Schillinger s Special Theory of Harmony: Hybrid 5- and 4-part harmony

Transcription

1 Hyrid 5- ad 4-part harmoy F.G.J. Asil Schilliger s Special Theory of Harmoy: Hyrid 5- ad 4-part harmoy F.G.J. Asil Feruary 9, 2008 Astract This documet descries to techiques as preseted y Schilliger [1] i his Special Theory of Harmoy ook. Chord structures, progressios ad applicatio aspects of the hyrid 5- ad 4-part harmoy techique ill e discussed ad illustrated ith score examples. Documet history: revised (typig errors corrected, mior edits) August 16, 2010, (score ad other errors corrected) August Itroductio Schilligers Book 5: The Special Theory of Harmoy [1] presets a great summary of toal 4-part harmoy. Toal harmoy implies that chord structures ad progressios are ased o a orderig i thirds of seve-pitch diatoic scales (Schilliger calls this the first expasio form of the scale). This meas that if e rite a 7-pitch scale o a toic p 1 as a ordered set (ascedig diatoic steps of a secod) E 0 = {p 1 p 2 p 3 p 4 p 5 p 6 p 7 }, the the first expasio (ito a sequece of thirds) is E 1 = {p 1 p 3 p 5 p 7 p 2 p 4 p 6 }. Chord structure is determied y eighourig susets: e.g., a triad is formed y takig 3 eighourig pitches, such as {p 1 p 3 p 5 } (the toic triad chord) or {p 5 p 7 p 2 } (the domiat triad chord). Root progressios are determied y pitch pairs ad are more atural he the pair lies close together ad the movemet is from right to left (from last to first pitch): e.g., root movemet y a third doard from p 3 p 1 is more atural tha p 6 p 2 (a fifth doard) or a third upard p 5 p 7 1. The hyrid 5- ad 4-part techiques are extesios to the stadard practice i toal harmoy. These ill ot e foud i other textooks. The hyrid 5-part techique is discussed i Schilliger s ook after the chapter o the seveth chord; as e ill see elo it comies the treatmet of the dissoat 7th ith the positive root cycles. The hyrid 4-part techique ivolves a geeralisatio of voice leadig i triadic chord progressio ith positive cycle root movemet. The hyrid 5-part techique allos the use of exteded chords ith acceptale voice leadig, hile prevetig the complete parallel voice movemet, so typical of sectioal harmoy settigs i jazz ig ad scores. Sice the availaility of the Schilliger ooks is limited, this documet presets a overvie of these techiques ith commets ad e examples. 2 Notatio We ill use the folloig otatio: 1 Note that this cocept deviates from covetioal theory of harmoy, here the domiat to root (a fifth do) chord progressio is the asis for chord cadeces. c F.G.J. Asil,.frasasil.l 1/19

2 Hyrid 5- ad 4-part harmoy F.G.J. Asil Tale 1: Hyrid 5-part chord structures Upper Structure S Root Chord structure S 5 S 7 S 9 S 11 S 13 p[i]: i-th pitch from a chord structure, ith i = 1,..., i max (ay pitch from a chord structure ith i max pitches). j = {1, 3, 5, 7, 9, 11, 13}: the fuctio of a pitch i a chord structure. Note that the fuctio group is ased o the orderig i thirds of a seve-pitch diatoic scale j = 1 is the root of the chord, j = 3 the third, j = 5 the fifth, etc. S k : chord structure ith k = {5, 7, 9, 11, 13} idicatig the tesio of the chord. The tesio is determied y the highest fuctio pitch i the chord structure. E.g., S 5 is a pure triad, S 7 a 7th chord, S 9 a 9th chord, etc. Special umers idicate chord iversios: S 6 is the first iversio, S 6 4 the secod iversio of a triad S 5. S 6 4 the first iversio, S4 3 the secod iversio ad S 2 the third iversio of a seveth chord S 7. R l, ith l = {±3, ±5, ±7}, idicates the root cycle, i.e., the root movemet i a chord progressio (goig from oe chord to the ext). R 3 meas the root is movig a third doard, R 3 meas a root movemet a third upard, etc. 3 Hyrid 5-part harmoy 3.1 Chord structure Valid hyrid 5-part chord structures are sho i Tale 1. The full chord structure is split ito a upper ad loer structure, that ill e treated differetly durig chord progressios. Note that the upper structure is a S 7 (4 eighourig pitches from the first expasio of the diatoic scale, yieldig a dissoat 7th iterval etee the loest ad upper fuctio). The S 5 i fact is a triad ith added 6th or 13th 2, i.e., S5 add6. Note that the itervals of the third etee fuctios are ot specified, so all kids of S 5 (major, mior, augmeted or dimiished triad), S 7 etc. are possile. What is e i this exteded techique is the fact that harmoic cotiuities, i.e., sequeces of these exteded chords may e used. This is differet from traditioal toal harmoy here sequeces of S 7 structures are the maximum tesio strig alloed; higher tesio chords (S 9 ad S 11 ) may occur oly as sigle, isolated chords (the must e properly prepared ad folloed y loer tesio chords). 3.2 Upper structure voice leadig The rules for the upper structure voice leadig are: 2 The added 6th is a traditioal ad frequetly used extesio to the toic or sudomiat triad. c F.G.J. Asil,.frasasil.l 2/19

