Schillinger's Special Theory of Harmony: Hybrid 5- and 4-Part Harmony

Hybrid 5- and 4-part harmony F.G.J. Absil

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

F.G.J. Absil

February 9, 2008

Abstract This document describes two techniques as presented by Schillinger [1] in his Special Theory of Harmony book. Chord structures, progressions and application aspects of the hybrid 5- and 4-part harmony technique will be discussed and illustrated with score examples. Document history: revised (typing errors corrected, minor edits) August 16, 2010, (score and other errors corrected) August 2015.

1 Introduction

Schillingers Book 5: The Special Theory of Harmony [1] presents a great summary of tonal 4-part harmony. Tonal harmony implies that chord structures and progressions are based on a ordering in thirds of seven-pitch diatonic scales (Schillinger calls this the first expansion form of the scale). This means that if we write a 7-pitch scale on a tonic p1 as an ordered set (ascending diatonic steps of a second) E0 = {p1 p2 p3 p4 p5 p6 p7}, then the first expansion (into a sequence of thirds) is E1 = {p1 p3 p5 p7 p2 p4 p6}. Chord structure is determined by neighbouring subsets: e.g., a triad is formed by taking 3 neighbouring pitches, such as {p1 p3 p5} (the tonic triad chord) or {p5 p7 p2} (the dominant triad chord). Root progressions are determined by pitch pairs and are more natural when the pair lies close together and the movement is from right to left (from last to first pitch): e.g., root movement by a third downward from 1 p3 → p1 is more natural than p6 → p2 (a fifth downward) or a third upward p5 → p7 . The hybrid 5- and 4-part techniques are extensions to the standard practice in tonal harmony. These will not be found in other textbooks. The hybrid 5-part technique is discussed in Schillinger’s book after the chapter on the seventh chord; as we will see below it combines the treatment of the dissonant 7th with the positive root cycles. The hybrid 4-part technique involves a generalisation of voice leading in triadic chord progression with positive cycle root movement. The hybrid 5-part technique allows the use of extended chords with acceptable voice leading, while preventing the complete parallel voice movement, so typical of sectional harmony settings in jazz big band scores. Since the availability of the Schillinger books is limited, this document presents an overview of these techniques with comments and new examples.

2 Notation

We will use the following notation:

1Note that this concept deviates from conventional theory of harmony, where the dominant to root (a ﬁfth down) chord progression is the basis for chord cadences.

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Table 1: Hybrid 5-part chord structures

Upper 5 7 9 11 13 Structure 3 5 7 9 11 S7 1 3 5 7 9 13 1 3 5 7

Root 1 1 1 1 1

Chord structure S5 S7 S9 S11 S13

p[i]: i-th pitch from a chord structure, with i = 1, . . . , imax (any pitch from a chord structure with imax pitches).

j = {1, 3, 5, 7, 9, 11, 13}: the function of a pitch in a chord structure. Note that the function group is based on the ordering in thirds of a seven-pitch diatonic scale j = 1 is the root of the chord, j = 3 the third, j = 5 the ﬁfth, etc.

Sk: chord structure with k = {5, 7, 9, 11, 13} indicating the tension of the chord. The tension is determined by the highest function pitch in the chord structure. E.g., S5 is a pure triad, S7 a 7th 6 chord, S9 a 9th chord, etc. Special numbers indicate chord inversions: S6 is the ﬁrst inversion, S4 6 4 the second inversion of a triad S5. S4 the ﬁrst inversion, S3 the second inversion and S2 the third inversion of a seventh chord S7.

Rl, with l = {±3, ±5, ±7}, indicates the root cycle, i.e., the root movement in a chord progression (going from one chord to the next). R3 means the root is moving a third downward, R−3 means a root movement a third upward, etc.

3 Hybrid 5-part harmony

3.1 Chord structure Valid hybrid 5-part chord structures are shown in Table 1. The full chord structure is split into an upper and lower structure, that will be treated differently during chord progressions. Note that the upper structure is an S7 (4 neighbouring pitches from the ﬁrst expansion of the diatonic scale, yielding a dissonant 7th interval between the lowest and upper function). The S5 in fact is a triad with added 6th or 2 add6 13th , i.e., S5 . Note that the intervals of the third between functions are not speciﬁed, so all kinds of S5 (major, minor, augmented or diminished triad), S7 etc. are possible. What is new in this extended technique is the fact that harmonic continuities, i.e., sequences of these extended chords may be used. This is different from traditional tonal harmony where sequences of S7 structures are the maximum tension string allowed; higher tension chords (S9 and S11) may occur only as single, isolated chords (the must be properly prepared and followed by lower tension chords).

3.2 Upper structure voice leading The rules for the upper structure voice leading are:

2The added 6th is a traditional and frequently used extension to the tonic or subdominant triad.

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1. At each chord progression the dissonant upper function of the chord (i.e., the 7th from the upper S7 structure) is properly resolved by downward stepwise movement or slurred into a common note in the next chord3. The need for proper preparation of the dissonant 7th is released4.

2. The lower three voices of the upper structure are free to move in any direction. However, in practice stepwise movement or small leaps are preferred. Crossing of voices is avoided (although not prohibited; we will see occasional voice crossing in the examples). If all voices move downward with minimal movement, the result will look like jazz music sectional harmony, but in general we now have more freedom in independent voice leading.

3.3 Root movement The root (lower structure) movement is determined by the following rules:

1. The lower structure has a constant function, i.e., chord root.

2. The root moves preferrably according to positive root cycles, i.e., R3 (3rd down or 6th up), R5 (5th down or 4th up) or R7 (7th down or, more likely, 2nd up). Occasionally negative root cycles (R−3,R−5 and R−7) may be used.

