Pitch-Symmetric Tetrachord Partitions

Polytrope

2009 vii 19

Contents

1 Introduction 2

2 Pitch-Symmetric Tetrachords 2

3 Pitch-Symmetric Tetrachord Partitions 3

4 Musical Significance 22

A PSTPs Realizable by Fully Combinatorial Rows 24

B Enumerating the PSTPs 37

This article outlines the underlying pitch-class structure of an original techniqueof twelve-tone musical composition which I frequently employ. I call this structurethe system of pitch-symmetric tetrachord partitions (PSTP). I use PSTPs becauseI consider that they afford systematically a sense of harmonic progression whileretaining the essential atonality1 of twelve-tone music.

1I.e. lack of tonal centres or with free and rapid shifting of tonal centres.

1

1 Introduction

It is not unreasonable to say that in serial twelve-tone music the classical role of keysand modes in support of unity and structural organization is played instead by tonerows, or more properly speaking, tone-row complexes—each comprising a ‘prime’ or‘original’ row form and other row forms derived from the prime form by combinationsof transposition, pitch inversion, and order reversal. Spelling this out a bit, each notein a tonal piece relates to a prevailing harmonic function (either as a chord factoror as a departure therefrom leading in a more or less expected way onward), andeach harmonic function (‘dominant’, ‘tonic’, etc) has a more or less specific role in aprevailing key and mode.2 In a serial twelve-tone piece, on the other hand, each notebelongs to a prevailing row form,3 and often significantly to a motivic subsegment ofthat row form.

In the PSTP system, each note in a piece belongs to (or is a temporary departurefrom4) a prevailing four-pitch-class set5 or tetrachord, and each tetrachord is a mem-ber of a prevailing set of three tetrachords whose union comprises all twelve pitchclasses; that is, each tetrachord is a member of a prevailing tetrachordal partition ofthe twelve. Notes from the prevailing tetrachord may appear in any order and mayreoccur arbitrarily often and in any registers: it is the set of four pitch-classes whichis prevailing, not any particular pitch-class sequence.6 The prevailing tetrachordalpartition may change fairly frequently during the course of a PSTP piece, just asthe prevailing key and mode in a tonal piece change fairly frequently at one level oranother due to modulations, secondary dominant formations, etc.

2 Pitch-Symmetric Tetrachords

One further specification completes an outline of the PSTP system: Each of thetetrachords of a PSTP must be inversionally pitch-symmetric (PS). That is, it mustbe a set like

2A given function or a given note may of course have an ambiguous role, for example in pivotmodulations, chord ‘reinterpretations’, etc.

3Occasionally more than one, for example when successive row forms are overlapped.4Preferably stepwise.5In this article, “set” denotes a collection considered without regard to any particular order of

its elements. This is the usual mathematical sense of the word, as opposed to a sense sometimesused in music theory, according to which “set” is equivalent to “row”.

6A simple sequential or simultaneous appearance of the four pitch classes is of course permitted.

2

{A, C, E, G}, in which A is nominally as far below C (three semitones) as G isnominally above E; or like

{C, D, E, G]}, in which C is nominally as far below D (a whole tone) as E isnominally above it, and G] is a tritone away from D, so equally distant fromC and E.

There are 15 types of PS tetrachords, as enumerated in Table 1. These and their enu-meration are further described in another article on this website: “Pitch-SymmetricTetrachords, etc.” The numbers in the first three columns of Table 1 indicate thespacing of the constituent pitch classes of each tetrachord as numbers of semitonesnominally above an arbitrary origin (0). This is construed musically in the “relativecontent” column. The numbers in the “content” column for a given tetrachord arearranged in circle-of-fifths order7 to emphasize that aspect of the tetrachord’s pitch-symmetry. For example, the entry “{0, 2, 9, 11}” for the V tetrachord correspondsto 0, 2, 3, and 5 perfect fifths above 0; i.e. two, then one more, then two more:‘2-1-2’. The “name” column of Table 1 assigns a fixed capital letter for each of the15 PS tetrachord types. In print, these should be italicized to minimize confusionwith pitch-class names and harmonic function symbols.

3 Pitch-Symmetric Tetrachord Partitions

0 72

9

41161

8

3

105

Figure 1: Circle-of-fifths order for Figures.

Figures 2-16 exhibit all the types of PSTPs, each figure showing all those thatinclude instance(s) of a particular PS tetrachord type. The dots in each circle inthose figures correspond to the twelve pitch classes in circle-of-fifths-order, as shown

7Cf. Figure 1.

