• • , • JOSEPH SCH I.LL INGER CORRESPOND E ·N.CE COURSE Subject: Mu�j.c With: Dr. Jerome Gross • Lesson LXV. SPECIAL THEORY OF HARMONY Introduction Special Theory of Harmony is confined to E, of the First Group of Scales, which contain·, all musical names (c, d, e, f, g, a, b) and without repetition. There are 36 such scales in all. The total number of seven-unit scales equals 462. The uses of E, refer to both structures and .. progressions in the Diatonic System of Harmony. The latter can be defined as a system which borrows all its pitch units for both structures and progressions from any one of the 36 scales. Whi'le the structures are limited to the above scales, the progressions develop through all the semi-tonal relations of the Equal Temperament. • The latter comprises all the Symmetric Systems of Pitch, i. e. , the Third and ti1e Fourth Group. Chord-stDuctures, contrary to common notion, \ • do not derive from harmo_ nics . ·I f the evolution of chord-structures in musical harmony would parallel the evolution of harmonics, we would never acquire the • developed forms of harmony we now possess . , • 2. • To · begin with, a group of harmonics, simultaneously produced at equal amplitudes, sounds like a saturated unison and not like a chord. • , • other words, a perfect harmony of frequencies and int ensities does not result i n musical harmony but in a unison. we • In �his means that through the use of harmonics, v«:>uld never have arrived at musical harmony. But we do get harmony, an d exactly for the opposite reason. ' The relations of sounds we use in Equal Temperament are not s.µnple ratios (harmonic ratios). When acousticians and music theorists advocate nJust intonationn , that is, the intonation of harmonic ratios, they are not aware of the actual ( situation . • On the other hand, the ratios they give . for certain trivial chords, like the major triad (4+5+6) , the minor triad (5+6+15) , the dominant seventh­ chord (4+5+6+7), do not correspond to the actual intona­ tions of the Equal Temperament. Some of these ratios, like¼, deviate so much from the nearest i ntonation, lik� the �inor seventh, which we have adopted through habit, that it sounds to us out of tune. Habits in music, as well as in all manifestations of life, are I more important tha.n the natural• phenomena. • If the problem of chord-structures i n harmony would be confined to the ratios nearest to Equal Tem perament, we could have offered 16+19+24 for the minor triad for example, , • • 3 .. L as it approaches the tempered.minor triad much better than 5+6+15. But this, if accepted, would "discredit the approach commonly used in all textbooks on harmony, and for this reason. If such high harmonics as the 19th are necessary for the construction o f· a minor triad, what would chords of superior complexity, , which are in use today, look like when expressed through ratios. When a violinist plays b as a leading tone to c and raises the pitch of b above the tempered b, his claims for higher acoustical perfection are nonsense, as the nearest harmonic in that region is the 135th • Facing faets, we have to admit that all • the acoustical explanations of chord-structures as being developed from the simple ratios, are pseudo- • scientific attempts to rehabilitate musical harmony, t o give.the·latter a greater prestige. l. Though the original reasoning in this field was caused by the ho11est ·spirit of investigation of Jean Philippe Rameau I • ("Generatio� Harmonique", Paris, 1737), his successors overlooked the -development . of acoustical science. , Their inspiration was Rameau plus their own mental I laziness and cowardice. • The whole misunderstanding in the field of musical harmony is due to two main factors: • • 4. (1) the underrating of habit; i (2) the confusion of the term "hermonic" in its I mathematical connotation, pertaining to simple ratios with "harmony" in its • mu.sical connotation, i.e., simultaneous pitch-assemblages varied in time sequence. Thus, musical harmony is not a natural phenomenon, but a highly conditioned and specialized field. • It is a material of musical expression, for which we, in our civilization, have an inborn inclination and need. This need is cultivated and furthered by the existing trends in our music and musical education. I. Diatopic System of Harmony. Chord-structures and chord progressions in the Diatonic System of Harmony have a definite interdependence: chord-structures develop in the direction • . • • opposite to their progression�. Thi-s s-t;;atement brings about the practical classification of.the Diatonic System into two forms: the posi�ive and the negative. As the term Diatonic implies, all pitch• units of a given scale constitute both structures and ' progressions, without the use of any other pitch-units (not existing in a given scale) whatsoever. l • • 5. In the form which we shall call positive, all chord structures (S) are the component parts of I the entire structure (E,) emphasizing all pitch-units of a given scale in their first tonal expansion (E,) ) • and in position Ci). , In the same form chord progressions derive from the same tonal expansion but in position In the negative form of the Diatonic 0 System, it works in the opposite manner. Chord­ structures derive from the scale in E, and in position (§) , wb�le the prog , ressions develop from E, @ . According to the qualities we inherited • and developed, the positive form produces upon us an effect of greater tonal stability. It is chrono­ logically true that the negative form is an earlier one. It predominates in the works where the effect o f tonality,_ as we know and feel it today, is, rather vague. ' . ' Such is the XIV and XV Century Ecclesiastic music, developed on contrapuntal and not on harmonic foundations. Many theorists confuse the negative form of the Diatonic System with "modal" harmony. As by Diatonic Tonality they mean, in most cases, Natural Major or Harmonic Minor scales moving in the positive form, they miss the tor1al stability when harmony moves backwards. Losing tonal orientation they mistake such .. • I 6. progressions for modes, which are merely derivative scales, and may also have the positive, as v1ell as the negative form. But as we have seen in the Theory of Pitch Scales, modes can be acquired from any original scale through the introduction of accidentals (sharps and flats). In the following table, MS represents "melody scale" (pitch-scale), and MR represents "harmony scale" (i.e. , the fundamental sequence of chord progressions) . Diatonic System Positive Form Negat:l,.ve For!D L = MSEr@ = .MSE,@ L HS= MSE, @ HS= MSE,@ • Figure I. • Example (Natural Major) Positive Form .g. (\ I , I .. -,. - • � -�· ,. • � ...- Negative Form ..- -e-- A .A. • -!I -9- t. .c. -. -... .- ·==========================================================----------- - - - ---= .. ' •• • HS - L. • , .. • ., .... -9- ♦ - -., - - � • - • 7. L In the positive form, chords are constructed upward, in the negative, on the cont rary, ' downward. The matter is greatly simplified by the fact that any prog ression, originally written as positive, becomes negative, when read backward. All the principles of structures and motion involved • are therefore reversible. No properly constructed harmonic continuity can be wrong in backward motion. Some composers without training in harmony (for example, Modest Moussorgsky) as v,ell as beginners, due to inadequate study, confuse the positive and the negative forms in writing their harmonic progressions. The resulting effect of such music is a vague tonaltty. The admire rs of Moussorgsky consider such styl� a virtue (in Moussorgsky's case it is about half-and-half positive and negative), and do not realize that all the incompetent students of a -harmony course incompetently taught possess full command over such style. t. • 8" Lesson LXVI. A, Diatonic Progressions (Positive Form) Expansions of the original Harmony Scale produce the Derivative Harmony Scales. The original HS and its expansions form the Diatonic Cycles. Diatonic (or Tonal) Cycles repre sent all the funda­ mental chord progressions. There are three Tonal Cycles in the Positive Form for the seven-unit scales. The First Cycle, or Cycle of the Third (C 3 ) , corresponds to HSE0; the Second Cycle, or Cycle of the Fifth (Cs-), corresponds to HSE, ; the Third Cycle, or the Cycle of the Seventh (C7), corresponds to H-SE ., Beyond 2 these expansions of HS lies the Negative Form o f Diatonic Pro gressions. ,,, In addition to both forms of progressions, there may be changes in a chord pertaining to the same root (axis). modified S of the Connections of an S with its s:ime root will be considered a Zero Cycle (co) • • In the follov1ing table notes are used merely for convenience: they indicate the sequepce of roots; their octave position was dictated by purely raelodic • considerations and by the neces.sity to moderate the range• • • • The respective interva·ls. .represening ' Cycles must be constructed downward for the Positive Form, regardless of their actual position on the musical staff. Figure II. Diatonic Cycles (Positive Form) •• • • • Cadences: Starting Ending Cycle of the Third·· (C3 ) . . �3 ..... I - • •. I .. c Starting . .. ' .... .. ,----; � - -. ,. • 0 0 •• ' · Starting ; Ci c �I = Ending • Combined c '. Combined --- ·--- In the above table·arrows indicate . cadences of the �espective cycles • Cadences consist of the axis-chord moving into its adjac��t chord and back. It is interesting to note, that what is usually kno\m as Plagal Cadences are the Star.ting Cadences and.that Cadences known as Authentic are the Ending Cadences. 0 Combined • Ending Z! The immediate seque11ce of Starting and • I - • 10. Ending Cadences produces Combined Cadences (the axis­ chord is omitted in the middle ) . Progressions of constant tonal Cycles (C3, or Cs, or C ., cons t·. ) produce a sequence of seven chords each appearing once and none repeating itself. The repetition of the axis-chord either completes the Cycle or star· t s a new one. The addition of Cadences to the Cycles is optional, as Cycles are self-sufficient. Considering constant Cycles as a form of Monomial Progressions, we can devise Binomia.l and Trinomial Progressions by assigning a sequence pf two or three Cycles at a time. In Binomial Progressior1s each chord appears twice and in a different combination with the preceding and the following chord� Thus, a complete Binomial Cycle in a seven-unit scale consists of 2 x 7 = 14 chords. Figure III. Binomial cycles Gs- + C3 Cs- + c1 (please see next page) , • • 11 .. • 2: - ... . :; -· - • ?