3.4 Application and examples Chord progressions of hybrid 5-part harmonies can be used in any of the three forms of harmonic continuity, that Schillinger discerns:

1. Type I: diatonic chord structures and root movements. Schillinger remarks that the scale used for building the chord structures and the root progression need not necessarily be identical, although the use of a single scale is considered (in his words) ideal and the latter case is presented in the examples.

2. Type II: diatonic root movement with symmetric chord structures5. Schillinger suggests using this type of progression to create higher tension and more harmonic variation than is possible within the Type I diatonic system.

3. Type III: symmetric chord structures and root movement6.

Chord structures may be of either constant (e.g., S9 only) or variable tension. Let us consider a ﬁrst example, demonstrating the three forms of harmonic continuity for constant chord structure.

Example 1: Chord progression using constant tension hybrid 5-part harmony. We will base the example on the following diatonic 7-pitch scale: E0 = {p1,...,7} = {d − e − f] − g − a − b[ − c]}. The chord structures are based on the ﬁrst expansion (into thirds) of this scale E1 = {d − f] − a − c] − e − g − b[}. The diatonic root progression is based on the following continuity

3Resolution of the dissonant 7th may be delayed until a few chords later, after a sequence of slurred notes. 4The characteristic preparation consists of introducing this pitch as a common tone in the preceding chord, where it is a consonant function, i.e., 1, 3, or 5. Next-best is an upward stepwise or leap movement into the dissonant 7th. 5Symmetric means that all chord structures are derived from the same diatonic scale and then transposed to the current root. 6Symmetric root movement means equal division of the octave into either two diminished 5ths, three major 3rds, four minor 3rds, six major 2nds or twelve minor 2nds.

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R~ = R3 + 2{R3 + R7} + 3R3 + R5 (positive root cycles only). The root cycle, starting on the tonic pitch d is7: 2{R3+R7} 3R3 z }| { z }| { R3 R3 R7 R3 R7 R3 R3 R3 R5 R~ = d − b[ − g − a − f] − g − e − c] − a − d.

The constant chord structure we will use is S9; on the root R = d this yields S9 = {d}−{f]−a−c]−e}, i.e., a minor 7th chord F ]m7 over a root d, also written as F ]m7/D. The diatonic progression, shown in Fig. 1.a, demonstrates the resolution of the dissonant 7th in the upper chord structure by stepwise downward movement. The other voices move stepwise or are slurred to the next chord (note the enharmonic equivalence). There is a general downward trend8, continuous close position voicing, and no crossing of voices. Occasional parallel perfect 5ths may occur in the inner voices. The 7-pitch scale and root progression were carefully constructed to obtain familiar extended chord structures on both the tonic d and dominant a. The diatonic-symetric progression is shown in Fig. 1.b. The upper chord structure is based on the constant S9 structure on the root d, i.e., F ]m7/D, which is then appropriately transposed for each subsequent root. Now we begin to see upward stepwise movement in the inner voices; see e.g., the f] − f\ − f] movement in the lowest voice of the upper structure. Symmetric√ progression is demonstrated in Fig. 1.c for various√ equal divisions of the octave, notated as R( n 2). (Equal division of the octave into chromatic steps, R( 12 2), is not shown here). 2 To construct similar examples for other 7-pitch diatonic scales and for other constant tension chord structures is left as an exercise to the reader.

Example 2: Chord progression using variable tension hybrid 5-part harmony. We will construct a harmonic progression with variable tension using the continuity

S~ = S5 + S9 + S13 + S9 + S7 + S9 + S5.

We will use the 7-pitch diatonic scale from the previous example, but now the root cycle sequence is

R~ = R7 + R3 + R5 + 2R7 + R5.

See Fig. 2.a, and again note the treatment of the dissonant 7th in the upper structure. There is either immediate resolution by stepwise downward movement, or a slurred note into the next chord (in case of a common tone). In the latter case, the requirement for downward resolution is dropped as can be seen in the ﬁgure, where the slurred g in the lower voice of the upper structure ﬁnally moves upward. This is allowed, since moving from the S9 chord on the root f] to the S7 chord on root g, the f] in the upper voice has become the dissonant note, that needs resolution (downward step into e). The example also demonstrates the voice leading for a partially open voicing, as can be seen in Fig. 2.b and the second progression in Fig. 2.c. We still see the general downward tendency, the occasional parallel perfect 5ths, and the avoidance of voice crossing. 2 Play this example on a keyboard to experience the combination of varying chord tension and smooth voice leading during this continuity. The next example will demonstrate the combination of various aspects of hybrid 5-part harmony,

7Note the use of grouping symbols {}. Multiplicating a group by an integer a means that this group has to repeated a times. Therefore, 2{R3 + R7} = {R3 + R7} + {R3 + R7} = R3 + R7 + R3 + R7, not 2R3 + 2R7 = R3 + R3 + R7 + R7. 8Schillinger discusses two techniques for countering this downward trend, i.e., exchange and octave-inversion of pairs of common tones at the interval of a third.

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(a) Schillinger Hybrid 5-part harmony F.G.J. Absil Type I: Diatonic harmonic progression (constant tension S9)

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w w w w #w w #w w w w w w & bw #w bw w bw w bw #w bw #w bw #w w w w w w w w w w ? w w w w w w Hybrid 5- and 4-part harmony F.G.J. Absil

Example 3: Hybrid 5-part harmonic progression with variable tension, progression type, root cycles and voicing. We will construct a harmonic progression with variable tension using the chord continuity

S~ = S5 + S9 + S5 + S11 + 4S9 + S5 + S13 + S7 + S9 + S5.