3

content* name* acoustic root* relative content guide tone

{0, 7, 2, 9} 0Q 0 {M2, P5, M6}7 {M2, P4, P5}2 {P4, P5, m7}

{0, 7, 9, 4} 9N 9 {m3, P5, m7}0 {M3, P5, M6}

{0, 7, 4, 11} 0M 0 {M3, P5, M7}4 {m3, P5, m6}

{0, 7, 11, 6} 0T 0 {A4, P5, M7} A411 {m2, P5, m6} P5

{0, 7, 6, 1} 0X, 6X 0, 6 {A1, A4, P5} P5

{0, 2, 9, 11} 2V 2 {P5, M6, m7}

{0, 2, 4, 6} 0W 0 {M2, M3, A4} A42 {M2, M3, m7} M3

{0, 2, 11, 1} 11S 11 {m2, M2, m3}

{0, 2, 6, 8} 2F, 8F 2, 8 {M3, d5, m7} M3

{0, 9, 4, 1} 9L 9 {m3, M3, P5}

{0, 9, 11, 8} 8K 8 {m2, m3, d4}

{0, 9, 6, 3} 0D, 3D, 6D, 9D 0, 3, 6, 9 {m3, d5, d7} m3

{0, 4, 11, 3} 4U 4 {P5, m6, M7}

{0, 7, 2, 1} 0R 0 {m2, M2, P5} P57 {P4, d5, P5} d5

{0, 2, 4, 8} 8A 0 {M2, M3, m6} m64 {d4, m6, m7} d48 {M3, A4, A5} A4

* see text for interpretation of numbers.acoustic root: nominally lower pitch class of a P5, M3, or m3 interval within the tetrachord.relative content: relative to the indicated acoustic root as P1.guide tone: relative to the indicated acoustic root as P1—see Hindemith, The Craft of

Musical Composition, ch.III.10, IV.5 et al.

Table 1: PS tetrachord types, showing some musical characteristics.

4

v v vvvvvvfff

fQQQ

0Q, 4Q, 8Q

v v vvvvv

v ff

f

fQQX

0Q, 6Q, 4X

v v vvvv

vv ff

f

fQQR

0Q, 1Q, 4R

v v vvvv vv

f

ff

fQMM

0Q, 4M, 6M

v v vvv

vv

vf ff

f QML

0Q, 11M, 1L

v v vvvv

vv

f ff

fQVS

0Q, 1V, 3S

v v vvvvvv

fff

fQWW

0Q, 4W, 11W

Figure 2: PSTP types with type Q tetrachords.

in Figure 1. In each circle the pitch classes of the particular tetrachord type whichis the subject of the figure are indicated in black, those of the other two constituenttetrachord types in red and white. Internal sub-captions (like “0Q, 11M, 1L” inFigure 2) name the constituent tetrachord types in the order “black, red, white”with the prefixed numbers (e.g. “0, 11, 1”) indicating the pitch-class relations whichobtain among the constituent tetrachords in any particular PSTP of this type.

The label in the centre of each circle is either a) a combination of the names ofthe constituent tetrachord types, or b) in square brackets, a label of type (a) followedby ‘+n’, where n is an integer from 1 to 11. A label of type (a) is used for the firstoccurrence of a particular PSTP type in Figures 2-16, and serves as a name for thePSTP type. A label of type (b) is used for subsequent occurrences of a particularPSTP type (under the headings of its other constituent PS tetrachords). The ‘+n’term in this case indicates the change in numbering from the original.

An example may make this clearer. The PSTP type labeled “QML” in Figure 2is the set of three PS tetrachord types

{{0, 7, 2, 9}, {11, 6, 3, 10}, {4, 1, 8, 5}} = {0Q, 11M, 1L}.

5

The same PSTP type appears in Figure 4 with the label “[QML+1]”, where it isshown as the set

{{0, 7, 4, 11}, {1, 8, 3, 10}, {5, 2, 9, 6}} = {0M, 1Q, 2L}.

A particular PSTP of this type is

{{D, A, E, B}, {D[, F, A[, C}, {E[, G[, G\, B[}} = {DQ, D[M, E[L}.

To match this with Figure 2, take 0 = D; to match it with Figure 4, take 1 = D.The difference accounts for the “+1” in the Figure 4 label. Note that this additionis modulo 12, so that e.g. 11 + 1 = 0.

It turns out that there are 47 distinct PSTP types (those with labels of type (a)in Figures 2-16). Tables 2 and 3 list these again, showing the pitch-class relationsamong their constituent PS tetrachord types.