= 9: � $ ... - !:? !) - ..... • - c, - ,.., '!: ,..,- - ,... - 0 . ' - ,... ...., - ;::;: � - e + Cr - ,... ., ,;; 0 .., I If • - ;;, � � - .. ,, C7 + C3 C7 p C � ., + C7 .,, -= ,... . � $ ,... • c,. $ !!:- + C7 .c� + • • c.- + c, • • ).: c ., .. e - e � :: ,. ;;, • � ...- - ( -;: a • '9, :: Q '; �- $ - ,... If I I i • $ ¢: 9 ... "2 • I • • • Q l II z; • • In Trinomial Progressions each chord appears three times and in a different combination with the preceding and . the following chord. Thus, a complete Trinomial Cycle in a seven-unit scale consists of 3 x 7 � 21 chord. t. • 12. Figure IV. TrinomiaJ Cycles C3 + C� + C1 c, + c, + c., C3 + C1 + Cf Gs: + CJ ± C7 - • {9: 12? C7 + c, + c� .... _. - : 9 l a G - C. � :2 >::::: !: :: Cs + C7 + C3 :::: a \9: s, Cs- + C 1 + C .,, 0 Q � s, ::::; C. C � C1 + CS' + C :,, =- 1'1=��� � � - 0 ..... ,;; ,., .., 0 ;, -? .., C. 9 0 - - ,::;; � � - ,,::: � -- 2 2 -- C $ - C :::::: 2 :;:;;; e - ,.. - ::: ,e:! � ,;;;; � � 11 - C ,... - � C7 ± Cs: + c, ,... ,.., -- c.,- + c, + c, $ s 9 � � - C. ,;, � � ..... >::::: � -- -- -- I 1i --- .....- ,::: ,0 a l I � � � � $ � � � IJ ,, I It • No. 230 Loose Leaf 12 Stave Style - Standard Punch • ,, 13 . • Lesson LXVII. Both Binomial and Trinomial Cycles produce the ultimate variety combined with the absolute • consistency of the character (style) of harmonic progressions. Being perfect in this respect they are of little use when a personal selection of character becomes a paramount factor. In order to produce an individual style of harmonic progressions, it is necessary to use a selective continuity of Cycles. This can be accom­ plished by means o f the Coefficients of Recurrence applied to a selected combination of Cycles. A combination of Cycles can be either a Binomial or a Groups producing coefficients of Tr,inomial. recurrence can be Binomial, Trinomial or Polynomial. t. The materials for these can be fo und in the Theory of Rhythm. Rhythmic resultants of different types and their variations provide various groups which can be used as coefficients of rec�rrence. Distributive Power-Groups as well as the different Series of Growth . and Acceleration can be used for the same purpose • • • 14. Figur�- Y� Binomial Cycles, Binomial Coefficients Cycles: C 3 + Cr; Coeffic ients: 2+1 I�: -• - -,; ,-;;;, � � � ,-::; ,:; � � .0 _, - .a C, ';! = 3t; Synchronized Cycles: 2C 3 +Cr ; 3 x 7 = 21 chords 0 9 0 .,, >• - .,,:! -s,. � ll \ Cycles: Cr + C7; Coefficients: 3+2 = 5t·' Synchror1ized Cyclos: 5 x 7 = 35 chords • - Q - - II - ... C Binomial Cycles, Coefficient-Groups with the nurober of terms divisible b 2. J9 • Cycles: c1 + C 3; Coefficients: r 4+3 = 3+1+2+2+1+3 = 12 t Synchronized Cycles: 3C7 + C 3 + 2C7 + 2c, + C1 + 3C3; 12x7= 84 chords ... - 0 - - � � 0 $9 - ,;; l!W-�!1,..:.�f .,, ;,;, - 9 � - 0 ,.. .., ..,. ,;; ..- ,: ,.., ., ,.. .,, � -..... - ,.. .,, :, No. 230 Loose Leaf �2 Stave Style - Standard Punch $ � :'2 1j s$ � � � - .., .. .. l • (Figure v, cont.) • 15. Binomial Cycles, Coefficient-Groups producing interference with the Cycles (not divisible by 2) Cycles: C.r + Coefficients: 3 + 1 + 2 C3 = 6t Synchronized Cycles: 3C; + c3 + 2Cr + 3C 3 + C; + 2c, Synchronized coefficients: 6t x 2 = 12t; 12 x 7 - -- - s 84 chords - .... Trinomial Cycles, Trinomial Coefficients Cycles: C3 +Cr + C7 Coefficients: 4 + 1 + 3 Synchronized Cycles: 4C3 + Cr + 3C7; .., 0 _, C 0 i:, - ,:, ·-----------. ���,.:.�( No. - ,., c >;;- - .... 8 x 7 - 56 chords - 9 - 230 Loose Leaf 12 Stave Style • Standard Punch - .... = 8t • 16. (Figure V, Cont.) Trinomial Cycles, Co efficient-G-roups with tl1e number of terms divisible by 3. Cycles: C 7 + c�., + Ce; � Coefficients: r5+2 = 2+2+1+1+2+2 = lOt . Synchronized Cycles: 2c7 + 2G3 +Cs-+ C 7 + 2C3 + 2CG; 10x7=70 chords - - - - .... ..... - ...- ....- - $ ,... ... 0 • Trino mial Cycles, Coe fficient-Groups �roducing interference with the Cycles (not divisible by 3). Cycles: c7 +C s- + C.) ; Coefficients: 3 + 1 = 4t • Synchronized Cycles: 3C7 + C� + 3C3 + c1 + 3C� + CJ Synchronized Coefficients: 4t x 3 = 12t; - •.. '--' 19: � 'J A ., 0 ·-·=-=-1"•• US!iCl0H HRA.l""llolD A ":l 1..1 ,.. 12 x 7 = 84 chords - ' A ...... QO 9 oa ..,, A C ,.._ ,;: a0 , ... o. 230 Loose Leaf 12 Stave Style - Standard Punch ;::1 ft o- - 9 11 • 17. The style of harmonic progressions depends entirely on the form of cycles employed. No composer confines himself to one definite cycle, yet it is the predominance of a certain cycle over others that makes his music immediately recognizable to the listener. In one case it may be that the beginning of a progression is expressed through the cadences of a certain cycle, in another case it may be a prominent coefficient group that makes such music sound distinctly different from the other. The style of harmonic progressions can be defined as a definite form of Selectiv� Cycles. Both the combination of cycles (their sequence) and the coefficient group determining their recurrence are the factors of a style of harmonic progressions. t. \ • C • • 18. u Lesson LXVIII. There is much to be said about the historical development of the cycles, as there are already some wrong notions established in this field. Though the common belief is that the progressions from the tonic to the dominant and back to the tonic (ending cadence in c ), is the foundation 5 of diatonic harmony, historical, evidence, as well as­ mathematical analysis prove to the oontrary. During the course of centuries of European musical history, parallel to the developwent of counterpoint, there was an awakening of harmonic consciousness. - can be traced, in its apparent XV . Century A.D. The latter forms, back to At that time harmony meant concord, an agreea.ble, consonant, stabilized sonority of • several voices- simultaneously sustained. Concordant prog_ressio11s could be accomplished therefore throug h consonant chords moving in consonant relations. Obviously such progressions require common tones, and the latter can be expressed as c3• As the -tonality, i.e., an organized progression of tonal cycles was at that time in the state of fermentation, it is natural to expect the cycle of the third to appear in both u positive (c3) arid negative (C-3) form. The following are a fevw illustrations tak en from the music of XV and XVI Centuries. C. • 19. Figure VI. C. '2. .� \ , -- ' -.J - -• • .....'-' 00 • • • • Opening of 11Ave Regina Coelorumn - Leonel Power, c. 1460 0, ..... " • ' - - • . . • - - - . .... -e,k. • . . . ,._.,_ '. ' ·- .-. • l ' 1 .. - "Benedicta Tu 11 MS. Pepysian 1236, Madrigal Collection, Cambridge, c • 1460 • �- I"I • t· r e • I' • , .., , ..... l -e- r .A. - � 11 Deutsches C3 I - - I • ... ,, ' .. - u r -. - '. ." . . Lied" - Adam von Fulda (1470) . ... • - e,k • � - • . .. ' - . - " , - • " Julio Cacchini (1550-1618) I....,, r, • ' .. • •' � - - ,.., q� '. ,, I f , l==sic==1·=J1£1 a BRAND ...- " � j n .. ..e,h.. ..... - - '" ., -<- �. [ , � I ' t , � I - . 3 ., ' " - t t ,. ..I < • No. 230 Loose Leaf 12 Stave Style - Standard Punch • -- . rI' , , - ' • 20. Cycle of the Seventh, on the other hand, has a purely contrapuntal derivation. When the two leading tones (the upper and the lower) move in cadence into their a respective tonics (like b.--+c and d > c) by means of contra.ry motion in two voices, we • , obtain the ending cadence o f c 7• Further development of· the third part was undoubtedly 11ecessitated by the d€sire for fuller sonorities. This introduced an extra tone (f in a chord of b) with which tones form S(6) i.e., a the remaining third-sixth-chord or a sixth­ chord, the first inversion of the root-chord: S(5). Figure VII. Ir- ' .... ..,; .- _-e- 0 e • domi11ance of It is only natural to ex�ect the pre­ the c7 in contrapuntal music. Cadences as in F!tgure VII are most standardized in the XIII and XIV Century European music. Machault (1300-1377) 11 See Guillaume de Mass for the Coronation o f Cha rles V11 (phonograph recording published by the Gramophone Shop).. u • • • 21 .. The appearance of the cycle of the fifth date, when c 3 and c 7 I of fer the following hypothesis must be referred to a later were already in use. of the origin of C6. The positive form might have • occurred as a pedal point development, where by sustaining the tonic and changing the remaining two tones to their leading tones, the sequence would represent c,. Another interpretation of the origin of Cs is the one which this system of Harmony is based • upon, i.e., omission of intermediate links in a series • This principle ties up musical harmony with harmonic structure of crystals, as used in crystallographic analysis. Figure VIII. Cs .. • The origin of the negative form of the cycle of th e fifth (C-5) is due to the desire of acquiring a concord supporting a leading tone. be a leading tone in the scale of c. Let b The most con­ cordant combination of tones in the pre-Bach - time, i.e., in the •mean temperament (the tuning system v. officially recognized in Europe before the advent of • 22. equal temperament) , harmonizing the tone b was the G-chord (g, b, d) . But when movi ng from G-chord to C-chord the form of the cycle is positive. In are the beginning and the ending cadences. Compare reality both forms, the positive and the ne gative, Figure IX with Figure VIII. Figure IX . • , • • • , • • • ,, J O SE P H S C H IL L I N G ER C O R R E S P O N D E N C E C O U R S E Subject: Music With: Dr. Jerome Gross Lesson LXIX. • The development of harmonic progressions · in the European music of the last three centuries c an be easily traced b ack to their sources. . The style of every composer is hybrid , yet the quantitative predominance of certain ingredients (like the cycles appearing with the different coefficients of recurrence) produces individual characteristics. In the following exposition I will confine the concept of nstylen to harmonic progressions in the diatonic system. I. Richard Wagner was the greatest representa­ tive of C 3 in th e XIX Century. This statement is backed by the statistical analysis of tonal cycles in • • his works as compared to his contemporaries and predecessors: c 5 v,as the universal vogue of a whole century preceding Wagner� In fact, it is not necessary to analyze all works of Wagner. The most characteristic progressions may be found at the beginnings of his preludes to musical dramas and also cadences. :in the various The beginnings of major works of any L • • ,.. • 2. • composers are important, fo r the reason that they composer. The importance of cadences as determinants cannot b e casual: it is the "calling card" of a of harmonic styles was stressed upon by our contempo­ rary, Alfredo Casella, in his paper, "Evolution of Harmony from the Authentic Cadence". Wagner, being German and intentionally ' • Germanic composer, undoubtedly has done some research of tt1e earlier German music, as he intended to deal with the subjects of German mythology, in which he was well versed. The XV Century German music discloses such an abundance of C 3 , that it is only natural to expect the influence of such an authentic source of Germanic mus ic upon Wagner's creations. In his time, Wagner 's harmonic progressions sounded revolutionary ' because many things were forgotten in four hundred years, and archaic acquired a flavor of modernistic. So far as the development of diatonic progressions in Wagner ' s music appears to the unbiased analyst, the whole mission of Wagner 's life was to develop a consistent combined cadence in C 3 • Starting with an early work like "Tannhauser", we find that already the very beginning of the Overture is typical in this respect. .. • 3. Figure x. • • " � ,I - -I • -· --, I • J... !