, pp. 136-145. Are altered chords Construction Resolution Upon super tonic Recapitulation. ALTERED CHORDS: FUNDAMENTAL CHORDS. CHAPTER XI, pp. 146- 163. Description Change of root To distinguish between altered chords and foreign fundamental chords The discovery of roots Ambiguous chords Altered chords in general use Neapo. 1 - itan sixth. FOREIGN CHORDS. CHAFFER XII, pp. 163- 170. Relation of dominant to tonic The system of "attendant 1 chords Their influence upon modern music Various forms, minor seventh, diminished seventh, etc. TABLE OF CONTENTS. vi MODULATION. CHAPTER XIII, pp. 170-184. How effected To connect any two triads Use of " attendant " chords To connect any two keys Formula for modulation By means of dominant seventh By means of closing formula By means of diminished seventh To any chord of new key Change of mode. PART III. VARIETY OF STRUCTURE. CHAPTER XIV, pp. 185 - 193. Suspensions Anticipations Retardations Syncopation. UNESSENTIAL NOTES. CHAPTER XV, pp. 193-203. Passing-notes Auxiliary notes Organ-point Invertec pedal General recapitulation Tabular view Essential and unessential dissonances. MISCELLANEOUS SUBJECTS. CHAPTER XVI, pp. 203-213. Cross relation The tritone Treatment of chord of six- four Licenses Sequences Related keys Naming the oc- taves The great staff The C clefs Chords of the eleventh and thirteenth Open harmony Five-, six-, seven-, and eight- part harmony. HARMONIZING MELODIES. CHAPTER XVII, pp. 214 - 223. The cantus firmus The chant Speed in writing Practi- cal application, ANALYSIS AND FORM. CHAPTER XVIII, pp. 224-235. Method of procedure Sonata-form How to trace the theme Harmonic analysis Rondo-Form Primary form Phrase Period Motive Thesis and antithesis. IMPORTANT. NOTE I. The student is urged to make frequent and persistent use of the keyboard for all appropriate exercises here given, for by this the practical efficiency of the study is greatly increased. Exercises in Scale, Interval, and Chord construction, in Chord connection, and Chord resolution, are suitable, and also the exercises in Part Writing, under proper conditions. NOTE II. For use in Class Drill the "Keyboard Diagram," published separately, is of value, for by its use a large class may receive the same practical and thorough keyboard drill as the single individual at the piano. NOTE III. Teachers (and those studying without a teacher) are referred to the author's Key to " Harmony Simplified" for nearly five hundred additional questions for use in the class-room, and for other hints in regard to the use of this work ; also for solutions of all the exercises contained herein. See p. 24. " Graded Lessons in Har- mony, by the author, contains further material for teachers and students. .HARMONIZING MELODIES. NOTE IV. For special work in harmonizing melo- dies see Chap. XVII. This work may be commenced after Chap. V., or even earlier, if simple exercises are chosen. HARMONY SIMPLIFIED PART L CHAPTER I. SCALES : SIGNATURES : KEYS : CIRCLE OF KEYS : HISTORICAL : THE PERCEPTIVE FACULTIES. Construction of the Major Scale. 1. A Major Scale is a succession of eight tones 4 placed at a distance of either a Whole or a //iz/^-step apart. A Half -step or Semitone, is- the smallest interval formed upon the Piano-keyboard ; that is, from any key to the next one, white or black ; e. g., C to Dfr: E to F : Afl to B, etc. A Whole Step is a step as large as two Half-steps ; e. g., C to D : E to FJ : GJ to AjJ : Bb to C. 2. The eight notes of a scale are called Degrees of the scale, and are numbered from the lowest, or Keynote, up ward to the octave of the keynote. 3 4 HARMONY SIMPLIFIED. 3. Notice, when playing the scale of C on the Piano, f .hat from the 3rd to the 4th "degree, and from the 7th to the 8th, are ^a//"-steps, while between all the other degrees are 'whole steps. This forms our rule for the construc- tion of any Major scale, ( also called Diatonic * Major scale,) 'without regard to the starting-place. There- fore, we will write the succession of figures, indicating the position of the half-steps by the sign - ', thus making a Formula, or general pattern, by which we can con- struct a scale starting from any note ; thus : i 2345678. Briefly expressed for memo. rizing, this formula is as follows : The ffalf-sivps are from 3 to 4 and from 7 to 8. All other steps are Whole steps. 4. To illustrate this formula, let us begin on the note G, and, following the above rule, form a scale : G A B C D E F G. Let us examine this step i 2 3_4 5 6 7^8. by step, comparing the notes with the formula : 1 to 2 should be a whole step, i. e., G to A is right. 2 to 3 should be a whole step, i. e., A to B is right. 3 to 4 should be a half- step, i. e., B to C is right. 4 to 5 should be a whole step, i. e., C to D is right. 5 to 6 should be a whole step, i. e., D to E is right. 6 to 7 should be a whole step, i. e., E to F is wrong, since E to F is only a ^a//"-step, where a 'whole step is required. To correct this, F# is used instead of F, giving the proper distance from 6. 7 to 8 should be a half-step, * The word Diatonic means literally " through all the tones." Its applied meaning is, that one ( and only one ) note is to be written upon each degree of the staff. It will be seen later that the word is also used to refer to scale-notes, to distinguish them from notes altered by accidentals. ( See 44.) HARMONY SIMPLIFIED. 5 i. e., F3 to G is right. (The FJ really corrects two faults, as without it the step 7 to 8 would have been too great. ) Expressed in notes with the formula, the corrected scale reads as follows : ^, /^ In this way the pupil should test each note in the following exercises. 5. In constructing scales, observe the folio wing points : I. Do not write two notes upon the same degree of the staff; e. g., A and A$. n. Do not skip any letter; e. g., /E ^ *? |. (The letter B is skipped.) ^Er~ NOTE. The word Scale is derived from Scala, meaning " ladder." The lines and spaces are used consecutively to form a regular series of steps, ascending or descending. If two notes should be written upon one degree of the staff (e.g., I), it would be necessary to omit the note on the next degree (e. g. , II) to make up for it. Such a method would make a very irregular looking scale or ladder; t g., Hi. To avoid the errors mentioned in I and II the beginner should always first make a skeleton, or outline, of the desired scale, i. e., the notes only, without sharps or flats, writing the formula of figures underneath. After- wards he may bring it to the required standard of steps and half-steps by using sharps or flats. For example : Fl *- 3 life O 6 HARMONY SIMPLIFIED. The next step is to "write in " the sharps necessary to make the notes correspond with the formula ; thus : 1 to 2 should be a whole step ; a whole step from F# is GJ: therefore, write a sharp before G. 2 to 3 should be a whole step ; a whole step from GJ is A$ : write a sharp before A. 3 to 4 should be a ^0//"-step ; a half-step from A# is B is right. Proceed in this manner till the scale is completed, result- ing as shown in Fig. 4. Exercises. 6. (.) Construct the Skeleton and Formula, and write Major scales starting from the following notes : C ; G;D; A;E;B;F8;C&. (.) Construct the same scales at the keyboard. Double Sharps. 7. Write the scale of Gft as above. N. B. It will be observed that the step 6 to 7, from E# a 'whole step up- ward, is not properly expressed by simply writing F#, as that is only a ^a//"-step from E#. It is here necessary to raise the FJ another half-step, to make the required dis- tance from E#, which is done by using a double sharp, written x giving "^ ^ =[4 *-> 6 7 Exercises, Write the scales of D#, AJ, Eft, and BJ, using double sharps where necessary. Repeat at keyboard. HARMONY SIMPLIFIED. * The Use of Flats. 8. Flats are introduced where without them notes would be a half-step too high. For example, in the scale starting upon F, (write it,) the interval from 3 to 4 is a whole step, while the formula requires a half-step. This is rectified by the use of a flat before B. Exercises. Write the scales of F, Bb, El?, Ab, Db, Gb, and Cb. Repeat at keyboard. Double Flats. 9. In the following scales, double flats, written bb, will be required. From the foregoing, the pupil should be able to find the reasons without further explanation. Exercises. Write the scales of Fb, Eft?, Ebb, Abb, and Dbb. Repeat at keyboard. Advanced Course. 10. From a consideration of the above it will be seen, that in one sense there is but one Major scale. The so-called various scales, F, D, C3, B&, etc., are but exact reproductions of each other, varying only in pitch. The name of the scale, therefore, merely indicates the name of the starting note or Keynote. There is a popular idea among Piano- pupils that the scale of C Major, having no black keys, is the one per- feet scale. But it will be at once seen that the Major scales are all alike in the manner of construction, the black keys upon the Piano simply serving to bring all the notes of the scale into proper relationship with each other, i. e., at the proper distance from each other. For exam- ple, it should not be said that there is a wide difference between the scale of C and the scale of D^, because one has no flats and the other so many. Rather should it be said, that these five flats serve to make the two scales alike, by keeping the series of steps and half-steps absolutely the same. Keys. Regular Course. 11. After writing a few scales as above indicated, the 8 HARMONY SIMPLIFIED. pupil will understand that the notes of the scale bear a certain relationship to each other. The starting-point of each scale is termed the Keynote ; the group of tones composing the scale, considered collectively, is called a Key. Signatures. 12. Exercises. () Returning to the exercises in 6 and 8, the pupil will gather the sharps or flats used in con- structing each scale, and place them in a group immedi- ately after the clef, thus forming the Signature of the key. Signatures are a result of this uniform construction of the scale, and not the cause or origin of the various keys. (<5.) Recite the order of sharps in signatures. (c.) Recite the order of flats in signatures. Circle of Keys with Sharps. 13. In forming the key-signatures as above, notice: ( a. ) That each successive scale has one more sharp than the one before it; e. g., C has no sharps, G has one sharp, D two, A three, etc. 14. (b.) That the note on the 5th degree of one scale is used as the first note of the next scale; e. g., No Sharp. Fig, 5. 15. ( c.) This succession continues till the note Bjt is reached. This note being the same as C natural, we may be said to have completed the Circle of Keys, startingfrom HARMONY SIMPLIFIED. 9 C and continuing till the same note ( though called B# ) is reached. This is called the Circle with Sharps. 16. ( d. ) The sharps or flats of a signature are always written in the order in which they successively appear in the Circle of Keys; e. g., FJ being the first to appear, is always written 'first, at the left, no matter how many sharps there may be in the signature. CJ, being second, always comes next to FJJ in any signature. Written in order, and numbered, they appear as in Fig. 6. Fig. 6.* i Notice, also, that if a certain signature has one sharp, that sharp will be the one at the left in Fig. 6. If a signature has two sharps, they will be the two at the left in Fig. 6. And no matter how many there are, those at the left will always be included. To learn the order in which \\iejlats appear, observe the order of their entrance ui the illustrations and exercises in 1922. 17. ( e.) It may be especially noticed, not only that the note upon the 5th degree is used as a starting-point for the succeeding new scale, but that the last half of one scale ( four notes ) is used as the first half of the next new one; e. g., Fig. 5. ( See also 32 and 45.) 18. (y.) But one note (or letter) is altered in passing from one scale to the next in succession. This altered note is always on the jth degree, and is shown by the added sharp appearing in the Signature.** * This order will be observed by reference to the entrance of each successive new sharp in the Exercises, 6. ** This fact may be used to find the Key indicated by any signature : The last new sharp being always at the right in the signature, we may say that tk* right-hand sharp is always on the yth degree of the scale. And, knowing the 7th degree, we may easily find the 8th degree or Keynote. ( N. B. The octave of the keynote is the same as the keynote itself.) 10 HARMONY SIMPLIFIED. Circle of Keys with Flats; Circle of Fourths. 19. A Circle of Keys using a gradually increasing number of flats, can also be formed, by using the 4th degree of each scale as the starting-note ( keynote ) of the next one; e. g., No Flat. Fig. r Exercises. 20. Write out the Circle of Keys with flats, using double flats where necessary. 31. It will be noticed that whereas in the Circle with sharps the last half of each scale forms the first half of the next, in flats this is reversed, the first half of one becoming the last half of the next. ( To understand this, write it out in notes.) The pupil will further notice, that the added or new flat will appear each time upon the 4th degree.* 22. In the Circle of Keys with sharps, the 5th note of the scale is used as the Keynote of the following scale. In the Circle with flats, the 4th note is so used. Now, counting four notes of the scale upward reaches the same note as counting five notes downward.** Therefore, these circles are called the Circle of Fifths, the sharps counting * Therefore, to recognize any key with flat signature, notice that the right- hand flat is on the fourth degree of the scale ; and to find the ist degree or key- note, count downward from 4 to i. ** In finding the fifth below, do not count i, 2, 3, 4, 5 ; but, instead, count 5, 4, 3, 2, i, remembering to keep the half -step between 4 and 3, in order to preserve the correct form in the new scale. HARMONY SIMPLIFIED. II upward, i. e., by ascending Fifths, anr 1 the flats down- ward, i. e., by descending Fifths. 23. These circles may be represented as follows, the figures opposite each key indicating the number of sharps or flats in the scale : Fig. 8. Read around to the right. Q Fig. 9. Read around to the left. N. B. In finding the above number of sharps or flats in a scale, remember that a Double sharp counts the same as two single sharps. 24. As the keys having more than six sharps or six flats are unnecessarily complicated in notation, it is cus- tomary to use the sharp keys for the first half of the circle, from C to FJ, and the flat keys to complete the round ; e.g., Fig. 10. Fig. 10. Read to right or left. In this way the change is usually made from F# to Gb, or vice versa ; though it may be made at any point in the circle, e. g., from G$ to AP, from Ft? to E, etc., and is called an Enharmonic change of key. See 78. 12 HARMONY SIMPLIFIED. Advanced Course. 25. There is an interesting way of learning the number of sharps in a scale where there are more than six: It will be seen at a glance that the key of C has no sharps, and the key of C$ has seven sharps. In other words, each of the seven notes has been raised by a sharp. Similarly, if the key of G has one sharp, the key of G8 will have i + 7= 8, since each one of the notes in its scale must be raised to change the key from G to G8. Similarly, the key of D having two sharps, the key of Df will have 2+7 =9. Similarly, the key of A having three sharps, the key of A8 will have 3 +7 = 10. Therefore, to find how many sharps there are in a key when the Keynote is written with a sharp, simply add 7 to the number of sharps in the signature of the key of the same letter without the sharp. 26. The same principle applies to flat keys having more than six flats ; e. g., B!> has two flats ; therefore Bkk will have 2 + 7=9 flats. 27. Another interesting point in this connection may here be devel- oped : In the Circles of Fifths in 13-24, the circle began each time with the key of C. This is not at all necessary, it being quite as easy to begin upon any other note and complete the circle back to that note again, proceeding in either direction. Let the pupil begin upon G& and form the circle by ascending fifths. This will decrease the number of flats by one each time till C is reached, after which sharps will appear and increase successively. Vice versa, a circle can be constructed beginning upon Fit and pro- gressing by descending fifths. Notice that in both cases the succession passes through the key of C and changes from flats to sharps, or vice versa, without altering the conditions in the least. 28. From this it will be seen that Flats and Sharps, in their rela- tion to each other, are like degrees above and below Zero on the ther- mometer, sharps being above and flats below the zero-mark. Or they might be compared to Positive and Negative quantities in Algebra. Keyboard and Written Exercises. Form examples of the above mentioned circles, starting in turn from CJ, D, D8, E, F, F$, G, G, A, Alt, and B, progressing first by ascending fifths, and afterward by descending fifths. 29. Resulting from the relationship of sharps and flats, keys are frequently compared with respect to their relative " sharpness," the key having the fewest flats or the most sharps being called the sharpest key. Or they may be placed in order, thus : Cb Gb Db Ab Eb Bb F C G D A E B FJ Cf 7 6 5 4 3 a i o I 2 3 4 5 6 7, and C mpared by Saying HARMONY SIMPLIFIED. 13 that one key is so many " removes " to the right ( i. e., sharper) or left ( i. e., flatter ) from another key, counting through the key of C regard- less of differences ; e. g., G is two removes to the right from F, or Bb is four removes to the left from D. ( See Weitzmann's "Musical Theory," page 90.) In a similar way the notes themselves may be compared, saying that D is a sharper note than G, since its key is represented by one more sharp, etc. This point is further noticed in 250. Exercises. Compare the sharpness of the following keys, i. e., tell how many degrees or " removes " from the first to the second in each pair, and state which is the sharper of the two : Keys of A and B ; A and D ; B and F; Ab and D ; Bb and AJ ; C and BJ ; Gb and Ab ; Db and Eb ; Gf and Ab ; F and G ; G and A ; A and B ; B and C. Exercises. Regular Course. 30. By means of the statements in foot-notes to 18 and 21, the pupil should be able to recognize at sight any key from its signature : What keys are represented by the following sig- natures ? - 31. It is also desirable to know the number of sharps or flats in the signature of a given key, without reference to a table. Exercises. Give the number of sharps or flats in the signatures of the following keys : A, Dfr, G, Bt>, Ab, D, B, F&, Gt>, Et>, E. N. B. If necessary to do so, write out each scale to find the number of sharps or flats. 14 HARMONY SIMPLIFIED. Related Keys. 32. Keys having most notes in common are said to be related to each other. In the Circle of Fifths, each key is related particularly to the one before it, since one half of it is found in that scale ; and also to the one following, since the other half will be found in that one ( see 45), e. g., the key of C is related to the key of G ; also to the key of F. This subject will be developed further. (See 17 and 334.) Exercises. Name the two keys related to the key of B : of F$ : of D: of Afl: of Eb: of A: of Gb: E: DJJ. Facility in Distinguishing the Various Degrees of a Key by Number and by Name. 33. To thoroughly prepare himselt for the subsequent chapters, the pupil should learn to recognize at a glance the various degrees of any scale, and indicate them by number or by name. Keyboard and Mental Exercises. Placing any desired scale before the pupils (for example, the scale of Bb), the teacher should ask various questions like the following : - Which degree of the scale is Eb? Ans. 4th degree. Which degree of the scale is G ? Ans. 6th degree. Which degree of the scale is D? Ans. 3d degree. This exercise should be carried through various keys, and continued till some proficiency has been gained. The exercise may be varied by such questions as the follow- ing: What is the 2nd degree in the scale of A major? Ans. B. What is the 3rd degree in E major? Ans. HARMONY SIMPLIFIED. 15 Specific Names. ( To be learned.) 34. Each Degree of the scale has also a Specific name, which is often used instead of the number, as follows : 1st degree, Tonic. 2d degree, Supertonic. 3d degree, Mediant. ( Meaning midway between Tonic ana Dominant.) 4th degree, Subdominant. 5th degree, Dominant. 6th degree, Submediant. ( Midway between Tonic and Sub- dominant, when the latter is written below the former.) 7th degree, Subtonic or Leading-Tone. 8th degree, Octave or Tonic. Mental Exercises. Apply test-questions, as shown in 33. Notice that the prefix " Sub" means " below," and " Super," " above :" e. g., Supertonic means the degree above the Tonic, and Subtonic the degree below the Tonic. The Tonic, Dominant, Subdominant, and Leading- note are especially important to know, and the pupil should be able to find them 'without hesitation in any key. The Minor Scale. 35. It was noticed that in the Major Scale the half- steps occur from 3 to 4, and from 7 to 8. The Minor Scale is formed by placing the half-steps between 2 and 3, 5 and 6, 7 and 8. 3 1 6 HARMONY SIMPLIFIED. This is called the Harmonic Minor Scale, to dis- dnguish it from the Melodic Minor Scale, which has a different and irregular arrangement of the half-steps, as shown in Figure 12. ( See also 46.) r- 12. tfe= ^^-^S^^^^ _..- ^^ n/^? -i J .42- -* -fi>- " Cr ^ (5> 45678 7 654 '~ " -- 1 3 N _^3 > ^ 3^JJ 1 36. The Harmonic Minor Scale is the basis of the chords in the Minor Mode,* while the Melodic Minor Scale is generally used in melodies. It may be consid- ered as a "free "form of the Harmonic scale, made necessary by the fact that the interval of i J steps from 6 to 7 in the Harmonic Minor Scale ( see Fig. u) is rather unmelodious, though not unmusical. 37. From the foregoing comparison of the Major and Minor scales, the pupil will realize that the character of a scale depends upon the position of the half -steps. Exercises. 38. Form Harmonic Minor scales, and write the figures under each note as shown in Fig. 1 1 , starting from the following notes : A, E, B, FJ, Cfl, Gtf, Djf, D, G, C, *', Bt>, Efr, At?, Di7. Relative Minor. 39. Every Major scale has what is called its "Relative Minor," which is the Minor scale having most notes in common with it, and having the same signature. This Relative Minor is always founded ( has its keynote, or Tonic) onthestxtfr degree of the Major scale. Thus, * The words " Major Mode" and " Minor Mode " are terms used when we do not refer to any particular key, but wish to speak of the character of Major or Minor in a general way. HARMONY SIMPLIFIED. 17 the sixth degree in the scale of C is A ; therefore, the Relative Minor of C Major is the scale ( or key ) of A Minor. ( In finding a relative minor, it may be easier for the pupil to look for the keynote I \ steps below rather than the sixth above, the result being the same.) Exercises. Find the Relative Minor ( and write the proper sig- nature) of C Major; of G, D, A, E, and B Major; of F, Bt>, Eb, Ab, Db Major. 40. Correlatively, each Minor has its Relative Major ? which is found on the third degree of the Minor scale. For example, the relativ-r major of A Min^r is C Major. In other words, A Minor is the relative Minor of C Major ; and C Major is the relative Major of A Minor. Mental Exercises and Drill. Find the Relative Majors of the following Minor scales : A, E, B, F& Ctf, Gtf, Dtf, D, G, C, F, Bb, Eb, Ab, Db. Signatures in Minor. 41. The pupil will notice that the Relative Minor of any Major scale has the same notes as the latter, except- ing the seventh degree, which is raised by an accidental. For example, A Minor has the same notes as C Major, excepting the G#. This accidental raising of the seventh degree is caused by the fact that the seventh degree, or "Leading-tone," should be only a half-step distant from the Tonic. ( See 46.) In collecting the sharps or flats to form the signature of a minor key, this fact should be considered : The accidental found before the seventh degree does not be- long to the signature. X 8 HARMONY SIMPLIFIED. Exercises. Write the signatures of the following Minor keys, proceeding as directed in 12: A, E, B, FJ{, CJ, GJf, Dtf, D, G, C, F, Bb, Eb. The Circle of Keys in Minor. 42. The Circle of Fifths can be made with Minor keys as well as with Major. Exercises. (a.) Form the Circle with sharps, beginning with the key of A Minor. ( <5.) Form the Circle with flats, beginning with the key of A Minor. (c.) Form the Circle beginning upon various other notes. The Chromatic Scale. 43. When the half-steps lying between the notes of the Diatonic scales are included, thus producing a scale of half-steps exclusively, it is called a Chromatic scale. It \s customary to use sharps in writing the intermediate hlf-steps in an ascending chromatic scale, and flats in the descending scale ; e. g., Chromatic Alteration. 44. When a note is raised or lowered a half-step by HARMONY SIMPLIFIED. an accidental, consequently 'without changing its post' tion upon the staff, it is said to be chromatically altered ; e. g. A Chromatic Half-Step is one expressed upon ont degree of the staff; e. g., A Aft. A Diatonic Half-Step is one expressed upon two degrees of the staff; e. g., A Bfr. In general, a Diatonic progression is one where the letter is changed in the succession of notes ; and a Chro- matic progression is one where the letter is not changed, but altered by the use of an accidental. At the close of each chapter the pupil should make a synopsis of the principal facts contained therein, class- ifying and arranging them in order. The following table is intended to assist the pupil in this. Synopsis of Chapter I. ' Formula : Half-steps 3-4 and 7-8. Keys. Signatures. Circle of Fifths: Ascending. Scales : Major : Relative Keys : \ Descending. Fifth above, or Dominant. Fifth below, or SubdominanL Relative Minor. Specific Names. Major scales all alike. Minor ; Chromatic : Formula : Harmonic; Half-steps, 2-3, 5-6, 7-8. Melodic; Half-steps, up, 2-3, 7-8 : down, 6-5, 3-2. Relative Major. j Same as Relative Major. '" ' \ Omit sign of raised Leading-note. Leading-note raised by an accidental. Position of half-steps. Notation. 20 HARMONY SIMPLIFIED. Historical. 45. The Modern Scale is a gradual development from the ancient Greek Modes, in which the semitones occu- pied varying places in the scale, according to the mode. See Grove's "Dictionary of Music ;" Vol. II, p. 341, and Baker's "New Dictionary of Musical Terms;" p. 88 et seq. The Major Scale may be considered as composed of two Tetrachords,* placed one above another; e. g., CDEF GABC Tetrachord. Tetrachord. Until the I3th century, the use and influence of the semitones in Music were not fully realized; therefore, in the music previous to that time, we find ( according to modern standards ) a lack of Tonal feeling, or sense of being in some particular key. In the time of Palestrina it became customary to sing the seventh degree as if it were only a half-step from the eighth, although this was contrary to the notation, showing the need of something beyond the scales then in use. In the seventeenth century the modern scale began to displace the Gregorian Modes ; the sharps and flats, instead of being dispersed through the composition or left to the discretion of the performer, were gathered to- gether to form the signature ; the dividing lines between the keys were thus more distinctly marked ; and Modern Music, as opposed to the Ancient Modes, soon made a distinct place for itself. 