We will use the 7-pitch diatonic scale from the ﬁrst example, but now the root cycle sequence is

Diatonic Symmetric Diatonic-Symmetric z }| { z }|√4 { z }| { R~ = R7 + R5 + R−3 + R5 + 3R( 2) +R7 + R−3 + R5 + R7 +R5, where after the 4th chord S[4] there is a change from a variable tension Type I Diatonic progression to a constant tension (S9) Type III Symmetric progression. The example returns to variable tension diatonic progression after S[8], then followed by 3 chords, S[10] to S[12], in the Type II Diatonic-Symmetric system, before the diatonic closing cadence S[12] → S[13]. A possible voicing of this progression is shown in Fig. 3. Dashed barlines indicate the change in chord progression type. Check the resolution of the dissonant 7th in the upper structure. Occasionally there are exact parallel 5ths in the inner voices of the upper structure (see S[3] → S[4] and S[12] → S[13]). There is twice the occurrence of a negative root cycle (from S[3] → S[4] and S[9] → S[10]). These have received special handling with respect to the subsequent root progression (not from the point of view of voice leading; note how the dissonant 7th in the upper structure is still properly resolved by downward stepwise movement). In both cases the subsequent root progression is based on the positive root cycle and the combination of both root movements leads to an overall positive root movement. This can be notated as: R3 z }| { R~ = R−3 + R5, or R3 R3 z }| { z }| { R−3 R5 R−3 R5 S~ = S[3] − S[4] − S[5], S~ = S[9] − S[10] − S[11] . Inspect the actual roots to conﬁrm this root cycle property; for S[4] → S[6] we have a − c] − (g[ = f]), and for S[9] → S[11] we have b[ − d − g. The example demonstrates another feature. In S[1] → S[2] and S[9] → S[10] there is the swapping of common tones (b[ )* d and later b[ )* g, both neighbouring pitches in close voicing); this technique can be used to counter the general downward trend in all voices. This example has been carefully constructed to show the combination of various aspects. Play it on the keyboard and experience the high tension, quite dissonant, atmosphere; this is caused by the high density of effects over a 13-chord sequence. A more relaxed distribution of effects over a more extended chord sequence might be preferred for a more musical effect. 2 This concludes our discussion of the hybrid 5-part technique. In the Schillinger book the 5-part technique is presented at the the end of the chapter on the S7 chord. The following section will discuss the hybrid 4-part technique, which differs quite a bit from the 5-part equivalent; in Schillinger’s book it is discussed at the end of the chapter on the S11 chord structure.

c 2008-2015 F.G.J. Absil, www.fransabsil.nl 7/19 2 Schillinger Hybrid 5-part harmony

#w w #w bw w w w & b ww # nw #Hybridw 5- andbn w 4-part harmonyw w #w F.G.J. Absil w # w nw w bw nw w ? Diatonicw w #w Symmetricw Diat-Sym w S5 S9 S5 S11 S9 S9 bw S9 Sbw9 S5 S13 S7 S9 S5 #w b w b w w w nw w b w w #w #w bww w w w b bw n w w #w w b#ww n w w b ww & # w ww w ww n w ww w # w #nww w w ##w w ? bw w bw w w w w #w √bw w w w w 4 R7 R5 R−3 R5 3R( 2) R7 R−3 R5 R7 R5 Figure 3: Hybridw 5-part harmonic#w continuity with#w variable tension,#w progressionw type, root cycles and voicing.& ## Notew thew swapping##w ofw common##w tonesw to counter##w the downwardw ##w trendw (see S[1]##w→ Sw[2] and S[9] → S[10]). w w w ? w bw w bw w bw w bw w bw w bw Table 2: Hybrid 4-part chord structures (those marked with an asterisk are less commonly used) bw #w #w #w bw Upper#w bw 5#w 5w 7 7#w 9w 9 11#w 13b w 13 #w b w & # w w Structure# w w 3# w 3w 5 3# w 7w 7# 9w 9w 11 # w 1 13bw 3 1 3 1 7 7 7 ? w w w w w w w Rootw 1 1w 1 1 1w 1 1 1w 1 w ∗ ∗ ∗ ∗ Chord structure S5 S5 S7 S7 S9 S9 S11 S13 S13 bw w #w #w bw #w w #w bw #w w #w bw & # w w # w w # w w # w w # w w # w w 4 Hybridb 4-partw harmony bw ? w w w w w w 4.1 Chordw structure w w w w w Valid hybrid 4-part chord structures are shownw in Table 2. The upper structure now contains 3 pitches, p[1], p[2]w, p[3]#.#w Three upperw structures arew regular#w triads Sw5 (see#w the 1st, 3rdw and#w 7th column),w two#w upper structures& bw containw the threebw essential#w functionsbw # w of a 7thbw chordw structurebw (root,# w third andbw seventh,# w j = 1, 3, 7, see the 4th and 8th column).# w #w w ? w #w w #w w #w w #w w #w w #w 4.2 Upper structure voice leading In the hybrid 4-part technique Schillinger allows greater freedom in voice leading; upper structure voice leading mayw bew based onw any# ofw the 6 fundamentalw #w transformationsw w possiblew w for 3-pitchw chord structures& (seebw the Appendix).#w bw Givenw this setbw of possiblew transformations,bw #w Schillingerbw #w proposesbw the following#w guidelines: w w w w w w w w w 123 1.? When twow neighbouring chordsw have identicalw chord structurew 9, do not usew the S[j] −→ Sw[j + 1] (all pitches constant) transformation, since this will lead to complete parallelism.

2. When a group of two subsequent chords is partly identical, use a constant single pitch transfor- ¯ ¯ ¯ mation (S[j] →1 S[j + 1], S[j] →2 S[j + 1], or S[j] →3 S[j + 1]), when this leads to a parallel

9Identical chord structure means that both chords are constructed from the same column in Table 2, and therefore contain the same chord functions.

c 2008-2015 F.G.J. Absil, www.fransabsil.nl 8/19 Hybrid 5- and 4-part harmony F.G.J. Absil

desirable interval with the bass (parallel third or sixth). Do not use these partly parallel transformations, when they yield a consecutive parallel seventh or ninth with the bass.