I can’t see that the brute combinatorial facts—15 PS tetrachord types, 47 PSTPtypes—have any mathematical or musical significance. The truth of the former factis argued in “Pitch-Symmetric Tetrachords, etc” (on this website). The latter factcan be ascertained through a computer enumeration, a program for which is includedin Appendix B.

The PSTPs are somewhat analogous to Josef Matthias Hauer’s ‘tropes’,8 whichcomprise the hexachordal partitions of the 12 pitch classes. However, only pitch-symmetric tetrachords are components of the PSTPs, whereas all hexachords areincluded as components of the tropes. This, together with 1) the intrinsic differ-ence between three-fold and two-fold partitions and 2) Hauer’s special treatment ofsemitone-adjacent pitch classes, makes the PSTP and trope systems very differentas compositional tools.

8See www.musiker.at/sengstschmidjohann/stichwort-trope.php3 and “Hauer’s Tropes andthe Enumeration of Twelve-Tone Hexachords” on this website.

6

vv

vv

v vv

v

f f ffNNL

9N, 8N, 10L

vv

vv

vv

vv

fff f

NND

9N, 3N, 2D

vv

vv

v

v

v

vff

ff NTR

9N, 11T, 1R

vv

vv

vvv

vff

f

fNVK

9N, 1V, 2K

vv

vvvvv

vff

f

fNKV

9N, 10K, 8V

vv

vv

vvv

v f

fff

NWA

9N, 11W, 2A

Figure 3: PSTP types with type N tetrachords.

7

v

vv

vv

vv

v ff ff MMX

0M, 6M, 2X

v

vv

vvv

v

v

ffffMVU

0M, 8V, 2U

v

vv

vvvvv

ff ff

MUR

0M, 6U, 8R

v

vv

vvvvvf

f

ffMRU

0M, 1R, 10U

v

vv

vv

vvv f

ff

fMAA

0M, 2A, 9A

v

vv

vvvvvf f

ff[QMM+6]

0M, 6Q, 10M

v

vv

vvvvv f

f

ff[QML+1]

0M, 1Q, 2L

Figure 4: PSTP types with type M tetrachords.

8

v

v

v

vvv vv

ff

ffTTL

0T, 9T, 10L

v

v

v

vv

v vvff f

fTTR

0T, 2T, 3R

v

v

v

vv v

vv fff

fTFA

0T, 4F, 9A

v

v

v

vvvv

vff ff

TKR

0T, 1K, 8R

v

v

v

vv

v

vv

fff f[NTR+1]

0T, 10N, 2R

Figure 5: PSTP types with type T tetrachords.

9

v

v

v

vv

v vv ff

f

fXXX

0X, 2X, 4X

v

v

v

vv vv v

fff f

XXD

0X, 3X, 2D

v

v

v

vvv v

vf ff

fXSS

0X, 2S, 8S

v

v

v

v vvv vf f

ff XFF

0X, 4F, 5F

v

v

v

vvv

vvff

ff

XUU

0X, 9U, 3U

v

v

v

vvvvvf

fff

[QQX+2]

0X, 2Q, 8Q

v

v

v

v vv vvf f

ff[MMX+4]

0X, 4M, 10M

Figure 6: PSTP types with type X tetrachords.

10

vvv

v

vvv vf f ff VVV

2V, 6V, 10V

vvv

v

vvv

vf fff

VVD

2V, 8V, 1D

vvv

v

vvvv

ff

f fVWW

2V, 4W, 1W

vvv

v

v vv

vf f

ffVLL

2V, 1L, 3L

vvv

v

vv

vvff

ffVRR

2V, 6R, 3R

vvv

v

vvvv f

f

f

f[QVS+1]

2V, 1Q, 4S

vvv

vv

v

vv ff f

f[NKV+6]

2V, 3N, 4K

vvv

vvv

vvf ff

f[NVK+1]

2V, 10N, 3K

vvv

v

v

vv

vff f

f[MVU+6]

2V, 6M, 8U

Figure 7: PSTP types with type V tetrachords.

11

v vvvv

vv v

f ff

fWWS

0W, 1W, 8S

v vvv

v

vv

vff ff

WFR

0W, 1F, 8R

v vvv

vvv

vf

ff

fWKA

0W, 7K, 9A

v vvvvv

vvf f

ff

[QWW+1]

0W, 1Q, 5W

v vvv

vv

vvf

ff

f[NWA+1]

0W, 10N, 3A

v vvvvv

vv

ff f

f[VWW+11]

0W, 1V, 3W

Figure 8: PSTP types with type W tetrachords.