46. The oldest form of the Minor scale was as shown in Fig. 14. A Tetrachord is a scale of four notes, having one half-step. Tetrachordi belonged to the musical system of the ancient Greeks. HARMONY SIMPLIFIED. 21 Flg. 1 4. I 4331 As the feeling of Tonality developed, a " Leading- note " was demanded which should point more decidedly toward the Keynote, and thus impart a greater feeling of satisfaction when the final chord was reached.* Thus the yth degree of the scale was raised by an accidental, giving the form as in Fig. 15, which is seen to be our present Harmonic Minor scale. FI..15.H& 3 3 4 4331 This form leaves an interval of i J steps between the 6th and 7th degrees; and for the sake of a smoother effect it became customary to raise the 6th degree also a half-step, where the harmony would allow it, giving the form shown at ( a ), Fig. 16, which is our present Melodic Minor scale. (a.) (b.) Fig. 16. y ^_, ^^ *f-^ "" W fD ^ U /f H5 2ZZ2 r3 |(TV & ^ & oil Vly 11 A " Leading-note '' being unnecessary in a descend- ing scale, the two notes raised by accidentals in the * The need of a " Leading-note " to give the feeling of satisfaction when the final chord is reached, is shown by comparing (a ) and ( b ) in Fig. 17. (a.) (b.) Fig. 1 7. 7 S3 ~l fay ^> 1 ^ra ^9 ff >^ 49 *-* 22 HARMONY SIMPLIFIED. ascending scale are usually restored in descending [(<5), Fig. 1 6], giving the complete Melodic minor scale now used. Exercises in Musical Dictation, for the Develop- ment of the Perceptive Faculties. 47. If we would rightly understand Music, it is indispensable that we become able to recognize what we hear, just as we recognize printed words upon first sight. The reason so few have the faculty of listening intelligently, is not that it is difficult, but because little or no attention has been paid to this most important subject. Briefly summed up, the steps of the process of development consist in gaining the power : 1. To distinguish Half-steps from Whole-steps. 2. To distinguish the various notes of the scale. 3. To distinguish Intervals. 4. To distinguish the Major from the Minor Mode. 5. To distinguish Chords and their inversions, and to realize their position in the key. 6. To trace simple Modulations. 7. To distinguish the Divisions of Time, Rhythm, etc. 8. To note the various features of Form, learning to recognize Motives, Themes, succession of keys, Periods, and the general plan of construction. 9. To be able to express all of the above in Musical Notation. * By taking this study step by step, and in connection with the study of Harmony, there will be added interest in the latter by reason of the ability to apply each point as soon as learned. There will also be a deeper and more practical comprehension of Harmony, and a more intelligent knowledge of Music as an Art and a Science. To Distinguish Notes of the Scale. 48. (a.) First teach the pupils, by experiment and careful concen- tration in listening, to distinguish between Half and Whole steps in both upward and downward progression. This may occupy parts of six or eight lessons. (6). The best and only really successful manner of teaching the notes of the scale and how to distinguish them, is through the medium of the voice. The foundation of the musical perceptions lies in the possession of a "working " knowledge of the Major scale. The first * The above represents the complete process, facility in all of which is attained only by gifted minds. But a moderate degree of proficiency is within the reach of any one possessed of ordinary perserverance. HARMONY SIMPLIFIED. 23 step is therefore to thoroughly practise singing the majoi scale, using the syllables Doh, Ray, Me, Fah, Soh, Lah, Te, Doh 1 .* This should be continued till the pupils can skip from any degree to any other, and can also recognize the same when sung or played by the teacher. (f.) In connection with the above, the teacher should sing, or play the ascending and descending scale, while the class, provided with a book of score-paper, write each note as it is sung. During this exercise ( of writing, or musical dictation ) the teacher should frequently ask, " What was the last note sung ?" requiring as an answer the name of the syllable, Doh, Ray, etc. 49. ( , SOH FAH ME the pupil should, by remembering the syllabic names of RAY the notes and the sounds connected with those names, try DOH to think how the passage would sound, afterward compar- Te ( ing with the effect when sung or played. Lah, Soh, 52. While exercising the pupil on Pitch, studies in Rhythm should be given, by means of notes of various lengths, suc- cessions of notes with rhythmic flow, etc. Rests should also be intro- duced. The inexperienced will find material for such exercises in any book on Sight-singing or Musical Dictation. Specimen Test Questions. Illustrating the Drill which should be given at the end of each chapter. Additional questions upon this and other chapters may be found in the author's Key to ''Harmony Simplified"; also answers for all questions and exercises in this book. 1. Where are Half steps in the Major Scale ? State two or three foundation principles covering its con- struction. 2. For what are Sharps and Flats used?* Also Double Sharps, and Double Flats ? 3. How many kinds of Major scales are there, and why ? 4. What is a Signature ? * Give its origin. * 5. Describe Tetrachords and their office in the Order of Keys. * 6. Give the Order of Scales with Sharps. Also with Flats. * But few reach the underlying thought in tnese questions. HARMONY SIMPLIFIED. 25 7. What is the difference between a Scale and a Key ? * 8. What is the difference between the Harmonic and Melodic Minor ? 9. Name the Keys related to G Major and give reasons therefor. 10. How would you discover a key from the Signa- ture in Sharps or Flats ? 1 1 . What is the Signature of any Minor Key ? 12. What is the office of the Half-step in scale con- struction ? * 13. Why is there an accidental in every Harmonic Minor scale ? 14. Was it there originally ? 15. How would you change a Major to its Tonic Minor key, or vice versa ? * 1 6. What do you understand by the term " Tonality " ? How is it developed, and how does it differ from " Key " ? * NOTE. The more obvious questions are here omitted, but should be included by the teacher, and possibly used to " lead up to " these questions. NOTE. The author introduced into his own work a system of Daily Drill upon each subject as it was completed in the course of study. The suggestion, it is hoped, will prove of as great value to others as it has proved in his own experience. Such drills may be found in the Key and Graded Lessons. CHAPTER II. INTERVALS. SPECIAL NOTE. On taking up this subject, it is well to observe that it is divided into four general sections, which at first are studied rather independently of one another, viz. : The Names of Intervals; the Specific Names ; Inversions ; and Consonant and Dissonant Try to keep these four lines of study distinct in the mind. * These questions are designed to stimulate original thought. 2 6 HARMONY SIMPLIFIED. 53. An Interval in Music is like an interval anywhere else, it is an expression of distance between two things. Consequently, it may be defined as the distances or difference in pitch, between two given tones.* 54. An Interval may be formed by two notes, eithei sounded together, or in succession. (a.) is called an Harmonic Interval. is called a Melodic Interval. General Names of Intervals. 55. An Interval is named according to the number of degrees of the scale included in its extent. Thus, the Interval from C to D * * is called a 2nd, because two Degrees of the scale are concerned in its production. Similarly, from C to E is a 3rd, from C to F a 4th, from E to B a 5th, etc. 56. To determine the name of an interval, count the degrees, including those upon which the notes of the interval stand (i. e., including extremes). For example, in determining the name of (fl^ & ", count the degrees *-> ~-<9- upon which C and A stand, as well as those lying between, giving the total of six ; therefore, the interval in question must be a 6th. N. B. Unless otherwise indicated, intervals are usually counted from the lower note upward Advanced Course. * The word Interval may also mean the relationship of two notes in respect to pitch ; or the effect produced by the two notes sounding- together or in suc- cession. * * The fow*r note of an interval or chord Is always mentioned first. HARMONY SIMPLIFIED. 27 Table of Intervals. Pig. 21. _- - - -- -- -- -- -- -- -- Unison 2d. 3d. 4th. 5th. 6th. ;th. 8th. gth. loth. or Prime.* Keyboard and Written Exercises.** 57. (a.) Form tables similar to the above, starting from the notes D, F, E, G, B, and A (all in the key of C). (.) Form similar tables in the keys of G, F, D, Bfr, etc. (c.) Write all the Seconds in the key of G ; e. g. FIR. 22. etc. (d.) Write all the Thirds in the key of F. (e.) Write all the Fourths, Fifths, Sixths, Sevenths and Octaves in the key of E ; in the key of Bfr ; A; Db; Fj. (f.) Repeat all of the above at the keyboard. Specific Names of Intervals. 58. Intervals are of various kinds, the names of which fairly express their meaning, as follows : Major : ? * * The normal or standard of measure- Perfect : j inent. The difference between the two will be explained in 60, 73 and 76. * When two voices sound the same note, there is no difference in pitch, and therefore no interval between them. Consequently, the Unison cannot strictly be called an interval. * * Pupils are liable to make mistakes when counting an interval upon the keyboard ; but when written, by counting: the lines or spaces upon which the notes stand and all the intervening lines and spaces, mistakes become impossible. Or better still, count the number of letters involved, including extremes. * * * For the present the pupil need only know that Unisons, Fourths, Fifths and Octaves may be Perfect, but not Major. (Some theorists call the Perfect intervals Per/erf Major, \g distinguish them from those which are limply Major.) 2 8 HARMONY SIMPLfflED. Minor: meaning "less" by a semitone than Major. Diminished: meaning still less, or less by a semitone than Minor or Perfect. Augmented: meaning increased, or greater by a semi- tone than Major or Perfect. The difference between the various kinds of intervals is illus- trated by the following, from Eugene Thayer: ' Let us take a pair of hand-bellows, and allowing them to take their natural position, find them to be nearly wide open the handles well apart. Let this position represent the Major interval. If the upper handle be pressed down a little, the distance between the two handles (or the interval) is lessened : this corresponds to the Minor interval. If we now raise the lower handle, pressing them still nearer together, the dis- tance (interval) is again decreased, representing a Diminished interval. Again, letting the handles spring back to their original (normal) posi- tion, representing the Major interval, if we raise the upper handle, or depress the lower one, we increase the distance between them, thus representing an Augmented interval.' The Standard of Measurement. 59. Consider the scale of C upon the keyboard. From C to any other degree of the scale of C, or from C to any white key upon the Piano, is a Major or Perfect, i. e., a Normal, interval. ( For example, see Fig. 21. All the intervals there given are Major or Perfect.) This gives us a practical standard of measurement by which ive can measure any interval; for, as we have seen in the above definitions, a Minor interval is a semitone smaller than a Major, an Augmented a semitone larger than a Major, etc. 60. Of the Normal Intervals (as shown in Fig. 21) the Unison, Fourth, Fifth, and Octave are called Perfect ; while the others, namely, the Second, Third, Sixth and Seventh, are called Major. The same statement in more general terms would be, " Normal Unisons, Fourths, Fifths and Octaves are called Perfect ; while Normal Seconds, Thirds, Sixths and Sevenths are called Major." Still another form of the statement is, " The Normal intervals are divided HARMONY SIMPLIFIED. 39 mto two classes, half of them being called Perfect and half Major." Memorize the first statement above, and do not seek to understand the reason for the above divisions until you study the Inversions. The reasons will become more apparent as you go into the subject. 61. It is not the mere elevation or depression of the notes that changes an interval, but the fact that the tones are either separated further from each other, or are brought nearer together; i. e., the distance is changed. 62. Let us make a practical application of the above. We found in 56, that from C to A (counting up- ward) is a Sixth ; and, according to 59 and foot-note, it is a Major Sixth : Let us apply some of the changes mentioned in 58 to this. Lowering the upper note a semitone, we have ;, which, being a semitone less than the Major Sixth, is called a Minor Sixth. Again, taking this Minor Sixth, by again decreasing the distance between the notes, this time raising the lower note by prefixing a sharp, we obtain a Diminished Sixth : * Again, returning to the Major Sixth as a starting- place, if the upper note be raised a half-step, the distance between the two notes will be increased, forming an Augmented Sixth : F((T) fl 1 ^ '- See also " Key," 62. Exercises. Name each of the following intervals. %5>- *The interval of the Diminished 6th is not commonly used (see Table, $ 64), but it is useful here for Illustration. 30 HARMONY SIMPLIFIED. Exact Measurement of Intervals. 63. An interval can be exactly measured, and its Spe- cific name placed beyond doubt, by counting the num- ber of Half- Steps contained in it (just as we counted the number of degrees to obtain the General name). For example, from C to A is a 6th. Counting the number of half-steps, we find it has nine. Therefore, as our Standard Sixth contains nine half-s*:eps, any other Afajor sixth, without regard to its position, must have the same number of half-steps. According to 58, an Augmented Sixth must have one half-step more, or ten half-steps : and a Minor Sixth one half-step less, or eight half-steps. In this way we may compare any interval with the Standard of Measurement, and learn whether it is Major, Minor, Diminished, or Augmented. 64. As no interval is commonly used in more than three of these forms, a table is subjoined, showing them in order as generally used. Table of Intervals. Showing the number of ^#^"-steps in any interval- [ For reference ; not to be memorized.*] Diminished. Minor. Perfect. Major. Augmented. o I I 2 3 3 4 5 * - 7 - 8 8 9 10 10 ii 12 13 14 15 * It is unnecessary to memorize this table, as the pupil can easily find the number of half-steps in a given interval by the use of the principles shown in Primes : Seconds : Thirds : 2 Fourths : 4 Fifths: 6 Sixths : Sevenths : 9 Octaves : ii Ninths : HARMONY SIMPLIFIED. 31 65. In working out the following exercises, the pupil jhould first find the note for the General name of the interval as shown in 56, afterward adding any sharps or flats necessary to bring it into correspondence with the requirements of the Specific name. For example, "What is the Major Sixth from E?" Process: Beginning to count with the note E, the sixth count will bring us to C ; therefore, a Sixth from E must be C ; C is thus the General name for the desired interval. Next, a Major Sixth must have (refer to the standard of measurement) nine half-steps : as there are but eight half-steps from E to C, it is evident that the latter must be raised by a sharp, giving, for the Major Sixth, C#. Advanced Course. 66. The standard of measurement, 59, showed the intervals from the note C to every other note in the scale of C to be " Normal "inter- vals. In a similar way, from the Keynote of any other key to any note of that scale would be also a " Normal " interval, e. g., from A& to any note in the scale of A^ would be just as " normal " as from C to any note in the scale of C. Therefore, instead of counting the half-steps in naming or forming a specific interval, the practised musician would think, " What is the ' Normal ' interval ?" counting from any given note (by transferring his thought for the instant to the key of that given note), and would raise or lower that normal note to obtain the required interval. For example : What is the Augmented Sixth from FJ ? Process : If we were in the key of F$, the Normal Sixth would be found by counting up to the 6th degree of the scale, giving the note D8. (Having found the desired interval, do not think further in the key of Fjf.) As the required Augmented Sixth is a half-step greater than the Normal, the Df must be raised another half-step, giving the note DX (double sharp). Regular Course. 67. Remember that the General name is obtained by counting the degrees of the scale, while the Specific name is found by counting the half-steps. Therefore, 32 HARMONY SIMPLIFIED. the use of sharps or Jlats can never change the General name of an interval ; a Sixth remains a Sixth even if there are several sharps or flats prefixed. The kind of Sixth it would be is quite a different question, coming under the head of Specific name. Exercises. 68. (a.) Form a Major Sixth from each of the follow- ing notes, counting upward: D, E, F$, B, A, Bb, Ab, Db, Aft, G, Eb, Gb, Cb, Git, CB, Fb, BJ, etc. (<5.) From the same notes, form Major Thirds, Minor Thirds, and Minor Sevenths. (c.) Form Diminished yths from D, E, FJ, B, A, Bb, Ab, AJt, C, Eb, Gft CJf, BJt, etc. (a?.) Form Augmented 4ths from D, E, F#,B, A, Bb, Ab, Db, Atf, Gtf, Eb, Gb, Cb, G*, Of, Fb, etc. NOTE. In accidentally raising or lowering a note, it is not custom- ary to raise or lower it beyond the pitch of the natural next above or below ; e. g., B would not be double-sharped, since that would bring it beyond C, the next natural ; nor would F have a double flat, since that would take it beyond the next natural. (e.) Repeat all of the above at the keyboard. Extended Intervals. 69. As an octave above any note is considered a repe- tition of that note and bears the same name, so intervals (with the exception of the Ninth), if they extend over more than an octave, are considered as repetitions of the smaller intervals formed by Ihe same notes an octave nearer together. Thus : I (fl\ which is an interval of an Eleventh, is considered as an extension of G HARMONY SIMPLIFIED. 33 which is a Fourth. Therefore, in rinding intervals, the notes should be brought within the compass of an octave. Exercises. Name the following intervals, lowering the upper note, or raising the lower, one or two octaves : O -,51- " " -^ -vr - ^ The interval of a Ninth is usually not contracted in this way, as the chord of the Ninth requires that interval to be nine degrees from the root. See Chapter VIII. Inversion of Intervals. 70. By Inversion of Intervals is meant that the notes change their relative positions ; the upper one, by being lowered an octave (retaining its original name) , becom- ing lower than the other ; or, the lower one, by being raised an octave, becoming higher than its fellow. Thus, the interval at (a) in the accompanying figure becomes like (3) by lowering the upper note, and like (c) by raising the lower one, which is the same thing as (3), but an octave higher. (a.) (6.) (c.) Keyboard and Written Exercises. Invert the following (a) by lowering the upper note one octave ; (b) by raising the lower note. 1 71. Subjoined is a Table showing a few intervals in- verted. The lower staff shows the result of inverting the intervals contained in the upper staff. Notice that in the 34 HARMONY SIMPLIFIED. tables the inversions are produced by raising the lower note one octave. It would have been quite as easy to lower the upper notes one octave, writing the inversions in the Bass clef. The quarter-notes in the lower staff show the notes which have been raised an octave. Fig. 23. -y- 1 1 i?H 1 1 Saz I I f3 \ _Q- -ee- is^ Prime Seco becomes becon 1 tid Third ics becomes 1 Fourth becomes , 1 sE * i * i * EEE IITV VMT y*3 gy Octave. Seventh. Sixth. Fifth. Sixth becomes Seventh becomes -<5>- Octave Augmented Diminished becomes becomes oecomes Fourth. Third. Second. Prime. Diminished. Augmented. 72. From the above let us notice the following : (a.) To learn what will be the inversion of an interval (that is, the interval which will result by inverting), sub- tract the number of the interval from 9, and the result will be the interval produced by the inversion. For ex- ample, what would the interval of a Sixth become by inversion? Process : 9 6=3; therefore, a Sixth, when inverted, becomes a Third. (See p. 42, Ad- dendum.) The following table shows the fact still more clearly : From 9 9 99 9 9 9 9 Substract 12345678 Result 8 HARMONY SIMPLIFIED. 35 From the first table (Fig. 23) we notice also : 73* 0^0 By inversion ( Major intervals become Minor. By inversion ( Minor intervals become Major. f Augmented intervals become By inversion Diminished. By inversion 1 Diminished intervals become Augmented. T, . . f Perfect intervals remain Per- By inversion < ( feet (and therefore Normal). This peculiarity of the Perfect intervals renders it necessary to class them differently from the Major, though in practical Harmony this distinction does not affect their use. A further difference between Major and Per- fect intervals appears in 76. Keyboard and Written Exercises. 74. (a.) Find the Perfect intervals in Fig. 2 1 . (There are four.) (<5.) From the note D form a series similar to Fig. 21, and invert each interval as shown in Fig. 23. (c.) Write examples of Diminished and Augmented intervals, and invert them. To learn what Diminished and Augmented intervals are in use, the pupil may refer to the Table in 64. Consonant and Dissonant Intervals. 75. In the preceding paragraphs, intervals were classed according to the number of half-steps contained. They are also classed, according to their musical effect, as : (a.) Consonant, meaning those intervals upon which it is agreeable to pause, and which do not need to be followed by another interval to produce a pleasant effect ; and HARMONY SIMPLIFIED. (6.) Dissonant, or those which are not satisfactory to dwell upon, or to use in the final chord of any composition. Consonances are further divided into Perfect and Imperfect Consonances, with reference to the degree of concord, as follows : All Perfect intervals, viz., Perfect Prime (or Unison), Perfect : * -} Perfect Octave, Consonances. \ Perfect Fourth, Perfect Fifth. ( Major Thirds and Sixths. Imperfect: j Minor Thirds and Sixths . Seconds and Sevenths, together with all augmented and diminished intervals ; i. e. , Dissonances. -| all intervals other than the Perfect inter- vals and Major and Minor Thirds and Sixths. Exercises. (a.) The pupil will refer to all the previous exercises and illustrations in this chapter, particularly to the Table in 64, and mark each interval as Perfect or Imperfect Consonance or Dissonance. (3.) Both at the keyboard and in writing, form first all the consonant intervals and then the Dissonant inter- vals from the note D. (c.) Proceed similarly from the other notes. 76. A furthur difference between Major and Perfect intervals appears at this place. When a Major interval * The distinction between perfect and imperfect consonances is of no importance to the general student, who will recognize an interval or chord either as a consonance or a dissonance. There need be no further distinction at present. HARMONY SIMPLIFIED. 37 is decreased by a semitone (see 62), it becomes a Minor Interval, but its classification as Consonant or Dissonant is never changed by this reduction. For ex- ample, a major 6th being consonant, the minor 6th will be consonant ; or, the major 2nd being dissonant, the minor 2nd will also be dissonant, as shown in the above table ; whereas, if a Perfect interval be decreased by a semitone, it at once loses its characteristic of being "Consonant" and becomes a "Dissonant" interval. For example, f(g) & ' is a Perfect Fifth. If we lower the upper note a semitone, the result is L/rrV b^? > which is a Diminished Fifth and a Dissonance. Thus we see that a Major interval can be made less (Minor) ivithout changing its classification of "Consonant"; while a Perfect interval cannot preserve its original classification when thus altered.* Confusion of Terms. 77. There is much confusion in the terms used in connection with the Theory of Music. Carefully notice to what each term refers. A few examples are given below of the various meanings of certain words : Degree may refer to the various steps of the Scale. Degree may also refer to the lines and spaces of the staff. Steps may refer to the various degrees of the Scale. * The word " Perfect " conveys but little meaning, as these intervals are perfect only in respect to their quality of remaining " Normal " when inverted, while Major intervals do not. A more descriptive name might be " Sensitive '' Interval, as such an ir terval cannot be changed in any manner without produc- Ing a dissonance. 38 HARMONY SIMPLIFIED. Steps and Half- Steps also to the distance between tones. Tones and Semitones may refer to the distance be- tween tones. Tones may also refer to sounds, regardless of dis- tance from other sounds.* Interval refers to distance between tones. Interval sometimes refers to the steps of the scale. The names of the Degrees of the Scale ( as Fifth degree, Third degree, etc.), are liable to be confused with the Intervals of the same name : therefore be careful to say whether you mean Degree or Interval. Definitions. 78. Diatonic Intervals. The word Diatonic refers to the scale ; a Diatonic interval would be, therefore, an in- terval formed by two notes of the scale without sharps or flats other than those indicated by the signature. Chromatic Intervals. The word Chromatic in Music refers to the half-steps lying between the notes of the scale, and which are produced by the use of acciden- tal sharps, flats, or naturals, to change the diatonic tones. A Chromatic interval, then, would mean one where at least one of the notes has an accidental sharp, flat, or natural before it. N. B. A Half-step can be either Chromatic or Diatonic; e. g., from C to C$ is a Chromatic half-step, because only one note ( C ) is concerned in the interval. ( See 44.) But if Q is called Dfr, the half-steo be- comes Diatonic, because two notes ( or two degrees on the staff") are concerned. The words Note and Tone are often used interchangeably, though a tone k properly a sound, and a note is a character to represent a sound to the tye, HARMONY SIMPLIFIED. 39 Enharmonic. This word refers to the notation only;* when the same tone is expressed in two different ways, there is said to be an Enharmonic Change; e. g., Ab when changed to G# is said to be enharmonically written, because the name has been changed while the tone remains the same. ( See foot-note, and 24.) This chapter should always be studied twice ( repeated very care- fully ) before proceeding, as it is impossible to understand the full meaning of the first part before the last part has been studied. Synopsis. 79. Before proceeding, the pupil should not fail to 'write a synopsis of the chapter as suggested at the close of Chapter I, and endeavor to gain an orderly view of the subject. Failure to do this is often the cause of very con- fused ideas in regard to Harmony. Historical. 