3. When two subsequent chords are totally different (i.e., no equal chord function in the upper structure. Examples are the combination of 1st and 8th, or 2nd and 9th column in Tab. 2), then the transformation that keeps all pitches constant, S[j] −→123 S[j + 1], is the most favourable. Schillinger summarizes these guidelines as: homogeneous chord structures are compensated by heterogeneous transformations (i.e., non-constant functions, clockwise or counterclockwise rota- tion of chord functions), and the reverse. The main goal is to prevent the occurrence of parallel octaves and ﬁfths in the voice leading.

This upper structure voice leading guidelines say nothing about preparation or proper resolution of the higher chord functions j = 7, 9, 11 or 13; in that respect there is more freedom than in the hybrid 5-part harmonic technique. Some of the chord transformations may lead to crossing voices, although in practice (see the examples) this happens rarely.

4.3 Root movement

Root movement is determined by positive root cycles only, i.e., R3,R5 or R7. When creating an overview of all possible voice leadings for a group of two hybrid 4-part harmony we must remember that there are ﬁve chord types (S5,S7,S9,S11 and S13), six transformations and three root cycles; this leads to a huge set of combinations, with a total nt of

chord structure type z }| { nt = 2(5 × 4 × 3 × 2) ×6 × 3 = 4320 possibilities. Schillinger’s book does not present all combinations as score examples, and most are left to the student. Here we will show the voice leading options for one identical and one non-identical group, and discuss the result. We will base the diagrams on the familiar diatonic 7-pitch scale E0 = {p1,...,7} = {d − e − f] − g − a − b[ − c]}. The result is shown in Fig. 4 and 5, where also crossing voices and parallelism are indicated. The clockwise and counterclockwise transformations maintain the close voicing, the three constant single pitch transformations change the voicing type from close to open voicing, while the triple constant pitch transformation obviously leads to complete parallelism in the upper structure. Parallelism between upper voices and bass will also occur; sometimes the parallel intervals are favourable (see the thirds in Fig. 4), sometimes these are unfavourable (see the fifths in Fig. 4, or the parallel seventh in the last transformation in Fig. 4.a). Remember Schillinger’s statement about homogeneous structures in combination with heterogeneous transformations (and the reverse), and see whether the figures provide sufficient evidence to confirm that fact. In that case Fig. 4 should have less objectionable features on the left, while Fig. 5 should yield better voice leading on the right for constant pitch transformations. The evidence is not convincing (author’s opinion), but the second set of diagrams is a somewhat extreme progression (S11 − S5) and we would really have to see the properties for a multitude of possibilities.

4.4 Application and examples After demonstrating the options for voice leading when moving from one hybrid 4-part chord structure to the next, we will now discuss a number of examples for harmonic continuities.

c 2008-2015 F.G.J. Absil, www.fransabsil.nl 9/19 2 Schillinger Hybrid 5-part harmony

#w w Hybrid#w 5- andbw 4-part harmonyw w w F.G.J. Absil & b ww # nw # w bn w w w #w 2 w # w nSchillingerw Hybridw 5-partb harmonyw nw w ? w w #w w w #w w #w bbw bw w w & b ww # nw # w bn w w w #w 2 w ##w nwSchillinger Hybridw 5-partb wharmony nw w b w w b w w w nw w b w w #w w #w ?& bww w w w #bwbw n w (a)w ##w n w b#ww n w #w b ww Transformation:# ww w w w n w w bw w bww # w w w ww # w w #w w #w constantb1¯w constantw 2¯ wconstant 3¯w constant 123 & b ww # nw # wbw bn w ww bwnw #ww w ?S − Sw !"w ##ww nw bw w 7 w7 b w w b #ww w w nw bww b w w w #ww #w bww w wCrossingw bw Paralleln w 3rdw Parallel#w 5thw b Crossing#ww n w Parallelismw b w &?# ww ww ww ww #bwn w ww w # w #nww w ww #w w bw bw w w # w #w w #w #w #w #w #ww #w bw #w w #ww ?& # ww ww #w# w w bw# w bww # w w # w w w w # w ww b w w b#ww w ww nww w w b w w #w w#w bww w w w b bw n w w #w w b#ww n w w b ww ?& # w ww w ww n w ww w # w #nww w w ##w w w bw w #bww w #bw w #bw w bw w bw #wR3 w #w w #w #w #w w #w &? # w ww # w w b#w w bww # w w w bw# w w w # ww w w w #w w w bw w w w #w #w w #w w (b) #w w #w #w #w #w ?S7 − Sw7 bw w bw w w w b w w b w & # ww bww # ww bww # ww bw # w bw # w bw # w bw w #w Parallel#bw 3rd Crossing#w w Parallelism & ##w w ##w w ##w w ##w w ##w w ##w w ? w w w w w bww w w w #w bww #w bww #w w #w w #w b#w #w b#w &? # ww w # ww w # ww w # ww w # ww w # ww bw bw bbww bw bw bw ?& #w w #w bw #w w #w bw #w w #w bw # ww bw # ww w # ww # w bw # w bw # w R w ww ww ww #ww #ww & ##w 5 bw ##w bw ##w w ##w w ##w b w ##w b w ? w w w w w bbww w w w w w w w (c) w w w S −#Sw #w bw #w #w bw #w #w bw &?7 # ww7 w # ww w # ww w # ww w # ww w # ww w Crossingbww w Cross./Par.w 3rd Parallel 5thw Parallelbww 5th Parallelismw w #w w w w w #w w w ?& bw # ww bw ww bw #ww bw ww bw #ww bw #ww w w ##w w # bww w #ww w # w w # w #w #w bw #w w #w bw #w ww #w bw ?& #ww #ww #ww #ww #ww #ww #ww #ww #ww #ww #ww #ww #bww w #w bw bw # w bw bw #w bw bw #w bw #w &? ww ww ww #ww ww # ww ww ww ww # ww ww # ww # w #w w w R7 w w #w w #w w w w w w ?& bww ##ww bww #ww bww #w bw ##w bw ##w bw ##w Figure 4: Hybrid 4-part transformations. Constantw tension S for root cycle R (a), R (b) andwR (c). w #w w w w 7#w w 3 5 w 7 All& six possiblebw # w transformationsbw #w are presentedbw ## (seew marksbw at thew top of thebw example).##w Parallelbw # movement#w is considered? internallyw in the# upperw structure orw with relation to#w the bass. ww w w w w #w w #w w w w w w &? bww ##ww bww #ww bww #ww bww ##ww bww ##ww bww ##ww w w w ? w w w w w w w w w w ww w w w w #w w #w w w w w w & bw #w bw w bw w bw #w bw #w bw #w w w w c 2008-2015 F.G.J. Absil, www.fransabsil.nl 10/19 ? w w w w w w w w w w w w 2 Schillinger Hybrid 5-part harmony