3 Hyrid 5- ad 4-part harmoy F.G.J. Asil 1. At each chord progressio the dissoat upper fuctio of the chord (i.e., the 7th from the upper S 7 structure) is properly resolved y doard stepise movemet or slurred ito a commo ote i the ext chord 3. The eed for proper preparatio of the dissoat 7th is released The loer three voices of the upper structure are free to move i ay directio. Hoever, i practice stepise movemet or small leaps are preferred. Crossig of voices is avoided (although ot prohiited; e ill see occasioal voice crossig i the examples). If all voices move doard ith miimal movemet, the result ill look like jazz music sectioal harmoy, ut i geeral e o have more freedom i idepedet voice leadig. 3.3 Root movemet The root (loer structure) movemet is determied y the folloig rules: 1. The loer structure has a costat fuctio, i.e., chord root. 2. The root moves preferraly accordig to positive root cycles, i.e., R 3 (3rd do or 6th up), R 5 (5th do or 4th up) or R 7 (7th do or, more likely, 2d up). Occasioally egative root cycles (R 3, R 5 ad R 7 ) may e used. 3.4 Applicatio ad examples Chord progressios of hyrid 5-part harmoies ca e used i ay of the three forms of harmoic cotiuity, that Schilliger discers: 1. Type I: diatoic chord structures ad root movemets. Schilliger remarks that the scale used for uildig the chord structures ad the root progressio eed ot ecessarily e idetical, although the use of a sigle scale is cosidered (i his ords) ideal ad the latter case is preseted i the examples. 2. Type II: diatoic root movemet ith symmetric chord structures 5. Schilliger suggests usig this type of progressio to create higher tesio ad more harmoic variatio tha is possile ithi the Type I diatoic system. 3. Type III: symmetric chord structures ad root movemet 6. Chord structures may e of either costat (e.g., S 9 oly) or variale tesio. Let us cosider a first example, demostratig the three forms of harmoic cotiuity for costat chord structure. Example 1: Chord progressio usig costat tesio hyrid 5-part harmoy. We ill ase the example o the folloig diatoic 7-pitch scale: E 0 = {p 1,...,7 } = {d e f g a c }. The chord structures are ased o the first expasio (ito thirds) of this scale E 1 = {d f a c e g }. The diatoic root progressio is ased o the folloig cotiuity 3 Resolutio of the dissoat 7th may e delayed util a fe chords later, after a sequece of slurred otes. 4 The characteristic preparatio cosists of itroducig this pitch as a commo toe i the precedig chord, here it is a cosoat fuctio, i.e., 1, 3, or 5. Next-est is a upard stepise or leap movemet ito the dissoat 7th. 5 Symmetric meas that all chord structures are derived from the same diatoic scale ad the trasposed to the curret root. 6 Symmetric root movemet meas equal divisio of the octave ito either to dimiished 5ths, three major 3rds, four mior 3rds, six major 2ds or telve mior 2ds. c F.G.J. Asil,.frasasil.l 3/19