-.. i; • , t � "?iwl ,. "' • Later on v,re find more extended pro­ gressions of C 3 , as in the Aria of Wolfram von • Eschenbach (the scene of Minnesingers contest): Figure XJ. • I • � C � ....... ·� � � - I l � _,,, • C3 than -�I C� C. � l • ..-11 "Logengrin" is even more abundant with 11 Tannhauser 11 • In "Farewell to S wan", as in many other places of the same opera, vre find the • ,. • 4. characteristic back-and-forth fluctuation: C 3+C-3. figure XJI. • • ,., ... r C7 0 (l • • • rJ �• • • .,. r .. w Forming his cadences, Wagner paid some­ time his tributes to the dominating "dominant'' of This produced combined b,ybrid cadences, which are characteristic of "Lohengrin". The first part of such a cadence is the beginning cadence in Ca, while the second part is the ending cadence in c5 : I - VI - V - I . Figure XIII . . ... I • ·- = i • • • t ... � ,I• , I • J. j_ • ' • • • • • 5. Dealing with other type s of progressions than diatonic in the cour se of his career, Wagner came back to diatonic purity in its complete and consistent form in his last work "Parsifal". The beginning of the "Prelude to Act I" reveals that the composer came to the realization of the combined cadence of C 3 : I - VI - III; Figure XIV. :t ., .. - ,' 4 J I ,, I - . ..... .., - � 3ZI �•1I -- ... A. .0. "' -e- . i - • The more extensive sequences of C3 are: I - VI - IV - II; Figur� xv_. • r-. I . , :l( . I· "- , u • • C3 c, C 3 ] • � - �- · 1� ,:: ., I Ii • j -ll =1t . • • 6• .A nd the complete combined cadence ( nProcession of the Noblemen of Graal"): I - VI - III - I •. • Figure XVI. -. " ' ,, ., I' - ]• • - ::c v, I � ., - ,. The second half of the XVIII Century and the first half of t:t1e XIX Century cover the period of the hegemony of the dominant and c5 in all its aspects in general. The latter are: continuous progressions o� c5; starting, ending and combined cadences ( I - IV - I; I - V - I; I - IV - V - I). The main sources of music possessing these characteristics are: the Italian Opera and tr1e Viennese School.. To the first belong : Monteverdi, · scarlatti, Pergolesi, Rossini, Verdi. The second is represented by Dittersdorf, Hayd n, Mozart, Beethoven, Schubert. Today th is style disintegrated into the least imaginative creations in the field of popular music. Nevertheless it is the stronghold of harmony in the educational music institutions. • • 7. Here are a few illustrations of c 5 style in the early Sonatas for the Piano by Ludwig van Beethoven: Sonata Op. 7, Largo; Sonata Op. 13, Adagio Cantabile. , ...- - I • I -- - . r·"' • .. � r"') � , ·� • • - • , l . I , . c -...... - • • � 7• � t/ - I• --"V '1 ,\ I ] - . I L• TI . • - � :i • ' • • 1w "4lw � F ,I •• , •... .. -•... - I r •. II �I .;, - ..- - I I • - C -s ') ,, Figure-XVII. • � • • j ] I ' Any number of illustrations can be found in Mozart 's and Beethoven 's symphonies, particularly in the conclusive p�rts of the last movements. Assuming that the historical origin of the cycle of the seventh can be traced back to contrapuntal cadences, it would be only logical to expect the evidence of c7 in the works of the great contrapuntalists. • 8. I choose for tl1e illustration of c 7, as characteristic starting progressions, some of the well knovm Prelud es to Fugues taken from the First Volume of 11 Well Tempered Clavichord" by Johann Sebastian Bach: Prelude I; Prelude III; Prelude V. Fig_ure XVIII. • ,. ... I /2 / � I • . r- .,. '1-- I ,.:,- �-- ...l- • • - ::, - , . • C7 I ::r - _1?. - ::, J; -- .. J ., I , •I -I --j •' I -., r-E!f� C'1 ,i ' (, '"' I: = ::s: • • ., � - I ,.I I • ' • • ] c, I ::=:� • • :-1i .. • � - - "'1 • :& - - � I -• • . � ·- • I • I I , • • Bach ' s famous "Chiacopna in D-minor n for r Violin, discloses the same characteristics, as the first chord is d and the s econd chord is e, which makes • -- , • 9. A consistent and ripe style of diatoni c progressi ons corresponds to a consis tent use of one form, either positive or negati ve and not to an indiscriminate mixture of both. Many theorists con£use • the hybrid of positive and negati ve forms wi th modal progressi ons, which the theorists have never defined clearly. In reality, modal progressi ons are in no respect different from tonal progressions, except for the scale structure. Both types (tonal and modal) can be eit her positive, or negative, or hybrid. Modes can be obtained by the direct change of key si gnatures, as descri bed in the "Theory of Pitch Scales 11 (transposi t i on to one axi s) . Here is an example, typical of Moussorgsky, from "Bori s Godounov 11 (opera): Figure J{IX. I ' I I I• /_ " I I � - � - • C � l • If . � � -� / ' � ] •► . .!. J .,.. e . � ' j � • •� .L � ts • ' I •� • J .!.. � � � • In the a bove example the mode (scale) i s Cd5 , the fi fth derivati ve scale of the Natural Major i n the key of c, known as Aeolian mode, while the progression of tonal cycles i s a hybrid of positive and negative forms. / • 10 • Lesson LXX. • B, Transformations of S(52 • . · In the traditional courses of harmony the problems of progressio�s and voice-leading are inseparable. Each pair of chords is described as sequence and a f orm of voice-leading . Thus each case becomes an individual case where the movement of voices is described in terms of melodic intervals (like: a fifth down, a second up, a leap in soprano, a sustained tone in alto, etc.). No person of normal mentality can ever memorize all the rules and exceptions offered in such courses. In addition to this unsatisfactory form of presentation of the subject of harmony, one finds out very soon that the abundance of rules covers a very limited material (mostly the harmony of the second-rateL XVIII Century European composers). The main defect of the existing theories of harmony is in the use of the descriptive method. Each case is analyz.ed apart from other cases and without an� general underlying principles. The mathematical treatment of this subject discloses the general properties of the positions and movements of the voices in terms of transformations of the chor dal fun ctions. Any chord, n-0 matter of what structure, from a mathematical standpoint, is an assemblage • of • • • 11. pitch units, or a (elements). gro up of conjugated functions These functions are the different pitch­ units distributed in each group, assemblage or chord according to the different number o f voices (parts) and the intervals between the latter. In groups with three · functio ns known as three-part structures (S = 3p) the fu nction s are a, b a nd c. These functi ons behave through general forms o f transformations and not throug h any musical specifications. As in thi s branch we are deal ing with so­ called fo ur-part harmony, we have to define the meaning • o f this expression more precisely • When an S(5) constitutes a chord-structure, the functions o f the chord are: the root, the third 8.Ild the fifth or 1, 3 and 5. In their general form they correspond to a, b and c, i. e. , a = l, b L = 3, and c = 5. The bass of such harmony is a constant root-tone, i.e,' co nst. 1 or co nst. a. Thvs the transformation of fu nctions affects all parts except the bass. Here, therefore,.we are dealin g with the gro ups consisting of three functions. formation.s: Such gro ups have two fundamental trans­ • • • 12. (1) clockwise (Z, ) a.nd (2) counterclockwise (� ) The clockwise transformation is : The counterclockwise transformation l.S : Each of these transform ations has two meanings: the first to be read -a is followed by b • b n C tt " n " C " a for the ,.-� and f:__.. a is followed by c C n " " b " a b " for the � • - ,,, d·iscloses the mechanism of the !)OSi tions of a chord; the second to be read -a transforms into b b C " " C " n a for the ;::! and a transf orrns into c C " " b " n for the 11:: """ ., � b a • 13. constitutes the forms of vo ice-leadi ng, Positions. The different positions of S (5) = 1, 3, 5 can be obtained by constructing the chordal functions dovmward from each phase of the transformations. a b c b c a c a b and a c b C b a b a c ......___ ____,➔ Substituting 1, 3 , 5 for a, b, c, we obtain 1 3 3 5 5 1 5 1 and 3 1 5 5 3 3 1 Ir;; ...... � • 3 1 - .., The cloclcwise positions are commonly _known as open, and the counterclockwise as close. Here are the positions for S(5) =4 + 3 = Bass is added for the doubling o f the = c - e - g. root. • • • • Figure 14. XX. P o s i +i o ,.,., 6 • \ ..... ,. I .,, ..• , • -- 4 - •, V .� • ', e rJ .' I V - •• - .! . .; ..... � ' I � 15" 'S' <• C g )a Counterclockwise form gives the • following reading: C ➔e g )C e ?a Let us take c5 in the sam e scale. The chords are: C = c - e - g and F = f - a - c. r ➔ C e F .,,,, ', a )c • C- ) C -- g -=, a -" ' IJ e -- ) f ('I - ,_' ;:, ... - - ➔ Let us take c7 in the same scale. chords are : C = c - e - g and D = d - f - a. The • • 17. r C )f e )a �d g ., "' ·• - .. -- r,. - .' I • Both forms of F,. are acceptable in r' � • this case, as the intervals in bo th directions are nearly equidistant. Jt:. • � C ,a e )d g- ) f -� 0 I u t. • • • u 18. Lesson LXXI. Each tonal cycle permits a continuous progression through one form of transformation. In the following table const. 1 in the bass is added . Apostrophies indicate an octave variation when the extension of' range becomes impractical. In c7 both directions are combined, offering the most practical form for the range. (please see Figure XXI. following page) I. • • • • 0 • - � I Ca ic: _, .' - $ , - I 0 A . 3 , . ... - .' . I'- , "... I - - � '' .. J - ,. I• •• .,., .... . .., .. . - . .. • .., -! r 0 � 0 - ;, - , .. � - - r, . � -- � , � - . - ..... - .., - -6- I - . - • - , .... - - - .. :;,s .... ..., •• ,, -.., '"' • -"?!" .,• ��j!'