#w w #w bw & b ww # nw # w bn w w w w 2 w # w nSchillingerw Hybridw 5-partb wharmony nw #w ? w w #w w w b#ww nw #w b bw bw w w & w ## w #nw n w w nw #w #w w bw w w b w w b w w w nw w b w w #w w #w ? bwww w ww w #bwbw n w w #w w b#ww n ww w b ww & # w ww w ww n w wwbw w # bww #nww w w w ##w w bw w bw w w ? w b w #w b#w w bw nw w b w w #w #w bww w w w b bw n w w #w w b#ww n w w b ww & # w ww w ww n w ww w # w #nww w w ##w w w w #w bw #w w #w bw w w w &? ##ww ww ##w #ww ##w bww w ##w w ##w w w ##w w w w w w w ? w bw w #bw w #bw w #bw w bw w bw & ##w w ##w w ##w w ##w w ##w w ##w w w w w Hybrid 5- and 4-part harmonybw #w F.G.J.#w Absil &? #w bw #w bw #w w #w w #w b w #w b w # w bw # w bw # w bw # w bw # w bw # w bw ? w w w w bw w #w w #w & #w bw #w bw #w w #w w #w b w #w b w # w w # w w # w bw # w w # w w # w bw w ? #w #w bw #w w (a) #w bw #w w #w bw &Transformation:# w w # w w # w w # w w # w w # w w bw constant 1¯ constant 2¯ constantbw 3¯ constant 123 ?S − S !"w w bw w w w 11 w5 w w w w w w w w w & #w w #Crossingw bw #w w #w bw # Crossingw w Parallelism#w bw # w bw # w w # w # w w # w bw # w w #w w w w w #w w w &? bww # ww bww #ww bww ##ww bww ww bww ##ww bww ##ww # w #w w ? w #w w #w w #w w #w w #w w #w bw ##w bw bw #w bw #w bw #w bw #w & w R3 w w #w w # w w w w # w w # w # w #w w (b) w ? w #w w #w w #w w #w w #w w #w &S11b−wS5 #w bw w bw w bw #w bw #w bw #w w Crossingw Parallelismw ? w w w w w w w w w #w w #w w w w w w w & bw #w bw w bw w bw #w bw #w bw #w w w w ? w w w w w w w w w w w w R5

(c) S11 − S5 CrossingSchillinger Crossing Hybrid 5-part harmony Parallelism 3 bw w w w w w w w w #w w #w & bw #w bw #w bw #w bw #w bw w bw w w b w w ? w bw w bw w bw w bw w bw w bw R7

Figure 5: Hybrid 4-part transformations. Variable tension S11 − S5 for root cycle R3 (a), R5 (b) and R7 &(c). All six possible∑ transformations are∑ presented (see marks∑ at the top of the example).∑ Parallel movement is considered internally in the upper structure or with relation to the bass. ? ∑ ∑ ∑ ∑

& ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

c 2008-2015 F.G.J. Absil, www.fransabsil.nl 11/19 ? ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

& ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

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? ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ Hybrid 5- and 4-part harmony F.G.J. Absil

Example 4: Hybrid 4-part harmonic continuity with constant tension for all progression types. Once again, we start from the diatonic 7-pitch scale: E0 = {p1,...,7} = {d − e − f] − g − a − b[ − c]}. However, now we will use a different mode of that scale, starting on the tonic e. We will use a constant tension S9 chord structure, that contains the functions S9 = {1} − {3 − 7 − 9} (see Table 2). On the 9 tonic root this yields the following chord: e − g − d − f], in conventional chord notation Emno 5. The diatonic root progression is given by R~ = R5 + 2{R7 + R3} + R3 + R7 (positive root cycles only). The voice leading for this harmonic continuity is shown in Fig. 6. The transformations in all progression types aim at smooth voice leading, i.e., slurring common notes and minimize the movement in each voice (stepwise movement preferred). Occasionaly this leads to crossing voices in the upper structure (indicated in the score, see Fig. 6.a and b). Such crossing can be prevented by a very wide voicing and leapwise movement, as is demonstrated in the ﬁrst symmetric root progression in Fig. 6.c. √ Note the general downward trend in the voice leading in the Type I, II and Type III for R( 6 2). Also note that the highest function in the upper structure (i.e., the 9, which creates a dissonant 7th interval with the 3) frequently is properly prepared. This means that it is either being slurred from the preceding chord or it is approached by an upward step. The dissonant interval is also frequently resolved by downward stepwise movement of the upper function. Doing so, the voice leading adheres to the principles of traditional tonal harmony. An exception is shown in the ﬁrst two chords in Fig. 6.a, where the b[ is approached by a downward leap from d. The transformations for the smooth voice leading attempt in the Type III progression are indicated in the score. These were not determined a priori; they turn out to be sequences of either clockwise or counter-clockwise circular permutations of chord functions. Since we use constant tension S9 chord structures, there is the danger of undesirable parallel 7th or 9th intervals between upper structure functions and the root. The voice leading shows that these have been prevented. 2 Next, we will present a somewhat more extended example with variable chord tension.