80. The beginning of Music was Melody, everything being in unison and without accompaniment. In some MvSS. of the loth century, examples of church-music are found, progressing at the regular interval of a Fourth. The meaning of this has been disputed, some claiming that it was intended to be sung in unison and then re- peated a Fourth higher, while others think the two parts were to be sung together, the effect of which would be disagreeable to modern ears. At about this time a " Drone Bass " was sometimes used i. e., aBass continuing upon one note regardless of the melody. In this way various intervals, such as Fourths, Fifths, and Sixths, were necessarily, though quite acci- dentally, formed. Soon afterward (nth century ) it was * Advanced students of theory may know that Enharmonic intervals havti a very slight difference in pitch ; e. g., G$ has a few vibrations more per secon4 than Ak though the Piano does not show it. 40 HARMON? discovered that two complete and independent melodies might be sung together and produce a pleasant effect. From this discovery came Counterpoint, and before the close of the I4th century music was written in four parts, though little was known of the effects of harmony. At this period the controlling principle was to invent several melodies which would not conflict when sung together, rather than to study the effect of the combination of three or four tones forming a chord. Consequently, at this time, till the close of the 14th century, the harmonic effects were accidental rather than studied. The Perceptive Faculties. ( Continued from page 24.) Intervals. 81. The perception of intervals, though more difficult than of single tones, need not cause any especial trouble if properly presented, and if the first steps have been thorough. It is probable that the student will advance more rapidly in Theory than in the development of the perceptions. Do not try to make the two keep exact pace, though in explaining each chapter, the ear as well as the eye and the under- standing should be actively interested. Process. 82. ist Step. This chapter should be taught as a direct continua- tion of the lessons on the degrees of the scale, not as a new subject. For example, taking up the subject at (c), 49, after singing or playing /r> j and the succession has been named Doh, Ray, tZvl/. _{_ O -<5>-^ by the class and written in notes, call attention to the fact that the pro- gression has been explained in Chapter II as an interval of a Second. (This forms a Melodic interval; see $4.) In a similar way, the teacher may proceed up the scale, the next time taking the notes D and E, the third time E and F, etc., being careful that the pupils do not lose sight of the syllabic names. As often as they forget or miss them, return to Doh, and let them sing (or recognize) up to the desired notes. HARMONY SIMPLIFIED. 4 1 N. B. Being a dissonance, the two notes of the interval of the Second should not be sung together, unless once or twice merely to show their dissonant character. 83. 2nd Step. Sing or play the notes I /k ; j , requiring the J Sf- 6f ~ syllabic names as before, and allow them to be written. Explain that this progression forms a Third, and proceed up the scale, taking the notes D and F, E and G, as shown above, requiring first the syllabic names, after which they should be written. Next, returning to C and E, ailow part of the class to sing the lower note, calling it Doh, while the remainder sing E, calling it Me. This illustrates the Harmonic interval, as singing in succession repre- sented the Melodic. Continue up the scale as before, but now allowing both notes to be sung together and properly written to express the harmonic interval. 84. yd Step. Treat Fourths, Fifths, Sixths, Sevenths, and Octaves ( not exceeding the limit of the voices ) in a similar manner, first Melod- ically, and then Harmonically. Carefully call attention to the musical effect of the different inter- vals as well as to the various distances apart. 85. 4/A Step. Sing or play successions of two single notes, requiring first the syllabic names and then the interval. 86. 5//4 Step. Play various intervals ( harmonic), first striking the notes in succession ("spreading") if necessary, requiring both the syllabic names and name of the interval. Let everything be written as soon as the pupil recognizes it, to gain the habit of expressing his impressions. (Begin this step [86] with Octaves and Fifths.) 87. 6th Step. Striking a Major Third, change.it to Minor by low- ering the upper note, calling attention to the different musical effect and the means of producing it ; explaining at the same time that some of the Thirds in the scale are Minor without any change, for example, from Ray to Fah, Me to Soh, etc. 88. jth step. Display Major and Minor Sixths in a similar manner. Introduce Diminished and Augmented intervals very cautiously, on account of their difficulty. 42 HARMONY SIMPLIFIED. 89. In general. Arrange the exercises carefully in point of pro- gressive difficulty. Do not let the pupil get confused in regard to the syllabic names. He must ha"ve a firm hold of the Tonality. Be patient. The pupil may now take two-part songs ( or the soprano and alto of hymns and choruses ), and try to think how they would sound afterward comparing with the effect when played or sung. Exercises in Rhythm should be continued. (Addendum to 72.) Complementary Intervals. Any two intervals which, when added together,form an octave, are called Complementary intervals, since each completes, or complements the other in the formation of the octave. This is simply another state- ment of Inversion, for any interval and its inversion form Comple- mentary intervals. Illustration : A Sixth and a Third are Comple- mentary, or the Sixth is said to be Complementary to the Third, and vice -versa. Similarly, Fourths and Fifths, or Seconds and Sevenths, are Complementary. HARMONY SIMPLIFIED. 43 PART II. CHAPTER III. TRIADS. The Foundation of the Harmonic System. NOTE. 90 is not to be studied. It is designed more especially for the teacher and for those inquiring minds who would know some- thing of the scientific basis of chord-formation, and observe the won- derful symmetry and simplicity of Nature's laws. Advanced Course. Harmonics. 90. Science has demonstrated that a musical tone is not one simple sound, but is made up of the combined sounds of many different tones, softly sounding with the principal or Primary tone. It has also been proved that these secondary tones bear a certain relation to the prin- cipal or Primary tone, and though they sound but faintly (being inaudible to untrained ears ), can be distinctly recognized by those who are trained in this direction. These secondary or accompanying tones are called Overtones or Harmonics. When a long string, tightly drawn, is put into vibration, it vibrates in its full length alone only an instant ; after a short time it vibrates also in sections ( without interfering with the full-length vibrations ) producing higher tones simultaneously with the principal or funda- mental tone. For illustration, if a string producing the tone KJ __ is put into vibration, this tone will be very distinct, but the presence of the following tones, sounding very faintly, can be proved. Flg.24.p2- --&-{- * These harmonics are not exactly true to pitch. 44 HARMONY SIMPLIFIED. This series is called the Harmonic Chord, or Nature's Chord Those who are already familiar with chords will observe, that the first six notes sounded together are simply an ordinary chord. If the next note, Bfr, is added, a Chord of the 7th is formed. If to the last chord the ninth note of the series is added ( the eighth note, C, is merely a duplicate of the lower octaves), a chord of the Ninth is formed. In these three chords, or rather in this one Harmonic Chord, is the basis of the Harmonic system, from which the various chord- formations can be logically developed. The above is designed to show four points, viz: ( a.) That a musical tone is made up of many tones sounding together as above stated.* ( 6.) That a chord, as commonly understood, is an imitation, at the hands of Man, of the great chord of Nature, or at least it has been made to correspond very closely with it. NOTE. Young students are liable to be troubled by the fact that some of the remoter harmonics are strongly dissonant with the funda- mental tone and triad. But this need not disturb them, as the harmonics are more indistinct as they are more remote from the Fundamental tone, and the finest ear cannot detect more than six or seven. Therefore, the upper ones are too weak to have much effect upon a tone, though Science conclusively proves their presence. ( c.) That chords are produced by a process of adding to, or building upon, a note called the Fundamental, or Root. The Chord of the 7th was produced by adding one note to the chord already formed ; and the Chord of the gth, by adding still one more. (d.) Conversely, that the chord built upon a Root is considered as derived from that Root. Regular Course. Triads. 91 . When any note is taken, together with the intervals of a Third and a Fifth above it, a Triad is formed. A Triad, then, is a chord of three notes : a Fundamental * There is a strong analogy between a single tone and a ray of light. When thrown through a prism, light is seen to be a compound of various colors, the prism serving to separate the ray into its constituent parts. Similarly, a tone can be shown by the laws of sympathetic vibration to consist of the Primary tone and Overtones, as shown in 90. HARMONY SIMPLIFIED. 45 or Root, together with its Third and Fifth, counting upward ; e. g., ^ As shown above, the whole Harmonic System may be said to rest upon this simple Triad. A distinguished musician has declared, "There is but one chord in the world, the Common Triad. All others are merely addi- tions to this." Exercises. ( a.) Write the scale of C, and upon each note, used as a Fundamental, construct a triad, without considering whether the intervals are Major or Minor (see Fig. 25). Fig. 25. , - ( 3.) Write similarly the scale of G, F, D, B, etc., not regarding sharps or flats except to place them in the signature, and construct a triad upon each note, as above. (c.) Repeat the above exercises at the keyboard, and, in addition, take each of the remaining major keys. Marking the Triads. 92. In 2 it was shown how each degree of the scale is numbered from the lowest to the highest. The Triads are described in a similar manner, by indicating upon which degree of the scale they are founded ; for exam- ple, "Triad on the 3d degree, " " Triad on the 6th degree," etc. For this purpose Roman Numerals are em- ployed, being written under the staff as shown in Fig. 26. Fig. 26. I II in IV V vi vn I Exercises. Mark the Triads formed in the exercises in 91. 46 HARMONY SIMPLIFIED, Specific Names of Triads. 93. Triads are divided into four kinds, Major, Minor. Diminished, and Augmented. These varieties corre- spond closely with the intervals of the same names, for they are named, according to the intervals of 'which they are composed* as follows : A Major triad has a Major 3rd and Perfect 5th. A Minor triad has a Minor 3rd and Perfect 5th. A Diminished triad has a Minor 3rd and Dimin. 5th. An Augmented triad has a Major 3rd and Aug- mented 5th. These four kinds of triads are indicated, in marking the triads, as follows : Major, by a large Roman numeral, for example : I. Minor, by a small Roman numeral, for example : n. Diminished, by a small Roman numeral with the sign affixed : vn. Augmented, by a large Roman numeral with the sign' affixed : III'. Exercises. ( a.) Write the Harmonic Minor scale of A, and form triads upon the various steps, as in 91. Next, describe each triad ( Major, Minor, Diminished, or Augmented), and mark as above indicated. (3.) Repeat the process in E, D and B minor. (c.) Repeat the above at the keyboard, adding other Minor keys. Principal and Secondary Triads. 94. The triads upon the Tonic, Dominant and Sub- dominant ( see 34 ) are called the Principal or Primary Triads, for the following reasons : (a.) They are most frequently used. (3.) They embrace every note of the scale. HARMONY SIMPLIFIED. 47 (c.) They are sufficient to determine, beyond doubt, {he key. The Triads upon the remaining degrees are called Secondary Triads. Exercises. Returning to the exercises in 91, 92, 93, the pupil will describe each Triad, indicating the Secondary- Triads by the proper Roman numeral, and the others by the first letter of their names ; thus, T, ( Tonic) ; D, ( Dominant ) ; and S. D, (Subdominant). Doubling. 95. In a Triad there are but three different notes. Therefore, if we write music in four parts, one of the three notes must be doubled, i. e., must appear in two parts. The Fundamental is the best note for doub- ling, and the Third che poorest. (See 162.) The four-part chord resulting from the doubling of one note of the triad is called a Common Chord ; e. g., or:< Position. 96. The three notes composing the Triad do not need to be always in the same order, with the Fundamental lowest and the Fifth at the top. The Fundamental of the Third may also occupy the highest place, and the term Position is used to denote which note is highest, as follows : ( a.) When the Fundamental or its octave is highest ( in the Soprano ) the chord is said to be in the Position of the Octave. y E___s I <= X sy \ in the three positions, using the proper (key-) signatures in each case. Four-part writing; Connection of Triads. 97. Each chord of four notes is considered as written for a quartet of voices, Bass, Tenor, Alto and Soprano. The Soprano and Bass are called the Outer or Extreme parts : the Alto and Tenor are called the Inner parts. In four-part writing the effect should be considered from two points of view : ( a.) The Melodic effect of each part (as it would sound if sung alone). ( 6.) The Harmonic effect of the four parts sounding together, and the connection between the successive chords. Before proceeding to practical exercises in connect- ing chords and leading the parts, the pupil should learn something of the difficulties in the way of making a good effect, as follows : HARMONY SIMPLIFIED. 49 Consecutive Fifths. 98. If we play a series of Thirds, for example, IJ gj_d etc., the effect is not' unpleasant. If we add a Fifth, changing each Third to a triad, thus : etc., we find the effect harsh and un- pleasant. This disagreeable effect was evidently not produced by the Thirds sounding in succession, for the , Q , | | 41., following: /f \& a ^ ^^ etc., is, if possible, ~ still worse. Therefore, we may conclude that the bad effect is produced by the succession of Fifths.* Consequently, Consecutive Fifths are not allowed. Consecutive Octaves. 99. Again, if in a four-part chorus two voices sing the same notes, either in unison or an octave apart, there would be in reality but three different parts, which would weaken the harmony. Therefore, Consecutive Octaves {and Unisons) are not allowed. 100. In order to learn how to avoid Consecutive Fifths and Octaves, the pupil should realize that in the progres- sion of the parts, three different movements are possible : ( a.) Parallel Motion, in which two parts move in the same direction ; see ( a ), Fig. 28. ( b.~) Oblique Motion, in which one part remains stationary, while the other moves; see ( b ), Fig. 28. (c.) Contrary Motion, in which the parts move in opposite directions: see ( c), Fig. 28. * The harshness of consecutive sths is caused by the suggestion of two different keys in succession without proper (modulatory) connection. If the second interval is a diminished fifth, a new key is not so strongly suggested: hence this exception is allowed. 50 HARMONY SIMPLIFIED. Fig. 28. In four-part writing, two or even three different kinds of motion can occur simultaneously between the different parts. Parallel motion between three parts is permitted, if no Consecutive Fifths or Octaves result from it. Parallel motion between all four parts is not good, and it is difficult to avoid the forbidden consecutives if the parts all move in the same direction. To Avoid Consecutive Fifths and Octaves. Let one or two parts progress in contrary motion to the others. This rule will cover all cases. Open and Close Harmony. 101. When the Soprano, Alto and Tenor all lie within the compass of an octave, the parts are said to be written in Close Harmony. If they exceed the compass of an octave, they are in Open Harmonv Close Harmony. Open Harmony. Fig. 29.< Close Harmony should be used in the following chapters unless otherwise indicated. To Connect Two Triads. NOTE. The following is of especial importance, and should be thoroughly mastered before proceeding. 1 02. Under this head two cases are to be considered : HARMONY SIMPLIFIED. 51 ( a.) When the two given chords have one or more nates in common. ( b.) Where there is no common tone to serve as a connecting-link. When the Chords have a Note in Common. Let us take C E G and A C E, for example, to connect. Having two notes in common, it is evident that there is a close connection between them, and it is only necessary to make this connection apparent to the ear. If we play the two chords thus : there is to the ear no connection whatever. But when r Q played thus : /^ the connection is very ap- parent. This is because the notes common to both chords are retained in the same parts. That is, the Tenor and Alto, which have the notes C and E in the first chord, retain them in the second. Therefore, notes common to both chords are to be retained in the same parts. It will be seen, that to follow this all-important principle, the " position" of the chords must be adapted to the necessities of the situation, sometimes one note being highest and sometimes another. The Process. 103. The following is given to illustrate the mental process by which the beginner should solve every prob- lem. Having written the first chord in notes : ist step. What are the notes of the second chord ?* This question, though unnecessary here, is of importance when the pupil begins to work exercises from a given Bass, as in in. 5 2 HARMONY SIMPLIFIED. ( N. B. If the pupil has trouble in keeping the notes of the sec- ond chord in mind during the following steps, he may write them on a separate slip of paper.) 2nd step. Is any note common to both chords ? What note is it? 3rd step. In which part (Soprano, Alto, etc.) of the first chord is this "common" note found? Ans. In the (Here mention whether it is Soprano, Alto^ Tenor, or Bass ) , therefore it must appear in the same part in the second chord. 4th step. Write it, and connect with the same note in the first chord by a tie. (Do not write any other notes yet.) $th step. Name the remaining notes of the second chord. 6th step. Which "position" of the second chord will enable the remaining notes of the first chord to pro- gress in the smoothest manner to the remaining notes of the second chord? Illustration. 104. To connect the triads C-E-G, written thus: tflT - v? As this makes a smooth leading HARMONY SIMPLIFIED. 53 of the Alto and Tenor ( no wide skips*) the effect is If the first chord is in this position : the connecting note, being G, is in the Alto in the first chord, and must appear in that part in the second chord. Now it is plain that we must so arrange the remaining notes of the second chord, B and D, that the Soprano and Tenor of the first chord ivill each have a note to which it may progress ; therefore, we cannot place both B and D below G, as was the case before, but one should be above and one below, and the choice ot position must depend upon the possibility of making a smooth progression. Let us try with D above and B below, giving : p^ tv,*^ 5 * , and compare it with the ~&~ effect when we place the B above and D below, thus : __ It will be seen that although the former ^ will answer, the latter gives the better effect, because there are no skips. Again, taking the first chord in this position : ~Y^ ^ ~? we find the connecting note in the lowest part ; therefore, both the remaining notes of the second chord^must be written above the connecting note, giving : Keyboard and Written Exercises. 105. (a.) Connect the triad of C with that of F Major, * In the early exercises the parts should not make very wide skips from note to note, but should progress by the smaller intervals ( ands and 3rds) wher- ever possible. In composition, where the parts progress by the smaller inter- vals, the effect is restful and tranquil. Where they progress by the larger intervals, such as 4ths, 5th, 6ths, and Sves, the effect is bolder and more aggressive. 54 HARMONY SIMPLIFIED. taking successively the various positions of the first chord, as illustrated above. Use one staff in writing. (3.) Connect ( in three positions ) the triad of C maj. with that of E min. ; with A min. Connect ( in three positions ) the triad of D min. with that of F maj. ; with A min. ; and with G maj. Connect ( in three positions ) the triad of E min. with that of C maj. ; with G maj. Connect ( in three positions ) the triad of E min. with that of A min. Connect ( in three positions ) the triad of F with triads upon C, A, and D ( all in the key of C ) . Connect ( in three positions ) the triad of G with triads upon C, E, and D Connect ( in three positions ) the triad of A with triads upon C, D, F, and E. Connect ( in three positions ) the triad of B with triads upon E, D, and F. Note that all the above are in the key of C Major. (c) Transpose ( b ) into other keys, and rej^eat. ( This transposition will not be difficult, if we remember that to transpose a note or a chord it is given the samo relative place in the new key that it occupied before being transposed. E. g., if a triad is on the second degree in the key of C, when transposed it must be placed upon the same degree of the new key : if on the fifth degree in the original key, it must be placed upon the same degree in. the new key. Likewise the ' ' position " and inversion of a chord must correspond when transposed. If we substitute the Roman Numerals ( as shown in 92 ) for the letters C^ D, E, etc., in exercise ( b ), it will be easy to find the notes corresponding to these numerals in any desired key. ( d.) Write ( b ) in four parts, as illustrated in Fig. HARMONY SIMPLIFIED. 55 -X- 1 J5 1 g 1 $ = 3* .x ^ 1 ^ 1 ^ 1 j. 17 3! 1" 1 ~ ' 30; the root of each chord being written in the Bass, which will remain the same for all positions. Fig. S0.< To connect two Triads when there is no Common Note. 106. Although two given chords belonging to the same key may not have a visible connection by means of a common note, there is a certain relationship through their being members of the same key, ( see the Author's "How to Modulate," 3,) and with a careful leading of the parts they may be used in succession. Especial attention must be given to avoid consecu- tive Fifths and Octaves, remembering that Contrary Motion is the means of so doing. It should be noticed that some Positions are much better than others for a given connection, and that some Positions cannot be used at all. The smoothest connection is usually where the three upper parts move in a direction contrary to the Bassr. The Process. 107. The mental process of finding the correct leading of the parts is somewhat as follows : Example for illustration. Connect the triad of C, in the position of the 3rd, with the triad of D. Ex- pressed in notes, thus : 56 HARMONY SIMPLIFIED. (1st step.) What are the notes of the Second chord? Ans., D F A. (2nd step.) In which direction does the Bass move in the example? Ans., Upward; therefore it would be well to have the three upper parts (or as many of them as possible) move downward ( to move contrary to the Bass) . (3rd step.) Which position of the second chord allows the proper progression of the parts, without Con- secutive Fifths and Octaves? ( Or, 3rd step.) Write the notes of the second chord, so that each part shall progress in the desired direction, avoiding Consecutive Fifths and Octaves. (4th step.) Would any other position give a better leading of the parts, by avoiding large skips or otherwise producing a better general effect?* N. B. All of the upper parts are not obliged to move contrary to the Bass. Sometimes it is better to have only one part progressing contrary to the Bass. Fig. 31 illustrates the connection of the triad of C (in its three positions ) with that of D. _() <*) (')_ Fig. 31. * There are other influences affecting: the leading of the parts, which are, however, as yet too advanced for the pupil. After having studied as far as 170. the pupil should review this section. HARMONY SIMPLIFIED. 57 108. At (a) it is necessary to double the Third to avoid Consecutive Fifths with the Bass, which would arise if the Alto should progress to A. Notice also that the Tenor should not progress downward to D at this place, as bad hidden Fifths with the Soprano would result. (See 134.) Exercises. 109. Copy the following, and fill up the vacant parts, aorjlying the " mental Process " to each of the ten sepa- rate examples. Fig. 32.. Key of C : n in in IV IV V V vi vi 9i= I ii in II in IV V in IV The above examples do not sound well unless used in connection with other progressions, when they lose much of their harshness. The teacher should give exam- ples in other keys, and as soon as the class can " figure " inversions (see 125-132), this section should be again taken up, using chords in their inversions. Exercises. ( a. ) In the key of G, connect the triad upon each eg HARMONY SIMPLIFIED. degree with the one upon the degree next above, trying the different positions to make the best effect possible. ( .) Repeat in the keys of Bb, A, and F. (c.) Repeat the above at the keyboard, adding all other Major keys. Review of the Connection of Triads, no. ( a. ) Avoid Consecutive Fifths and Octaves. ( .) Contrary motion is the means of avoiding them. (c.) If there is a connecting note, keep it in the same part in both chords. ( d.) If there is no note in common, adopt contrary motion and avoid wide skips, especially guarding against consecutive Fifths and Octaves. ( e.) In doubling notes, the Fundamental is the best note, the Third the poorest. The Leading-note should be doubled only under exceptional circumstances : though doubling any part is better than open consecutives. (_/".) Avoid wide skips. Let each part be melo- dious. (,".) Avoid progressions of Augmented intervals, as they are not melodious. Part-writing. in. Having learned to connect two given triads, the pupil should proceed to put his knowledge into practical use by writing exercises on given Basses. In these exer- cises is nothing new ; each exercise may be considered as a little series of examples illustrated in 102 to no. N. B. A figure over \hejirst Bass note ot an exer- cise, indicates whether the Third, Fifth or Octave of the Bass note is to appear in the Soprano. Should the pupil need further guidance, the follow- ing mental process," illustrating Fig. 33, will help. HARMONY SIMPLIFIED. 59 33. m 112. Process: The Figure 8 over the first note indi- cates, that we are to begin with the octave (or double octave ) of the Root as the highest note, giving the chord in this position : The first problem then is, to connect this chord with the chord founded on F, as indicated by the second Bass note in Fig. 33. Now let the pupil go through the process shown in 103, giving as a result: I i The next problem is to connect the chord last found with the chord founded on C, as indicated by the third note in the given bass. Following th? same process brings one more chord. Continuing in the same way gives the completed example : Fig. 34.< IV 113. In the first exercises the Soprano part is given a well as the Bass, leaving the pupil to find the names of the remaining notes in each chord and to place thena 80 that they will progress as smoothly as possible. 