4 Hyrid 5- ad 4-part harmoy F.G.J. Asil R = R 3 + 2{R 3 + R 7 } + 3R 3 + R 5 (positive root cycles oly). The root cycle, startig o the toic pitch d is 7 : 2{R 3 +R 7 } {}}{ R = d R 3 R 3 g R 7 a R 3 f R 7 g 3R 3 {}}{ R 3 e R 3 c R 3 a R 5 d. The costat chord structure e ill use is S 9 ; o the root R = d this yields S 9 = {d} {f a c e}, i.e., a mior 7th chord F m 7 over a root d, also ritte as F m 7 /D. The diatoic progressio, sho i Fig. 1.a, demostrates the resolutio of the dissoat 7th i the upper chord structure y stepise doard movemet. The other voices move stepise or are slurred to the ext chord (ote the eharmoic equivalece). There is a geeral doard tred 8, cotiuous close positio voicig, ad o crossig of voices. Occasioal parallel perfect 5ths may occur i the ier voices. The 7-pitch scale ad root progressio ere carefully costructed to otai familiar exteded chord structures o oth the toic d ad domiat a. The diatoic-symetric progressio is sho i Fig. 1.. The upper chord structure is ased o the costat S 9 structure o the root d, i.e., F m 7 /D, hich is the appropriately trasposed for each susequet root. No e egi to see upard stepise movemet i the ier voices; see e.g., the f f f movemet i the loest voice of the upper structure. Symmetric progressio is demostrated i Fig. 1.c for various equal divisios of the octave, otated as R( 2). (Equal divisio of the octave ito chromatic steps, R( 12 2), is ot sho here). To costruct similar examples for other 7-pitch diatoic scales ad for other costat tesio chord structures is left as a exercise to the reader. Example 2: Chord progressio usig variale tesio hyrid 5-part harmoy. We ill costruct a harmoic progressio ith variale tesio usig the cotiuity S = S 5 + S 9 + S 13 + S 9 + S 7 + S 9 + S 5. We ill use the 7-pitch diatoic scale from the previous example, ut o the root cycle sequece is R = R 7 + R 3 + R 5 + 2R 7 + R 5. See Fig. 2.a, ad agai ote the treatmet of the dissoat 7th i the upper structure. There is either immediate resolutio y stepise doard movemet, or a slurred ote ito the ext chord (i case of a commo toe). I the latter case, the requiremet for doard resolutio is dropped as ca e see i the figure, here the slurred g i the loer voice of the upper structure fially moves upard. This is alloed, sice movig from the S 9 chord o the root f to the S 7 chord o root g, the f i the upper voice has ecome the dissoat ote, that eeds resolutio (doard step ito e). The example also demostrates the voice leadig for a partially ope voicig, as ca e see i Fig. 2. ad the secod progressio i Fig. 2.c. We still see the geeral doard tedecy, the occasioal parallel perfect 5ths, ad the avoidace of voice crossig. Play this example o a keyoard to experiece the comiatio of varyig chord tesio ad smooth voice leadig durig this cotiuity. The ext example ill demostrate the comiatio of various aspects of hyrid 5-part harmoy, 7 Note the use of groupig symols { }. Multiplicatig a group y a iteger a meas that this group has to repeated a times. Therefore, 2{R 3 + R 7} = {R 3 + R 7} + {R 3 + R 7} = R 3 + R 7 + R 3 + R 7, ot 2R 3 + 2R 7 = R 3 + R 3 + R 7 + R 7. 8 Schilliger discusses to techiques for couterig this doard tred, i.e., exchage ad octave-iversio of pairs of commo toes at the iterval of a third. c F.G.J. Asil,.frasasil.l 4/19

5 Hyrid 5- ad 4-part harmoy F.G.J. Asil (a) Type I: Diatoic harmoic progressio (costat tesio S 9 ) Schilliger Hyrid 5-part harmoy F.G.J. Asil Schilliger Hyrid 5-part harmoy F.G.J. Asil Schilliger Hyrid 5-part harmoy F.G.J. Asil Schilliger Hyrid () 5-part harmoy Type II: Diatoic-symmetric harmoic progressio (c) Type III: Symmetric harmoic progressio R( 2 2) R( 3 2) R( 4 2) R( 6 2) F.G.J. Asil,.frasasil.l F.G.J. Asil Figure 1: Hyrid 5-part harmoic cotiuity. Costat tesio S 9 progressio i the diatoic system (a), diatoic-symmetric system () ad symmetric system (c). Note the stepise doard movemet of the dissoat 7th iterval i the upper structure ad the voice leadig i the other upper voices (stepise movemet, o crossig voices, geeral doard tedecy). c F.G.J. Asil,.frasasil.l F.G.J. Asil,.frasasil.l 5/ F.G.J. Asil,.frasasil.l F.G.J. Asil,.frasasil.l

6 Hyrid 5- ad 4-part harmoy (a) Type I: Diatoic harmoic progressio (variale tesio) S 5 S 7 S 13 S 9 S 7 S 9 S 5 () Type II: Diatoic-symmetric harmoic progressio (c) Type III: Symmetric harmoic progressio F.G.J. Asil,.frasasil.l F.G.J. Asil,.frasasil.l 2 R( 3 2) Schilliger Hyrid 5-part harmoy R( 6 2) F.G.J. Asil,.frasasil.l F.G.J. Asil Figure 2: Hyrid 5-part harmoic cotiuity. Variale tesio progressio i the diatoic system (a), diatoic-symmetric system () ad symmetric system (c). Note the stepise doard movemet of the dissoat 7th iterval i the upper structure ad the voice leadig i the other upper voices (stepise movemet, o crossig voices, geeral doard tedecy). c F.G.J. Asil,.frasasil.l 6/19