Example 5: Hybrid 4-part harmonic continuity with variable tension and mixed chord progression type. We will construct a harmonic progression with variable tension using the chord continuity

S~ = S9 + 3S5 + 2{S9 + S11} + S7 + 3S13 + S11 + S9.

There is a deliberate gradual build-up of chord tension towards the end of the progression, with a return to a more stable S9 on the ﬁnal chord. We will use the 7-pitch diatonic scale (on tonic e) from the previous example, but now the root cycle sequence is

Diatonic Symmetric Diat.-Symm. z }| { z }|√4 { z }| { R~ = R3 + R7 + R5 + R7 + 3R( 2) +R7 + R3 + R3 + R7 +R5 + R7, where after the 4th chord S[4] there is a change from Type I Diatonic progression to Type III Symmetric progression. The example returns to variable tension diatonic progression after S[8], then followed by 3 chords, S[10] to S[12], in the Type II Diatonic-Symmetric system, before a diatonic closing cadence S[12] → S[13] → S[14]. The root progression contains positive root cycles only and has been carefully constructed to start and return on the tonic root of this modal scale. The transformations (not indicated in the score in Fig. 7) have been chosen to yield smooth voice leading (slurred common notes, stepwise movement preferred, maximizing proper preparation and downward stepwise resolution of dissonant upper functions). An occasional improper resolution of the upper

c 2008-2015 F.G.J. Absil, www.fransabsil.nl 12/19 Hybrid 5- and 4-part harmony F.G.J. Absil

Score Schillinger Hybrid Harmony Score SchillingerHybrid 4-partHybrid examples Harmony Frans Absil (a) Crossing Hybrid 4-part examples Frans Absil Score Score #w Schillinger Hybrid Harmony & 4 w b#wwSchillingerw bHybridww w Harmonyww #w w #w wHybrid# 4-partw examples#w w # ww Fransw Absil 4 w b#ww wHybrid b4-partww examplesw w #w Frans Absil & 4 w w bw # w #w w # ww ww ? 4 w #w w w R w R bw R R w R R R ? 4 #ww 5 7 3 w 7 3 #w 3 w 7 w 44 #w b#ww w bww w w #w & 4 w b#ww w #bww(b) w ww ##ww ww & 4 w w w # w #w w # ww ww #w Crossing bw ? 4w w ww w wbw b ww nw w #w w w ?& 4w w w bbw n w w w #w#ww ww w #nwww 4#w ww w b w nw w & w w bw n w w #ww ww #nww ? w w b w w w #w w w w bw w ? #ww w #w w w #w ww bw nbww nw w w & w R5 ww R7 bbw R3 nbww R7 nw R3 #ww R3 ww R7 #nww & w bw w b w w w #w ww #nww #w #w #bww b w #w #w b w nw #ww ? w bw w n w bww bbww wb w(c)w w w #w#ww b ww bw wn w ?&Transformation:w w w w nw w #w wb w nw w #w w n#w #w bww bb w #w #w #ww b ww w n#ww & w bw w w b w w nw w w bw bw ? w!"!!!!!!!w w bw w w w w bw w #w bw ? #ww bw #ww #ww bbww #ww #ww b w nww #www #w bbww n#w #w bbwww bbww #w #w ##www bbww nbw n#ww & w bw n w w b ww b ww nw w #ww b ww bw n w #w bw ? w ww w w bww w wbw bw w w ?& w √ bw #www w ##√w w bw w b w√w #w nww w#w w #w 2 bw w 3 w b w n w4 #w w w & Rw( 2) #ww R(#w2)w bw Rb(w 2) w ##w b w bn ww nww #w ? w #w bw w w w ? """"""#w #w #ww w bww w w w & #w #ww ##ww bw bw w #w & w #ww ##w bbw bn ww nww #w b w n w w bww w ? #w #w #ww w bw#w w bbw # n w #w nw w #w #w #w ?& w w ##ww bw w#w nw wbw b w ww w ww nw ww w #w √w w bwbw n w #ww w bww #w w #w w 6#w w bw bw nw b w #b ww w nw w #w w & wR( w2) #w w w w w w bw w nw w w ? w #w w w #w bw w w w w w Figure 6: Hybrid 4-part harmonicw w continuity. Constant tension S9,b forw Typew I diatonicw (a), Type II diatonic-symmetric? #ww #ww (b),w and Type III symmetric#ww (c)bbww harmonicn ww progression.#ww w bw #w ww #ww & #w #w w bw bw nw bbw #bnww #w nw bw #nw #w #w & w #w #w bw w nw w #b ww w nw w nw #w w bw ? w w w w #w bw w bw w w w w ? w #w w w #w bw w w w w w ©2015 F.G.J. Absil, www.fransabsil.nl

#w 4 w b#ww w bww w w #w & 4 w w w # w #w w # ww ww w bw w w ? 4 w #w w w

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S9 S5 S5 S5 S9 S11 S9 S11 S7 S13 S13 S13 S11 S9 #w w w w bw w bbw n w #w w bw #w w #w & w #w #w bw w nw w #b ww w nw w nw #w w ? w w #w bw w w w w √ bw w w w #w 4 w R3 R7 R5 R7 3R( 2) R7 R3 R3 R7 R5 R7

Figure 7: Hybrid 4-part harmonic continuity. Variable tension, for mixed type harmonic progression. structure function 9 is shown in the progression S[7] → S[8], with the upward stepwise movement f[ % f\. ©2015 F.G.J. Absil, www.fransabsil.nl 2 Let us have a look at another variable tension example.