6o SIMPLIFIED. The parts should not cross ; for example, the Altr should not go higher than the Soprano or lower thar the Tenor. Write the exercises in close harmony. 114. The various parts should not exceed the compass of a good voice of corresponding pitch, as shown in Fig 35- Soprano. Alto. Bass. The pupil should always mark the Roman numerals in the exercises, as shown in Fig. 34. Always write them before beginning to form the chords. 1 15* Exercises. A I. 2. L/ i 1 1 .^ ^ -^ I?T\ ^i ^^ ^2 E(p ^Z. \\.|^ X3 G** 1 ^^ 1 1 ^^ \ ^^ ^^ 5 5 ^ ^ * & & II -J--& , 1 fj <^ I.I a 3 - J{ ^2 ^ J 3 c~v ^s ^s 1 1 * 1 22 2i & > ( I 4. i i 5. HARMONY SIMPLIFIED. 6. 61 V XT ,-= =^ u Hfi . fTV ^> fiy & _^r , ^j | f >, v^jy t -*' <=> [ ' "* II <5> O 2 ^^ J 5 8 p\ Bf? ss . & ^ /5 sp 5 - G> 1 ^ 7. 3 z> II o II tez II o 3 RT$ ffl ^^ & ' j^> & 8. Z 3 (5 ^ es ^^ ^Ti ^-i & II fr\ -~ e> *-~ *-^ II saj j 3 CVtf jt ^ t T**j* f? {3 ^ it fT) 1 1 6. In the following exercises the pupil will write the Soprano as well as the other parts. Where the figure 3 is found over the first Bass note in an exercise, it indicates that the first chord should appear in the posi- tion of the 3rd. The figure 5 shows that the position of the 5th is desired. Where no figure is given, the posi- tion of the octave is to be written. This applies to the first chord only of each exercise. 1. Jadassohn. 2. Richter. I 1 ^ -\ \ ^ II ^ o* -^ V rj \ -1 \^-\\^^ _S! I IV I 3. R. 3 VI IVIIVVI 4. R. f\* 1 1 22 ^ 8. J- $ 8 SS 1 1 9. J- JJ 3523-1 1 - ^^ II tz 5 l|. JJ o 1 ^/ c? |J 11. J- 3 1 i *l 1 4-~"\ * It L, ^^ . 1 1 1 [/ /fl J xJ 22 z II [' P V-' J F - ii 12. *' 3 * 1 I f\ *tf ir j*^ 1 , lff^ /[i ^^ J ^ a \ If \\S & Synopsis. 124. Form a synopsis of the chapter as usual. CHAPTER IV. Inversions of Triads. 125. It is not necessary that the Fundamental note (or root) of a Triad should always occupy the lowest place.* The Third or the Fifth can also occupy that place, and when this occurs, the chord is said to be inverted. When the Fundamental is lowest, the chord is in its Direct form. When the Third is lowest, the chord is in its ist Inversion. When the Fifth is lowest, the chord is in its 2nd Inversion. ( See Fig. 38.) Notice that " Position " relates to the Soprano or highest part, while " Inversion " relates to the Bass or lowest part. * By inversion the Root is not changed, but transferred to a higher part The root of a chord is the Bass note only when the chord is not inverted. HARMONY SIMPLIFIED. fy Keyboard and Written Exercises. ( a.) Form various Triads, and show their Inrer- sions, as illustrated in Fig. 38. Avoid doubling the Third. Fig. 38. a 1 ii A S3 <2 /? 4 BTS ^-> HZ 5 Z | _^ rl ' ~\ . f^ 1 . ^ -^ ^y I Direct Form. I ISt Inversion. I 2nd Inversion. 126. ( .) Write various Triads in their several In- versions and Positions, using two staves. The pupil should not forget that Fig. 38 represents not three differ- ent chords, but three forms of one and the same chord. We could not say ( because E is in the Bass ) that the form marked " ist inversion" in Fig. 38 is the triad on E. It is the triad on C in an inverted form. The note C is the fundamental or root from which the chord is derived, which note may be placed lowest or highest. Therefore, in marking the triads, the inversions are to be marked like the direct form (the same Roman numerals), as shown in Fig. 38. The pupil should carefully distin- guish between the actual Bass note, and the root of the chord. The Bass note changes with each inversion, while the real root of the chord remains the same for all inversions and positions. Figuring Triads. 127. In 56 the pupil learned to recognize intervals according to their distance from a lower note, and to indicate the same by figures. In a similar manner, whole chords can be figured, by indicating the interval 68 HARMON-* SIMPLIFIED. which each note forms with the Bass or lowest note. For example, if we have a note with the figures 5 and 3 over it : we understand that the interval of a Third from the note C is required, and also the interval of a Fifth, from the same note. Thus : If we have the same note with the figures 6 and 4 over it, we should form the intervals of a Fourth and a Sixth from that note : 128. These intervals are not necessarily in the same octave as the Bass note, nor in the exact order indicated by the figures, as their arrangement depends upon the pro- gression of the parts in preceding chords. Exercises. (a.) Figure the chords in Fig. 38. ( 6.) Write the scale of C major, and form a triad upon each degree. Write each ti'iad in its direct form and both inversions, using two staves, and writing in four parts. Figure each chord thus produced. (c.) Write on an upper stafF ( treble ) the chords indicated by the figures over the following Bass notes HARMONY SIMPLIFIED. 69 remembering the caution above given in regard to the notes being neither in the same octave as the Bass note, nor in the order expressed by the figures : (.) 6 means the same as f; a " chord of the Sixth" is the same, therefore, as a " chord of the Six-three." ( c.) A sharp, flat, or natural, placed after a figure, is the same as if placed before a note, meaning that the note indicated by the figure is to be made sharp, flat, or natural, as the case may be. If a sharp, fla 4 ; or natural is given without any figure, the Third from the Bass is intended. A line through a figure, e. g., &, is the same as a sharp after it. ( d.) The doubling of the parts, positions, leading of the parts, etc., are not indicated by the figuring. ( r^-- ^Ff II 6 J i 7. J- 8. Harmonizing the Scales. 135. An excellent exercise, at every stage of advance- ment, is the practice of harmonizing the scales in every HARMONY SIMPLIFIED. 75 key, and using as many different chords and as much variety as the pupil may have studied at the time. It will be noticed that every note of the scale may belong to three different chords, and either one of these three chords may be used to harmonize that note if a smooth connection with the preceding and following chords can be made. The scale to be harmonized should be written sometimes in the Bass and sometimes in the Soprano, (see examples below). [For advanced pupils it may also be written in the Alto and Tenor.] When written in the Bass, it should be observed that there can be no common notes to connect two successive chords, unless chords of the 7th are used, for which see later chapters. Exercises. ( a.) Fill out the four parts in the following: I. 2. f /( ^ n Q-\\ ^i I irTi -j & l-l - & & 51 II } Z 1 CV f a o II II m ^ L ^ -. '^ ,*3 II II ^ *y ^y ^y t-^ f^ I | ^y _- fp V -^5 2 6 66 A 3 * 4 ' 3 ^ , Y-*- !V II IS /k^/^l-n \\ & rt ^ II r - \ =>* & n 5.Z2ZZI5 S|Z & & (.3 II g 2 , a* ^ {? II & a II <^ 66 &- -&- 5- 6 6 6 3 6 ^ -<9- 66 1 6 ~ J & ^ II 1 -j ^^ ^^ \ \ ^-) ^3 ^^ II ~J ^^ ^"* \ \ ^ "^ II II 1.1 HARMONY SIMPLIFIED. ( .) Harmonize the ascending scale of C in as many ways as possible, using only the triads with their inversions. (c.) Harmonize the descending scale similarly. ( oT.) Harmonize similarly the ascending and de> ecending scales in all other keys. ( e.) Advanced Course. Harmonize similarly the Minor scales. (_/".) Repeat all of the above at the keyboard. Synopsis, 136. Write a synopsis of the chapter as at the end of Chapter III. The Perceptive Faculties. Continued from page 40. Triads. 137. After teaching the pupils to recognize two tones sounding together, it is but a step further to recognize three tones. This section is merely a continuation of the foregoing, and may be treated somewhat as follows : ist Step, (a.) Teacher sounds the note C, and says : " This tone is Doh. Write it in the key of " ( mentioning any key, not necessarily the key of C ). ( 6.) Teacher sounds E and asks, " What is this tone?" Ans> Me." Write it." Two pupils sing Doh and Me. ( c.) Teacher sounds G and asks, "What is this tone ?" Arts. Soh. " Write it." Third pupil sings Soh. Three pupils sing the three notes together. ( called a Chord of the Seventh. ( It may be noticed in reference to this building-process, that each note is at the interval of a 3rd either Major or Minor from the note next below. In the following chapters it will be seen that this g, HARMONY SIMPLIFIED. same rule of placing the successive notes a 3rd apart is followed in forming chords of the Ninth ; also in what are called, by some theorists, the chords of the Eleventh and the Thirteenth.) As the character of Triads differs according to the character of the component intervals, (Major, Minor, etc., see 93,) so the character of the Chords of the Seventh must differ. Exercises. 148. ( a.) Write chords of the 7th upon every note of the Major scales of C, G, F, and D, describing the character of the triad as in 93, and indicating, on a sep- arate line, the character of the yth. For example, form- ing the chord of the yth on the third degree of the scale of C, it would be described as in Fig. 40. Fig. 40. 1 in with Minor Seventh. The Roman Numeral, being small, indicates that the triad is Minor. The character of the 7th is plainly expressed. ( 6.) Write chords of the 7th upon every note of the Minor scales of A, E, D, and B, indicating the char- acter of the triad and the 7th as above. (c.) Repeat all of the above at the keyboard. 149. The pupil, while writing the above, should notice the following : ( i.) That some of the chords, particularly in the Minor keys, sound so badly that they could not be used. (2.) That the chord of the 7th formed upon the 5th degree, the Dominant, is alike in Major and Minor, HARMONY SIMPLIFIED. 83 and is not only the most agreeable one, but is the only one having a Major 3rd and Minor 7th. (3.) That none of the chords of the 7th are satis- factory to rest upon, but, like the Augmented and Di- minished intervals and triads, seem to require something to come after them to create a feeling of repose, (a.) (6.) Fig. 4 1 . For example, the chord (), in Fig. 41, although it sounds well, is evidently not satisfactory to dwell upon, or to use as the final chord of a composition, as it seems to suggest something which should follow to make it complete. Notice that the chord marked ( b ) gives this sense of completeness and repose. 150. This leads us to consider that all chords may be divided, with respect to this quality ( repose or the lack of it), into two kinds: Independent chords, or those which are satisfactory to pause upon ; and Dependent chords, or those which demand that some chord should follow to establish repose. This classification corre- sponds with the division of intervals into Consonant and Dissonant ( see 75 ) ' ^ or those chords containing conso- nant intervals exclusively are Independent, while those containing even one dissonant interval are De- pendent chords. 151. This demand on the part of a Dependent chord to be followed by something reposeful, is satisfied if a consonant chord succeeds it. The process of passing 84 HARMONY SIMPLIFIED. from a Dependent chord to one that is consonant ( Inde- pendent ) is called "resolving"'*., and the chord to which it progresses is called the " chord of resolution" It is necessary to resolve dissonances, not only because they are unsatisfactory to rest upon, but also be- cause there are tendencies on the part of certain inter- vals contained in them, and of certain notes of the scale, to progress in definite directions* The Principle of Tendencies: Melodic Tendencies. 152. ( a.} This tendency of certain notes of the scale to progress in definite directions may be illustrated by sing- ing the Major scale up to and including the 7th degree, then suddenly pausing without singing the regaining (8th) degree. By thus pausing, a sense of incompleteness will be felt, a desire for the delayed note. Thus it is clear that the 7th degree has a strong tendency to progress to the 8th degree, which is the Tonic, or most perfect resting-place in the whole scale. On account of this marked tendency toward the tonic (or its octave), the 7th degree of the scale is called the Leading-note. (6.) Similar experiments will show that the 3rd degree of the scale has a distinct tendency to ascend, though the tendency is not so strong as in the case of the 7th degree ; and that the 4th degree tends downward. ( c.) An accidental sharp tends to continue upward, and an accidental flat downward. ( N. B. When a nat- ural is used to raise a note already flatted by signature or otherwise, it is like a sharp in its effect, and has the same tendency to ascend. Likewise, when a natural is * In fact, the reason that some chords are unsatisfactory to pause upon is simply because certain intervals and notes have the above-mentioned tenden- cies ; for while perfectly agreeable to listen to, they point unmistakably toward something which is to follow. HARMONY SIMPLIFIED. 85 used to depress a note already sharped, it is like a flat iu its effect, and has the same tendency to descend.) Notes having a tendency to progress in a particular direction, are called Tendency-notes. Tendency of Continuity. 153. A tendency to progress in any desired direction may be given to a note, or a natural tendency counteracted, by approaching that note from a contrary direction. Thus, if it is desired to have the yth degree progress downward, it can be done by approaching it from above : ^Tfc-t & This might be called the Tendency oj Continuity, i. e., to continue in a given direction after having started. Harmonic Tendencies. 1 54. An Harmonic Tendency is the tendency of a dissonant interval, or of the notes forming it, to progress in certain definite directions. It is apparent that the nat- ural (Melodic ) tendencies of the above named Tendency- notes are not so strong but that they may be overcome by the tendency of Continuity. But, when this natural tendency is heightened by the presence of dissonant in* ter-vals, the demand for progression is quite unmistakable. Let us examine the effect of dissonant intervals upon the tendency to progress. ( i ) Play the following and pause: T the ear will then demand that GJ shall progress to A This is caused, first, by the fact that in touching G# we have started on the road from G to A, and having com* 86 HARMONY SIMPLIFIED. pleted half the distance ( theoretically a little more than half) , it is only natural to desire to continue to the desti- nation. It would be useless to go half-way, and then turn back. (See 152, c.) Secondly, it is caused by the fact that we have at x a perfect 5th, followed by an aug- mented 5th. The perfect 5th has been made larger by a sharp, and it would be expected to develop into something else instead of retreating. Thus it is apparent that the combined influences at work must create a demand for a chord to follow any dependent chord. (2.) Again, striking the interval at (), Fig. 42, we find a strong tendency to progress to the interval at (<$.) This is caused by the same tendency of an augmented interval ( here an augmented 4th ) toward a further di- gression of the parts. (a.) (b.) Fig. 42. ( 3.) On the other hand, if we take a Normal inter- val and diminish it, there is a strong tendency to still fur- ther contract, as in Fig. 43, where ( a ) is the Normal interval, ( b ) the diminished and ( c ) the result of the tendency toward further contraction. (a.) (b.) (c.) Fig. 43. These are illustrations of Harmonic Tendencies. To formulate the illustrations in Figs. 42 and 43, the following is given : All Augmented intervals tend toward further expansion. All Diminished intervals tend toward further contraction. NOTE. This law is a direct result of the principles stated in 152 ( a ^^ 6 6 & 2. -&- C 67 6 4 7 J- 4. HARMONY SIMPLIFIED. The Principles of Part-leading: " Influences," Combined and Opposed. 161. It has been said that the rules of Harmony were made only to be broken, and that every rule has more ex- ceptions than applications. It would seem better, there- fore, to review the principles from which the rules are derived, and thus gain a sound judgment in regard to the leading of the parts, which must ultimately replace any rules that could be given. The sources of the rules which are commonly given, are found in the necessity of considering the following points in order to produce good effect in part-writing: ( i.) The Harmonic effect of the four parts together. (2.) The Melodic effect of the indi- vidual parts. (3.) The Tendencies of certain notes of the scale, and of various dissonant intervals; i. e., the Melodic and Harmonic Tendencies. (See 152 to 155.) (4.) The bad effect of Consecutive 5ths and 8ves. (.5.) The bad effect of doubled 3rds.* (6.) The Prominence of Outside Parts.** ( 7.) The desirability of Connection between suc- cessive chords. (8.) The arrangement of the notes in a chord; i. e., their distance apart.*** (See 97.) r **: *** See the following paragraphs. 94 HARMONY SIMPLIFIED. The above-mentioned points may be called, for con- venience, " Influences " which affect or control the lead- ing of the parts. Sometimes these various Influences agree, or combine to demand the same progression ; some- times they oppose one another. Notes upon the Preceding. 162. *By doubling the 3rd a certain dissonant overtone (See 90; and Note, p. 44) is brought into prominence, making the chord some- what rough in effect. Therefore it is not well to double the 3rd with- out some definite reason. Furthermore, in the Tonic triad, and also in the Dominant triad or Chord of the 7th, chords which appear very frequently, the 3rd is a tendency-note. (See 152.) Now, it will be seen that tendency-notes should never be doubled, if possible to avoid it, as the result must be either consecutive 8ves or the contradiction of the tendency by one of the notes. Therefore, where the 3rd is a tendency-note, it should not be doubled. Where it is not a tendency-note, it may be freely doubled if thereby a better leading of the parts is obtained. ( The chief Ten- dency-notes ( Melodic ) of a scale are the 3rd and 7th. When the 3rd of a chord happens to be one of these notes, it is better not doubled.) 163. **It will be observed, that the Soprano and Bass parts are more conspicuous than the inner parts. Therefore, that which might be allowed in the inner parts may be found very disagreeable and consequently be forbidden, when occurring in the outer parts. In- cluded in the above are found most frequently the two points of ( a.) Disregarded tendencies : e. g., (a.) Bad. (b.) Good. Fig. 46. G> L 1 At ( a), Fig. 46, the upward tendency of the Leading-note, B, is disregarded, it being led down to G, with very bad effect. At ( b ) the same thing is done, but in an inner part. The effect here is very good, as the Alto, an inner part, is less prominent than the Soprano, and as the note to which the Leading-note would have progressed is still HARMONY SIMPLIFIED. 95 found in the last chord. Again, by the progression of the Alto Lead- ing-note down to G, the last chord has the 5th which would other- wise be lacking. (t>.) Hidden Consecutives : e. g., (a.) Fig. 47. 1 The progression at ( a ), Fig. 47, is too harsh to be effective, the hidden consecutives appearing in the outer parts ; but the progression at ( b ) is much more agreeable, as the hidden consecutives are be- tween one inner and one outer part. Also, where the natural ten- dency of a note is disregarded, the effect of a Hidden Consecutive is less likely to be agreeable than where the tendency has not been dis> turbed. When considering the introduction of a Hidden Consecutive, this point should be considered. In the example, the downward tendency of F ( the 4th degree of the scale ) is disregarded, with bad effect where the neglect is made prominent by being in the Soprano. In the Alto it is less disagreeable, though it is easily seen that the effect of such progressions might be made still better by observance of the Tendencies. Distribution of the Parts. 164. ***To produce the best effect, the notes of a chord should be at about an equal distance from each other. If necessary to distribute them unequally, the larger intervals should be in the lower parts. Excepting between the Bass and Tenor, there should not be more than an octave between two neighboring parts. Play the following : Bad. Good. 06 HARMONY SIMPLIFIED. Opposition of Influences. 165. An illustration of this opposition is given in Fig. 48. In this ixample there is a tendency on the part of the Leading-note, B, to as- nd. If this tendency is followed, the next chord will have no 5th. Fig. 48.< I As in some cases ( for example in a full chorus ) this would weaken the effect of the four voices singing together see " Influences i and 8" it is sometimes better to sacrifice the upward tendency of the Leading-note in order to gain a full effect in the following chord, giving the progression : 1 In disregarding an Influence as was just shown, the pupil should guard against violating some other Influence ; for example, if the Leading-note were in the Soprano or Bass, it could not progress downward on account of Influence 6. The effect would be very bad, as shown in Fig. 46, (a\ Again, if the Bas? note G, in Fig. 48, should progress downward to C, instead of upward, the leading-note could not pass downward, on account of the bad Hidden 5ths ( both parts moving by a skip, see 34); e.g., 166. Another illustration of the manner in which these influences may oppose each 'Jther is shown in Fig. 49. HARMONY SIMPLIFIED. 97 49. / z <^, H II I?T\ rd ^ $j X3 II v-LJ g 6? 2 ^ f* i x J^T? (*^ Bv 1 J B r I -^ ^ 1 At x the 3rd of the chord, E, is doubled, iu opposition to Influ- ence 5. The reason for this is shown in Influence 2, namely, the advantage of a smooth progression of the parts : also in Influence 4, for if the Tenor note, E, in the chord marked x, be changed to C in order to avoid doubling the 3rd, the result would be Consecutive 5ths with the Alto, which are much worse than a doubled 3rd. Contrary motion and the Tendency of Continuity combine to prevent any bad effect which might be expected from doubling this Tendency-note.* 167. One more illustration may be given : Influence 7 recom- mends the retention of a common note in the same part ( see also 102 ). But it occasionally happens, that other considerations, par ticularly Influences 2 and 8, are more important, and demand that this Influence be sacrificed for them. This is shown in Fig. 50. Here the note C, which in the first chord is taken by the Tenor, is in the second chord taken by the Alto.** Fig. SO. * A Melodic Tendency may be disregarded far more freely than an han ttionic tendency, since the former can be removed by Continuity. (See 153.) ** If circumstances should allow the rearrangement of the first chord, it would still be possible to retain the common note ; e. g., 1 This would illustrate the fact that in writing exercises, if the pupil finds it diffi- cult to make a certain connection, by going back a few chords and working in a different position, a way may be opened. p8 HARMONY SIMPLIFIED. Many other illustrations might be given, showing how circum- stances alter cases, and that what is good in one place may not be best in another. The pupil should understand that part-writing is not a question of following rules, but is a matter of judgment, controlled by the considerations above mentioned. In general the pupil will find that .the more prominent of the above Influences are Nos. 3, 4, 6, and 7. General Directions for Part-writing. 1 68. In summing up the above, and formulating di- rections for Part-leading which shall be simple and yet adapted to all cases, the following may be given : ( i.) Avoid Consecutive 5ths* and 8ves. (2.) Avoid Hidden 5ths and 8ves only when they make a bad effect. (3.) A note common to two chords is to be retained in the same part, unless some other Influence requires another progression. (4.) Smooth progressions are better than wide skips in the parts. (5.) Study the Influences. If they agree, there will be no question in regard to the progression. If they disagree, let the stronger rule unless consecutives are pro- duced. (6.) Listen to the effect. If it is bad probably some Influence has been disregarded. ( 7. ) Consider the range of the voices. (See p. 60.) 169. From this time forward, the teacher, when cor- recting exercises, should designate which Influence has * A single exception may be given. A Perfect sth may be followed by a Dimin. sth, thus, ES^^^:Er| but not reversed, thus, ^T^""""^"!], / Good. ,/ Bad. because the latter prevents the diminished interval from contracting. In the opposite direction, the tendency to contract causes a return to the first har- mony (good). See footnote, p. 49. HARMONY SIMPLIFIED. 99 been disregarded in each case ; or he may simply draw 'A line through the wrong note and mark the number of the influence which, if followed, will rectify the error, leaving the pupil to change it. This will awaken the critical powers, and cultivate the judgment. Also allow the pupils to correct one another's work according to the same plan, in each case giving the reason for the correction. NOTE. The pupil should distinguish carefully between the chord of the Dominant seventh on G and the chord of the Dominant seventh in the key of G. The former has the note G for its root ; while the latter is built upon the sth degree ( the dominant ) of the scale of G, i. e., D. Exercises. 170. Mark the Roman Numerals under the Basses before proceeding. 1. J-367 6 4687 -3=* 2. R' 5 ^ 67 4 35 ;e .&. 4. -z? h 100 5. HARMONY SIMPLIFIED. 7 57 6. J- 3 6 $ 066,666 $~ B=P EHI ^dil 7. J. ^H 5 7 8. J- 36*, 6 $ 6 7 < ~ 9. 7 6 P v-fe-zt- ^i^ ^ z= l = ^ ^-M 2 r4 ' 8 7 B - 10. J- Exercises fn Harmonizing the Scale. 171. Harmonize the Major and Minor scales, using chords of the 7*h where possible, and the triads with inversions, working both at keyboard and in writing. Synopsis. Write the usual Synopsis of the chapter. HARMONY SIMPLIFIED. ioi CHAPTER VI. INVERSIONS OF THE CHORD OF THE SEVENTH. 172. We have repeatedly seen the different Positions of the Chord of the Seventh. We will now consider the Inversions, which are very similar to the Inversions oi Triads, though a little more complicated, owing to the presence of four notes in the chord. Compare the fol- lowing with 125. (a.) When the Root is in the Bass, the chord is in its Direct form. (6.) When the Third is in the Bass, the chord is in its 1st Inversion. (c.) When the Fifth is in the Bass, the chord is in its 2d Inversion. (d.) When the Seventh is in the Bass, the chord is in its yd Inversion. The Inversions are figured and named as follows : Direct, ist Inversion. 2nd Inversion. 3rd Inversion. 7 66 64 6 5 or 7 5 or 5 4 or 3 4 or 2 33 3 2 Six-Five-Three, Six-Four-Three, Six-Four-Two, or or or Six-Five. Four-Three. Second. Example :< V 1 i II 1 ^ 1 & S3Z ? II j ^ o c~\ * (\ IT3 || * 1 rt \ ^ \ 1 ^ *^ 1 II 7 6 6 S r 5 3 64 6 4 or 3 4 or 2 3 2 , 02 HARMONS SIMPLIFIED. Exercises. (a.) Taking each of the 12 keys in turn, write Chord of Dominant Seventh in its several inversions, and figure them. Vary the positions in the different exercises. (.) Repeat the above at the keyboard, in all keys. To find the Root of a Given Chord of the Seventh. Proceed as shown in 129. When the chord is in its " Direct form," it is said to be placed in 3rds, since each note is a 3rd above the one next below. It should be noticed that when placed in its Direct form, a chord is always figured I, or such part of these figures as may be necessary. If either of the figures 2, 4, or 6 appears, an inversion, and not the direct form, is present. Exercises. (a.) Write the chords indicated by the following fig- ured Basses, and mark the appropriate Roman numeral : (b.) Play the chords indicated by the above figured Basses. Resolutions of Inversions of the Chord of the Dominant Seventh. 