7 Hyrid 5- ad 4-part harmoy F.G.J. Asil Example 3: Hyrid 5-part harmoic progressio ith variale tesio, progressio type, root cycles ad voicig. We ill costruct a harmoic progressio ith variale tesio usig the chord cotiuity S = S 5 + S 9 + S 5 + S S 9 + S 5 + S 13 + S 7 + S 9 + S 5. We ill use the 7-pitch diatoic scale from the first example, ut o the root cycle sequece is R = Diatoic Symmetric {}}{{}}{ Diatoic-Symmetric R 7 + R 5 + R 3 + R 5 + 3R( 4 {}}{ 2) +R 7 + R 3 + R 5 + R 7 +R 5, here after the 4th chord S[4] there is a chage from a variale tesio Type I Diatoic progressio to a costat tesio (S 9 ) Type III Symmetric progressio. The example returs to variale tesio diatoic progressio after S[8], the folloed y 3 chords, S[10] to S[12], i the Type II Diatoic-Symmetric system, efore the diatoic closig cadece S[12] S[13]. A possile voicig of this progressio is sho i Fig. 3. Dashed arlies idicate the chage i chord progressio type. Check the resolutio of the dissoat 7th i the upper structure. Occasioally there are exact parallel 5ths i the ier voices of the upper structure (see S[3] S[4] ad S[12] S[13]). There is tice the occurrece of a egative root cycle (from S[3] S[4] ad S[9] S[10]). These have received special hadlig ith respect to the susequet root progressio (ot from the poit of vie of voice leadig; ote ho the dissoat 7th i the upper structure is still properly resolved y doard stepise movemet). I oth cases the susequet root progressio is ased o the positive root cycle ad the comiatio of oth root movemets leads to a overall positive root movemet. This ca e otated as: R 3 {}}{ R = R 3 + R 5, or S = R 3 {}}{ S[3] R 3 S[4] R 5 S[5], S = R 3 {}}{ S[9] R 3 S[10] R 5 S[11]. Ispect the actual roots to cofirm this root cycle property; for S[4] S[6] e have a c (g = f ), ad for S[9] S[11] e have d g. The example demostrates aother feature. I S[1] S[2] ad S[9] S[10] there is the sappig of commo toes ( d ad later g, oth eighourig pitches i close voicig); this techique ca e used to couter the geeral doard tred i all voices. This example has ee carefully costructed to sho the comiatio of various aspects. Play it o the keyoard ad experiece the high tesio, quite dissoat, atmosphere; this is caused y the high desity of effects over a 13-chord sequece. A more relaxed distriutio of effects over a more exteded chord sequece might e preferred for a more musical effect. This cocludes our discussio of the hyrid 5-part techique. I the Schilliger ook the 5-part techique is preseted at the the ed of the chapter o the S 7 chord. The folloig sectio ill discuss the hyrid 4-part techique, hich differs quite a it from the 5-part equivalet; i Schilliger s ook it is discussed at the ed of the chapter o the S 11 chord structure. c F.G.J. Asil,.frasasil.l 7/19

8 2 Schilliger Hyrid 5-part harmoy Hyrid 5- ad 4-part harmoy F.G.J. Asil Diatoic Symmetric Diat-Sym S 5 S 9 S 5 S 11 S 9 S 9 S 9 S 9 S 5 S 13 S 7 S 9 S 5 R 7 R 5 R 3 R 5 3R( 4 2) R 7 R 3 R 5 R 7 R 5 Figure 3: Hyrid 5-part harmoic cotiuity ith variale tesio, progressio type, root cycles ad voicig. Note the sappig of commo toes to couter the doard tred (see S[1] S[2] ad S[9] S[10]). Tale 2: Hyrid 4-part chord structures (those marked ith a asterisk are less commoly used) Upper Structure Root Chord structure S 5 S 5 S 7 S 7 S 9 S 9 S 11 S 13 S 13 4 Hyrid 4-part harmoy 4.1 Chord structure Valid hyrid 4-part chord structures are sho i Tale 2. The upper structure o cotais 3 pitches, p[1], p[2], p[3]. Three upper structures are regular triads S 5 (see the 1st, 3rd ad 7th colum), to upper structures cotai the three essetial fuctios of a 7th chord structure (root, third ad seveth, j = 1, 3, 7, see the 4th ad 8th colum). 4.2 Upper structure voice leadig I the hyrid 4-part techique Schilliger allos greater freedom i voice leadig; upper structure voice leadig may e ased o ay of the 6 fudametal trasformatios possile for 3-pitch chord structures (see the Appedix). Give this set of possile trasformatios, Schilliger proposes the folloig guidelies: 1. Whe to eighourig chords have idetical chord structure 9, do ot use the S[j] 123 S[j + 1] (all pitches costat) trasformatio, sice this ill lead to complete parallelism. 2. Whe a group of to susequet chords is partly idetical, use a costat sigle pitch trasformatio (S[j] S[j + 1], S[j] S[j + 1], or S[j] S[j + 1]), he this leads to a parallel 9 Idetical chord structure meas that oth chords are costructed from the same colum i Tale 2, ad therefore cotai the same chord fuctios. c F.G.J. Asil,.frasasil.l 8/19