.��f) 1 , � ' .... C I:;) ,... '' .. .;.. :.> --.... ,:, •. - ,, - --- - :I .... , -- - ·- -.... , - , 0 v, • - --.• - r ...... -- - . . � !.- ,..... No. 230 Loose Leaf 12 Stave Style - Standard Pun r • - -,� ' - - - . - .... • - -- � j ro - � -- ? • - - • .... , - ,c i.. � � � -... .... ,, . C - , - j J • -& - .• :-=,... ¾ - - , ' • C5 .., >, , , ... - "' ' -- -. . ,... ,� - ., C l >5 b· )b 3 �3 C )a 5= ) 1 It is ·best to have 3 in the upper voice for such purposes, as in some positions voices cross otherwise. Function 3 from close to open followi ng chord. Reverse the procedure from open to position moves upward to the function 3 of the close. L • • 21. Figure XXII., Const. 3 Transformation • .. � � '" � v i ,,:,,- .. ,,. , I • •• , - -• I ,� ,......,-- ,r:� ,' -- -· -. � ,'f V i1 -, . -r .' ' :: I ➔-9-f ..•·- •• •I F -• - • r ;,. 0. ... ..... ,., I �-a- •S° ' V .' .... � _ -, '=• , ... I• � .,, 01' ..... -, < .... ..Q. ..., ,. ,. ·� .. :;.. I s- - Continuous application of const. 3 transformation produces a consistent variation of the 2 and the � positions, regardless of the sequence of tonal cycles. The following table offers continuous progressions through canst. cycles and const� 3 transformation. Fig ure XXIII, (please see next page) • • C3 Const .. 3 "' �· ,' .. •• " . � " ,..... I •.. ¼ .: ,. - ,, - ,, - -..., - - --.... -,. C -..... ..... ,, ..... ,. - ' � - - _, , - ' . ,., • ..... fr � - ·- - - ....... -v -- •a - c-..,, - - -.., ' ., _,, 0.., •• • - Const. 3 .,, -5- . ..... r ' t - • .... .., ..,; ') .£. ,.., ,_ • ' .' -- - .. -- ..... � , 22. Figure �I I I . Constant 3 Transforma tions • - :a. ..... ·- � ..0.. ? � • - •' - . c7 Const. 3 � '! � ... ,.... I,. . - .a.. .., - - � , . .: ,' a-.... • ,. - -- 'l -- - ') , - � • .... -5- - .,, • i':l�i!�W:( No. 230 Loose Leaf 12 Stave Style - Standard Punch - ,..., C . ,. {l • ---,.., • •:, � - , 0 - ,. - - ..25.Cr • • J OSE P H S C H I L L IN G ER C ORRES P O N D ENCE CO UR S E Subject: 1viusic With: Dr. Jerome Gross Lesson LXXII, There are four forms of relationship between the cycles and the transformations with regard to the variability of both. (1) const. -cycle, const.-transformation; (2) const.-cycle, variable transformation; • (3) variable cycle , const�-trans formation; (4) variable cycle, variable transformation. The forms of transformation produce their own periodic groups, which mpy be superimposed on the groups of cycles. Monomial forms of transformations (const. transformations):·- ( 1) (2) (3) const.- 3 Binomial forws of transformations: (1) � Here Const. 3 is excluded on account of tl1e crossing of inner voices. • Coefficients of recurrence being applied to the forms of transformations produce selective transformati9n-groups. L • • 2. For example: 2 ; ! + ! ; ; 3 � ; + 2 ;� ; = ! + 2 �; ; 4 i� + � + 2 ; °! + �; + -;.. '= "' rJ 4 -':: .. <' 3 � ! + 2� � + ➔ Jc. · 5 -J 2 .. ,, ; ., ,_, + i,.. + 3 .. , + t=:-- + + 2 ;.: + 3 ....� + �, + «"' 4r 'j + 2 l:;; ,. .... ,.... ➔ + I:: "" . + 8 .. ,, ; f"+ .. ..., .c. p .... , Though the groups of tonal cycles, as well ...1 as the forms of transformations, may be chosen freely with the writing of each sub sequent chord, rhythmic planning of both guarantees a greater regularity and, therefore, greater unity of s tyle. . Examples of variable transformations applied to constant tonal cycles. Figure XXIV. ' - • ' ,J .l:1. • c, ,... � const ., 4 , - • , � ,. .. - -' + 2 .t T � ,_ <> • .....• -� - ..,.., , •• u .. • -·. �.,, - - . ' . , ' . .., - ' -,SI ' • ,. .., . ,: - ,. - ' -€ ..., - .,;. ,J ' + .. ? + 2 � t C r const. 3. ..j;.,' + ,,. ,- .,, nI • • .. - ,, • .a. .., - - - • - ls t. , ·- + 2 .=: ,. + .. > II: � 0 ,.,. � • • - .... " _, r' , -9- .... - - ..... ... ic::' added for the ending. -e,.... -� ' ' 0 ,' •• 3 ., '' -,:, --- � • • • Examples of variable transformations applied to variable tonal cycles. Figur� XXV . -- CO + C1 + C 3 ; 2 � + �� r"\ I • ' ' ..a.. • ,• • ,' - . a .... , ..., -· ...., - - - - ( ..QI. 6 � -. • • . . - . � - • -- • . • - - .... 0 • -- n - . • . - - ; 4� .. + 2 ;3 + -,.. � I ,., . - .. (' • ... "' .. . .. - - ..%1. • ���r-:.�ff � .- · , •• - � -......- . - -- .... 0 - - � C - - r .., � - -- ;;, ' .... r� ..... . . ,... .... - - - . r -..... -,�- - • ', --St- - -... - � - No. 230 Loose Leef 12 Steve Style - Standard Punth • - .c • --....� . ..... • >"Cz . - - - C - .., . - � ,, ➔ . - - • 4. All forms of harmonic continuity, due to their property of redistribution, modal variability and convertibility, are subject to the following modifications: (1) Placement of the voice representing constant function, and originally appearing in the bass, into any other voice, •l. . e . , • tenor, alto or soprano . There are four forms of such distribution: s A T B s A T B S· A T B s A T B Red letters re present the voice functioning - as const .. 1. (2) General redistribution (vertical permuta­ tions) of all voices according to 24 L variations of 4 elements. (3) Geometrical inversions : @ , @, © and @ for any or all forms of distribution of tlie four voices . (4) Modal variation by means of modal trans­ positj.on, i.e., direct change of key signature, without replacing the notes on the staff. , • Original �. � - ,..,. , . . ' I> ,, - • ., ... A ..., _, � � ,. -..... ..,, •• - i �J • a 0 -.... • ... ,.. -e- , r,. - 7 • • • 5. Example of variations . Figu;re XXVI. • ,... . ...,, ·, • - - � . ·- -'?Ji • .' .... • - ,. • � • -8 JS. ""' - � -?-- _, . ,� � - . . .... 0 0 .- I , - flt -..,, 0 .., � ....0.. ;..; --.- -...... r; ,, _ s -..... ,.... ,. .. -.-- , .,,..., � - ,_. - � ..... -� A, ,.... - .., � , .,,, v . "" V .t".• - "- , , _ � � -- - - . . r, - - ,. V C - � ,;, ,. ,' r> - - � -(:r ,J. '. ,,. -� I .. r' ,. I• -- , � - I•� C' r Jl -::::=-:;:== 1•a�•­ =s=ic ANo -<-C • - -- --6- - ..., -- ,.- Original : @ --s TC ,::J, - (3) ' -:ii •• -& ·� .' I ·i .. ,_ .s;.. ., ,. � � ,-. .... � - - - . � ,,., r,; - -- • .... r., -- - � I., � I:h • • No. 230 Loose Leaf 12 Stave Style - Standard Punch ... � • . - - .e, ..,,, .., - • • I , -- r ... II-rA - - - � - .., I • - � . . " - - , _, -e•• . • I V - - ·- • (Fig. XXVI cont.) , \ • .·-.,I • I �- • .� Origi nal + 3 l;i • ,, .., • .' I - .. - • _. - .... . .. . ,., - ..Q... (4) - �· $ • -,.. • - ,, --- � • 6. , a - ,r,• -- -e- . - � .... • • - ·- .... • • '. •,. , � ... Original + (b \;, + r tf) - G roel ., minor: d3 \ - 7 -, .., 7 . ' \...._,, I' ·� .• , ... ,.... - ,... •-a . .. 2!! •_,, " , . ., • - � ,, == [.,.,usia·a BRA!"'IIIID - -- ' . � � - -- - - -e- -.... - -. -= -, • - �---,·� ,, .-. - -.., ·. , , No. 230 Loose Leaf 12 Stave Style - Standard Punch -- � '. . . .., � - -- ' - - - ' i • .a.. .., ... ... , -... --- � - - •, , ·- • • 7. tesson LXXIII. c. The Negative Fprm. As it was previ ously defined, the negative form of harmony can be obtained by direct reading of the positive form in position @ • Here, for the sake of clarity in the entire matter, I am offering some technical details which explain the theoretical side of the negative form. According to the definition given to the harmony scale in the negative form, we obtain the latter by means of further expansions of HS. In the positive form we have used : H SE0 (= C 3 ), HSE, = ( Cs ) and HSE2 (= C 7). Novv by further expanding HS, we acquire the cycles of the negative form: H SE 3 (= C - 7) , HSE = ( C - 5), HSE (= C - 3) . � � Figure x;xv_r�. (please see ne�t page) • • • ... • a. C - • •• C -J -e- Cadences Cycles 1 0 - > ,, II .Q, '\ :: � •• C e = R.= I 0 .. C C · =I JI ,..,. ::, ; ll ;s 0 -- . .' As chord-structures are built downward fro m a given pitch unit, such a pitch unit becomes L the root-tone of the negative structure : the negative root ( - 1) . All chord-structures of the negative form, according to the previ ous definition, derive from HS @ • Thus in order to construct a negative S (5), - it is necessary to take the next pitch-unit dovmwa rd, which becomes the negative third ( - 3) and the next tt T unit do wnv.rard fro m the latter, which bec omes the negative fifth ( - 5) . • • • • For example, starting from c as a - 1, we obtain a negative S (5), where a is - 3 and f is - 5. Figure )Q.CVIII . Natural C- Major. ,.. � """ / • / ' lo.. J �, -J Positions of chords, as they were expressed through transformations, remain identical in the negative form, providing they are constructed upv,ard. • In such a case, the addition of a cons t. 1 in the bass must be, strictly speaking, transferred to the soprano. Here is how a negative CS (5) would appear in its fo ur-part settings. Figure XXIX . • ... I ,, .· ' '' I• ' -� -T ,-, .0... � . , . - - -e- . . '� -'2 , .,.- -I -e--- • . . -3 -I • � - _, . ,. , � .....- - - '2 - I • • • • =; 10. If, u11der such cor1ditions, the chord were constructed downward, the reversal of ;;.! and reading would take place. Transformations as applied to voice­ leading possess the same reversibility : if everything is read downv.rard, the 'ic-..? and the ': ...., tra.nsformations correspond to the positive form, while in the upward ,+ Jr: reading the ,:::::� becomes the ...., � and vice-versa. Let us connect two chords in the negative cycle of the third: CS (5) + C 3 + ES (5). = C - a - f. ES (5) = -1, -3, -5 = e - c - a. CS (5) = -1, -3, -5 Figure XXX. I , ., ' ... --- �,; .. ...... I _-$ - - -- ., ,.__ - -- - • � - x -• , � • It is easy to see that in the upward reading chord C corresponds to F, and chord E corres�onds to A. Transposing this upward reading to C, we notice that th is progression is c ---) E. This proves the reversibility of tonal cycles and the correctness of reading the positive form of progressions in positi on @ , when the negati�e form L • • is desired. 