773. If the simple tendencies shown in 157 are fol- lowed, the pupil will have no difficulty in resolving the inversions of the Chord of the Seventh. Remember, that the Leading-note tends upward, the 7th from the root downward, Augmented intervals tend to increase, while Diminished intervals contract. ( See 152 to 155.) Exercises. (a.) Following the above principles, resolve the inversions shown in Fig. 51, and place the proper Roman Numeral under each chord. HARMONY SIMPLIFIED. 103 Fig. 5 1 . (3.) Write inversions of the chord of the Dominant 7th, and resolve them, in the keys of G : F ; D ; Bb ; A ; Eb; B; Ab; F. (r.) Repeat the above at the keyboard, adding all other Major and Minor keys. Exercises. 1. R' 66326 47 T-l* j j <5> * <-^ f 3 m ^ II ~s ffi <& 2 (5 1 <=? 2 R- 3 f e e f i -& & R. Y^ ~ ^2 '^ II ~3L n * v P_ G II 4. 3636436 &-3--=l 8 7 6 f- 6. R. f 6 e 6 \ 6 4 f 1 *^\" I r" '~~ i ' rr?'(j> - ? * c> 1 ' -& *! - | j i : 1 1 1 i jj^* '^^- ' *-* 104 HARMONY SIMPLIFIED. 7. e 7 e 4 ft is; m 8. 6 7 6 4 ft 6 626 326 4= 6 87 26 45- 10. J- 365 266 36647 3664 t~ ' I ' c ^ &~ ^ C. 11. J. 8 66 326 P*F^ -ir &-f= 12. J- HARMONY SIMPLIFIED. 105 175. Exercises in Harmonizing the Scale. Harmonize the scales, using the chords of the 7th with their inversions, and the triads with their inversions. CHAPTER VII. SECONDARY CHORDS OF THE SEVENTH. 176. The chord of the Dominant Seventh, because it plays such an important part in the key, is also called the Principal chord of the seventh.* The chords formed upon the remaining degrees (for they are nearly all found in Harmony) are called Secondary or Collateral Sevenths. Formation of Secondary Chords of the Seventh. As seen in 147 and 148, they are formed by the addition of a 7th to the triads upon the various degrees of the scale. As the triads are of various kinds, viz., Major, Minor, Diminished or Augmented, the Secondary seventh-chords will have the same variety of formation, * It is also called the Fundamental Seventh, since its intervals are formed like those of Nature's (Harmonic) chord, with Major triad and Minor yth from a Root-tone. (See 90.) I06 HARMONY SIMPLIFIED. thus contrasting with the Dominant Seventh with its Major 3rd and Minor 7th. This irregularity of construction should not be considered a fault, for the chord of the Dominant 7th points so strongly to the Tonic, that if all the Chords of the Seventh were like it the sense of Tonality would be disturbed. ( See 266.) As it is, the characteristics of the key are much better preserved than would otherwise be the case. Again, as we need Major, Minor, Diminished, and Augmented triads to make up the complete list of triads in a key, so do we need the same variety in the structure of the Chords of the Seventh. Resolution of Secondary Chords of the Seventh. 177. As the Secondary Chords of the 7th are formed in a manner similar to the chord of the Dominant Seventh, so their resolution follows in a general way the same pat- tern; viz., the Chord as a whole tends to resolve to the Triad situated a 4th higher than the root of the Chord of the Seventh. This is the same as from the Dominant to the Tonic. (See 158, foot-note.) More accurately expressed, the chord D-F-A-C would tend to resolve to the triad on G, for G is a 4th higher than D. So also E-G-B-D would resolve to the triad on A, since A is a 4th higher than E. The individual notes in a Secondary Seventh-chord have a tendency, though not so pronounced, to progress as in the Dominant Seventh-chord; viz., the 7th from the root may descend, and the 3rd from the root may ascend. Pig:. 52. 1 For example, in Fig. 52 the general tendency is to the triad on G (the Cadencing Resolution ) . The 7th, C, being a minor 7th and therefore a dissonance with the HARMONY SIMPLIFIED. I0 7 a /L ^ ff\\ t^> 532 s t A ~a. II '1. ^ ^ ^^ M root, tends downward ; F tends upward, not so much on account of any dissonance or " Influence," as for the rea- son that it is the shortest way to a place in the next chord, and that we are accustomed, in the chord of the Domi- nant 7th, to hear the corresponding tone pass upward. According to Influence 8, it could also pass downward to D, making the second chord fuller. For many cases this would be better than the upward progression, provided that it made no bad hidden 5ths with the Bass. If the Bass should move upward to G, this would be quite satisfactory ; e. g., 53. Exercises. 178. (0.) Form Chords of the 7th upon all degrees of the scale of C Major, and resolve them as shown in Fig. 5 2 > or 53- NOTE. The resolution of the seventh-chord upon the 4th degree of the scale to the triad upon the 7th degree, is not commonly used, for the following reason : A Dependent chord demands a resolution to an Independent chord. Now, as the triad on the 7th degree is a Di- minished triad, it is not Independent, and is therefore not suited to be a chord of resolution. But it is possible to use this progression if the triad upon the 7th degree should in its turn be followed by an in- dependent triad; e. g., Fig. 54. Another restriction in the use of the chord of the 7th on the fourth degree of the scale is shown in the foot-note'to 187. (6.) Write the Secondary Seventh-chords, with resolutions, in the keys of F, G, D, Bb, A, El? and Fjf. (c.) Repeat the above at the keyboard. roS HARMONY SIMPLIFIED. Chord of the Seventh upon the 7th Degree in Major. 179. As a Secondary chord of the 7th, the natural res- olution of this chord is : VII 07 This is quite correct. But a more common resolu- tion is found in a consideration of the following We have seen how the Leading-note (7th degree of the scale) has a strong tendency to progress to the Tonic. (See 152.) The triad formed upon this note has also a strong tendency to progress to the Tonic triad ( see Fig. 56) , resulting from this tendency, while the chord of the fth upon the same note is even more strongly in- clined to progress in the same direction. E.g., ( plav it): (a.) (6.) Fig. 56. 1 Triad. Seventh. There are two reasons for this tendency; viz., (a.) The tendency of the Leading-note, mentioned above. (3.) The similarity of construction to the chord of the Dominant Seventh, which progresses naturally to the. Tonic. For example, G-B-D F (play it) is the chord of the Dominant 7th, resolving to the Tonic triad C E-G (play it) . If now the Root, G, is omitted, we have the triad B-D-F remaining, which resolves just as if the Root HARMONY SIMPLIFIED. 109 were present. This chord without the root (B-D-F) , is seen to be the same as the triad formed upon the 7th de- gree of the scale. Now, as the triad upon the 7th degree has such a distinct tendency toward the Tonic triad, it will be readily understood that the chord of the seventh upon the same degree has a similar tendency, which is in- creased, rather than diminished, by the addition of the 7th. This similarity to Dominant harmony will be further explained in Chapter IX. 1 80. From a consideration of the above, it will be seen that the Chord of the Seventh upon the 7th degree may be looked upon in two ways : ( a ) As an incom- plete form of Dominant harmony, ( in which case it would resolve most naturally to the Tonic triad, as in Fig. 56, b ) ; or ( b ) As an ordinary Secondary Sev- enth-chord upon the 7th degree, resolving most naturally to the triad a 4th higher, as in Fig. 55. Preparation of Dissonant Intervals. 181. A dissonance may be either agreeable or disa- greeable. This anomaly is explained by the fact, that although a chord may sound well, it is technically called a dissonance if it demands that another chord should follow to give a feeling of completion or repose. (See 150 and 151.) It was formerly the rule, that all dissonances should be " prepared." At the present day it is the custom to "prepare" only harsh disso- nances. The chord of the Dominant seventh was the first to be freed from the restriction, and the chord of the Diminished seventh is also free,* while the Secondary Chords of the Seventh, and the Chord of the Ninth (par- * Though not requiring preparation, it is well to approach the milder disso- nances by a Diatonic step rather than by a skip. no HARMONY SIMPLIFIED. ticularly the Minor Ninth, because it is a harsh disso- nance) , are usually prepared. " Preparing " a dissonance means, that the note which causes the dissonance shall have been present as a consonance in the chord immediately preceding; e. g., The note C, having appeared in the first chord as a con- sonant note, is thus " prepared " in the second chord where it is a dissonant note. 182. Instead of being " prepared," all dissonant notes may enter diatonically ; i. e., from the next step above or below. E. g., (The dissonant note C enters from the next step above.) In general, therefore, we should not skip to a dissonant interval, but either " prepare" it or lead stepwise into it. 183. A dissonant note, i. e., a note which forms a dissonance with another, should not be doubled. Being a tendency-note, as all dissonances are, if it were to be doubled either consecutive Sves would result, or a contra- diction of its natural tendency by one of the notes. ( See 162.) Exercises. 2. R. HARMONY SIMPLIFIED. lit 3. Fw^ffp^ t 5. J- 626 7 57 -g 6. J- 3 6 7 7. J- 9 :?s- 8. J- 3 6 7 = 1 9. J- 1 10. J- -^ t- F! 11. J- 367 m &ARMONY SIMPLIFIED. Succession of Chords of the Seventh ; Resolu- tion of one Seventh-Chord to another Seventh- Chord. 185. Instead of resolving to a triad, as shown in the preceding chapters, a Chord of the Seventh may progress to another Seventh-Chord; e. g., Fig. 57 might be called a contraction of Fig. 58 ; for, Fig. 58. since the Chord of the Seventh is merely an enlarge- ment of a triad (see 147 ), we are allowed to progress directly from one Seventh-Chord to another, considering that the Tonic Triad, or regular resolution, is implied in its enlarged form. Advanced Course. See "How to Modulate," page 42. HARMONY SIMPLIFIED. 3. IE 62677 F Secondary Sevenths in Minor. Advanced Course. Exercises. 187. ( a.) As in 148, the pupil will form Seventh- Chords upon each degree of the scale of C Minor, and describe them. He will notice that some of these Seventh-Chords, like the triads of the Minor scale, are too harsh for practi- cal use, owing to the various extremely dissonant intervals contained. It will be noticed that, beside the Dominant Seventh ( 'which is alike in Major and Minor ) , the most agreeable of the Secondary Chords of the Seventh in Minor are those upon the 2nd and 7th degrees. The others, either on account of their harshness or the forced leading of the parts in their resolution, are but seldom used. ( 6.) The pupil should try to resolve the Seventh- Chord upon each degree of the minor scale i. e., the Cadencing resolution to the triad a 4th higher and he will see the difficulty of resolving some of them without bad leading of the parts.* * The pupil will find, in resolving the seventh-chord upon the 4th degree, that the Bass, if moving upv ' g g ' mi^x^. ^ 41^ i J^? ^^ CS 1 CS ^ ^? ^*^ 1 1 f2 2 1 C V 7 CV IV V 7 V? CW The possible combinations with the Non-cadencing resolutions of the Chords of the Seventh are almost lim- I20 HARMONY SIMPLIFIED. itless, as will be shown in the next exercises. The above examples marked N. B. show the connection of the Dom- inant seventh-chord with the Dominant seventh-chords of various foreign keys: such connections will be further explained in the chapter on Modulation. Keyboard Exercises. Advanced Course. 195. Non-cadencing Connections with Triads in the Key. (a.) Starting upon the Chord of the Dominant 7th in the key of C, try to resolve it to ( or connect with ) the triad upon each degree of the key of C. If the effect is not good, try a change of position in the first chord: if the different leading of the parts does not produce an agreeable effect, reject the triad and try the next one. ( b.) Repeat in various keys. Non-cadencing Connections with Triads Foreign to the Key. ( c.) Starting upon the Chord of the Dominant 7th in the key of C, try to connect it with the Major triad upon each degree of the Chromatic scale. Reject the unsatisfactory progressions. ( d.) Try to connect the Chord of the Dominant 7th in C with the Minor triad upon each degree of the Chromatic scale, as above. ( e.) Starting upon the Dominant Seventh-Chord in other keys, try to connect with the Major and Minor triads as before, rejecting all progressions that cannot be made effective. Non-cadencing Connections with Dominant Seventh-Chords in Foreign Keys. (f.) Starting upon the chord of the Dominant 7th in C, try to connect it with the chord of the Dominant 7th in all other keys ( pro- ceeding Chromatically as before). (g.) Starting upon the chord of the Dominant 7th in other keys, try to connect with all other chords of the Dominant 7th as above. In the above exercises it will be found that those connections are best which have a note common to both chords, and that few con- nections can be made without it. The exercises at (/) and (g) will be treated further in Chapter XIII. HARMONY SIMPLIFIED. 121 Ig6. Exercises. 1. R' 3 76 587 677 -p=__i_ (S ,__^=c===pi=^ 2. R. 5 6 7 2667 647 2 5655 765 66 7 Open Position. 5. 6. 6 7 4 $ 6, ^ n (gi- i 7. 6 I 6 4*60 6 ,5 ii 8. R. 5 7 . 6 4 7 133 HARMONY SIMPLIFIED. Non-Cadencing Connections of Secondary Chords of the Seventh. 197. We have seen how the Chords of the Domi- nant Seventh are frequently connected with chords other than those forming the Cadencing resolution. The Secondary Chords of the Seventh are capable of being treated in a similar manner. Many of them, especially in Minor, which cannot be used in the Ca- dencing resolution, may be connected with other chords with very good effect. As in the free resolution of the Dominant Seventh-chord, the yth from the root may pro- gress downward, remain stationary, or progress upward, as desired. Keyboard Exercises. 198. ( a.) Try in succession the Secondary Seventh- Chords in the key of C major, and find as many agreea- ble connections with other chords as possible ( even con- necting with chords in other keys ) , proceeding in detail as shown in 195. (<$.) Proceed similarly with the Secondary Seventh- Chords in C minor ; also in other keys. Rules for Figured Bass. 199. Short horizontal lines following figures denote the retention in the following chord, or continuation, of the notes indicated by the figures. E. g., the notes in- dicated by 6 and by 3 are continued into the following chord. In notes, thus : HARMONY SIMPLIFIED. Even when the Bass note changes, the horizontal lines denote the continuance of the notes already sound- ing, whether indicated by figures in the preceding chord or not ; e. g., 200. Exercises. 766 76 564 5 1 '-i E 23Z 4. 66 66 & , 54 777 6 |_6_X2 L_l_ 7 _ 6. J- e. J. b 6 6 5 J- HARMONY 8. J- 6 7 5 22: 7b 8[J 9. J. 70 I m 7t> etj eQ 8 7 S - 1 s^~ -"g 1 -&- t^- Analytical and Comparative Review. 201. The pupil should strive to keep his knowledge collected and classified. To this end it is desirable to tabulate some of the facts already learned, the student being expected to find the definitions and commit them to memory if he is not already familiar with them. (i.) Hoiv the terms Major, Minor, Augmented, and Diminished are used. I. Intervals: -, there are 'Major, Minor, Augmented, II. Triads : j Diminished. there are J III. Chords of the ) Seventh: (.Major, Minor, Dimin- the ;th may be ) ished. HARMONY SIMPLIFIED. 235 (2.) How the term Principal is used: I. Triads : Tonic, subdominant, and Dominant II. Chords of the Seventh : Dominant. (3-) How the term Secondary is used: I. Triads : Upon all degrees not occupied by Principal triads. II. Chords of the Seventh : Upon all degrees not occupied by Principal Seventh. (4-) Of Tendencies : I. Of Leading-note to the Tonic. II. Of the Third upward. III. Of the Fourth downward. Melodic : -j IV. Of Continuity to continue in either direction. V. Of an accidental Sharp to ascend. L VI. Of an accidental Flat to descend. I. Of a Diminished Interval ; to be- ,, come still less. Harmonic: -! TT , . ._ 11. Of an Augmented Interval ; to be- come still greater. ( 5) Natural Resolutions : I. Of Dominant Seventh ; to triad a 4th higher, i. e., the Tonic. II. Of Secondary Sevenths ; to triad a 4th higher. III. Of Seventh-Chord on yth degree ; to Tonic or to triad a 4th higher. ( 6.) Non- Cadencing Resolutions : I. Of Dominant Seventh ; to secondary triads in the key. II. Of Dominant Seventh ; to foreign chords. III. Of Secondary Sevenths; to various triads in the key. IV. Of Secondary Sevenths ; to foreign chords. 2 6 HARMONY SIMPLIFIED. (7.) Figuring Inversions: I. Of Triads ; According to distance from actual Bass. II. Of Chords of the Seventh ; Same as triads. Synopsis. Write the usual Synopsis of the chapter. Historical. Concluded from page 39. Triads and Chords of the Seventh. 202. With Palestrina (early in the i6th century) the Harmonic effects began, though unconsciously, to appear upon the horizon of musical development. First the Common chord was used in its direct form, then with its inversions. Next we find the alternation of consonances and dissonances, and after a time Suspensions and Reso- lutions. The use of the Chord of the Seventh ( Domi- nant seventh ) met with much opposition at first. For many years its dissonant notes were " prepared," but in recent times gradually increasing freedom has been al- lowed, until now the chord can be used without especial caution. Following in the path of the Chord of the Seventh came the Chords of the Ninth, the Chord of the Diminished Seventh, and the chords of the Augmented Sixth ( to be described in subsequent chapters ) , all of which have been shown to be various forms of Dominant ( or Dependent ) harmony. Afterward came the various forms of ornaments, and devices for imparting variety, shown in Part III. The development of the Harmonic System, and of the modern scale as opposed to the Gregorian Modes, were to a great extent coincident and mutually dependent ; for, whereas the Gregorian Modes were formed in refer. HARMONY SIMPLIFIED. 127 ence to the Melody, the modern scale was designed with direct reference to the requirements of chord-construction. ( See 46. ) This brings the history to the close of the i6th cen- tury, when it was substantially as it is to-day. The boundaries of the keys had been well defined, and the use of the more ordinary chords had become common. Since then more freedom in the use of the Dependent chords has been gained, and a knowledge of those closely related chords which lie just beyond the limits of a key, but are used as if they belonged to it. ( See Chap. XII.) During the last two centuries progress has been more ir the line of development than of discovery. ( End of Historical Remarks.) The Perceptive Faculties. 203. The teacher will not need further detailed instructions, as the same manner of hearing the tones individually, of singing them by syllable, of writing them, and hearing them collectively, is here followed. The teacher should be careful to grade his instruction in this department well within the abilities of the pupil, and to pro- ceed very slowly. Exercises in Rhythm, and in Altered intervals ( Aug. and Dim.), may properly be introduced or continued at this period. CHAPTER VIII. THE CHORD OF THE DOMINANT SEVENTH AND NINTH. 204. The formation of chords has been repeatedly shown to be a process of building, or adding to a Rootoi Fundamental note. (See 90 and 147.) It has alsc 12 8 HARMONY SIMPLIFIED. been noticed that each note added is at the interval of a 3rd from the next lower note. If, according to this plan, a note be added to the Chord of the Seventh, there will be produced a chord of the Seventh and Ninth, called also the chord of the Ninth. As the one most commonly used is derived from the Dominant, we will consider only that one at present. In Fig. 62 is shown, at (0), the chord of the Seventh, and at ( b ) the same with the 9th added. (0 In a Major key the 9th will be Major; and in a Minor key the 9th will be Minor, as shown in Fig. 63 ; the 9th, A, being made flat by the signature. Fig. 63. The pupil should not look upon this as a new and strange chord, but as a Chord of the Dominant Seventh with an interval added. The Chord of the Seventh was produced by adding a note to the triad, and the Chord of the Ninth is formed by a further addition of a note to the chord of the Seventh. 205. The characteristics of the chord ( the dissonant intervals and the Tendencies ) are not changed by add- ing the new interval, as may be seen by tracing the dissonant intervals in the same manner as shown in 157. It is apparent that the added note merely creates two new dissonant intervals, the 9th from the root, and the 7th from B.* As both these intervals would be re- * In the chord of the Minor Ninth there is also the dissonant interval of a Diminished sth, D-Ab, in Fig. 63. HARMONY SIMPLIFIED. 129 solved by allowing the pth, A, to descend in the resolu- tion of the chord, it. is apparent that the addition of the new interval does not alter the natural resolution of the underlying chord of the fth, or in any way change its nature. We merely need to be careful to avoid consecu- tive 5ths, which may occur in adding the new note. The Tendencies of the various notes and- intervals are not changed. Therefore, the chord of the Dominant Seventh and Ninth is seen to be only an enlarged form of Dom- inant Harmony. 206. Fig. 64 illustrates the resolution of the chord of the Dominant seventh and ninth according to the above, the first chord being used to prepare the dissonance ( see 181 ), which is particularly harsh when entering abruptly. As there are Jive notes in this chord, one must be omitted in four-part writing. The 5th, being the least essential and characteristic, and also the tone with which the ninth might create consecutive 5ths, is usually the one left out. Major. Minor. Fig. 64. mi , i i * 1 E Keyboard and Written Exercises. 207. From the chord of the Dominant Seventh in every key, both Major and Minor, form the chord of the Seventh and Ninth; find and describe their dissonances and Tendencies as shown in 157; prepare and resolve them as shown in Fig. 64. The consideration of the above is of great importance and should be thoroughly understood, as the following chapters are de- rived directly from this section. , 30 HARMONY SIMPLIFIED. Inversions and Figuring. 208. The inversions of this chord are used, excepting those in which the root and the 9th come too close to- gether. The figuring is similar to that of the Chords of the Seventh, the added note simply adding a figure. Exercises. Form examples of inversions of the Chord of the seventh and ninth. Secondary Chords of the Seventh and Ninth. 209. Secondary chords of the Seventh and Ninth are occasionally used, though not often. Not belonging to Dominant harmony, the 9th and the 7th ( the dissonant intervals) must both be prepared. In the Dominant Seventh and Ninth-Chord the preparation is not obliga- tory, though customary. Synopsis. Write the usual synopsis of the chapter. CHAPTER IX. THE CHORD OF THE DIMINISHED SEVENTH. 210. The Diminished Triad and Chord of the Seventh upon the 7th degree in Major have already been mentioned as partaking of the qualities of Dominant harmony (179). The Chord of the Seventh upon the 7th degree in Minor partakes of these qualities even more strongly. (See 188.) They are both considered as incomplete forms of Dominant harmony. The one formed upon the 7th degree in Minor is especially important, as it occurs very frequently, gives a smooth effect without being prepared, and is of great value in modulations. (See 300.) HARMONY SIMPLIFIED. Construction of the Chord of the Diminished Seventh. 2ii. This chord is derived from the Chord of the Dominant Seventh and Ninth in the Minor mode, by simply omitting the root. (a.) <*.) Fig. 65. ' " ' In Fig. 65 at ( a ) is given the Chord of the Domi- nant 7th and 9th as shown in Fig. 63. If the root is omitted, we have the chord shown at ( b ), Fig 65, which is a chord of the Diminished Seventh, but it is consid- ered as derived from the root G (indicated in Fig. 65 by W ) an d therefore having the same resolution as if the root were actually present. Therefore we say that the chord of the Diminished Seventh is an incomplete form of Dominant harmony. In the chord of the Dominant 7th and 9th the disso- nant intervals are the Minor 7th from the root and the Minor 9th. In the chord of the Diminished Seventh, the same notes, F and At?, form the dissonances, appearing as a Diminished 5th and a Diminished 7th from the Bass of the chord. These dissonances are re- solved in the same manner as if the root were also sounding, e. g., Fig. 66. -fr ^ -~^& |] i NOTE. The same rules for doubling notes apply here as in the simple Dominant form ; i.e., do not double the real jd and ;th, even though they appear to be the root and 5th ( 157). 132 HARMONY SIMPLIFIED. Keyboard and Written Exercises. 212. ( i ) Form Chords of the Minor 7th and 9th upon all notes from C to C, i. e., upon C, Cj, D, Djf, etc ; also using flats instead of sharps, as Di? for Cj, EP for D$, etc. ( 2 ) From each chord of the Minor 9th just written, form a chord of the Diminished 7th by omitting the root and writing the sign W in its place. ( 3 ) Resolve each chord of the Diminished 7th according to the tendencies in 157. N. B. It will be found that the resolution is the same as if the root were still sounding; see Fig. 64. Use of the Chord of the Diminished Seventh in Major Keys. 213. In 204 it was apparent that the Chord of the gth is Major in Major keys, and Minor in Minor keys. The Chord of the Minor 9th and its derivative, the Chord of the Diminished 7th, are, however, often used in Major keys; the 9th from the root being lowered by an acci- dental ; e. g., Flg. 67. As the Chord of the Dominant Seventh is alike in Major and Minor, we may say that it resolves equally well to Major or Minor triads ; and the same holds good of all forms of Dominant harmony, whether Chords of the 7th, of the 7th and 9th, or of the Diminished 7th. Keyboard and Written Exercises. 214. ( i ) From the Chord of the Dominant 7th, in all major keys, form Chords of the Major 9th as shown HARMONY SIMPLIFIED. 133 in Fig. 62. From these chords of the Major pth form chords of the Minor 9th by lowering the Ninth by an accidental. Omit the root of the Minor ninth-chords, producing Chords of the Diminished 7th in Major. ( 2 ) Resolve these chords of the Diminished 7th as in Fig. 66 or 67. N. B. The chord of the Dimin- ished 7th resolves to either a Major or a Minor triad, as mentioned in 213. Similarity of Sound of the Diminished Seventh- Chords. 215* Write the chords of the Diminished Seventh as in Fig. 68.* Now play them upon the piano, and it will be seen that there are apparently but three differ- ent chords, if we consider that inverting and changing the notation do not alter the sound. This is shown in Fig. 68 where the chords are divided into four groups, w, #, _y, z ; and, by trying at the Piano, it will be seen that No. I of group iv is the same as No. i of group x, or y, or z, in that the same notes are struck on the key- board. The difference consists in the fact that the chord is inverted and differently written. Therefore, any chord of the Diminished Seventh can, by changing its nota- tion, belong to four different keys. This subject will be explained further in 300. (>) (*) (y) (*) Fig. 68. Roots : ( G I GS t A I AS ( B I C J CS j D ( Df ( E ( F j F Keys : } C \ C$ } D } DS \ E } F \ FJf \ G \ GS \ A \ & \ B * The pupil should write a series (Chromatic) to represent the roots of the chords, as shown in the line marked "Roots "in Fig. 68 and try to build the required chords from these roots (as shown in 212) without referring to Fig. 68 unless necessary. ,34. HARMONY SIMPLIFIED. The chord of the Diminished 7th, being Dominant harmony, does not require preparation. Inversions and Figuring. 