9 Hyrid 5- ad 4-part harmoy F.G.J. Asil desirale iterval ith the ass (parallel third or sixth). Do ot use these partly parallel trasformatios, he they yield a cosecutive parallel seveth or ith ith the ass. 3. Whe to susequet chords are totally differet (i.e., o equal chord fuctio i the upper structure. Examples are the comiatio of 1st ad 8th, or 2d ad 9th colum i Ta. 2), the 123 the trasformatio that keeps all pitches costat, S[j] S[j + 1], is the most favourale. Schilliger summarizes these guidelies as: homogeeous chord structures are compesated y heterogeeous trasformatios (i.e., o-costat fuctios, clockise or couterclockise rotatio of chord fuctios), ad the reverse. The mai goal is to prevet the occurrece of parallel octaves ad fifths i the voice leadig. This upper structure voice leadig guidelies say othig aout preparatio or proper resolutio of the higher chord fuctios j = 7, 9, 11 or 13; i that respect there is more freedom tha i the hyrid 5-part harmoic techique. Some of the chord trasformatios may lead to crossig voices, although i practice (see the examples) this happes rarely. 4.3 Root movemet Root movemet is determied y positive root cycles oly, i.e., R 3, R 5 or R 7. Whe creatig a overvie of all possile voice leadigs for a group of to hyrid 4-part harmoy e must rememer that there are five chord types (S 5, S 7, S 9, S 11 ad S 13 ), six trasformatios ad three root cycles; this leads to a huge set of comiatios, ith a total t of t = chord structure type {}}{ 2( ) 6 3 = 4320 possiilities. Schilliger s ook does ot preset all comiatios as score examples, ad most are left to the studet. Here e ill sho the voice leadig optios for oe idetical ad oe o-idetical group, ad discuss the result. We ill ase the diagrams o the familiar diatoic 7-pitch scale E 0 = {p 1,...,7 } = {d e f g a c }. The result is sho i Fig. 4 ad 5, here also crossig voices ad parallelism are idicated. The clockise ad couterclockise trasformatios maitai the close voicig, the three costat sigle pitch trasformatios chage the voicig type from close to ope voicig, hile the triple costat pitch trasformatio oviously leads to complete parallelism i the upper structure. Parallelism etee upper voices ad ass ill also occur; sometimes the parallel itervals are favourale (see the thirds i Fig. 4), sometimes these are ufavourale (see the fifths i Fig. 4, or the parallel seveth i the last trasformatio i Fig. 4.a). Rememer Schilliger s statemet aout homogeeous structures i comiatio ith heterogeeous trasformatios (ad the reverse), ad see hether the figures provide sufficiet evidece to cofirm that fact. I that case Fig. 4 should have less ojectioale features o the left, hile Fig. 5 should yield etter voice leadig o the right for costat pitch trasformatios. The evidece is ot covicig (author s opiio), ut the secod set of diagrams is a somehat extreme progressio (S 11 S 5 ) ad e ould really have to see the properties for a multitude of possiilities. 4.4 Applicatio ad examples After demostratig the optios for voice leadig he movig from oe hyrid 4-part chord structure to the ext, e ill o discuss a umer of examples for harmoic cotiuities. c F.G.J. Asil,.frasasil.l 9/19

10 Hyrid 5- ad 4-part harmoy F.G.J. Asil (a) Trasformatio: costat 1 costat 2 costat 3 costat 123 S 7 S 7 Crossig Parallel 3rd Parallel 5th Crossig Parallelism 2 Schilliger Hyrid 5-part harmoy R 3 () S 7 S 7 Parallel 3rd Crossig Parallelism 2 Schilliger Hyrid 5-part harmoy R 5 (c) S 7 S 7 Crossig Cross./Par. 3rd Parallel 5th Parallel 5th Parallelism 2 Schilliger Hyrid 5-part harmoy R 7 Figure 4: Hyrid 4-part trasformatios. Costat tesio S 7 for root cycle R 3 (a), R 5 () ad R 7 (c). All six possile trasformatios are preseted (see marks at the top of the example). Parallel movemet is cosidered iterally i the upper structure or ith relatio to the ass. c F.G.J. Asil,.frasasil.l 10/19

11 Hyrid 5- ad 4-part harmoy F.G.J. Asil (a) Trasformatio: costat 1 costat 2 costat 3 costat 123 S 11 S 5 Crossig Crossig Parallelism R 3 () S 11 S 5 Crossig Parallelism R 5 (c) S 11 S 5 Crossig Crossig Parallelism 3 Schilliger Hyrid 5-part harmoy R 7 Figure 5: Hyrid 4-part trasformatios. Variale tesio S 11 S 5 for root cycle R 3 (a), R 5 () ad R 7 (c). All six possile trasformatios are preseted (see marks at the top of the example). Parallel movemet is cosidered iterally i the upper structure or ith relatio to the ass. c F.G.J. Asil,.frasasil.l 11/19