11. The mixture of positive and negative forms i.n continuity does not change the situation, but merely reverses the characteristics of voice­ leading with regard to positive and negative forms. For example, C 3 in ;:-.! in the positive system produces two sustained comwon tones. In order to obtain an analogous pattern of vo ice-leading in C- 3 , it is necessary to reverse the transfor mation, i.e. , to use the : � form in this case. • • • • 12. Lesson LXXIV. II. Symmetric S, y stem. Diatonic harmony can be best defined as where chord-structures as well as chorda system •• progressions derive· from a given s�a�� - Structural consti tution of pi tch assemblages, known as chords, as well as the actual intonation of the sequences of root-tones, knovm as tonal .cycles, are enti rely conditioned by the structural constitution of the scale, which is the s:Jurce of intonation. Symmetric harmony is a system of pre­ • selected chord-structures and pre-selected chord progressions , one indepe ndent from an other. In the symmetr ic system of harmony scale is the result, the conseguence of chords in motion. The selection of L intonati on for structures is independent from the selection of intonati on for the progressions. A. Structures of 8(5). In this course of harmony only such three-part structures vvill be used, whic� satisfy • the definition of "special theory of harmonyn. The :ingredients of chord-structures here are limited to 3 and 4 semitones r Under such limitations only four forms .of_ 8 (5) are possib le. It should be remembered, though, that the number of all possible • • • three-part structures would amount to 55, which is the general number of three-unit scales from one axis. Table of S(5) s , (5) = 4 + 3, knovm as major triad; S2 (5) = 3 + 4, kno,m as minor triad; ·s 3 (5) = 4 + 4, known as augme nted triad; S'f (5) = 3 + 3, known as diminished triad. " s,(s) -J - Figure XXXI . - -i- - i:;s M . I .,, - .- � So long as S(5) will be the only structure I.. for th e present use, we shall simplify the abov e • expressions to the • following form: Whatever th e ch ord-progression may be, structural constitution of chords appe aring in such progression may be either constant or variable � Constant structures will be considere d as monomial progressions of structures, while the variable structur es will be considered as binomial, trinomial and polynomial structural groups. • • • 14. Monomial forms of S(5) • ••• •••• •••• •••• Total: 4 forms Binomial forms of 8_(5) Sa + S "I s , + Sa s , + s'f , 6 combir1ations, 2 permutations each. • Total : 12 forms • Trinomial forms of 8(5) • S ' + S ' + S3 s, + s, + s ,.. • S., + Sa + S a s , + s,, + s� 12 combinations, 3 permutations each. Total: 36 forms • • 15. s, + S2 + Sa s, + S2 + Sy S2 + Sa + S &f • • 4 combinations, 6 permutations each. Total : 24 form s . The total of all trinomials: 36 + 24 = 60. S ' + S t + S t + S2 Quadrinomial forros of 8(5). • L 12 combinations, 4 permutations each. Total: 48 forms • 6 combinations , 6 permutations each. Total: 36 forms • • 16. s , + S, + S2 + Sa s , + s , + S 2 + S� s , + s , + Sa + Sy s , + S 2 + S 2 + Sa s , + S2 + S2 + � S , + Sa + S 3 + S� s , + S 2 + Sa + Sa • s , + s2 + s� + s� s , + S 3 + S� + 8¥ • 12 combinations, 12 permutations each. • • Total : 144 forms • • 1 combination, 24 permutations. Total: 24 forms. • The total of all quadrinomials: 48 + 36 + 144 + 24 = 252. In addition to all these fundamental forms of the groups of S (5), which represent a 11eutral harmonic continuity of str�ctures, there are groups with coefficients of recurrence, which represent a selective harmonic • • 17. continuity of structures. individual selection. The latter are subject to Any rhythmic groups may be used as coefficients of recurrence. Examples (1) 2S, + Sa (2) 3Sa + S2 (3) 3S , + 2Sa + S2· (4) 2S 2 + s , + S2 + 2S 1 (6) 3S f + S2 + 2S ' (8 ) 2S 1 + S2 + S, + S2 + S, + S2 + 2S 1 + 2Sz + s , · + S2 +I.· S 1 + (9) + S2 + S , + 282 \ 4S, + 2s 2 + 2s� + 2s , + s 2 + s� + 2s , + s 2 + s� (10) • B. Symmetric Progressions . • Symmetric · zero CYcle (C0 ) • A group of chords with a common root-tone but positions and variable structures produces with variable • a symmetric zero cycle (C0 ). • • 18. Such a group may be an independent form of harmonic continuity, as wel l as a portion of other symmetric forms of harmonic continuity. Coefficients of recurrence in the groups of structures, when used in a continuity of C0 , acquire the following meaning: a structure with a coefficient greater than one changes its positions, The change of until the next structure appears. structure requires the preservation of the position of the chord. This can be expressed as a form of interdependence of structures and their positions in the C0 : position var. S const• • S var. ------- position const. = s, + s, • t. For instance, in a case of 3S, + Sa + 2$ 2 = + s, + Sa + S 2 + S 2 , the constant and variable positions appear as var. var. s , + s' + s, follows: con st. con st. + + Sa • • Ex.amples of harm onic cont inuity in C 0 • Figure XXXII • • .. I .· • ., ,... • - ,, .... .... � z� ,., � • • . h4�-... - � , , 7. ,• . '-" ".,.. • J .a. ..... ' , , - ii � V •L -& -- ���r.:.�f � . , -e- . • . . --. � ..,, .. .... " ... -, - ... ..0.. .., -tr • -e- -e.. • ': i -e- . • � . I, 0 .0.. ... II., : -..,, • - • ....,. 9 • �-• • V - Ii - r • ,. -& I •• • • 0L r - ... -e- -9- , I• r� . , I • -9 . .§.. - I ,. ' ,. ,.. - - " .a. ... . .,,.. - � • ..... - - ..0. , • (' -9- ...... -� . I . , - - • � ,� - ,_ .-. ..g.. ·� • .- • • 20. Lesson LXXV. Diatonic-Symmetric System of Harmony (Type II). • Diatonic-Symmetric system of harmony must satisfy the following two requirements: (1) all root-tones of the diatonic-symmetric system belong to one scale of the First Group; (2) all chord structures must be pre-selected ; they are not affected by the intona tion of scale formed by the root-tones. • In this system of h armony structur al groups must be superimposed upon the progressions of the root-tones belonging to one scale. This form of h armony has some advan tages over the Diatonic System ( to which I will refer as Type I). Like the diatonic system, the diatonic-symmetric system produces a united tonality, which is due to the structural unity of the scale. Unlike the diatonic system, the diatonic-symmetric system is not bound to use the structures wh ich are considered defective in the Equal Temper ament [ like S � (5) , for example ] , as the individual struc tures and the structur al groups . are a matter of free choice. Unlike the di atonic system, the di atonic- • • L • 21 . symmetric system has a greater variety of intonations, as the pre-selected structures unavoidably introduce new accidentals (alterations), 1,mich implies a modulatory character without destroying the unity of the tonality. Examples of Harmony TYJ?e ;r. Figure XXXIII. (please see following pages) C • • • v ., , r .... I:} Pitch-scale : + - 0 ,;:: ? 0 �3 !'? - ,- . r,' •• -.... "£ . • . -.. '--" .. I.J ,.. -e- ·- .., • .' ::::I :..I • . - .., -.,, • ., _.., � . ' ' .' • 9 .... . �c r;.;. . •• �· - -. ...'. ., "'L ba ... t,J 0 + C3 + - -- ,., .... _,, - � q� - ..... . " - - ul - - '· � -- - • - -- -' . .., .J I 0 _. ,' � - .... .__ • • • � •• -- _. -fV Structural group: S , . + S 3 + 2S 2 ... •• t. : .. l' I ,,, '... - - .., L. • � � -.- - --- •�o - -r - - . ·- '. � • r- . . V ,. "7 .- -s,, ...: .- j ·- � . - I - et.: 0 i�'' ' Tonal cycles: 2C 3 + Cr •• c, I - -- - [,... " . - - . .,, ., , --... �� ,.. ..... 1._ _i L..,. � - � 2c1 Structural group : S , (5) co11st. -. - -- .... -e- � � ,. .... '' =- - .... � ,.._ I 0 o�� ,... Pitch-scale : ' ,., Q � � Tonal cycles; ., .' 22 • f � . i:ti� • • No. 230 Loose Leaf 12 Stave Style - Standard Punch ����� ... - 0 b� r . ,- I I &-� .� tF.;:. ,.. - - - b:b:?:I�" . 1,, ,,. ., • (Fig. XXXIII cont . ) • i - -I J ,_.' � ' • I ., . •v '.,; I I I ... r �I I' , - J. ,.' - I .. #: ... - I I� ' tc • I - � � , , •.; • ,J , � -t ·� I 0 ,, ,fr I .' . - •• ' . ' • ' ... _, • .. 3 ., �J I.'. •" .' I• • ' �' ' ·• . . � - • - - I- • r.' '- I" JI I _. •• ....I ' JI V .., .i - I < •. • No. 230 Loose Leaf 12 Stave Style - Standard Punch ff ��j��� 1� ,., • .. • ;it - , I� I C ' �-- h - ... , -... - ' - • �:. r - ,"'! � � -'1 ' , - ..' ·�.Ji"� ".. . - tr-· ;1 ' ' I • • .,; ._, r� . • S C H I L L I N G E R J O S E P H C o ·u R S E C O R R E S P O N D E N C E Subject: Music With: Dr. Jerome Gross Lesson LXXVI. • Symmetric �ystem of Harmony 0 (Type III) Symmetric System of harmony must satisfy the following requirements: (1) the root-to nes and their progress ions are the roots of two (i.e. ,ff,, 3,12, "./2, "../2, 1':/2) , that is the points of symmetry of an octave. (2) chord structures are pre-selected . As a consequence o f motion through symmetric roots, each voice of harmony produces one of t.he pitchscales of the Third Group. 1. Symmetric C0 represents one tonic; • ./2 represents two to nics; " three " './2 "J2 � '':/2 " " " four • Sl.X " " twelve n The correspondences of the tonal cycles and ti1e symmetric roots are as follows: • • • 2. One ton ic: C Two tonics : 0 Co Cr C Four tonics : C C Six tonics: f' F , • Three tonics: C • C C C c-, E} C Alz C C.3 3 c, E- C-3 A Ci C-3 C7 C- 7 Twelve tonics: C C ER D B)? C7 C-3 c, A!? ·c ·-3 F it C EiP c, F,: C-1 A C- 1 C '1 C-7 -D� E Ai? C7 C- 7 c, C-J C -c Ft V A C7 C7 BJz Ftt E D C- 7 C-7 . D� E� C7 C7 E� • • • C7 B B� C A' A C-7 C- 7 C-7 C-7 .. Transformations with regard to positi ons and voice-leading remain the same as in the diatonic system. In case of do ubt cancel all the accidentals. Two Tonics. Two tonics break up an octave into two uniform intervals. The second to nic (T�) being the .[§, produces the center of an octave. makes the t wo-to nic system reversible. 0 This property All points of intonation in the � � as well as in the � transformations are identical, i.e�, both the clockwise and the c, C-7 C • • 3. I counterclockwise voice-leading produce the same pattern of motion. This is true only in the case of two tonics. Two tonics form a continuous system, i.e., the recurring tonic does not appear in its original position. Two tonics produce a triple recurrence-cycle before the original position: falls on the first tonic (T ,) for the �� and the � Const. 3 produces a closed system. 1. Figure XXXIV_. • � I,,, 5 1 ' �--. A• ., ' « � ' • � I.- • • 0I ,1 i , ,, , � - ff • .,, T1 ,, ' .� ' � � � •• ,� • . '- � -e -- < I -·- ---- ------ The upper voice of harmony produces the following scale: c .__.- d'- e - r"­ L= g - a - (c) = � • 4. = (1+3 ) + 2 + (1+3 ) + 2. All other vo ices of the above progression pr oduce the same scale starting from its different phases. I t is easy to see that this scale belongs , to the Thir d Gro up and is constructed on two tonics. By selecting other structures and structural groups of 8(5) one can get some other scales of the Third Group. For example, the use of S 2 cor1st. produces the follov1ing scale: c - d'7 - e � - fM'- g - a - (c) = = (1+ 2) + 3 + (1+ 2) + 3 . Structural gro ups may be used in two ways: (1) S changes with each t o nic; ( 2) the groups of S produce C0 on eact1 tonic. Illustrations of the first method Figure -XX.XV • -----------� -----------------I • ., ff!i:.' ·� . ..p At .:e- - :i! ,_ • =e � ' "I... -s I I- • '( . fi . • .I .. •• -� • • 5• Illustrations of the second met hod • Figure XXXVI. • Combinations of the preceding two methods with regard to the structural selection for · each tonic of one symmetric system are applicable t o all symmetric systems • • • 6. Example f�gtn' e ;xxxvrr . • • Longer progressions can be obtained through the use of longer structural groups, such as rhythmic resultants, power-groups, series of growth, etc. In some cases the number of terms in the structural group produces interference against the number of tonics in the symmetric system. Example ' • 7. Three Tonics. Three tonics produce a closed system for ; � and 1: "' ., , and a continuous sy stem (two recurrence-cycles) for const. 