216. The chord of the Diminished yth is used in all inversions, which are figured by counting from the actual Bass note, as for other chords. The sign is used to indicate Diminished. Exercises. ( a.) Form a series of Diminished seventh-chords similar to that shown in Fig. 68, but with the sharps changed to flats; e. g., instead of using FJ for the Root of a chord, write it GP, which will cause the whole chord to appear without sharps. Divide the series into groups as shown in Fig. 68, and number them. Write also the Roots and Keys under the chords as there shown. 217. It will now be observed, that by changing the notation of the Root ( i. e., from a sharp to a flat, or vice versa ) , the notation of the whole chord is changed, al- though the notes on the keyboard remain the same. 218. It will also be seen, the first chord of each group ( see Fig. 68 ) being the same, that, by a change of Root (and therefore of notation), the same chord ( i. e., upon the keyboard ) may become Dominant harmony in four different keys, as shown by the series of keys in Fig. 68. As Dominant harmony resolves naturally to its Tonic, it is clear that by proper notation these chords of the Diminished seventh can resolve to any one of four different keys. Exercises. 219. (<5.) Completing Fig. 68 as required in the foot-note, 215, take the first chord of each group in Fig. 68, and resolve it to its proper Tonic triad as indi- cated by the notation. HARMONY SIMPLIFIED. J 35 ( c.) Take the second chord of each group, and pro- ceed as before. ( d.) Take the third chord of each group, and pro- ceed as before. (e.) Name the Root of the Diminished seventh- chord which shall resolve to the triad of D major. ( N. B. The Root of the chord is desired, not the Bass note. Remember that the Root of the chord of the di- minished seventh is the same as the Root of the Dominant harmony from which it is derived ; therefore, to find the Root of a chord of the Diminished seventh, the pupil may proceed as in 159, and, having the Root, the chord may be developed as shown in 212.) Write the chord, and indicate the root by the proper sign. (jf.) Name the Root, and write the chord which shal! resolve to the triad of D minor; of Ab major; of At> minor ; of FJ; of G$; of A$; Btf; Bb; Db; Eb. (^.) Repeat (e) and (f ) at the keyboard. 220. Exercises. e e ,7 e 1 1 i HARMONY SIMPLIFIED. . 8. Exercises in Harmonizing the Scale. 221. Harmonize the scales, using chords of the Di- minished seventh where possible, together with the chords previously learned. Try this exercise also at the key- board. Synopsis. Write the usual synopsis of the chapter. CHAPTER X. CHORDS OF THE AUGMENTED SIXTH. 222. Most decided differences of opinion still exist with regard to these chords. They will here be shown to be forms of Dominant harmony, or derived directly from it. This exposition will be found by far the simplest HARMONY SIMPLIFIED. 137 and most practical, giving a more intelligible derivation, and a wider application, than is possible in any other way.* 223. The chords of the Augmented Sixth are Chro- matically Altered chords, i. e., chords in which some note has been changed without radically modifying the chord or its progression. ( See 246.) As the Chords of the Dominant Seventh, the Domi- nant 7th and 9th, and the Diminished 7th, belong to Dominant harmony, though each appears in a different form ( one note more or less ; with the Root or without it; etc.), so the chords of the Augmented Sixth are no exception, but may be developed from Dominant har- mony, as will be shown. Construction and Resolution. 224. These harmonies appear in three forms, viz., Augmented Six-Three, Augmented Six-Four-Three, and Augmented Six-Five-Three chords, e. g., Fig. 69. To Construct the Augmented Six-Three Chord. 225. Let us take a Dominant seventh-chord, for ex- ample : , place it in its 2nd inversion, * Although the full application of the theories here advanced is original with the author, there is abundant authority to support his views. The investi- gations of the last half-century seem to converge, but the results of research had not yet been systematized and the practical application shown. While not claiming the discovery of new principles, it is here attempted to arrange and apply the truths brought out by Day, Ouseley, MacFarren, Parry, Piutti and other theorists. ,-jg HARMONY SIMPLIFIED. omit the Root, P/fc-s , and Chromatically lower the ~ 5th from the original Root, giving the chord : f(o) ^ This is called the Chord of the Augmented Six-Three. The Root being G, and the original chord an ordi- nary Dominant 7th, the natural resolution is to the triad on C : REre g ^ ga ~ just as it 'would be if the note D ' were not altered. Notice that the Leading-note progresses upward, the Minor 7th downward ( as in the ordinary progression of a Dominant 7th chord) , and that the interval of an Aug- mented 4th is resolved naturally by further expansion, as in chords of the Dominant 7th, Diminished 7th, and Minor pth ; while the 5th lowered by an accidental follows the natural tendency downward. The characteristic interval of the Augmented 6th, Db-B, from which the chord is named, resolves by further expansion. Exercises. Taking in turn the chord of the Dominant 7th in every key, place it in its second inversion, omit the Root, lower the 5th (from the Root) by an accidental, thus form- ing a chord of the Augmented Six-Three, and resolve it as shown above. To Construct the Augmented Six-Four-Three Chord. 226. If the same Dominant seventh-chord is taken in its second inversion as before, but this time without omit- ting the Root, and the 5th lowered as above, we shall have the same Augmented Sixth-chord as before, with I Q the addition of the Root, G : EScizfc:. This is called HARMONY SIMPLIFIED. the Chord of the Augmented Six-Four-Three. For pre- cisely the same reasons as the Augmented Six-Three chord, the natural resolution is to the triad on C : Exercises. (a.) Taking in turn the chord of the Dominant yth in every key, place it in its second inversion, not omitting the Root, lower the 5th (from the Root), thus forming a chord of the Augmented Six-Four-Three, and resolve it as shown above. (3.) Repeat the above at the keyboard. To Construct the Augmented 6-5-3 Chord. 227. If we take the same Dominant harmony as be- fore, this time with \heMtnor qth from the Root added^ ?- ? , place it in its second inversion, omit the Root, and lower the 5th ( from the Root ) by an acci- r-Q- dental, we shall have the chord : F?q\-bg g ~T called the Augmented Six-Five, which has the characteristic of sound- ing like a Dominant seventh-chord. This chord, being derived from the same harmony as before, though in a fuller form, has the same natural resolution to the triad But here are consecutive 5ths, which may be avoided in various ways. Among them may be mentioned : ( a ) Resolving first to an Augmented | or i, which, being pre- cisely the same harmony, does not affect the character of the final resolution: or (3) delaying the resolution of I40 HARMONY SIMPLIFIED. some of the parts, thus forming a chord of the f on trfe Subdominant before the common chord enters. Both ways are exemplified in Fig. 70. i ^i i_ ; .*- i ___r.^ r%-fii*'^ / w \ ^ ' l~ I V IV I Exercises. (a. ) Taking in turn the chord of the Dominant yth in every key, add the Minor 9th, place it in its second in- version, omit the Root, lower the 5th (from the Root) by an accidental, thus forming a chord of the Augmented Six-Five-Three, and resolve it as above. (6.) Repeat the above at the keyboard. 228. The chord of the Augmented Six-Three is called the Italian Sixth ; the chord of the Augmented Six-Four-Three is called the French Sixth ; and the chord of the Augmented Six-Five-Three is called the German Sixth. In the Italian Sixth, there being but three notes, it is necessary to double one of them. The best one to double is the 7th from the true Root. ( N. B. It is quite proper in this case to double the yth, since by the omis- sion of the Root the downward tendency of the 7th is less marked than if it were present.) Another reason is, that the lowering of another note by an accidental disturbs the feeling of Tonality, so that the 7th does not seem to have the full tendency downward. (See "How to Modulate," 44 and 45.) The ten- dencies thus having been removed or modified, can hardly be said to have been violated. N. B. The pupil should now review chapters V to HARMONY SIMPLIFIED. 141 X, especially comparing 157, 173, 177, 179, 180, 205, 211, and 224-228. He must not fail to understand practically that, as asserted, the Chords of the Dominant yth, Diminished yth, Major and Minor gth, and the three forms of the Chord of the Augmented 6th, are nothing more than different forms of the same Fundamental harmony, derived from the same Root, having the same dissonant intervals, and the same resolution. NOTE. All chords of the Augmented sixth are properly classed among the Altered chords. ( See Chapter XI.) Chord of the Augmented Sixth derived from the Supertonic. 229. There is another chord of the Augmented Sixth, which, although it is not strictly in the key, is in such common use that it will be mentioned here. The chord in question is the one which resolves to the Chord on the Dominant. Therefore, its Root should be found a 4th lower than the Dominant, i. e., on the Supertonic. In order to have exactly the form of a Dom- inant seventh and ninth-chord ( which must be exact in all its intervals if it is to serve as the basis of an Aug- mented Sixth-Chord), the 3rd from the Root must be Major. Therefore, in Fig. 7 1 * F must be made sharp, though the signature does not require it. The Minor 9th from D, which is necessary for the Six-Five form, is El?, which also is not indicated by the signature. Thus this chord LjsL Triad. ^ TONE, ^ Major 9th. *>Aug.6 or 5 ROOT. ( ( Secondary 7ths. 3 236. It will be further observed that (a.) The natural resolution of all Dependent chords is gov- erned by the same Tendencies and Influences. ( b.) The same laws of Part-leading control the connection of all chords, Independent or Dependent. (c.) Dependent chords may sometimes progress without ca- dencing resolution, in which case they are governed, not by the laws of natural resolution, but by the laws of part-leading in chord-connections as in Independent chords. 237. From a consideration of the above it will be seen, that the different chords are but different forms or manifestations of the same Primary chord. It is, there- fore, but logical that, as above shown, the same laws should govern all the forms. The Harmonic System is wonderfully simple, yet complete. HARMONY SIMPLIFIED. CHAPTER XL ALTERED CHORDS : FUNDAMENTAL CHORDS. How to Distinguish them; Their Roots and Keys. 238. Any note of a chord may be Chromatically raised or lowered; e. g., When this occurs, certain changes take place which render it necessary to consider the chord from a new point of view. To enable the pupil to understand the changes which take place, it is necessary to study the following. Preliminary Premises. 2 39- ( *) Fundamental chords (i.e., chords like Dominant chords, also like Nature's chord), can be built upon any and every note. (See 91.) Funda- mental chords may appear as triads, Chords of the 7th, of the Diminished 7th, of the Major 9th, or the Minor 9th. They must have, counting from the Root, a Major 3rd, a Perfect 5th, a Minor 7th ( if a chord of the 7th ), and a Major or a Minor 9th ( if a chord of the 9th) . Or, for convenience in comparing, the chord may be described by describing the successive 3rds when the chord is in its Direct form, as follows : From i to 3 is a Major 3rd, from 3 to 5 is a Minor 3rd, from 5 to 7 is a Minor 3rd, and from 7 to 9 is either a Major or a Minor 3rd according to the key. Placed HARMONY SIMPLIFIED. 147 one above the other as in the chord, it may be expressed as follows : * 9-. > Major or Minor. 7 y Minor. 5 [ Minor. 3) > Major. i ' This might be called the Formula for constructing Fundamental chords, since they must correspond exactly with it in order to be Fundamental. Exercises. 240. Form Fundamental chords in the four forms mentioned, up- on all notes of the Chromatic scale, and compare them with the formula. 241. (2.) Fundamental Dependent chords, like Dom- inant chords, whether appearing as Chords of the yth, Diminished 7th, or of the pth, resolve naturally to the triad a 4th higher. ( See foot-note, 158.) 242. (3.) All Fundamental chords are considered as built, each upon a particular Root. The chord of resolution is a 4th highei than this Root, in every case. 243. (4-) Change of Root. This can best be ex- plained by illustration. By reference to 241, it becomes apparent that the natural resolution of any dependent chord is to the triad a 4th higher. Reference to Fig. 68 and the accompanying text shows that the same notes ( on the keyboard ) may be * If the Root is omitted, as in the chord of the Diminished seventh, the Major jrd from i to 3 will not be present in the formula. 148 ffARMONY SIMPLIFIED. derived from different Root-notes, the only difference be- ing in the manner of writing the chords, i. e., the nota- tion. It may be said, conversely, that the different nota- tion shows that the chords spring from different Roots. To illustrate, ( a ) and ( b ) of Fig. 72 are alike in sound. But, if a chord of the Diminished yth is built upon the Root G, it will be like ( a ), while a similar chord erected upon the Root Aj will be like ( b ) when inverted. The two chords, which sound alike, have different notation because they are erected upon different Roots. Reference to 211 will show that their resolutions differ radically. This is on account of the law of Tendencies shown in 152, viz., the Leading- note tends toward the Tonic, the fourth degree of the scale tends downward, and the chief dissonance, the Di- minished yth between the Leading-note and the 9th from the original Root, tends to contract. Therefore we may say that, by the change of notation, the Root is changed, and, consequently, the resolution is also changed. Therefore, if we would have a proper resolution, the chord must be so written as to show which note is the Leading-note, which the gth from the original Root, etc., in order to know how to apply the law of Tendencies. 244. In 243 is shown how a change of notation, or Enharmonic change, as it is called, implies a change of Root, even where the notes on the keyboard remain the same. In cases where one or more notes are really HARMONY SIMPLIFIED. ^ altered by accidentals, the change of Root is even more clearly apparent. For example, ~fa~%^n. is the chord of the Dominant yth on the Root G, resolving to the triad of C. If the note G in this chord is chromatically raised, thus : ifeiggzn, the chord is like the chord of the Diminished 7th built upon the Root E (which is of course omitted ), resolving to the triad on A. Therefore, the Root of the chord, as well as the resolution, may be said to have been changed by the alteration of the single note. Consequently : 245. ( 5.) By a change of Root a change in the res- olution is necessarily caused. Now we will proceed to consider the Altered chords. 246. By reference to the foot-note, 158, it becomes clear that the natural resolution of any dependent chord is to the triad a 4th higher than the Root of that Depen- dent chord ; and we have just seen that when through a change of notation, or other causes, the Root is changed, the natural resolution of the chord is completely changed in consequence. This fact is illustrated in Fig. 68 and the accompanying text. A change in a chord, whether of a note or simply in the notation, which produces a change of Root ( and therefore of resolution), is called an Harmonic change. Where the change simply affects one part transiently, not producing a change of Root and resolution , the change is called a Afclodic change. Where a chromatic change in a note is made as sug- gested in 238, the result must be one of two things : , 5 HARMONY SIMPLIFIED. either the Harmonic change just mentioned, or the Me- lodic change. By an Harmonic change a completely new chord is formed, which is outside the key, speaking strictly, since it contains a note foreign to the scale of the key. Such changes will be considered under the head of Foreign Chords, Chapter XII. By a Melodic change the alteration has more to do with a single part, rather than effecting any change in the character of the chord. Such changes produce Al- tered chords, if they have sufficient duration to be consid- ered as chords ; or Passing-notes, if of insufficient duration. Such alterations may occur in Chords of the 7th as well as in triads. But the pupil will desire to distinguish between Al- tered chords and Foreign chords, and to discover the Roots and resolutions of the Foreign chords. The fol- lowing is the method : To Distinguish between Altered Chords and Foreign Fundamental Chords. 247. ( I.) For convenient survey, place all the notes within the compass of one octave, striking out all dupli- cates. ( 2.) Place the chord in 3rds. ( See 172.) ( 3.) Construct a descriptive formula of the 3rds as shown in 239, and compare it with the formula of a Fundamental chord there shown. If they correspond, the chord in question is a Fundamental chord. If not, it is clear that either it was not originally a fundamental chord, or that some interval has been altered. ( If the Root of such a chord is unknown to the pupil, he must discover the altered note or notes by comparison with the formula before proceeding to find the root by the method HARMONY SIMPLIFIED. outlined in the following paragraph.) But, before pro- ceeding, let us illustrate the above. 248. For example, to find whether is a Fundamental* or an Altered chord : Placing all the notes within the compass of one octave, gives : j-g^- ^ . Inverting, to obtain the required figuring, we have successively : the last being the required form. Describing the yds as required in 239, we have the formula 7 5 V minor. > minor. H > minor, i ' * It should be noticed that the Dominant Is the only Fundamental chord of the Seventh which is to be found in any key. The Secondary Sevenths do not correspond perfectly in their intervals with the intervals of the Fundamental. ( This accounts, in part, for the prominence given to the various forms of Dom- inant harmony.) Chords which are not Fundamental chords may be Secondary chords. Therefore, if the formula does not correspond with the formula for a Funda- mental chord, we should compare it with the Secondary chord having the same Root (provided that the given Root represents a Secondary chord) before deciding that it is an Altered chord. I HARMONY SIMPLIFIED. Comparing with the standard formula : Standard Formula of the Formula : Given Chord : 9 ) * > major or minor minor }mi, minor minor 5 V |- minor minor 3 I > major i ' we find it agrees with it in every particular, as far as it goes. It is therefore a Fundamental chord without its root, i. e., a chord of the diminished yth. Again, to learn whether the chord E^ztggz:^ is a Fundamental chord or not : Proceeding as before gives the formula : > major. ( minor. 3 I > major. i ' Comparing this with the standard formula, we find that the intervals of the given chord cannot be made to correspond with three successive intervals in the standard formula. Thus Standard Formula of Formula : Given Chord : 9) ( 7 > major or minor . . . major -j : corresponds. 7 5 J- minor minor -j : corresponds. 5' i ) ( ** > minor major < : does not cor- 3 \ '-i respond. HARMONY SIMPLIFIED. '53 Therefore, even if the Root of the given chord were found, whatever note it might be, it could never form a Fundamental chord in connection with the notes as given. Comparison with the chords of the yth upon the various degrees of the scale, by comparing the formulae, shows that this chord might be the Chord of the 7th upon the ist degree of the scale of Bi? major, resolving naturally to the triad upon the 4th degree ; e. g., Exercises. State whether the following chords are Altered chords, or Fundamental chords, or whether they might be secondary chords in some key : To Discover the Root of any Fundamental Chord. 249. ( i.) Write all the notes in the compass of one octave, striking out duplicates.* ( 2.) Place the notes in 3rds, as shown in 247. (3.) If it is a triad (three notes), the Root will be the lowest tone. ( This is merely the result of the defi- nition of the Direct form of a chord. See 125.) It will now be apparent whether the chord is ( I ) an ordi- nary Major or Minor triad ; ( 2 ) an Altered triad ; or (3 ) an incomplete form of a Fundamental Dependent chord. * Sometimes a note is omitted in a Chord of the 7th, or 7th and 9th. The pupil should refer to 248, and observe how the intervals in the Fundamental chord would occur if the Root were omitted; for without the Root a different order of intervals would result, which might lead the pupil to think a chord to be an Altered chord when in reality it is an incomplete form of a Fundamental chord. 154 HARMONY SIMPLIFIED. N. B. Remember that a Diminished triad may be considered as an incomplete form of a Chord of the 7th, and resolve accordingly. (See 179-) 250. If it is a Chord of the Seventh (four notes ), we must first be sure that it is a Fundamental and not an Altered chord. How to accomplish this is shown in 247. If shown to be a Fundamental chord, either with or without the Root, we may proceed as follows : Com- pare the notes as shown in 29, to discover which note is relatively the " sharpest" and which the "flattest." In comparing the notes, the sharpest one will be the Leading-note. ( The Jlattest note will be the 9th, if it is a Chord of the 9th, otherwise it will be the 7th.) The Leading-note being a Major 3rd above the Root of a Fundamental Dependent chord, to find the Root when the Leading-note is known simply count a Major 3rd down- ward from that Leading-note. ( N. B. The Root may not be present in the chord. It never is in chords of the Diminished 7th.) When the Root is found, it can be proven by the " flattest" notes, which should be the 9th or the 7th from the Root as above shown. ( For further explanation of this point, see " How to Modulate," p. 18.) 251. Illustration of -preceding Section. To find the Root of pijazi^:^. Comparing the notes to find the "sharpest" note, we see that B is represented by five sharps ; D by two sharps ; F by one flat ; and Ab by four flats; consequently B is the "sharpest" note, and there- fore the Leading-note. As the Root of the chord should be a Major 3rd below the Leading-note, by counting downward a Major 3rd from B we find that G is the Root HARMONY SIMPLIFIED. '55 of the chord. Building u Fundamental chord upon the Root G, we have G-B-D-F-Afr, which is a chord of the Minor 9th, and corresponds to the notes of the given chord. Therefore, the chord in question is a Chord of the Diminished yth upon the Root G, resolving to the minor or major triad on C. ( See 213.) Again, to find the Root of the chord Comparing as before, we find that Cfr is represented by seven flats ; D by two sharps ; F by one flat ; and A]? by four flats; consequently, D is the "sharpest" note. A Major 3rd below D is Bi?, which is consequently the Root of the chord. Placing the chord in 3rds, and writing the Root in its place, the full chord is seen to be a Chord of the Minor pth upon the Root BJ?. 252. To discover in 'what key such a foreign chord is written, simply remember that the "sharpest" note is the Leading-note, or yth degree of the scale. There- fore, the chord hffkHTgp 13 ma y be said to be written in the key of C minor, and the chord Kiftr-bz? - m the key of minor. ( See also " How to Modulate" 20.) Exercises. 253. Name the Roots and Keys of the following chords : Ambiguous Chords. 254. .Sometimes a chord may occur which might be either an Altered chord or a Foreign chord. E. g., i 5 6 HARMONY SIMPLIFIED. F#-C-D# might be either a chord derived from the Sec- ondary 7th on the 2nd degree of C Major ( by raising F and D by accidentals ; notice that the chord appears without the 5th ; write it), or it might be considered as derived from a new Root, B, being an incomplete form of the Chord of the Diminished 7th ( write it). To learn which of two Roots is intended, examine the resolution : for if the resolution is the same as it would have been without the alteration, it proves that the chord is Altered ; whereas, if the resolution is differ- ent, it shows that the chord is a Foreign chord. . For example, in the above, if the Altered chord derived from the Root D is intended, the progression would be to the chord G C E, which is considered as inter- polated* between the chord on D and its natural reso- lution which follows. (See a, Fig. 73.) If the Root B is intended, the resolution would be to the minor triad on E ( a 4th higher than B) . ( See b, Fig. 73.) Fig. 73. PE5- Root: * By an interpolated chord is meant a chord placed between two chords which naturally belong together. For example, the natural resolution of the seventh-chord upon D is to the triad on G. But the chromatic alteration of the - *&5i , inclines it away from its place in the chord of note D, thus : G, and would cause an awkward effect should it return after starting else- where. Consequently, the triad on C is interpolated for smoother effect ; but the true resolution is only delay 'ed, for it enters immediately after. (See Fig- 73, <*.) HARMONY SIMPLIFIED. '57 Treatment of Altered Chords. 255. As mentioned, any note of a chord may be al- tered by an accidental ; and when the resulting change does not cause a change of Root, it is called simply an Altered chord ; e. g., pEKH^zz: is the common triad on C ; if the note G is raised chromatically, thus : we say that the note G has been altered from its original condition, and the whole triad might be called an altered triad. The triad has not been essentially changed (we still look upon C as the root) , but the note G, having been raised, is strongly inclined to progress to the next note above, A. Such alterations may occur in seventh-chords as well as in triads. 256. The pupil needs little guidance in the treatment of Altered chords, other than to remember that the ten- dency of a chromatically raised note is to ascend, and the tendency of a chromatically lowered note is to descend. The general rule that accidental sharps tend upward, and accidental flats downward, is good to remember, but it does not convey the whole idea, for a natural may have the effect of raising a note previously flatted by signature or accidental ; e. g., p^fr *. \ . The natural here raises the El? chromatically, and is similar to : K8 In the same way, a natural may chromatically lower a note : e. g., U/lffit" ^ Qi* Thus it is clear that flats, naturals and sharps are relative rather than specific terms. i 5 8 HARMONY SIMPLIFIED. A chromatically altered note, being a tendency-note, should not be doubled. Altered Chords in General Use. 257. Of the many altered chords, those most in use are : ( a.) The Triad with raised 5th ; ( b.) The Chord of the yth with raised 5th ; ( c.) The Chords of the Augmented 6th ; ( F - F^ p2=fca=|=^^=y 3. R- 3 I 2* 6 7 g I 6 7 7 HARMONY SIMPLIFIED. 4. _5l_5 _ 3 58 5 - 6_ 3 5$ 6 6 6 ^^^f S =f f ^f= -f=^=(=f= 6 6 * 7 50 7 3 502 6 -^fr g flzJ - 1 6. J- 5 507 Jfl 7. J- ^ 75508767505055ljf 50 8. J- *. 