12 Hyrid 5- ad 4-part harmoy F.G.J. Asil Example 4: Hyrid 4-part harmoic cotiuity ith costat tesio for all progressio types. Oce agai, e start from the diatoic 7-pitch scale: E 0 = {p 1,...,7 } = {d e f g a c }. Hoever, o e ill use a differet mode of that scale, startig o the toic e. We ill use a costat tesio S 9 chord structure, that cotais the fuctios S 9 = {1} {3 7 9} (see Tale 2). O the toic root this yields the folloig chord: e g d f, i covetioal chord otatio Em 9 o 5. The diatoic root progressio is give y R = R 5 + 2{R 7 + R 3 } + R 3 + R 7 (positive root cycles oly). The voice leadig for this harmoic cotiuity is sho i Fig. 6. The trasformatios i all progressio types aim at smooth voice leadig, i.e., slurrig commo otes ad miimize the movemet i each voice (stepise movemet preferred). Occasioaly this leads to crossig voices i the upper structure (idicated i the score, see Fig. 6.a ad ). Such crossig ca e preveted y a very ide voicig ad leapise movemet, as is demostrated i the first symmetric root progressio i Fig. 6.c. Note the geeral doard tred i the voice leadig i the Type I, II ad Type III for R( 6 2). Also ote that the highest fuctio i the upper structure (i.e., the 9, hich creates a dissoat 7th iterval ith the 3) frequetly is properly prepared. This meas that it is either eig slurred from the precedig chord or it is approached y a upard step. The dissoat iterval is also frequetly resolved y doard stepise movemet of the upper fuctio. Doig so, the voice leadig adheres to the priciples of traditioal toal harmoy. A exceptio is sho i the first to chords i Fig. 6.a, here the is approached y a doard leap from d. The trasformatios for the smooth voice leadig attempt i the Type III progressio are idicated i the score. These ere ot determied a priori; they tur out to e sequeces of either clockise or couter-clockise circular permutatios of chord fuctios. Sice e use costat tesio S 9 chord structures, there is the dager of udesirale parallel 7th or 9th itervals etee upper structure fuctios ad the root. The voice leadig shos that these have ee preveted. Next, e ill preset a somehat more exteded example ith variale chord tesio. Example 5: Hyrid 4-part harmoic cotiuity ith variale tesio ad mixed chord progressio type. We ill costruct a harmoic progressio ith variale tesio usig the chord cotiuity S = S 9 + 3S 5 + 2{S 9 + S 11 } + S 7 + 3S 13 + S 11 + S 9. There is a delierate gradual uild-up of chord tesio toards the ed of the progressio, ith a retur to a more stale S 9 o the fial chord. We ill use the 7-pitch diatoic scale (o toic e) from the previous example, ut o the root cycle sequece is R = Diatoic Symmetric {}}{{}}{ Diat.-Symm. R 3 + R 7 + R 5 + R 7 + 3R( 4 {}}{ 2) +R 7 + R 3 + R 3 + R 7 +R 5 + R 7, here after the 4th chord S[4] there is a chage from Type I Diatoic progressio to Type III Symmetric progressio. The example returs to variale tesio diatoic progressio after S[8], the folloed y 3 chords, S[10] to S[12], i the Type II Diatoic-Symmetric system, efore a diatoic closig cadece S[12] S[13] S[14]. The root progressio cotais positive root cycles oly ad has ee carefully costructed to start ad retur o the toic root of this modal scale. The trasformatios (ot idicated i the score i Fig. 7) have ee chose to yield smooth voice leadig (slurred commo otes, stepise movemet preferred, maximizig proper preparatio ad doard stepise resolutio of dissoat upper fuctios). A occasioal improper resolutio of the upper c F.G.J. Asil,.frasasil.l 12/19

13 Hyrid 5- ad 4-part harmoy F.G.J. Asil Score Score Crossig R( 6 2) Score Score Crossig Schilliger Hyrid Harmoy Schilliger Hyrid Harmoy Hyrid 4-part examples (a) Hyrid 4-part examples Schilliger Hyrid Harmoy () (c) Hyrid Hyrid 4-part 4-part examples examples R 5 R 7 R 3 R 7 R 3 R 3 R 7 R 5 R 7 R 3 R 7 R 3 R 3 R 7 Fras Asil Fras Asil Trasformatio: R( 2 2) R( 3 2) R( 4 2) Fras Asil Fras Asil Figure 6: Hyrid 4-part harmoic cotiuity. Costat tesio S 9, for Type I diatoic (a), Type II diatoic-symmetric (), ad Type III symmetric (c) harmoic progressio F.G.J. Asil,.frasasil.l c F.G.J. Asil,.frasasil.l 2015 F.G.J. Asil,.frasasil.l 13/ F.G.J. Asil,.frasasil.l 2015 F.G.J. Asil,.frasasil.l