3 . Figure XXXVIII. • •• • Four Tonics. .. Four tonics produce a continuous system (three recurrence-cycles) for closed system for const. 3 . ..-=; ,:.,, and (please see next page) • le , .� , and a • .... s , const • , .,., ' .1 Const. j � � 3 , .f / -- -� , ' . r�I - � � 1- i Six Tonic s . s• Figure XXXIX. • • • I rI r,1 <� - � • r.. r- Six tonics produce a closed system for as well as for the const. 3. f;lg11tp S , const. -� ,c:,... I:; - and _ � , x;r... I � I�• • ' tit � • Const. 3 - -i - : ' � ; ..,,. rI � . � ,I rI . r;7 2 ff No. 230 loose leaf 12 Stave Style - Standard Punch s1c1·a • �. . E(RAND ' ' , - - • • • 9• Twelve Tonics. l:. Twelve tonics produce a closed system for ,� __ and ...__:::, as well as for the const. 3 . Figure, X}:,I. s , const. � �-­ t-,....., < •• Const. 3 ��---...9-"-----------------'------�_,. l!t:���f No. 230 Loose Leaf 12 Stave Style - Standard Punch , • • 10 . I Lesson LXXVII. Variable Doubl ings • Harmony, 1n many cases conceived as an accompaniment, may be given a self-suf ficient character by means of variab.le doublings. This device attributes to �hord progressions a greater versatility of sonority and voice-leading than the one usually observed. Variable doublings comprise the three functions of S (5) . Thus · the root, the third or the The corresponding notation fi ft h can be doubled. to be used is: 8(5)© , 8(5)@ and 8(5) © . As the root-tone remains in the bass, S ( 5 ) u) is the only case of doubling where all three functions (1, 3, 5) appear in the upper three parts. The followi ng represents a comparative table of functi ons in the three upper parts under . various forms of doubling. 8 (5) © = 1, 3, 5 8 (5)© � 3, 3, 5 S (5)© = 3 , 5, 5 • Figure XLII. . V In cases S (5 ) @ and 8 (5) @ only three positions are possible for each case. Black notes • • • 11. represent variants where unison is substituted for an octave. , Positions Jf I) &I �- .. - r1 -I •• - *' .. , I I • • -- •• • j --- --- ·- ,. • ,I • TI • • •• I, Fi&!!_re •XLIII . Tr.ansf ormati ans 8(5)© 5 ( .- 5 3� )3 c� � 8 (5 ) @ 5( ...): 3 5( � 3 3( � 5 3 �e t3 lf > 3 1 ( :)3 ti - ::,;. A- •• - • ., ,i � .. ,l, � l < >5 . c s- � t7 � _ _ ____ __ __,_ - - - . ,,.. . , ._ ' • • 12. S (5) (!) < ➔ 8(5)@ 5( ) 3 3( 3 f�� 3 ,� • • >5 8 (5 ) (i) < >3 5' )5 5f 3 ( .> 5 3( ) 5 1( >i 3 1( i� e1 � - cf��- e 3 �- )5 > S (5) {j) 5( �5 3( ➔ 3 1( ) 5 e,, e� • • • s (5 ) • + ---�> s { 5 ) © ev...- 3( > 5 • • • • • • ( -------t) 8 ( 5 ) (l) 8 (5 ) @� 3( ➔6 3 !-} 3 .. • When r eading these tables, consider identical directions of the arrows for the sequen�e of structures and for the c orrespondi ng transforma­ tions. Notice that there always are three transformations when S(5) © participates and only one when it does not . • • • 14 . Musical tables in the above Figure are devised fro m the initial chord being in the same position. frow all Similar tables can be• constructed positions as well as in rev�:rse sequence and also in the cycles of the negative form. Variable doublings are subject to distributive arrangement and can be superimposed on any desirable cycle-gro up. Figure XLIV. Example : 2C.3 + C , + C 7 ; 8 (5)© + 28(5) (1) Ir = 8 (5)0 + C 3 + 8 (5) + C5 ® + 8 (5) � + C 3 + 8 (5) (V + + S(5)(i) -. C ? + 8(5) (D • (S) • • Example: 2C� + C 3 + C, + 2C 7 ; + 8 (5) @ + 8 (5) © . Ir = 8 (5) @ Ci) 8 (5) + 0 8 (5) + + c, + �(5)(D + c, + S (5) @ + + c, + 8 (5)@ + c, + 8(5)© + c ,, + 8 (5) 0 + + c 7 + s (5)@ + c, + 8(5)© + er + 8 (5)@ + + C3 + 8 (5)0 + c, + 8 (5)© + C '7 + S (5)@ + + c, + 8 (5)� • • 15. I I I I I Variable doublings are applicable to all types of harmonic progressions , thus including types II and III . Figure XLV • • Type II (diatonic-symmetric) . Ir as in the preceding example. :: 2S2 + Sa + s , I I I I • Figure XLVI,. Type III (symmetric ) . ©+ (i) + T S,Q + T� S - - T, S1(i)+ T�S� � (6T) 3 3 + TsS at + T. S1G) + T, S -a,(j). © • • • • 16 • • • • • • . • • .. • 17. Lesson LXXVIII, Inversions of 8(5) The usual tech nique of i nversions, • The strictly speaking, is unnecessary to a co mposer. reason for this is, that by vertical per mutations of the pos itions of parts i n a ny harmonic continuity of 8(5) , the inversions appear automati cally, as inner or upper parts beco me the bass parts u nder such conditions. This teclmique was fully described in my "Geometrical Pro jections of Mllsic", in the branch dealing with the co ntinuity of geometrical i nversions. For a n analyst or a teacher, however, a thorough systematization 0£ the classical te�hnique of inversi ons is a necessity. There is no other branch of harmony I know of, where- confusion is great�r and t he information less reliab le • The first inversion of 8(5) is know n as a "sixth-chord" or a "third-sixth-chord" a nd is expressed i n th is notati on by the symbol 8 (6). The only condition under which 8 (5) becomes an 8 (6) is when the th ird (3 ) appears in the bass. The' positions • of the upper voices are not affected by suc h a cha nge, . th e foFms of do ublings -- are.. Which do ublings are appropriate in each case, will be discussed later. Assuming that �ny S(6) may be eit her 8 (6) ©, or 8 (6) @ , or 8(6) @ , we obtai n the fo llowi ng Tab le of Positions: • • 18. u - 1. � , -- . - - • ,_ V. 1 I :.; ,Ir .. - - .,.. .,. ,,i r� , • Figure XLVII. . ,. - � � ·� •• • � s . 'I i � - i!' $ r• � . It is easy to memorize the above table, as $(6 ) © and 8 (6 )© positi ons are systematized through the followi ng cliaracteristics: (1) the doubled function appears above the remaining function; (2) the doub led fu nc tion surrounds the remaining function; (3 ) the doubled fu nction appears below the remaining function. 8 (6 )@ is identical with 8 (5) positions, except for the bass having constant 3 • . Harmonic progressions (Ir ) consisting of , 8 (5) and 8 (6 ) are based on the followi ng combinations by two: ,, - • • 19. (1) 8(5) (4 ) s (6) ) 8 (5); , s (s ) . ( 2) 8 (5) > S (6) ; (3) S (6) J S (5) ; As the first case is covered by the previous technique, we are concerned, for the present, with the last three cases. All the following transformations, being applied to vo ice-leading, are reversible, as in the case of Variable Doublings of 8 (5) . always measured thro ugh root-tones• • Figure XLVIII . S(5) �-----4 • 5( � 5 3 • ll l• ) 1 • es- 8(5) 3• -. 5 1.- � 1 S (6) © 5 � ►l 5�1 l< > 5 • S (6) (l) 5E�l 5< �5 5• )5 lt·) 5 1� ) 5 lf •l 3• )5 • 5< ll Tonal cycles are 3� ) 1 3( ) 5 • 20 • • 5 (5 ) 8(6) ® 5'(" � 3 5� )l 3.-• , l 3 fl ) 3 lf � 5 1.- ) 5 Const .,. 3 Const- 3 � = C,5' • • • 21 • • ,,, ' - � J - + ,� - ... ..,. . ,. • ·� rI , ,I ,,I t. 8(6-fD 5( • l" 1� e� .., 1 S(6) @ )5 �s-- e, S' e, • , S (6)(q _____,.;,...___ > 8 (6)© 5 ( )1 5( ) 5 1( ) 5 • 22 • • © S(6) 5 f-) 5 .( -----� � S (6) (D 5f-�l 5(""3 l'( ) 3 1( >t 5 1� > 5 l" ) 3 lf' > 1 �s �.----.----.----,--o---:_r====:Jc:::;;==== c� S (6) (D S(6) (D 5� 1 5( )Z 3 < )1 3� ) 5 l< ) 5 l'f ·) 3 ➔ ,. .; F .... 4 ..._, I , • • - ,j j - � , - r. - =- _;:;,, - 'I __, _r � I a � ;: - � � � • V � 8 �I $ � • @ 8(6) ( )5 5( =. 1 5( ) 5 5( > 5 5 t' l< > l l< ) 3 ) 23. © S (6) 5< ) 3 5( ) 1 1( � 5 0 • .Any variants conformed to identical transformations (like the black notes in some of the preceding tables) are as acceptable as the ones in the tables. • • • 24 • • , v Lesson LXXIX. Doublings of 8(6) rapidly. Musical habits are formed comparatively Once they assume a form of natural reactions, they infl uence us more tha n the purely acoustical factors . This is particularly true in the case of doublings of 5(6 ) . The mere fact that identical doublings in the different musical contexts affect us in a different way, sh ows that our auditory reactions i n music are not natural but conditioned. The principles offered here are based on a comparative study of the respective forms of music. There are two technical factors affecti ng • the doubling i n an 8 (6 ) : (1) the structure of the ch ord; ( 2) the degree of the scale (on which the chord is co nstructed) . These two influences are ever-present regardless of the type to which the respective harmonic continuity belong s., Yet, while in harmonic progressions of type II and III the structure of the chord is the most infl uential factor, i n the diatonic progressio ns (type I) it is exactly the reverse. The influence of a constant pitch-scale is so overwhelming, that each • • 25. chord becomes associated with its definite position in the scale. Thus, one chord begins to sound to us as a do minant and another or a leading tone. as a tonic, a mediant This hierarchy of importance of the various chords calls for the different forms of doubling, particularly when the respective cl1ords appear in the different inversions. The fo1-lowing is most practical for use in diatonic progressions • • Figure XLIX. Stro ng Factor The degree of the scale I, IV, V, VI Regular Doubling CD , © Irreg. Doubling Weak Factor The structure of the chord s, (6) Regular Doubling G) , @ II, III, VII S 14 (6) Regular doublings are statistically pre­ dominant. Irregular doublings, in most cases are the result of melodic tendencies. In reading the above table, give preference to the strong factor, except in the case of S 3 (6) and 6 � (6) . It is customary to believe that an s, (6) must have doubled root or fifth. But in reality it seldom happens when such a cl1ord belongs to II, III or VII. Irreg . Doubling @ • 26. Naturally, all our habits with regard to doublings are formed o n more customary maj or and minor scales. The above table will work perfectly when applied to such scales. There will be no discrepancy when 8 3 (6) and S� ( 6) will be compared with the data on the left side of the table, as such structures do • not occur o n the main degrees of the usual scales • When using less familiar scales, one or another type of doubling will not make as much difference. • in such oases the structure may become a more Yet influential factor, though the sequence - is diatonic. In the types II and III the most practi cal u forms of doublings are: Structure Figure L • Regular Doubling s , (6) CD,® S 2 (6) (D, @ 8 3 (6) @ (y , @ , @ s.. ( 6) Irregular Doubli ng @ @ Continuitz of 8(5) and 8(6) �he comparative characteristic of S (5) is its stability, due to the presen�e of the root-tone i n the bass. The absence of the root-tone in the bass of S (6) deprives this structure of such stability. • • • 27. Composition of continuity consisting of S (5) and S ( 6 ) results in an interplay of stable and unstable units or groups. The following fundamental forms of co nti nuity with ut ilization of the above­ mentioned structures are possible : (1) ( 2) (3) ( 4) 8 (5) const.