30 =4 Bii E = P^ tf.0 ~ = Open Position. Jft 4 $ 8 If 6*5022 i6o HARMONY SIMPLIFIED. Close Position. e 1O. J* 8 e t - t 65 7087 706 ft] 87 PS Open Position. 11. J. a 5 5(5 ~^5 P*^ 35 5 5fl *= F &t G> -sr- Advanced Course. Neapolitan Sixth. 259. ( Usual explanation. For author's exposition of the chord, see 261.) Among the altered chords is one in such common use as to receive a distinctive name. When the triad on the 2nd degree of the Minor ' ' """^ ~ ~ : * u its Root lowered by an accidental is used in its first inversion, a very soft effect is pro- duced. The chord is considered effective only in this inversion, and is called the Neapolitan Sixth ; e. g., Fig. 74. =F f- This alteration of the note on the 2nd degree is purely arbitrary, like the lowering of the 5th in the Chord of the Augmented 6th ; and it is frequently used, probably on account of the fact that the natural ( un- altered ) triad on the 2nd degree in Minor is a Diminished triad, and HARMONY SIMPLIFIED. therefore has tendencies of too pronounced character for effective use in ordinary chord-connections ( not resolutions ). It was found, how- ever, that by lowering this note the apparent tendency was hidden, making the chord more manageable. Keyboard and Written Exercises. 260. Form chords of the Neapolitan 6th from the triad on the 2nd degree of every Minor key, and resolve them. Derivation of the Neapolitan Sixth-Chord. 261. (The following is submitted entirely upon the author's respon- sibility.) The Neapolitan Sixth is believed to be a form of the Augmented sixth-chord, with sufficient license in its treatment to admit of the smoothest effect in Minor. The following examples will illustrate the assertion and the grounds for the belief. BANISTER. Fig. 75. EMERY. Fig. 76. -&- 3 BEETHOVEN. Fig. 77. j62 HARMONY SIMPLIFIED. The license mentioned above is this : That the notes comprising the full chord of the Augmented 6th are often divided between two chords. The chords marked X in the illustrations are the chords in question. If the chord marked x in Fig. 75 is a chord of the Augmented 6th, it is the 5th from the Root which is altered by an accidental. The al- tered note being Eb, the root should be A. Let us assume that the Root is A, and develop the chord from it. The chord of the Minor 9th upon A, (from which the Chord of the Augmented 6th is devel- oped,) is (ct) ^>~~~~^- Omitting the Root and lowering the 5th, we have Now, this chord is the same as the chord marked x in Fig. 75, ex- cepting that the Leading-note, Cf, which appears in the next chord, is absent, leading us to think that the notes have been divided between the two chords. 262. Again, in 224, the chord of the Augmented 6th is shown in Major, with the dominant of the key as the Root. Notice that, when the Dominant is the Root, it is the 2nd degree of the scale which is the chromatically altered note. If the above assertion is wrong, is it not rather strange that the note which is altered by an accidental to pro- duce the chord of the Augmented 6th in Major should happen to be the same note that is so altered in the Minor key ? And is it not still more strange that the resolution of the two chords should be the same ? And is it not strange that the process of building a Funda- mental chord upon the chosen Root should result in the desired Chord of the Neapolitan Sixth ? In further proof, the example in Fig. 76 is offered. Here the Nea- politan 6th, marked x , which is built upon the Root E (since the chro- matically altered 5th above the root is &), resolves directly to the triad 'on A ( a 4th higher than E ) without the help of any other chord. No- tice, however, that the next chord comes in to supply the Leading- note, for the cadence has not been quite strong enough without it. The next example, Fig. 77, from Beethoven, refutes the idea that the chord is good in only one inversion. Here the chords marked X have the chord of the Augmented 6th divided between them, and the notes, though identical with those of the other examples, are in a different inversion, giving an excellent effect. hstRMONY SIMPLIFIED. 163 It is submitted that the example from Beethoven is as effective as the examples in Fig. 75 and 76. The pupil is recommended to read 4250 in " How to Modu- late." Exercises. 263. Form chords of the Neapolitan 6th, from the Dominant as a Root, in ever}' Minor key, and resolve them. 264. Attention is again called to the wonderful simplicity of the system of developing the chords shown in this volume. By bringing the chords of the Dominant 7th, Minor and Major gth, Diminished 7th, Augmented j|, Augmented J. and Augmented ,' and the Neapol- itan 6th all urjder one head, derived from the same Root, having the same dissonant intervals, and the same natural resolution, one is inclined to accept the statement that " There is but one chord in the Universe, the Common Chord. All others are merely additions to this chord." Synopsis. Write the usual synopsis of the chapter. CHAPTER XII. ATTENDANT CHORDS. 265. The object of this chapter is to enable the stu- dent to recognize some of those chords which, though technically foreign to the key, so constantly intermingle with chords which belong wholly to the key. These foreign chords have such a peculiarly close relationship to the chords of the key, that we cannot well say that we are in a foreign key when they occur, but that a foreign key is suggested or touched. ( Se.e Grove's Dictionary of Music, Vol. II, p. 351.) 164 HARMONY SIMPLIFIED. The following chapter will be developed from a prin- ciple which is already familiar to the pupil, viz., The Natural Resolution of Dominant Harmony to the Tonic. 266. By Dominant harmony is not meant the chord of the Dominant yth alone, but also the chord of the Dom- inant pth (both Major and Minor), the Chord of the Diminished 7th. and the various forms of the Augmented Sixth-chord, which are all forms of Dominant harmony, and resolve to the Tonic. Keyboard and Written Exercises. Preliminary to the following, and to enable the pupil easily to grasp the subject, he should form chords like the Dominant Jtk, upon every ( chromatic ) degree of the scale, and resolve them, like the Dominant Jth, to the triad a 4th higher. Do not write any signatures, and do not call them Dominant and Tonic chords. Sim- ply notice that the Chord of the yth upon any note re- solves to the triad a 4th higher, and observe that the ten- dency of the seventh-chord toward the triad a 4th higher is so strong that there is clearly a close relationship be- tween the two chords. This relationship is the same as the relationship of Dominant to Tonic, but they should not be called Dominant and Tonic unless they are con- sidered as belonging to some key, and that is not now desired. The object here is to shoiv the relationship of the two chords, whether they are in a key or are consid- ered by themselves. The intervals of a chord of the Dominant 7th are a Major 3rd, a Perfect 5th, and a Minor 7th. Therefore, in forming these chords, the pupil will see that these intervals are present, and will use accidental sharps and HARMONY SIMPLIFIED. 165 flats to secure them. ( These chords are the same as the Fundamental chords described in the last chapter.) 267. The next step is to learn the reverse of the above, viz., To find that chord of the 7th which shall resolve to any given triad, Major or Minor. Process. As a chord of the 7th resolves naturally to the triad a 4th higher, to find the triad which shall resolve to a given triad, we simply need to look a 4th lower than the Root of the triad. Illustration. To find the chord of the 7th which shall resolve to the triad ( Major or Minor ) upon A : Looking a Perfect 4th below A, we find E to be the Root of the desired chord of the 7th. Completing the chord of the 7th upon E, by the addition of a Major 3rd, Per- fect 5th and Minor 7th, we find the full chord to be : ~, resolving to : N. B. Remember that the chord of the 7th resolves to either Major or Minor, since the chord of the Domi- nant 7th of A Major is the same as in A Minor. Keyboard and Written Exercises. 268. Taking each ( chromatic ) degree of the scale, in turn, find the Chord of the 7th which will resolve to the triad upon that degree. Complete the chord of the 7th, and resolve it to the proper triad, as above shown. 269. The pupil has now learned, that there is a Chord of the 7th closely related to every Major and Minor triad. Therefore it would not be strange to find, that these re- lated chords are sometimes used, although they are not, strictly speaking, in the key. 1 66 HARMONY SIMPLIFIED. - tf ^H^ ' & L rSi _J ^ IJ LN^^' -^ ^-_^d_ _^D_ -X^- - U -*Q- Fig. 78. I VI IV Notice that the chord marked x is not strictly in the key of C, but is apparently like the Dominant yth in the key of A. It does lead to the chord of A, and is in so far like the chord of the Dominant yth in the key of A. But the chord on A is in the key of C (on the 6th degree) . Now let it be noticed that the chord marked x is like the chord of the Dominant yth : but as there can be but one chord of the Dominant ^th in a key, we must adopt some other way of describing the relation of this chord to the triad on A, and will call it the "Attendant " chord of A. ( The reason for thus naming such chords is more clearly de- scribed in the author's " How to Modulate.") 270. From a consideration of the above, 265 to 269, it is clear that each major and minor triad in any key has its attendant chord.* As shown in the following example, these attendant chords can be used with good effect. They are indicated by [A]. Fig. 79, [A] * The triads upon the 7th degree in Major, and the 2nd and 7th in Minor, are prohibited from having [ A ] chords. The reason for this prohibition is, that being Diminished triads, and therefore not consonant, they could not be the resolution of . a dissonance ( see 151), and therefore could not stand in the relation of Tonic, which would be required if they were to have [ A ] chords. (It has been shown that although not Tonic and Dominant, a triad and its [ A ] chord stand in the relationship of Tonic and Dominant.) For the same reason, the Augmented triads in Minor are prohibited from having [A] chords. HARMONY SIMPLIFIED. 167 vi [A] in [A] IV [A] V N. B. In practical composition, [A] chords would not be so frequently used as in the above example, which is given to show how the [ A ] chord of every Major and Minor triad in the key can be used. Keyboard and Written Exercises. 271. (a.) Taking the key of G, find in succession the [ A ] chords which shall resolve to the triads on n, in, IV, V, and vi, proceeding as in 267. ( .) In a similar way, take all the Major and Minor keys in turn. Much repetition and persevering practice are neces- sary to give the required proficiency. Before proceeding, the pupil must be able to give instantly the [ A ] chord of any Major or Minor triad. 272. It is remarkable what frequent use of the [A] chords has been made by composers, beginning with Beethoven. In the following example, from Mendels- sohn's Spring Song, are five [ A ] chords in seven meas- ures. The explanation is found in the marking under the staff. For example, [ A ] of n means the [ A ] chord re- solving to the triad on the second degree of the scale. Therefore, after the [ A ] of n we may expect to hear the chord on n. In the second measure we do hear it, but as it has a major 3rd D$, it becomes also the [ A ] of V, For further explanation of this example see " How to Modulate," p. 7. 1 68 HARMONY SIMPLIFIED. Fig. 80 ^ ^TTu r IV HARMONY SIMPLIFIED, 169 v 7 v 7 273. The pupil should examine some of Beethoven's Sonatas, and also examples from Mendelssohn, finding the [ A ] chords and indicating by proper marking to which degree of the scale they are attendant. He should also be on the alert to find examples of [A] chords in the music in daily use. Exercises. 274. (a). Write little successions of chords, intro- ducing one or two [A] chords. Be careful not to wander away from the key, but see that each [A] chord resolves to some triad in the key. There need be but three or four chords, after which a close may be reached by a Closing cadence. (6.) Repeat the above at the keyboard. (Continue this keyboard drill indefinitely, becoming familiar with all keys.) 170 HARMONY SIMPLIFIED. 275. A remarkable feature of [A] chords is that they give great variety by enlarging the boundaries of the key, so to speak, instead of confining everything to the chords upon the seven degrees of the scale, and their in- versions. Another highly practical use of the [ A ] chords is their wonderful power in modulating. This will be ex- plained in the following chapter. Synopsis. Form as usual. CHAPTER XIII. MODULATION. 276. Modulation is the passing from one key to nother ; and is effected by the use of one or more chords characteristic of ( belonging to ) the key to -which it is desired to modulate. There are innumerable ways of modulating, but the very multiplicity of the means employed has always made it most difficult for the beginner to grasp them, and the usual result is utter confusion of ideas, and little practical skill in passing from key to key. The method here presented is held to be simple, systematic, and compre- hensive. 277* Modulation is effected by connecting some chord of the " old key " with some chord in the "new key.'* ( N. B." Old key " and " new key " refer, respectively, to the key from which, and the key to which, it is desired to modulate. ) Therefore, if we can find a method of connect- ing any two triads, the difficulty is easily solved. HARMONY SIMPLIFIED. 171 Notice, we do not say that Modulation is effected by connecting the " old " Tonic triad with the " new " Tonic triad ; but by connecting any ( Major or Minor ) triad of the " old" key with any of the " new" key. Our range of possibilities in variety and delicacy, and means of hid- ing the modulation, is therefore very large if we can mas- ter this one point, viz., to connect any tivo triads. To Connect any Two Triads. 278. It has been shown at the beginning of study how chords are connected by means of a common note. ( See 102.) We have also studied in the last chapter the sys- tem of Attendant chords, and learned that any Major or Minor triad may have its appropriate [AJ chord. Upon trial it will be found that if there is no direct connection between two given chords by means of a com- mon note, the connection can be made by the use of one or both of their Attendant Chords. Thus it becomes possible to connect any two chords without considering whether they belong to the same key or to different keys. For example, let us connect the chord of C with the chord of FJJ. As there is no common note to connect the two triads, we will write them with their Attendant chords, which we will indicate by [A], The second chord in Fig. 8 1 is the [A] chord of C, the third chord that of Ftf. Fig. 8 1 . [AJ of C. [A] of Ft. 172 HARMONY SIMPLIFIED. 279. Usually only one [ A ] chord is necessary, as for example in connecting the triads of C and D Major, shown in Fig. 82. Fig. 82. [A] of D. Thus it will be seen that although two chords may not have a common note to connect them, when we con- sider their Attendant chords a connecting-link will become apparent. 280. In the following exercises the pupil will connect two given Major or Minor triads.* The mental process, given below, will be of much assistance. The example given to illustrate the process is : To connect the triad of C major with the triad of Bfr major. Process. NOTE. Follow this process with the hand upon the keyboard, playing each chord as mentioned. Given, to connect the triad of C with that of fy\ 1st Step. What are the [ A ] chords of the triad from which and the triad to which we would pass?** Ans. The [ A] of the triad on C is G-B-D-F. The [ A ] of the triad on Bt? is F-A-C-Eb. (Write the notes for reference) . * Should any two triads have a common note, the connection may be made without the help of the [ A ] chords. But in many cases it will be observed that the use of the [ A ] chords gives a smoother connection and more repose when the filial chord is reached. * For the present we will use the [ A ] chords in the form of a chord of the 7th. HARMONY SIMPLIFIED. '73 2nd Step. Is there any note common to the triad of C and the [ A ] of Bt>. Ans. Yes, C is common to the two chords, and will enable us to make the connection. 3rd Step. Of the four chords before us, viz., the triad on C and its [ A ] ; and the triad on Bfr and its [ A ] ; how many do we need to make a good connection? Ans. Three, the triad on C, the [A] of Bi? and the triad on Bfr. 4th Step. Write them, trying to secure a good lead- ing of the parts. Fig. 83. -fe- I fl C [A] of Bb Bb C [A] of Bb Bb Could this connection be made in any other way? Ans. Yes, both [A] chords could be used instead of one, as there is a note common to both [ A ] chords. F is that common note. The connection using both [A] chords is shown in Fig. 84. Fig. 84. C [A] of C [A] of Bb Bb C [A] of C [A] of Bb Bb Keyboard and Written Exercises. N. B. While working out these exercises, the pupil should constantly refer to the notes in 282-284. ^^ i- I _ ^ -& &^ %. . z?H z? ttE, i '74 HARMONY SIMPLIFIED. 28.1. ( #) Connect the major triad on C with the majoi triad on C#. Connect the major triad on C with the major triad onD. Connect themajortriad on C with the major triad on DjJ. Connect the major triad on C with the major triad on E. And so on, till the triad on C has been connected with every other triad. Then ( <5.) Connect the triad on Qf with the major triad on C. Connect the triad on C# with the major triad on D. Connect the triad on C$ with the major triad on D#. And continue through the chromatic scale as before. (c.) Starting from the triad upon each remaining note of the scale, connect with every other triad. (a?.) Connect as above each Minor triad with all other Minor triads ; and with all Major triads. 282. In doing the above exercises, it may be possible to make many connections in two or more ways, viz., (a.) Without any [ A ] chord. ( b.) Using the [A] chord of the triad to which we pass. (c.) Using the [ A ] chord of the triad from which we pass. ( d.) Using both [ A ] chords. N. B. The Enharmonic change is often employed, changing sharps to flats, and vice versa. 283. If only one [ A ] chord is used, that of the triad to which we progress will usually be the better one, for the following reason : The natural tendency oi an [ A ] chord is strongly HARMONY SIMPLIFIED. '75 toward its triad, like the tendency of a Dominant seventh- chord towards its Tonic triad. Therefore, in connecting two triads, if the [A] of the one from which we go is used, the natural tendency would be to return to that triad ; whereas, if the [ A ] of the triad to which we go is used, there is a natural tendency to continue to that desired triad. This explains why some of the connections made by the pupil wiH be harsh and forced. ( The next para- graph will show how the above-mentioned tendency to return may be hidden, and the harshness avoided.) The difference in effect between the [ A ] from which, and the [ A ] to which we go, is illustrated in Fig. 85. (a.) W Fig. 85. [ A ] of C. [ A ] of B. ( a ) is not positively bad in effect ; but the superior- ity of ( b ) , using the [ A ] of the triad to which we pass, is manifest in its smoothness and repose. 284. To remove the Tendency to return shown in the [ A ] of the triad from "which we progress. It will be found that by inverting this [ A ] chord, the natural tendency toward its triad is to a great extent hidden. In composition, chords are inverted not only to give variety, but also to induce a smoother leading of the individual parts. Thus the melodic tendencies of individual parts become more prominent, and the harmonic ten- dencies less so. From this we learn that : (a.) Inverting an [ A ] chord reduces the force of its characteristic tendency toward its triad. !^6 HARMONY SIMPLIFIED. 1 ( 6.) Melodic tendencies of the individual parts also serve to cover the same tendency. This is illustrated in Fig. 86, where the same con- nection as in ( a ), Fig. 85, is given, using the [ A ] of the triad from which we pass, and producing a very sat- isfactory effect. Fig. 86. [ A ] of C. Therefore : In using the [ A ] of the triad from which you progress, always invert it, and consider the melodic tendencies, making the individual parts progress with as little skipping as possible. To Connect any Two Keys. 285. Having learned to connect any two triads, we proceed to connect any two keys ; for it is evident, that the connection ( or modulation ) is effected by selecting a triad from the old key and one from the new key, and finding the connection between these tivo triads, as shown above. And when the two triads are connected, the keys are thereby connected, and the modulation is effected. Therefore, the connections shown in Figures 81 to 86, might be taken as a method of passing from one key to another, instead of from one chord to another. Keyboard and Written Exercises. 286. ( a.) From every Major key modulate to every other Major and every Minor key. ( 3.) From every Minor key modulate to every other Minor key and every Major key. HARMONY SIMPLIFIED. 177 287. Note I. It should be observed that the [ A ] chords resolve equally well to Major and Minor triads. Therefore, the Major and Minor triad of any degree ( for example, the Major triad of G and the Minor triad of G ) would both have the same [ A ] chord. 288. Note II. Notice that the [ A ] of the Tonic chord ( or key ) to which we modulate is nothing more or less than the chord of the Dominant Seventh resolving to its Tonic. 289. Note III. To thoroughly establish the new tonality (or con- sciousness of the new key), the Closing Formula should follow the connection of the two triads, particularly if the triad to which we pr&- gress appears in an inversion. The sense of incompleteness without the Closing; Formula is illustrated in the following: F, g .87.^ -&& I"-"** 1 r_z?_ ? i 6 W' IV I* V I 290. In the preceding pages, we have learned to con- nect any two triads, and, in a similar way, any two keys. The process, being founded upon a principle which is folloived implicitly in all cases, might be represented by a formula which shall give a visible plan of procedure, and show between which chords the [ A ] chords are to be in- troduced, if at all. The chord-connections shown in 278 to 288, would be represented by the formula : Old Chord, [ A ] , New Chord. The method of connecting two keys by connecting the tonic triad of the old key with the tonic triad of the new key would be : _L_, [A], * , Old Key New Key 291. The terms Old key, and New key, are used to indicate briefly that the chords designated by the Roman j^S HARMONY SIMPLIFIED. Numerals belong to the key from which, or the key to which, we modulate. The Roman Numerals indicate upon which degree of the scale the chord ( a common triad when not other- wise indicated ) is to be taken. [ A ] indicates that an Attendant chord is to be in- serted if necessary. Sometimes two [ A ] chords may be employed to advantage. 292. Observe that the [ A ] of _ is simply the New Key chord of the Dominant Seventh in the new key. . As the progression of an [ A ] to its triad is precisely the same as that of a Chord of the Dominant Seventh to its Tonic triad, we may draw the logical conclusion that if we can pass to the Tonic of a Foreign key through its Dominant chord, -we can pass to any other Major or Minor triad of a foreign key by using Attendant chords. As these Attendant chords are so easily found, and have a most intimate relation with their Primary chords, they will prove a simple, practical and correct means of connecting the original key with any desired chord of the new key. 293. With the assistance of the Attendant chords it becomes possible to formulate the principal methods of Modulation, giving a most thorough and comprehensive view of the whole subject. If we modulate by means of the Dominant Seventh- chord of the new key, we must connect the original key and the New Dominant ; if we modulate through some other chord of the new key, ive must connect with that chord. Upon this plan the Formulas are con- structed. HARMONY SIMPLIFIED. 179 Modulation by Means of the Dominant Seventh- Chord of the New Key. 294. According to the heading of this section, we V 7 must pass through __ ; therefore, the first prob- New key I V 7 lem is to connect __ and _ . Should there Old key New key be a note common to both chords, we can proceed at once to the desired chord. If not, the Principle of Attendant Chords will supply the connection. Thus, the formula becomes _ _, [ A ] _ZIl*_ JL- Observe that [ A ] Old key New key may indicate the [ A ] chord of either the Old Tonic or the New Dominant, or of both if necessary. To illustrate, let us modulate from C to F$. Now the formula becomes more specific : Old key represents the triad on C : represents that on F#, New key and _____ _ the Dominant Seventh-chord on Qf. As New key there is no connecting-note between the chord on C and that on CjJ, we resort Lo the Attendant Chords, and dis- cover that we can use either the Attendant chord of C or that of C#. Writing the chords and the formula together shows plainly the connection, using first the [ A ] chord of and then the [ A ] chord of , as rep- Old key New key resented in Figs. 88 and 89. An [ A ] chord can resolve to a Seventh-chord instead of to a simple triad, on the ground that one Dominant Seventh-chord can resolve to another (See 185.) i8o SIMPLIFIED. Fig. 88. Old key ' New key Fig. 89. Old key New key 295. In every case of Modulation through the Dom- inant Seventh of the new key, there will be a feeling ot incompleteness. This will disappear if, after the new Tonic has been reached, the "Closing Formula" is added. This is illustrated in Fig. 90, where the same Modulation as in Fig. 89 is given, with a slightly differ- ent leading of the parts on account of the Closing For- mula following. r. 00. Old key New key Closing Formula Keyboard and Written Exercises. 296. For the first exercises, start from the Tonic triad of C and pass to all other keys through the new Domi- HARMONY SIMPLIFIED, j.8i nant Seventh-chord, using the [ A ] chords if necessary to make the connection. Next, proceed from Qf to every other key ; then from D ; and so on, till every key has been used as a starting-point from which to modulate to every other key. To gain the fullest benefit, the ^lupil should practise modulating both at the keyboard and in writing. 297. Attention must be paid to the correct leading of the parts. A Modulation which is harsh in one posi- tion and with a certain leading of the parts, may often be much improved and softened by a change of position and different movement of the parts. It will be found that while many of these Modula- tions are harsh in spite of a good leading of the parts, when made directly through the new Dominant Seventh, they may be made very pleasant by the use of one or both [ A ^chords. The student must not fear to take the chords in their different inversions to induce a smooth leading of the parts. A good effect depends also upon a proper arrange- ment of the accents, as shown in 190. ( See also "How to Modulate," 15.) 