14 Hyrid 5- ad 4-part harmoy F.G.J. Asil S 9 S 5 S 5 S 5 S 9 S 11 S 9 S 11 S 7 S 13 S 13 S 13 S 11 S 9 R 3 R 7 R 5 R 7 3R( 4 2) R 7 R 3 R 3 R 7 R 5 R 7 Figure 7: Hyrid 4-part harmoic cotiuity. Variale tesio, for mixed type harmoic progressio. structure fuctio 9 is sho i the progressio S[7] S[8], ith the upard stepise movemet f f F.G.J. Asil,.frasasil.l Let us have a look at aother variale tesio example. Example 6: Hyrid 4-part harmoic cotiuity ith variale tesio. I this example e ill use to modal variats from the diatoic scale, the first o the toic root g, the secod o e. The variale tesio chord structures are idicated i the score (see Fig. 8). The first modal variat is ased o a Type I Diatoic progressio, the secod o a Type III Symmetric progressio. Agai, e aim for a fairly smooth voice leadig. Both examples are free of vocie crossig, or do they cotai parallel 7th or 9th movemet etee upper fuctios ad roots. Dissoat itervals i the upper structure are regularly resolved properly. Note the exceptio i Fig. 6.a i S[2] S[3] ith the upard leap of the dissoat major 7th f a, or at the ed S[7] S[8] ith the upard stepise movemet d d. The secod modal variat o e delierately alterates S 7 S 11 ; these chord structures have oly oe upper structure commo fuctio, i.e., 7. Therefore, this heterogeeous group should yield a preferrece for costat fuctio trasformatios, i case of smooth voice leadig. The marks aove the staff idicate that there is ideed some evidece to support this rule (see the costat 123 trasformatios). Hoever, slurrig commo otes leads to a circular permutatio type of trasformatio. Note the toic root chord, a half-dimiished 7th chord E, that opes the example. Fortuately the closig chord o the same root has a differet structure (i.e., tesio S 11 ), sice the toic triad S 5 is highly usuitale for actig as a stale structure that ill cofirm the toality of the example 10. The fial example icludes a fe additioal aspects. Example 7: Hyrid 4-part harmoic cotiuity ith fully heterogeeous variale tesio chord structures. I this example e ill ivestigate the effect of trasformatio types o the voice leadig, he dealig ith completely heterogeous chord structures. Therefore, the harmoic cotiuity is ased o alteratig S 5 S 11 chord structures Rememer, that i hyrid 4-part chord structures the upper structure of the S 5 tesio chord cotais the fuctios {1 3 5}, hereas the S 11 upper structure cosists of {7 9 11}. So, ideed, these structures are completely dissimilar (o overlappig fuctios, although oth are perfect triads). The root sequece is give y R 3 {}}{ R = 2R 3 + 2R 5 + 2R 7 + R 3 + R 5 +R 3, 10 This example is of academic ature. I case of ritig real music, I ould have adapted the structures to yield a more stale chord o the toic root. c F.G.J. Asil,.frasasil.l 14/19

15 Hyrid 5- ad 4-part harmoy F.G.J. Asil (a) 2 Trasformatio: costat 1 costat 1 costat Schilliger 1 Hyrid Harmoy S 5 S 9 S 13 S 13 S 9 S 13 S 13 S 5 2 Schilliger Hyrid Harmoy R 3 R 3 R 5 R 7 R 7 R 5 R 5 () costat 123 costat 123 costat 123 S 7 S 11 S 7 S 11 S 7 S 11 S 7 R( 6 2) Figure 8: Hyrid 4-part harmoic cotiuity. Variale tesio, for Type I diatoic (a), ad Type III symmetric () harmoic progressio. c F.G.J. Asil,.frasasil.l 15/19

16 Hyrid 5- ad 4-part harmoy F.G.J. Asil a highly artificial sequece, sice it delierately cotais repeated positive root cycles (e.g., 2R 3 ), ut also a composite group of a egative plus a positive root cycle (the R 3 + R 5 group). The first solutio (see Fig. 9.a) is ased o variale trasformatios, aimig for smooth voice leadig. The resultig trasformatio type is either clockise or ati-clockise circular permutatio of chord fuctios. It is ot a costat fuctio type of trasformatio, the guidelie give y Schilliger for hadlig of heterogeeous chord groups. The costat 123 trasformatio (ot preseted here) ould lead to a set of parallel triads i the upper structure ith uacceptale parallel voicig. Applyig (or etter, forcig) a costat fuctio trasformatio is demostrated i Fig. 9. ad c. Note the alteratig close ad ope type voicig i the upper structure, ad the occasioal crossig voices. There are may ide leaps i the upper voices; usig this type of trasformatio (as opposed to the smooth voice leadig solutio i Fig. 9.a) ould affect the orchestratio i the case of ritig real music. I ould apply the variale trasformatio solutio to a harmoic (log otes, chorale type) settig for oed strigs (or for tromoes at the octave elo), hereas the costat pitch solutio ith its ide leaps suggests pizzicato strigs or oodid accets. The fial example cotaied a egative root cycle i a hyrid 4-part harmoic cotiuity. This does ot appear i the Schilliger ook, ad therefore as ot preseted as a root movemet optio at the egiig of this sectio (there, as a rule, e alloed positive root cycles oly). Hoever, voice leadig diagrams like Fig. 4 ad 5 may also e costructed for all six trasformatios ad for all three egative root cycles (this is left as a exercise to the studet). Whether this additioal destailisig effect to toality i the case of hyrid 4-part higher tesio chords is acceptale musically, is a matter of taste. The last example as meat to make the reader coscious of that optioal extesio to the techique. 5 Coclusio This documet presets to extesios to the traditioal theory of diatoic harmoy, as proposed y Schilliger [1], i.e., hyrid 5- ad 4-part harmoy. Rules ad guidelies for these techiques ere folloed y detailed discussio of examples. Note, hoever, that oly the chord progressio has ee cosidered. The examples sho herei do ot yield musical eauty; they oly demostrate voice leadig aspects, he usig the hyrid structures i a harmoic cotiuity. I order to create real music other elemets such as melody ad rhythm must e icluded i the score. Here e have discovered a techique to delierately create varyig tesio over a chord progressio. The composer may use this toolox of techiques, derived from traditioal harmoy, ad play ith ad cotrol chord tesio, chord progressio types ad trasformatios; this may lead to useful ad iterestig musical results. Try it! c F.G.J. Asil,.frasasil.l 16/19