- ---­ stable 8 (6 ) const. ---- unstable ( 8 (5) + S (6 ) ] + . . . alternate 2S(5) + 8 (6 ) + S (5) + 2S (6 ) 3S(5) + S (6 ) + 2S(5) + 28(6) + S ( 5) + 38(6) 48 ( 5) + S (6 ) + 38(5) + 28(6) + 2S(5) + 3S (6) + + S (5) + 48 ( 6 ) • • increasing stability increasing instability (5) 4S(5) + 28(6) + 28(5) + S { 6 ) ---� proportionately decreasing ratios proportionately increasi ng ratios 1-------­ (6) 8 ( 5) + 28(6) + 38(5) + 5S (6) + 85 (5) + 138 (6) progressive over-balanci ng .pf unstable e,lements S ( 6) • + 28 (5) + 3S ( 6) + 5S ( 5) + 8S ( 6) + 13S ( 5) progressive over-balancing of stable elements Many other forms of distribution of 8 (5) V and S(6) may be devised o n the basis of t he "Theory of Rhythm " . • • Examples of Progressions 28. Figure LI. Diatenie S ( 6) Const . ; 2C '1 + 2C !' + Ca + Cs� t '!, • l. • Figure LII. Diatonic-�nnmetri� a • • - 2C S' + C"1 + C '° + 2C 7 2S 2.. ( 6) + S 1 ( 6) + S 3 ( 6) + S 1 ( 6) + S" ( 6) ; ,: I� I ,, r l=f - " .-- • - , ,.,. I:,J ,� ,� Ii) /.i'I I -.... -� • /.i'i I� I - /?) (.i) - ' � • ✓ ,� -:,.,, '-� • §YHH:!!!#tric + S: ( 6) + S Ji ( 6) + 2S 1 ( 6) ; Six tonics 1�==i�t= ::i 3 • ) •.,.., -aCl :�� �0 No. 230 Loose Leaf 12 Stave Style - Standard Punch ...,.. , • 29. Figure LIII. D!2tonic 38(6) + 8 (5) + 28(6) + 2S(5) + S(6) + 3S(5) ; 2C 1 + C 1 • •• lo Diatonic-Symmetric + s , (6) + 2S 2 (5) ; 2·c· -r + c,; Scale- of roots: Aeolian .. I Symmetric + s2(6)] T, + [sq(6) + S,(5)] T2} + . . . == ... ·o�...a u:si!-= BRA,.._.D No. 230 Loose Leaf 12 Stave Style - Standard Punch Fo111· Tonic�. • J O S E P H S C H I L L I N G E R C O R R E S P O N D E N C E With: Dr. Jero me Gro ss Less9n LXXX. C O U R S E Subject.: Mu�ic Groups with, fa ssip.g Ch9rds A. Passing Sixth-chprd;; A group with a passing S (6 ) is a pre-.set c ombinati m of three chords, namely: 8 (5) + 8 ( 6 ) + • + S (5) . Every papsi ng c�prd occupies. the_�enter of its group, appears, on a �eak beat and has � doubled bass. • The complete expression for a group (G) with passing sixth-chord is: G = 8(5) + 8 ( 6 )@ + 8 ( 5) . 6 This formula is not reversible in actual intonation. between the extreme chord,s of G6 is This relationship reruain� constant in all cases The relati onship C-5. of classical music .. all cycles. We - shall extend this principle to Under such conditions G6 r�tains the following characteristics: (1) The transf'orma tion between the extreme 'chords of the group is always c�ockw�� e for both the positive and. the negative cycles (2) The . bass progression is: 1-�> 3 �> l, which necessitates the first condition. • • 2. In the classical form of Gs, bass moves by the thirds. Thus, 3 in the bass under S (S) is a third above its preceding position under the rirst S (5), and a third below its following position under the last S(5) . • In order t.o obtain Gs , it is necessary to connect S (5) with the next S (5) through C-5 and add the intermediate third of the first chord in the bass, without moving the remaini ng voices. G Figure LIV. q I = 8 (5) + S (6) 3 + S(5) 1:: G • There are three melodic for ms for the bass movement. Figure LV. 0 • • • • , • 3. Combinations of these three forms in sequence produce a very flexible bass part and, being repeated with one G6 , make expressive cadences of Mozartian flavor . fig:ure LVI • • • • • Continuity of G6 • Continuity of such groups can be obtained by connecting them through the tonal cycles. • while C8 Connecting by c5 closes the sequence, and c produce a progression of 70 • 7 6 Figure J..,Y.I.I_. (please see next page) • • • • ' • • • � l C ,I ILI • It= • •' '. •• ,, 4• , . � . , • ' C1 .. . . - - - -·- Further versatility of G6 progressions can be achieved by varying the cycles betv,een the groups. Any time a decisive cadence is desirable C5 must be introduced, as this cycle closes the progression. u • 5. Figure LVIII. P + • = G6 + c7 + G6 + c 7 + Gs + c3 + Gs + C 7 + Gs + C3 + Gs + C3 + Gs + C5 c, c, • Generalization of G6 • In addition to . the classical form of G6 , other forms can be developed through the use of other Of course, each cycle than C-5 cycles within the gro up. produces its own eharacteristic bass pattern. Figure LIX. Various forms of Gs: ,�� (c.� ) �I · u � r, r� ... • �b (t,$ ) .,. "7 � , - �L, (e, r1 - -... ) - 2 �lo (�-3) • .. r ' 7 4£.( t- �' (e-s) i ... F - ', ,. $ fil - $8 (If) • 6. u The respective variations of the bass pattern will be as follows: • Figure LX. • , - • 7 r , ' Q �1.. ( � - 5) -- • � 4lr> ( � -1) ����������1��!������������3�;��� G • Continuity of the generalized G6 Such a continuity can be developed through the selective progressions of the various forms of G6 combined with the various cycle connections between the groups. 8· • 7• • Example: figure LXI. � = G6 (C-5 ) + C7 + G6 (C3) + c5 + G(C-7) + c3 + + G(C-5) + C5 + G(C7 ) + C5 I. • Generalization_ of �pe Passing Th�rd It follows fro m · the technique of gro ups • • • with a passing sixth-chord, that the first two chords, i.e. , S (5) and 8 (6)@ , belong to C0 , and that as the position of the three upp er parts does not change until the last chord of the gro up appears. This last chord , 8 (5) , can be 1n any r elation but C0 with the preceding chord. If we think about the appearance of the thi rd in the bass dur ing 8 (6)'.V merely as a passing third, it is easy to see that this entire technique can be general ized. providing the The passing 3 can be used after any 8 (5) , transformation betwe en the latter and • 8. the following S {5) is clockwise for all the cycles. Such a device can be applied to any progressions of root-tones in the bass. S (5) w1 th the Figure LXI):, Ex.ample : • • I , 'd , r tI -· ' - . r� I, • l!Si • , � I - I r:;I C c I • ' � I • • • , i I L [ ] r • . J � I r .I • -• • The effect of such harmonic continuity is one of overlapping gro ups of G6 , as marked in the above Figure. u ' - ' -. • - ---- ------ -- . ··- � • -i- • :j -i- -'t (ii -- ,. U· • , • Lesso n LXXXI, Applications of Gs to Diatonic-Symmetric • (Type II) and Symmetric 1Typ e JII) Progressions, The use of structures of S(5) and 8 (6)@ in the groups with a passing sixth-chord must satisfy the following requireme nt : �he a�ja�en� 8(5) and S(6)(j) of one group, must hav� identical structures • This re,quirement does not affect the form • . of the last S(5) of a group; neither does it influence the selecti on of the forms of S (5) in the� adjacent groups. V • As each G6 consists of three places, two of which are identical, the number of structural combinations for the individual groups equals 42 = 16. S1 + Sl S2 + S1 81 + S 3 S 2 + S3 S1 + 8 2 S1 + S4 S2 + S2 S 2 + 84 . S3 + S 1 S4 + Sl S3 + S3 S4 + S 3 S3 + S 2 83 + S4 S4 + S 2 S4 + S4 Thus we obtain 16 forms of G6 with the following distribution of structural co mbinations • • • • 10. G = S1 (5) + sl (6)® + 81 (5) 6 G6 S1 (5) + s1 (s)® + s 2 (5) = Ga = s1 (5) + S 1 (6)® + s3 (5) G6 = 81 (5) + s1 (6)® + s4 (5) G6 = s2 (5) + s 2 (6)® + s 1 (5) G6 = 82 (5) + s2 (6)@ + s 2 (5) • G 6 = s 2 (5) + s 2 (6)@ + s3 (5) G = s (5) + s (6) + ·S 2 (5) Ga = s3 (5) + s 3 ( a)@ + s3 (5) G = s3 (5) + s3 ( a)� + 84 (5) 6 G6 = s4 (e,) + s4 (6)® + 81 (5) Gs = S4 (5) + S4 (6)® + S 2 (5) G6 = 84 (5) + 84 (6)@ + 83 (5) G6 • u = 84 (5) + 84 (6)® + s4 (5) As the melodic interval in the bass, while moving from the root (1) in 8 (5) to the third ( 3 ) i n • 11. S(6 )@ is identi cal for the forms s 1 and s 3, as well as s2 and s4, the total qUanti ty of intonations in . the bass part for one type of 0 is ½ = 2 . 6 81 + 8 1 S1 + 82 S2 + S 1 82 + 62 As each intonation has 3 melodic forms • and there are two different intonations, the total number of intonations combined with melodic forms in the bass part is 2 x 3 = 6 . • ,_ -. ,� -s -G Progressions 9f. �e ty-pe t. - IJ. Figure LXIII. Example: Forms of S: s 2 (5) + s 2 (6)® + 81 (5) r = G6 (C-5) + c3 + G6 (C-5) + C7 + G6 (C-5) + C5 • • fl .,,. - • 12. Example : Forms of S : [s1 (5) + s1 (6)® + s2 (5) ] + [S3(5) + + S3(6:® t S2 (5) ] � = as in the preceding example • • • • u • Example : Forms of S : s2 (5) + s2 (6)@ + s2 (5) r = as in Figure LXI. -� II - " -. � !' ,_ � ' - �· I, � - ·-- · _.. • . • •• .... I I r � ' � � , +- j,,i t"t ' JI I • I I - , l3l Ir -(l + ', • •• . • - ..� "" �I • - • .. • 13 • • Generalization of the passing third 1s applicable to this type of harmonic progressions as The following is an application of the well. structural group Figure LXII. 2s 1 + s2 + 2s1 + s2 2s1 + to the Figure J;XJ.V. • ,. • '- ::.--� ,, r -,.j! � fil �f r, I pi l� I - r-• i L ,. •• ' • • • • u • 14. • Lesson LXXXII. Progressi�ns of_ the type IJI. Applications of G6 to symmetrical systems of tonics disclose many u nexplored possibilities, among which the two-tonic S)'stem deserves a particular As intervals formi ng the two tonics are • attention. equidistant, the passing tones of S ( 6 'fiJ, which i n turn may also be equidistant from T 1 and T2 , thus produce, in the bass movement, diminished seventh­ • chords in symmetric harmonization, a device heretofore unknown. • The justification for the use of G6 in the symmetrical systems of tonics is based on the following deduction from the original classical form, i.e., • --- -- ---- (Symmetric) (Diatonic) • The abovementioned equidistancy of the two tonics permits t o obtain r = 3G6 until the cycle • 15. • Selecting closes. • ,. -y � II � � - • • � ,I I "" . -t: � Ii, I • J s1 for the entire G6 , we obtain: Figure LXV. 1� I • $,_ Ia: 7'7 ,7 t; ., ,� I r I • l ., I .·• h r• t. u The overlapping of groups, indicated by • the brackets in the above Figure, is an invariant of the symmetrical systems. Thus, the passing third can be considered a general device for progressions of the type • III. The number of bass patterns for the cycle of the two tor1ics equals: 22 = 4. The number of intonations in each cycle of the two tonics equals : 22 :;:: 4. The latter is due to the use of the different forms of S (5) . The interval between 1 and 3 equals 4 and is identical for The interval between l and 3 equals It • • 3 and is identical for s2 (5) and s4 (5) . Thus, by • • 16. distributing the different structures through two tonics, we obtain the following co mbinations : 81 (T 1) + S 1 ( T2) Sl (Tl ) + S 3 (T 2 ) �3 (T 1) + 81 (T 2) • S2 (Tl ) + S4 (T2 ) S4 (T l) + S 2 ( T 2) • • identical intonations in the bass part S3 (Tl ) + S 3 (T 2 ) S 2 (Tl ) + S2 (T 2) • .