298. When we use the [ A ] chord of the new Dominant, we touch the key of the Dominant of the new key, as we make use of the Seventh-chord on its ( the Dominant's ) Fifth degree. Thus, in Fig. 89. the new key is FJ and the key of its Dominant is CJ. Now it will be seen that the [ A ] chord, having Bf, is like the Dominant Seventh- chord in the key of C$. Dr. Stainer says, in his " Composition," that a new key should be entered through related chords or related keys. Here it is plain that we have entered through a related key, that of the Dominant. Thus it appears how the System of Attendant Chords fills the requirements of related chords or related keys in Modulation. Change of Mode. 299. The change from a Major key to the Minor key of like name (e. g., C Major to C Minor) cannot be lS2 HARMONY SIMPLIFIED, called a modulation, since the key-note is not changed, but merely the mode. Notice that the chord of the Dominant yth is the same in both Major and Minor, and that the two triads may follow each other without the interposi- tion of any modulating chord ( Fig. 91, a) \ or the com- mon Dominant yth may be interposed (Fig. 91, 6) . Many examples of this interchange between Major and Minor may be found in the works of the masters. (a.) C=BL Fig. 9 1 . 300. In the preceding paragraphs we have entered the new key at the Tonic triad or the Chord of the Dom- inant. It is equally convenient to enter at any other (Major or Minor) triad of the scale. To construct the formula for such a case, we should merely substitute the desired degree for the term . New key It is also possible to leave the " old" key at points other than the Tonic triad. The [ A ] chords can be used, not only in the form of seventh-chords, but also in the form of Diminished yths, Augmented 6ths, or Chords of the 9th. These different methods, together with the possible diffeient points of leaving the old and entering the new key, offer great variety in the means of modulation. The chord of the Diminished seventh is especially useful in Modulation, since it has a direct and natural resolution to four differ- ent chords. ( See 215.) Having just seen that it is pos- sible to enter the new key at various points, each one of the above-mentioned chords of resolution might be con- sidered either the Tonic, Dominant or Supertpnic of a HARMONY SIMPLIFIED. ^3 key.* In this way, each one of the four chords might repre- sent not one, but three different keys. The four different chords would then together represent twelve different keys ; i. e., all the different keys. It is therefore possible to modulate from any chord of the Diminished seventh directly into any one of the twelve Major and twelve Minor keys. By means of the above-mentioned methods, it is pos- sible to pass directly from any key to any other. This is a most desirable accomplishment for organists, concert- players and accompanists, who are frequently called upon to bring two wholly unrelated keys into immediate prox- imity in successive selections. But it must be understood that such promiscuous intermingling of keys is never allowed in constructing any single piece of music. In Composition the range of selection is usually limited to the "Related keys;" viz., the keys of the Dominant, Subdominant, and their Relative Minors, and the Rela- tive Minor of the key itself. ( See 39 and 334.) Modulation by Means of a Common Triad. In connecting two related keys, it will be found that instead of a single common note serving as a connecting- link, there is a complete chord which is common to both keys, offering the closest possible connection. E. g., in connecting the keys of C and G, the following triads will be found the same in both keys : C : I and G : IV ; C : in and G: vi; C: vi and G: n. Any one of these chords may be used as the connecting-link, the chord be- ing approached as belonging to the key of C and left * Each of these chords could just as well be taken as a Mediant, Subdomi- nant or Submediant, as for Supertonic or Dominant. The three selected are merely more prominent, and suffice to enable one to modulate to all keys. 184 HARMONY SIMPLIFIED. as belonging to the key of G, as shown in the marking under the illustration. (a.) (b.) (c.) t^-l L ^ J = zt5 V? I (7:1 in vi Keyboard and Written Exercises. Starting from various keys in turn, modulate, by means of a common triad, to each of the related keys, as mentioned above. There are also many other ways of modulating, which are not so comprehensive in their application as those already described, but are useful where circumstances happen to favor their introduction. Being of good effect and in common use, a few of them are men- tioned : ( a ) Compound modulation, passing through a series of keys to the one desired : ( b ) Single Note Con- nection ; ( c ) By means of the False Cadence ; ( d ) By means of Enharmonic Change. All the above-named means of modulation, together with the principles of artistic modulation, are described in detail in the author's " How to Modulate." Synopsis. Write as usual. UAttAIONY SIMPLIFIED. 185 PART III. CHAPTER XIV. VARIETY OF STRUCTURE : SUSPENSIONS : ANTICIPATIONS : RETARDATIONS. 301 . For the purpose of giving variety to the harmonic structure of a composition, many devices are employed. Among them may be mentioned Suspensions, Anticipa- tions, Retardations, Passing- Notes, Passing- Chords, Changing-Notes, Appoggiaturas, Organ-Points, Sus- tained Notes, and Syncopations. These devices should not be looked upon as altering the principles of chord-construction already learned, but as means of giving greater variety to a Disposition. They are to Musical Composition what interior decora- tion is to Architecture, merely a means of ornamenting and enriching a substantial structure. Suspensions. 302. In a succession of chords, when one tone is de- layed, or held over till after the next chord has entered, a dissonance is formed, called a Suspension. This delayed and therefore dissonant tone moves but one step clown or up, usually down, to its tone of resolution in the next chord. i86 HARMONY SIMPLIFIED. The essential features of a suspension are : the Prep- aration, the Dissonance, and the Resolution. The Preparation consists in the suspended tone being pre- viously heard as an essential part of a chord. The Disso- nance, technically called the Percussion, is caused by the progression of a single part being delayed while the remain- ing parts proceed. The Resolution is effected by allow- ing the delayed tone to proceed to its place in the following chord. In Fig. 92, the Suspension is in the Alto ; the first note is the preparation ; the second, connected with the first by a tie, is the Dissonance, or Percussion ; and the third note the note of Resolution. Fig. 92. I 303. Let the pupil notice the following conditions im- plied by the definition and illustrated in Fig. 92 : (a.) One note is held over and prevented from pro- gressing with the others. This is accomplished by the use of the tie. (.) By being heard in the first chord, the sus- pended tone is prepared. The Preparation should be as long as the Dissonance, else the Preparation would not be sufficiently marked. ( c.) The Preparation, Dissonance, and Resolution should be in the same part. Otherwise we could not have ( particularly in vocal music ) any effect of Preparation or of Resolution. ( 307. Suspensions may occur in two or more parts at once, in which case they are subject to the same rules as when occurring in only one part. (Fig. 93, a.) Suspensions may also occur with a progressing Bass, i. e., while the tone of resolution is sounding, the Bass progresses to another tone, thus producing a new chord- formation ( Fig. 93, <5), or another inversion of the same chord. Fig. 93. Suspensions may also be resolved ornamentally, i. e., by the use of interpolated notes between the suspended note and its resolution. The note of resolution must be the same as if no ornaments were introduced ( Fig. 93, c). Exercises. 190 HARMONY SIMPLIFIED. R - 2. - s a a e 6 7 47 I 3. R. " -^ CX .^TJ- 4. R. 8 7 9 6 - 6 - e 7 ). J- 5 - 765982 659 65f --&- Open Position. 65 i * \ 6. J- ?___ 3 H I m HARMONY SIMPLIFIED. 76 70 6 1 - 4 6 33 i- C\' 1 1 /O 1 1 *-J. j & 1 O si cJ \ ? ii 7. J- 366 96 6 9 t . f8 7 * - ^\ * ^ /*r? ^? *r- | | 1 . ^ j P^ -~ 1 j~ ^^ -^ sy -\ ^ K | | J^ -^- ^*^ ^^ y^- <5^ 5* ^^ ,^- ^^ -- J65 4370 ^^C 8 6 4,5 5fl 6, -*S^ 5, 5b $ 3, B 6 8 7 fA- r^^l^ P ! *-!., / ^ * P f? , It 1 Z HD 2 EZ2Z .. 7 7b Jf^ 987 1 987 7656 6D 5 9^ J 866 9 657 ^765 7 433 4 - -48 Anticipations. 309. In a Suspension one tone of a chord is held over till after the next chord has entered. Anticipation is in one sense the reverse of this, for, instead of being delayed, a tone is advanced, or heard before the rest of the chord. Differently expressed, it is where a tone of one chord is anticipated in the previous chord. This is shown by the notes marked x in Fig. 94. . 94. , I J'J 192 HARMONY SlMPLiFtts.0. 310. Notice the following in reference to the above example : ( a. ) Unimportant positions : Suspensions occur upon the accented parts of a measure ; Anticipations, upon unaccented parts. Anticipated notes are also usually short, never taking more than half the value of the pre- ceding note, and usually less. Anticipations, therefore, are seen to occupy unimportant positions, in respect to both rhythm and duration. ( .) Anticipations are usually restruck, i. e., not tied to the note which they anticipate. ( c.) Anticipations do not need to be prepared and resolved like Suspensions. They may enter freely by skips, and proceed by skips if desired. 3"- Keyboard and Written Exercises. Form examples of Anticipations in various keys. Retardations. 312. Retardations are the opposite of Anticipations. A tone of the chord is held over while the remaining tones progress to the next chord. Retardations differ from Suspensions in being treated freely like Anticipa- tions; i. e., they require no preparation, but may enter by skips ; and ( b ) they are allowed to progress by skips, not being forced, like Suspensions, to progress to the note only one degree higher or lower. Fig. 95. < EE EZ r -r -r ~r 1 0ARMONY SIMPLIFIED '93 Keyboard and Written Exercises. 313. ( a.) Form examples of Retardations, in va- rious keys. (3.) Form examples, mingling Anticipations and Retardations. Syncopation. 314. Syncopation is a kind of irregular Rhythm, where the more important notes are placed upon unimpor- tant beats or parts of beats ; or where the notes fall between the beats. It may be produced by Anticipation or by Re- tardation; i. e.,by pushing forward one part ahead of the others, or by holding it back till the others have moved. Fig. 95 is an example, the note marked x serving to form a Syncopation, which is continued by the retarded notes marked o. Synopsis. Write as usual. CHAPTER XV. UNESSENTIAL NOTES : PASSING-NOTES. Those notes which, coming after a chord, ar* not essential to it, but lie between the essential noces, are called Passing-notes. (a.) Fig. 96. ffi=^l- tt*- ^=M IM) & W^~f ! m nr 1 -- 1 TT , j ^^~ 7-j c\* u Or 1 5, i y,p m t ^ \> f ^^ 1 H '94 HARMONY SIMPLIFIED. Notice the following: ( i.) In Fig. 96 the notes marked x do not belong to the chords. (2.) These marked notes serve to connect the chord-notes melodically with each other. In Fig. 96, ( b ) , the chords are like those at ( a ) , but in ( a ) the notes marked x serve to lead very smoothly from one chord to the next. (3.) These Passing-notes may occur in any part. They are usually found upon the unaccented portions of the measure, when they are called Regular Passing- notes ; but are occasionally found upon the accented parts, when they are called Irregular Passing-notes. (4.) The harmonic structure of a composition is of the first importance, forming the basis or skeleton. The passing-notes and other ornaments are to be added after- ward. 316. Passing-notes may be chromatic as well as dia- tonic. In Fig. 97 the notes marked o are chromatic; those marked x are diatonic. Rff. 97. HARMONY SIMPLIFIED. '95 The pupil should find and write the chords forming the harmonic structure of Fig. 97, as at (3), Fig. 96. Care must be exercised in securing a correct leading of the parts in the structure of the harmonies, i. e., in the chords before the passing-notes are added, since concealed 5ths and 8ves may, by the use of Passing-notes, become open consecutives. Keyboard and Written Exercises. 317. (a.) Return to the first exercises, Chapters III and IV, and insert passing-notes where possible, either in the given Bass or in the upper parts. (.) Try this exercise at the keyboard. Exercises in Harmonizing the Scale. Harmonize the scale, using Passing-notes where pos- sible, together with the chords previously learned. 318. Two or more Passing-notes may be used simul- taneouslv, or even all the notes in a chord, thus forming a Passing-chord. Occurring upon unaccented parts of a measure, Passing-chords are not expected to always har- monize perfectly, but may be looked upon rather as a number of Passing-notes leading melodically to the next chord upon an accented part of the measure ; for upon the principal beats the harmony should be quite correct, thougrh liberties are allowed on the \veak beats. Fig. 98. 96 HARMONY SIMPLIFIED. 319. The pupil may not clearly distinguish between altered chor.ds and chords with chromatic passing-notes. The following constitutes the difference : ( i.) To be an Altered chord, the tempo should be slow enough, and the accents such as to allow the al- tered note to be heard as part of a chord. A chromatic or diatonic scale-passage, accompanied by a single chord, would be said to consist principally of passing-notes ; e. g-, (2.) Only chromatic alterations can be considered in connection with altered chords. If another note of the scale is substituted for a note of a chord ( making a dia- tonic instead of a chromatic change ) , it is, of course, a passing-note. A note may be said to belong to a chord even if it has two flats or sharps before it, but as soon as it changes its name, it loses its membership in that par- ticular chord; e. g., Fx belongs to the triad: but if we call it G, it could not belong to the triad of Dj. Auxiliary Notes. 320. An Auxiliary note is one used for ornament or embellishment, and is found one degree above or below its principal note, which belongs to the chord. It pre- cedes the principal note, and is heard either with or be- fore the remaining notes of the chord; e. g., HARMONY SIMPLIFIED. 197 (S , L. -__. Fig. 100. The peculiarity of the Auxiliary note \$>, that while it may enter by a skip (i.e., need not be prepared), it must progress by a single step to its note of resolution. \ See Fig. 100.) These notes are also called Changing - .votes, Appoggiaturas, and Free Suspensions. 321. Trills, Shakes, Turns and all similar ornaments are classed with Auxiliary notes. This principle is well expressed in " Musical Composition," Goetschius (N. Y., G. Schirmer), as follows: "Every harmonic interval is attended by four Neighboring tones, consisting in the next higher and lower Letters, in their notation as whole step and half-step. Thus: Fig. 101. ^ I) The Neighboring tone cannot be chromatic ( as at Fig. 01, <5), because the Letters must differ. " The Neighboring tones may occur in almost any connection with their own harmonic interval ( Principal tone ) as Unessential or Embellishing notes. "All the common forms of Embellishments or Grace- notes (the Turn, Trill, Appoggiaturas, Mordent, etc.), are based upon the association or alternation of a Principal tone with one or another of its Neighboring tones, thus ; 198 HARMOA T Y SIMPLIFIED. tr Fig. 102. " o signifies ' Neighboring note.' " Keyboard and Written Exercises. Construct illustrations of the above. Organ-Point. 322. An Organ-Point, or Pedal-Point, occurs when a note in the Bass is sustained through a succession of chords in the higher parts, part of which chords only are in harmony with the Bass note. Fig. 103. Notice that the chords marked x do not harmonize with the Bass, but, alternating as they do with chords of which the Bass note is a part, the effect is still good. Essentials of Correct Organ-Point. (a.) The first and last of the series of chords should harmonize with the sustained note. (6.) The first chord should be heard upon an accented beat. (c.) Chords harmonizing with the sustained note should pre- dominate, though they may occupy either accented or unaccented beats. ( rf 2 I~~^T -i ex 1- 1 ^S> -P* ^5- -I SH-H EBSiSyEi 6. Emery. p~7-rF S3 ' " 3^ "3 "11 cz ID 1 (^ S>__ J II Y^ Em ery. 1 6 4 7 i Q^ , (*? i" ri L-Ji/* ^d a r^^/ S. ^ 3ZJ i fi' 1 ^ II N. B. The Tritone is here allowed, for otherwise the sequence would be broken. Advanced Course. [Quoted from Banister's " Music."] 333. "A Sequence is termed Real when all the chords, or intervals, are major, minor, etc., at each recurrence of the pattern-progression as at the original occurrence of it. "A Sequence is termed Tonal when the chord or intervals, at each recurrence, are according to the key in which the passage occurs, and therefore do not strictly resemble the original pattern. This is the more frequent kind of sequence. Fig. 109 is a Tonal sequence ; two of the ascending 2nds are major, one (from D to Eb) minor; more- over, some of the chords are major, others minor. "The preservation of a sequential progression, will often lead to and justify exceptional intervals, doublings, etc.; the symmetry of the sequence outweighing the objections which might otherwise lie against such exceptional arrangements. Design, using the word in its artistic sense of intelligent aim at a defined and desirable effect, especially with regard to form, reconciles and more than reconciles the mind to HARMONY SIMPLIFIED. 209 details which, taken by themselves, would be questionable or even positively objectionable. "In Fig. no, for example, the Tritone 4th in the Bass, from C to FJ, and the non-resolution of the Diminished 5th in the Tenor, at *, till the next chord but one, are both justified by the sequential form of the passage. " Such exceptional progressions, however, though permissible BANISTER. Fig. 109. J L_U L in the course of the sequence, must not occur in the original pattern, in which the writing must be perfectly pure." BANISTER. Fig. 1 1 O. rz?-gq_gj_^!ipi^:p Related Keys. 334. In 32 the keys related to a given key were stated to be the key having one more sharp ( its Domi- nant ) , and the key having one less ( the Subdominant ) . To these may be added the Relative Minors of the key itself, of its Dominant, and of its Subdominant. Thus the relative keys of the key of C are : the key of G ( the Dominant ) , the key of F ( the Subdominant ) ; the key of A minor ( Relative Minor of C ) , the key of E minor 2IO HAxMONY SIMPLIFIED. (Relative Minor of G), and D minor (Relative Minor of F). To this may be added, as it is so frequently used though not allowed by all theorists the Tonic Minor, or Minor key of the same name, in this case C minor. The related keys of a Minor key are the Minor keys of its Dominant and Subdominant, and the relative Majors of all three, i. e., of the key itself, of its Domi- nant, and of its Subdominant, to which we may also add, as above, the Tonic Major. Thus, the related keys of C minor are G minor and F minor, Eb major, Bb major, Ab major and C major. Naming the Octaves. 335. Musicians speak of Three-lined A, Great-octave B, Small-octave F, etc. The system of naming the vari- ous octaves is as follows : This note and the six notes below are called the Sub- Contra- Great Small Octave. Octave. Octave. Octave. Marked j^ Once-accented or One-lined Octave. C Twice-ac- cented, or Two-lined Octave. B_ Three- lined Octave. *** JSl- ^ -*L_L Four-lined Octave. 8va M Marked c b" c \. K c be" T> ( d hi C2 b 3 c3 b c* b* or:| e , h' c" b " C'" ^rrr^tftr \/ ffi - SIMPLIFIED. 211 The Great Staff: the C Clefs. 336. In very old music, instead of two staves of five lines each and an added line above the Bass or below the Treble for middle C, a great staff of eleven lines was used ; and the various parts, Bass, Tenor, Alto, and So- prano, were placed high or low upon this staff, accord- ing to the pitch of the voice : Fig. 111. The notes in the great staff were written just as in the present system, G being the lowest note in the Bass, and leading up step by step to the 5th treble line, which is F. Notice that the 6th line is C, corresponding to our middle C. In fact, our staff is the same as the old one, except that to help the eye the middle Iin2 is omitted un- less actually in use, when it is written as an added line, and the two sections are separated a little. The sign ISt is called the C clef, and always de- -f^H~ * notes middle C, or the 6th line of the great staff. In forming a Tenor staff, for example, it is considered in which part of the great staff the chief notes of the Tenor 'lie (all staves consisting of five lines and four spaces ) . ^ow, the Tenor sings most easily from the 3rd line of iie great staff to the seventh line, or from small D to jne-lined E. It not being necessary to employ all of the Great staff for the limited compass of the Tenor, it became customary to take out the proper section of the great staff, leaving the clef to denote which part had been taken. Reference to Fig. us, (a), and 112, (<), will 212 HARMONY SIMPLIFIED. make it clear how the Tenor, Alto and Soprano staves were formed. The C clef, then, instead of moving about for the different staves, in reality remains stationary, different parts of the great staff being used with it to suit the com- pass of the different voices. Fig. 112, a. Treble or Tenor Alto Sop. Violin Tenor. Alto. Sop. Clef. Clef. Clef. Clef. Middle TT I I I 4= Middle 4H-H . T^T; ^rr ^EB c- ]Ui flE iflf c "- - c.- Flg. 112, ( 6.) Tenor. Alto. Sop. Treble. Middle C. c- v 1 - - Exercises. These clefs should be brought into use, either by writing future exercises in them, or by copying past exer- cises, hymn-tunes, etc., employing a separate staff for each part, thus forming what is called Vocal Score. Chords of the Eleventh and of the Thirteenth. 337. According to the principle of forming chords by tae addition of a note a 3rd above the last note, we may form chords of the nth by the addition of a note to the Chord of the 9th; e. g., 5!" ; and if to this Chord of the nth we add still another 3rd, we shall have a Chord of the i3th; e. g., HARMONY SIMPLIFIED. 2 1 3 These chords have no practical application in Har- mony, since so many notes must be omitted in four-part writing, and the dissonant intervals prepared, that they become practically nothing more than suspensions. Exercises in Open Position, or Dispersed Harmony. 338. The pupil is now sufficiently experienced to write in Open position, placing the Tenor part upon the Bass staff. It is not required that every chord shall be in open position ; when more convenient, close position may be used. In distributing the parts, try to keep the larger inter- vals between the lower parts. Avoid, if possible, hav- ing more than an octave between the Tenor and Alto, or between the Alto and Soprano. Exercises. Refer to the exercises in the preceding chapters, and, ignoring the figure over the first Bass note ( i. e., trying various positions ) , write them in Open position. The results will not always be satisfactory, but the comparison of the effect in the various positions will be helpful. Five, Six, Seven, and Eight-Part Harmony. 339. Having studied the principles of Harmony rather than a series of set rules, the pupil will be able to write in more than four parts, without special directions. The Ten- dencies and Influences will need to be interpreted with rather more freedom, on account of the increased compli- cation resulting from the larger number of parts. Exercises. The pupil will attempt to compose phrases of eight measures, introducing five, six, seven or eight parts. 214 HARMONY SIMPLIFIED. CHAPTER XVII. HARMONIZING MELODIES. 340. The pupil has learned to build chords upon a given Bass, and to connect them. It is now necessary to find appropriate harmonies for a given melody, or to supply the remaining parts for a given Tenor or Alto. Hitherto the chords have been chosen for the pupil ; now he must choose them for himself. Especial care is required in this, one of the practical applications of the previous study. The pupil has used chords in their various inver- sions. He has also learned that any particular note may belong to several chords, a fact which renders the first attempts somewhat confusing. For example, the note C may belong to any one of the following chords : C E G, F-A-C, A-C-E, D-F-A-C, or F-A-C-E, all of which are strictly in the key of C, besides the list of altered, diminished, and [A] chords. The best harmony for a given note will depend principally upon the chords pre- ceding and following. In the exercises below, the appro- priate harmony will be indicated. Exercises. 1. C G C F d 7 \_yr (_, h ? V ,K ^j ^S 5 e i L /* ** r S3E i d 7 HARMONY SIMPLIFIED. 215 2 J. F Bb F Bb C 7 Bb C 7 F g F 7 F g 7 :fe&_^ * 1 g 1 ~fc g H F, Bb Eb Bb C T F^ -ifc* ^7- " ^ I.C) V" \ 1 r TBZSZ. Of : * 4. D 7 G a, D 7 G D XI " /'I* n ~^ & <5 i?t\ \ / z & f3 G C D 7 G e a 7 D 7 G J. Eb Ab Eb c Ab Bb c 7 f 7 Eb 7 Bb Eb /Lb K/ 4 n <2 ^^ ri ^^ ^.^^^^ tf<\9 v\\j & t* ^ ^ & -5 J. Gb Db eb bb eb Db D^ Gb Zjaanz3 S3 fm p i?fi v y ^ Gb eb 4 D 7 Gb 1 2l6 HARMONY SIMPLIFIED. A E A D A m r-<5>- m E 7 f 7 b-y E^ 341 . In the following exerctees, the melody to be har- monized ( also called the Cantus Firmus ) is placed in the Alto, the parts to be supplied being the Soprano, Tenor and Bass. Write the exercises in Open position. J. C 1. ^ J G 7 \f> F 7T- s? Efc: ey & & B 2. J. F C 7 F g7 C 7 F, F g F C F %Fg-| & d g e A 7 d g A 7 Bb g A 7 d In the next exercise, the Cantus Firmus is in the Tenor. Supply the other parts, writing in Open position. P Eb Ab Bfc Eb Ab Eb -G>- I HARMONY SIMPLIFIED. f 7 B Eb f 7 Eb f 7 217 E l',V g f JO g C 2J- /^ "^" <7_ -a. CJr* D 7 f^' U i E L ^ --^^y (]* */ * g a g D Eb a D 7 g -- a :Bi;i2= Zffi d A 7 d A d g A 7 d cto d e A d ^- -Or- 5tf?~ -"^- --*- 5^" *" * "" H g D 7 Eb a g J' -<2. _- g a<> Eb a g D g 342. In the following exercises, in which no assis- tance is given, the pupil should endeavor to find chords which progress smoothly from one to another, constantly looking ahead to see if the following chord will easily succeed the one under consideration. The following hints will be found helpful : (i.) Use simple harmonies. Do not attempt to be original at first, but be content with commonplace effects. (2.) The Principal triads are used more than the others, but the Secondary triads should not be neglected. (3.) Inversions are conducive to smooth progres- sions. 218 HARMONY SIMPLIFIED. (4.) Contrary motion is like oil, it helps the smooth running of the parts. ( 5) Do not let too many parts skip at one time. ( 6.) Avoid consecutives : not only 5ths and 8ves, but also 4ths, 2nds and 7ths. (7.) Keep the parts at about an equal distance from each other. ( 8.) Do not let any part exceed the limits of a good voice of corresponding pitch. (9.) Use the 2 chord in the middle of an exercise with caution. This chord usually indicates a close too keenly for use except in a cadence, or under special con- ditions. (10.) Secondary chords of the seventh resolve, like the chord of the dominant, most naturally to the triad a 4th higher. ( n.) Apply the principles of Influences and Ten- dencies. ( 12.) When the Soprano is low, the chords should be in close position. With a high Soprano, the chords should be in open position. Exercises. Dr. CROFT. WARSAW. L. M. HARMONY SIMPLIFIED. 219 Other chorals and slow hymn-tunes should be se- lected and used as melodies for harmonization. They may be used in the Alto or Tenor as a Cantus Firmus, when transposed to a key suited to the voice taking them. Observe the Soprano Clef below. See p. 212. ggapnirs Q -^ '- ex 1 ifli j * .... & ^ *^ i Jr * ^ ' ^ j*b X^3 /r j & &^ &^ * fl & ^^