17 Hyrid 5- ad 4-part harmoy 2 Schilliger Hyrid Harmoy 2 Schilliger Hyrid Harmoy F.G.J. Asil (a) Trasformatio: variale S 5 S 11 S 5 S 11 S 5 S 11 S 5 S 11 S 5 S 11 R 3 R 3 R 5 R 5 R 7 R 7 R 3 R 5 R 3 () Trasformatio: costat 1 S 5 S 11 S 5 S 11 S 5 S 11 S 5 S 11 S 5 S 11 Crossig R 3 R 3 R 5 R 5 R 7 R 7 R 3 R 5 R 3 (c) Trasformatio: costat 2 S 5 S 11 S 5 S 11 S 5 S 11 S 5 S 11 S 5 S 11 Crossig Crossig Crossig R 3 R 3 R 5 R 5 R 7 R 7 R 3 R 5 R 3 Figure 9: Hyrid 4-part trasformatios. Variale tesio S 5 S 11, Type I diatoic harmoic cotiuity, for variale (a), costat 1 (), ad costat 2 (c) trasformatio. Note the egative root cycle R 3 i the harmoic progressio. Also ote the alteratio etee closed ad ope voicig, ad the voice crossig for the costat 1 ad 2 trasformatios. c F.G.J. Asil,.frasasil.l 17/19

18 Hyrid 5- ad 4-part harmoy F.G.J. Asil Appedix: Trasformatio of chordal fuctios We ill cosider chord structures S ith 3 pitches or chord fuctios p[i], i = 1,..., 3 ad ill preset the voice leadig optios durig the trasformatio from oe chord S[j] 11 to the ext S[j + 1]. We ill desigate the i-th fuctio i the j-th chord as p[i, j]. We therefore ca rite the triadic chord structure as: S[j] = p[3, j] p[2, j] p[1, j] Sice the chord structure cotais 3 pitches this yields 3! = = 6 optioal trasformatios:. 1. A clockise circular permutatio of chord fuctios S[j] S[j + 1], here p[3, j] p[1, j + 1], p[2, j] p[3, j + 1], p[1, j] p[2, j + 1]. 2. A couterclockise circular permutatio of chord fuctios S[j] S[j + 1], here p[3, j] p[2, j + 1], p[2, j] p[1, j + 1], p[1, j] p[3, j + 1]. 3. Keep the 1st chord fuctio costat, hile sappig the 2d ad 3rd fuctio. This is otated as 1 S[j] S[j + 1], here p[3, j] p[2, j + 1], p[2, j] p[3, j + 1], p[1, j] p[1, j + 1]. 4. Keep the 2d chord fuctio costat, hile sappig the 1st ad 3rd fuctio. This is otated as 2 S[j] S[j + 1], here p[3, j] p[1, j + 1], p[2, j] p[2, j + 1], p[1, j] p[3, j + 1]. 5. Keep the 3rd chord fuctio costat, hile sappig the 1st ad 2d fuctio. This is otated as 3 S[j] S[j + 1], here p[3, j] p[3, j + 1], p[2, j] p[1, j + 1], p[1, j] p[2, j + 1]. 6. Keep all chord fuctio costat. This is otated as S[j] 123 S[j + 1], here p[3, j] p[3, j + 1], p[2, j] p[2, j + 1], p[1, j] p[1, j + 1]. I this documet trasformatios types 1 ad 2, i.e. (couter)clockise circular permutatio of chord fuctios, are referred to as heterogeeous trasformatios (usig Schilliger parlace). 11 Note that e use iteger suscripts (e.g., S 7) to idicate the tesio of the chord, hile the iteger idex [j] idicates the positio of the chord i a sequece (i.e., i the time domai). The iteger idex [i] idicates the simultaeous soudig of pitches p[i] i a chord structure (ordered from lo to high pitches). c F.G.J. Asil,.frasasil.l 18/19

19 Hyrid 5- ad 4-part harmoy F.G.J. Asil Refereces [1] Joseph Schilliger. The Schilliger System of Musical Compositio, volume I ad II of Da Capo Press Music Reprit Series. Da Capo Press, Ne York, fourth editio, ISBN ad xxiii pp. c F.G.J. Asil,.frasasil.l 19/19