Example 6: Hybrid 4-part harmonic continuity with variable tension. In this example we will use two modal variants from the diatonic scale, the first on the tonic root g, the second on e. The variable tension chord structures are indicated in the score (see Fig. 8). The first modal variant is based on a Type I Diatonic progression, the second on a Type III Symmetric progression. Again, we aim for a fairly smooth voice leading. Both examples are free of vocie crossing, nor do they contain parallel 7th or 9th movement between upper functions and roots. Dissonant intervals in the upper structure are regularly resolved properly. Note the exception in Fig. 6.a in S[2] → S[3] with the upward leap of the dissonant major 7th f] % a, or at the end S[7] → S[8] with the upward stepwise movement d[ % d\. The second modal variant on e deliberately alternates S7 )* S11; these chord structures have only one upper structure common function, i.e., 7. Therefore, this heterogeneous group should yield a preferrence for constant function transformations, in case of smooth voice leading. The marks above the staff indicate that there is indeed some evidence to support this rule (see the constant 123 transformations). However, slurring common notes leads to a circular permutation type of transformation. Note the tonic root chord, a half-diminished 7th chord E∅, that opens the example. Fortunately the closing chord on the same root has a different structure (i.e., tension S11), since the tonic triad S5 is highly unsuitable for acting as a stable structure that will confirm the tonality of the example10. 2 The final example includes a few additional aspects.

Example 7: Hybrid 4-part harmonic continuity with fully heterogeneous variable tension chord structures. In this example we will investigate the effect of transformation types on the voice leading, when dealing with completely heterogenous chord structures. Therefore, the harmonic continuity is based on alternating S5 )* S11 chord structures Remember, that in hybrid 4-part chord structures the upper structure of the S5 tension chord contains the functions {1 − 3 − 5}, whereas the S11 upper structure consists of {7 − 9 − 11}. So, indeed, these structures are completely dissimilar (no overlapping functions, although both are perfect triads). The root sequence is given by

R3 z }| { R~ = 2R3 + 2R5 + 2R7 + R−3 + R5 +R3, 10This example is of academic nature. In case of writing real music, I would have adapted the structures to yield a more stable chord on the tonic root.

c 2008-2015 F.G.J. Absil, www.fransabsil.nl 14/19 Hybrid 5- and 4-part harmony F.G.J. Absil

(a) Transformation: 2 constant 1¯ constant 1¯ constantSchillinger1¯ Hybrid Harmony S5 S9 S13 S13 "!!"S9 S13 S13 S5 2 bw w Schillingerw Hybrid Harmony w #w bw w b# w w bw nw & w w w w w w w bw #w w w # w w w w bw w b w w bw nw ?& w w #w #w w w w w R3 R3 R5 R7 R7 R5 R5 ? w w #w w w w #w (b) w 123 123 123 wconstant constant constant & bSw7 #Sn11w " #Sw7 Sbb11w " Sw7 Snw11 " S7 w w bw w bw bw nw bw ?& w ##w #w bbw w nw w w bw bw w bw ? w #w bw w √ bw w w R( 6 2) #w w #w Figure& b 8:w Hybrid#w 4-part harmonicw continuity.#ww bwVariablew tension,#w for Type# Iw diatonicw bw (a), and# Typew III symmetric (b) harmonicw progression. w #w w w w #w w #w ?& bw #w w #ww bw w #w # ww bw #w w #w w w w #w bw w w w ? w w w w w #w w w w bw bw w #w w bnw bw w bw w & w #w bw w #w w #w b w w w w b w #w w bw w #w w bnw bw w bw w ?& w #w bw w #w w #w b w w w w bbw #w w ? w w w #w w w w w bw w bw #w w #w bw w #w w bw w b#w w bw #w c 2008-2015& w F.G.J.#w Absil, www.fransabsil.nlw w w w w w 15/19 bw #w w #w w w w w w #w w w #w ?& bw #w #w w bw w b w w bw w w #w w w w w bw w ? w w w #w w w w w bw w Hybrid 5- and 4-part harmony F.G.J. Absil

a highly artificial sequence, since it deliberately contains repeated positive root cycles (e.g., 2R3), but also a composite group of a negative plus a positive root cycle (the R−3 + R5 group). The first solution (see Fig. 9.a) is based on variable transformations, aiming for smooth voice leading. The resulting transformation type is either clockwise or anti-clockwise circular permutation of chord functions. It is not a constant function type of transformation, the guideline given by Schillinger for handling of heterogeneous chord groups. The constant 123 transformation (not presented here) would lead to a set of parallel triads in the upper structure with unacceptable parallel voicing. Applying (or better, forcing) a constant function transformation is demonstrated in Fig. 9.b and c. Note the alternating close and open type voicing in the upper structure, and the occasional crossing voices. There are many wide leaps in the upper voices; using this type of transformation (as opposed to the smooth voice leading solution in Fig. 9.a) would affect the orchestration in the case of writing real music. I would apply the variable transformation solution to a harmonic (long notes, chorale type) setting for bowed strings (or for trombones at the octave below), whereas the constant pitch solution with its wide leaps suggests pizzicato strings or woodwind accents. 2 The final example contained a negative root cycle in a hybrid 4-part harmonic continuity. This does not appear in the Schillinger book, and therefore was not presented as a root movement option at the beginning of this section (there, as a rule, we allowed positive root cycles only). However, voice leading diagrams like Fig. 4 and 5 may also be constructed for all six transformations and for all three negative root cycles (this is left as an exercise to the student). Whether this additional destabilising effect to tonality in the case of hybrid 4-part higher tension chords is acceptable musically, is a matter of taste. The last example was meant to make the reader conscious of that optional extension to the technique.