- - identical intonations i n the bass B_art S4 (T l) + S4 (T 2) 61 (T l) + S 2 (T 2) Sl (Tl) + S4 (T2 ) S 3 (T l) + S 2 (T 2) ident ica l intonations in the bass part S 3 (T l) + S4 (T 2) S2 (T 1) + S1 (T2 ) S2 (Tl ) + S3 ( T2 ) S4 (T l) + S l (T l) S4 (Tl) + S 3 (T'2 ) • identical intonations in the bass part • \ • 17 . The following is a table of intonations and melodic forms in. the bass part on two tonics. Total : 4 2 = 16. Figure LXVI•. • s,. • .r 54- I., L The above combinati ons can be incorpor . ated into a versatile continuity of 06 on two tonics. ' • • 18 • • LXVII. Fig11£e -. Example : • .) � , -- � ,. , • d, '. � � ,= � - :p I • � ] -, - tfi • I. produces V Application of G to three t9pic§ 6 8 melodic forms in the bass par t : 2 3 = a . Fi&1re LXVII I . a a • , • 19. Figure LXVIII (cont . ) .r • I •• 'I �1 -- : - - •� - � --------- - • different S A ,� • ----. • , .- ' - -----------------,.,----- The number of distributions of the through thr ee tonics is 43 = 64, while the number of non-identical intonati ons is 23 = a. . Non-identical intonations: S1(T1 ) + s1 (T2) + s1 (T a ) Sl ( T1) + Sl (T...� ) + S 2 ( T3 ) S l( Tl) + S 2 ( Tz ) + S l (T a ) Sa ( T ) + S ( T2 ) + S (Ta � 1 1 1 (j . • - - • u · 1 (T a) Sa ( Tl ) + S2 (T2 ) + S • Sa (T l) + Sl (T 2 ) + S2 {T 3 ) Sl (Tl) + S\( T2 ) + S2 (T 3 ) S2 (Tl) + S2 (T 3 ) + S2 (T 3 ) The total number of different intonations and melodic forms in the bass part is 8 2 = 64 • Examples of continuity of G6 on three tonics • Figure LXIX, u ? I • • -' .,, • ,. .,, ) I �· 1 - -� • ,, ., ::,. J � -I I ,, • 'I• 'I I " •I ,,, • ·-- 1I " * -- s, -- -Ir � '' • V I �- V . • .-� I •I . �" , i ,·:, ,, ifi: � '1-tJ- • • • "'· ' :a; • * n -- S:i.. j I • � • "' - •I " /} • • • • • -+- ,I I �7 • I� •• , • Application of 06 to four tonics If produces 2 = 16 melodic forms in the bass part . 21. The number of distributions of the four forms of S through four tonics produces 4� = 256 intonations� The number of intonation$ in the bass part is limited to 2'4 = 16. Thus the total number of intonations and melodic forms in the bass part is 16 2 • Examples of continuity of G6 on four tonics. • Figure LXX. (ple.ase see next page) • • , • 0 = 256 • • 22. Figure LXX, � (J L ' "" l ,. I " 1 r- �· . • 1f I l • .. :j , t-8-1 • ,Ir, I I s � ,� ... - I • • - s • �sici•an BRAND No. 230 Loose Leaf 12 Stave Style - Standard Punch - .., t. , r ,r� -• ,_ • p__ ,_ �' � � " � , -- • •• • # � I I•• J " • , 2' Application of G6 to six tonics produces = 64 melodic forms in the bass part. The number of distributions of the four ° 4 forms of S through four tonics produces intonations. 2 4 = 64. = 409"6 The number of intonati ons in the bass part is The total number of intonations and melodic forms in the bass part is 64 2 = 4096. Examples of continuity of Ga on s·ix tonics. ¥1gur� LXXI • • L S1 �- I S4 I " e. i ' .I 0 , s 411 ·s ,� 11'1 ' l'.) r I I 5 • � - s . I\ I • r,1 ff • I � � lI • •I J' - I T -I I • • • 24. 1 produces 2 � Applicati on of G6 to twelve tonics = 4096 melodic forms i n the bass part. lhe number of distributions of the four forms of S through four tonics produces 4 It. • = 16,777, 216 • The number of intonations in the bass part is 2 1 � = 4096. The total nu mber of intonations and melodic forms in the bass part is 40962 = 16,777, 216. Examples of conti nuity of G6 on twelve tonics. 'Figµre LXXII. (please see next page) • • • u • 25. s , const ., ,� I • ,I � - 1 i.. "' � � "' �·� :n- � � • -� • • --I - -I • _ , ,• I I I ' �- .J u I' • '- , n- S 0/I � I • 1• -,, 118 ' cons t . I ,, 1�'7 ..� � 1;;• :a .. II . I I - ·- • ,, - I ' t;.11 , • L -- , � It. 'I: �"ljj == == (>========== =========== =================== • I 26. Lesson LXXXIII. B. Passing Fourth-Sixth Chords: S (4) . of S (5) is a fourthThe second inversion • This name derives from the old sixth chord: s(!). basso continuo or generalbass, where intervals were measured from the bass. 7 • - -,� � I-I 8 ( !) has a fifth (5) in the bass, while� t he three u upper parts have the six usual arrangements� • The use of s(:) in classical music is a very peculiar one. This chord appears only in definite pre-set combinatiODS r group with a p?ssing One of theru is the fourth��ixth chord : G�. As in the case of G 6, the passing chord itself' appears on a weak beat, being surrounded by the two other chords, and has a doubled fifth: s6 4 ®• The two other chords of G: are : 8 (5) and 8 (6). The latter can have two forms o f cbubling (regardless of the chord-structure): S (6) 0 an� 8 (6)® . The group v,ith a passing fourth-sixth 0 chord, contrary to G , is reversible. 6 • 27. u , This property being COD)bined with the • choice of two possible doublings produces four •• variants. • + 8 (6) CD • • l • The arrows in the above formulae specify • the directions of the bass pattern which is always scalewise, and therefore can be either ascending or descending. The bass pattern is developed on three adjacent pitch-units, which correspond to the three • a chords of 9: • C) � C l 8 (5) - *;, 5 3 ...... s(!) \\ 8 (6) Arabic numberals :represent the respective chordal functions. , • - 28. Transformations between 8 (5) and s (:) in the G4 : as the bass moves from 1 to 5, when read in upward motion, the three upper voices must move clockwise, in order to get the transformation of 1 into 3. n ' ., , • ...., - • $ rJ • l. G· • The transition from s (:) into 8 (6) © or S (6){f) follows the forms of transformations, where two identical functions participate, as in the cases of S ( 5) < ) S (6) © and S (5) ( > S ( 6) © However, classical technique adopted . definite routines concerning this transition: • (1) one part must carry out a melodic form reci;proc,!11_ to the bass (i.e., position � of the bass melody); (2) it is ·th is reciprocal part that deviates from its path in order to supply the (j d9Jl,bling of the fifth in an 8 (6). ' • • 29. f '-' Under such conditions G! acquires the followi ng appearances: • " I ' • .,.� - . '-- .J • . ...._ ,..., ..., , ,� r -- - --c:::,. 0 •. , - -- - - """! _,.,,,,. -s. -t. The following sequence of operations • is recommended : (1) bass (2) part reciprocating the bass • (3) coxumon tone (4) part supplying the thir d for s ( !) The relations between the chords of G4 6 are as follows: Co ' l, 8(5) . + C-5 + s (!) + c 5 + S (6 ) • Co + 8( 6) + C-5 + $ ( 4 6) • 30. Each group can be carried out in 6 positions which· depend on t he starting position. a! forms of • l ... - - '� � 0 '.. ,I • 0 -- I - The following is the table of all four in one position. Figure LXXIII. ' - .. , ::,. 0 - - - � � � I• . , :.., - , ,.; � .' , - - • 0 - . , 6 can be The dif ferent forms o f G4 connected by means of tonal cycles and their coefficients of recurrence can be specified. It is desirable to make the following tables: c7 c3 (2) const. ; " " lf " " n (3) const . ; " " 11 " " n const. ; ff n " ,, " " (4) 0 c5 const.; (1) 6 G4 ! © - const., const., const. • 31. • (5) • (6 ) (7) (8) (9) • G4 rr © a!!© 6 T G 4 ati © (0 6T 0 G4 COilSt . ; c+ = C3 + C5 + C7 const.; c-t = C3 + C5 + C7 const .; cot = C 3 COils t . ; :!© + G = c"" + G� T C 3 (f) + c + C " " " " " " TT " " n " (11) (12) 5 6 G + " (10) + c 7 5 + 0 C 7 , ,• C3 c 5 COilSt • const .. ,• C7 const . •' � = C3 + C5 + C7 � is the symbol of a group of cycles (cycle COiltinuity ) . Continuity of G64, when connected through a constant tonal cycle, consists of seven cycles: � = 7C. e Figure LXXIV, Example : G4 r • .,, .., -e -s,-.. ' I • - - 0 - • • V .. ...,, ,-.. -8-- � © const. • i..... .._, ., • .... ,� ' - .' ,.i .., • '.J ., • � - � .0.. •• - - -e- - - . � - � � I ,,i q •• � • -i ;a;- .a=:. • ,c ,; •• � • • 32. ·v a: Continuity of of different forms and co11nection through different cycle-groups can be applied in its present form to Diatonic progressions. a: in symmetric progressions of the types I I and III require identical structures for the two " extreme chords of one group. - This requirement does not affect the middle chord of the group, s (:), nor do es it influence the selection of • structua,es for the following groups. Examples of continuity with i • e •, a: in progr�ssions of the types• I a.nd I I . Ir' = • , � " 'I• • - - � .,, .a.. ..... - � •• -s• - ·� , -9-..,, C' _, 6' 6 2G4t + G4 ..... • --- .,, C • r � � ! � r• !; � ::: 61T' 6 + G4 J + 2G4 .... - 9-- .w. -- .' � FiBur e LXXV •. -.... ..0- ' - ..-: ., ,. • • -- - .., , i ,. • ', ._, -!r -e- ., ... � - ' A 0 - - � ,. r, Figure LXXVI. 1-r and � as in the preceding example• • -------- - .... ,.., I • - -s- .... -5-- . ' � , -e. -- � ,., ,.... - t.29£t £ .-- ..0. - • ,,.._ - - I .,.. � r -'' .0.. - -6- . ·- ....., .. _ - 11.._,,, __• r, - • • J O S E P H S C H I L L I N G E R C O R R E S P O N D E N C E With : Dr. Jerome Gross C O U R S E Subject: Music Lesson µcxxiv, Application of G� to symmetric systems requires the following seq uence of tonics: GW � (T, + T 2 + T , ) + (T 2 + T 3 + T2 ) + • • • For example, the three-tonic system must be distributed a s follows : 0 G� = (T + T 2 + T ) + (T 2 + T 3 + T 2 ) + , , • + (T 3 + T + T 3 ) . , The quantity of to nics in tl1e respective system specifies the cycle. either 8(5) or S (6 ) . Eac11 gro up may begin with Each group acquires the following distri­ • bution of inversions: Under such conditi ons, each tonic appears in all the three inversions. • 6 Table of G4 • applied to all symmetric systems " Figure tXXv!l. I 1 ., , ..I I ,, � - �- • , ,I �- I ,� ' . CI 11- ,,, � a, L1 r � � • • 5 IC> rouR To N l C. � . 1iI r l'l I , -r� , :i , � . • - � .. :f � 5 s, � ,-;j /.., Cl - t 01'{ , e $ f?- -r1 i'';l. -(.� -r,. -ri --, i "' 4 f4 .,...� � f's- . - f 2.., 4 ,_.,, - � - Tt, - ,1 � - Tt ,I - -e,- - (p 4 --r � r.1 ,J • � - .. -r, "' 4· • "f< rI � - ' - J t I ,. , I ' ' n.,,., I• r:, ,1 -..,- , r.I ' "' I rI -r� r.,, ' # a,. >< + �t: _---tl • • • c. Cyc�es. �d ��oups Mixed. Tonal cycles can be introduced into the continuity of groups, as well as groups can be intro­ duced into the continuity of cycles. It is convenient to plan the mixed form of cycle-group continuity by the bars (T). Bars of cycles and bars of groups can be • - • • assigned to have different coefficients of recurrence. , When planning such a continuity i n advance, it is i mportant to co nsider that there is always a cycle-connection between the bars • • Examples: • Figure LXX}:;x. Ir = 2TC + TG + TC + 2TG = (C� + C3) + C 7 + (Ca + C 7) + + c, + G6 + C 7 + (C3 + C3 ) + Cr + G:1© + C 1 + G! _l,(f)+ C3 • L •• • • T'(PE It • , ...__.,,, , • •