5 Conclusion

This document presents two extensions to the traditional theory of diatonic harmony, as proposed by Schillinger [1], i.e., hybrid 5- and 4-part harmony. Rules and guidelines for these techniques were followed by detailed discussion of examples. Note, however, that only the chord progression has been considered. The examples shown herein do not yield musical beauty; they only demonstrate voice leading aspects, when using the hybrid structures in a harmonic continuity. In order to create real music other elements such as melody and rhythm must be included in the score. Here we have discovered a technique to deliberately create varying tension over a chord progression. The composer may use this toolbox of techniques, derived from traditional harmony, and play with and control chord tension, chord progression types and transformations; this may lead to useful and interesting musical results. Try it!

c 2008-2015 F.G.J. Absil, www.fransabsil.nl 16/19 2 Schillinger Hybrid Harmony

bw #w w w # w w 2 w w bw Schillingerw Hybridb Harmonyw w bw nw & w w w w w w w ? bw w w #w w w w w #w #bwHybrid 5-w and 4-partb# w harmonyw bw F.G.J.nw Absil 2& w w Schillingerw Hybrid Harmonyw w w w ? w #w w w w bw #w #w w # w w w w w bw w b ww w bw nw & bw w#nw #w w bbw w w nw w w w w bw w bw w ? w w#w bw #w (a) w w ww Transformation:w variable #w bw w w & bw #nw #w bbw w nw w w bw w bw S5 !!"!!!!!!S11 S5 bSw11 S5 S11 S5 S11 S5 S11 ? w #w bw w w w bw nw w & bw #w # w #w ##www bw bbw w #ww ##wnw bw #w w w w ww w w #w bww ww w bw w ? w #w bw w w w w #w #w w bw w bww #w w & bw R w R w R R bw R w R w R R bw R w w 3 #w 3 w 5 #ww 5 w 7 #w 7 #w −3 # ww 5 w 3 #w ? w (b) w w Transformation:#w constant 1¯ w bw w w #ww ww wnw w #ww w w & bSw5 Sw11 #Sww5 S11 bbwSw5 Sbw11w wS5 S11 bbwSw5 Sw11 & w ##ww w #bwww w #ww #w#w # www w #w b w w w w b Crossingw #w w ? w #w w w ww w bw #w #ww w bwnw bww bwbw w bw w & w #w w w #w w #w b w w bw w w b w #w w w ? w b#ww #w bw #ww w R w R ww R w R nw R ww R #w R ww R w R #ww & bw 3 w 3 ##ww 5 w 5 bww 7 wbw 7b w −3 w 5 bbww 3 ww & w b##ww bw w ww b#w #ww w #w w w (c) w w ?Transformation:w bw constant 2¯ #w #w ww w w ##ww ww w w ww #bwbw ww w #w & bSw5 S11 #Sw5 S11w bwSw5 Sw11 b Sw5 Sw11 bwS5 S11w Crossingw #w Crossingw Crossing ? w w #bw #w w #bw #w #w bw w #w w bw w b w w bw w & w #w w w w w w w w ? w w w #w w w w w bw w R3 R3 R5 R5 R7 R7 R−3 R5 R3

Figure 9: Hybrid 4-part transformations. Variable tension S5 )* S11, Type I diatonic harmonic continuity, for variable (a), constant 1¯ (b), and constant 2¯ (c) transformation. Note the negative root cycle R−3 in the harmonic progression. Also note the alternation between closed and open voicing, and the voice crossing for the constant 1¯ and 2¯ transformations.

c 2008-2015 F.G.J. Absil, www.fransabsil.nl 17/19 Hybrid 5- and 4-part harmony F.G.J. Absil

Appendix: Transformation of chordal functions

We will consider chord structures S with 3 pitches or chord functions p[i], i = 1,..., 3 and will present the voice leading options during the transformation from one chord S[j]11 to the next S[j + 1]. We will designate the i-th function in the j-th chord as p[i, j]. We therefore can write the triadic chord structure as:  p[3, j]    S[j] =  p[2, j]  . p[1, j] Since the chord structure contains 3 pitches this yields 3! = 1 × 2 × 3 = 6 optional transformations: 1. A clockwise circular permutation of chord functions S[j] S[j + 1], where p[3, j] → p[1, j +! 1], p[2, j] → p[3, j + 1], p[1, j] → p[2, j + 1].

2. A counterclockwise circular permutation of chord functions S[j] S[j + 1], where 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 function constant, while swapping the 2nd and 3rd function. This is notated as ¯ S[j] →1 S[j + 1], where p[3, j] → p[2, j + 1], p[2, j] → p[3, j + 1], p[1, j] → p[1, j + 1]. 4. Keep the 2nd chord function constant, while swapping the 1st and 3rd function. This is notated as ¯ S[j] →2 S[j + 1], where 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 function constant, while swapping the 1st and 2nd function. This is notated as ¯ S[j] →3 S[j + 1], where 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 function constant. This is notated as S[j] −→123 S[j + 1], where p[3, j] → p[3, j + 1], p[2, j] → p[2, j + 1], p[1, j] → p[1, j + 1].

In this document transformations types 1 and 2, i.e. (counter)clockwise circular permutation of chord functions, are referred to as heterogeneous transformations (using Schillinger parlance).

11 Note that we use integer subscripts (e.g., S7) to indicate the tension of the chord, while the integer index [j] indicates the position of the chord in a sequence (i.e., in the time domain). The integer index [i] indicates the simultaneous sounding of pitches p[i] in a chord structure (ordered from low to high pitches).

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References

[1] Joseph Schillinger. The Schillinger System of Musical Composition, volume I and II of Da Capo Press Music Reprint Series. Da Capo Press, New York, fourth edition, 1946. ISBN 0-306-77521-2 and 0-306-77522-0. xxiii + 1640 pp.

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