12

v

v

vvv

v vvf f

f

fSSS

11S, 7S, 3S

v

v

vvv

v

v

vff ff

SSR

11S, 4S, 8R

v

v

vvv

v

vv fff

f SLU

11S, 3L, 9U

v

v

vv

vv vvf f

ffSUU

11S, 8U, 10U

v

v

vv

vvv v fff

f[QVS+8]

11S, 8Q, 9V

v

v

vvvv v

v f

f

ff [XSS+9]

11S, 9X, 5S

v

v

vvvvv

vff f

f[WWS+3]

11S, 4W, 3W

Figure 9: PSTP types with type S tetrachords.

v

vv

vvv

vv

f fff

FFD

2F, 5F, 1D

v

vv

v

vv

vv

f

f ff[TFA+4]

2F, 4T, 1A

v

vv

vvv v

vf

f f

f[XFF+9]

2F, 9X, 1F

v

vv

v vv

vvff f

f[WFR+7]

2F, 7W, 3R

Figure 10: PSTP types with type F tetrachords.

13

vv

v vv

v

vv fff f

LLD

9L, 3L, 2D

vv

v vv

vv

vff

f

fLKK

9L, 7K, 2K

vv

v vv

vvv

ff

f fLAA

9L, 2A, 11A

vv

v vvvv

v f

ff

f[QML+8]

9L, 8Q, 7M

vv

v vvv vv

f ff

f[NNL+11]

9L, 7N, 8N

vv

v vv

v

v

vf

fff

[TTL+11]

9L, 11T, 8T

vv

v vvv

v

v

fff

f[VLL+6]

9L, 8V, 7L

vv

v vv

v

vvff

ff

[SLU+6]

9L, 5S, 3U

Figure 11: PSTP types with type L tetrachords.

14

v vv

v vvv

vf fff

KKD

8K, 2K, 1D

v vv

vv

v

vv

f ff

f[NVK+6]

8K, 3N, 7V

v vv

v vv vv

fff f[NKV+10]

8K, 7N, 6V

v vv

v v

vv

vff f

f[TKR+7]

8K, 7T, 3R

v vv

v

vv

v vf fff

[WKA+1]

8K, 1W, 10A

v vv

v

vvv

vf

f

ff[LKK+6]

8K, 1K, 3L

Figure 12: PSTP types with type K tetrachords.

15

vv

vv

v vvv ff

f fDDD

0D, 1D, 2D

vv

vv

vv vv

f ff f[NND+4]

0D, 7N, 1N

vv

vv

v

vv

vf

ff

f[XXD+1]

0D, 1X, 4X

vv

vv

v vv

v

fff

f[VVD+11]

0D, 7V, 1V

vv

vv

v

v v

v fff f

[FFD+2]

0D, 1F, 4F

vv

vv

vvv

vf ff

f[LLD+10]

0D, 7L, 1L

vv

vv

vvv

vfff

f[KKD+11]

0D, 7K, 1K

Figure 13: PSTP types with type D tetrachords.

16

vv

vv v

vvv

f

f ffUAA

4U, 2A, 1A

vv

vv v

vv

vf f ff[MVU+2]

4U, 2M, 10V

vv

vv

v

vv

v f ff f[MRU+6]

4U, 6M, 7R

vv

vvv v

vv

ff

ff[MUR+10]

4U, 10M, 6R

vv

vv

v

vv

vf f

ff[XUU+1]

4U, 1X, 10U

vv

vv

v

vv vf

fff

[SLU+7]

4U, 6S, 10L

vv

vv

vv vv

ffff

[SUU+8]

4U, 7S, 6U

Figure 14: PSTP types with type U tetrachords.

17

v v

vvvv

v

vff ff RRR

0R, 4R, 8R

v v

vvvvvvf

fff

[QQR+8]

0R, 9Q, 8Q

v v

vv

v vv

v

ff

ff[NTR+11]

0R, 8N, 10T

v v

vvvv v

vff

ff[MUR+4]

0R, 4M, 10U

v v

vv

vv

v

v fff

f[MRU+11]

0R, 11M, 9U

v v

vvvv vv f

f

f

f[TTR+9]

0R, 9T, 11T

v v

vvv

vv

vf

fff[TKR+4]

0R, 4T, 5K

v v

vv

vvvvff f

f[VRR+9]

0R, 11V, 3R

v v

vvvvv

vf

ff

f[WFR+4]

0R, 4W, 5F

v v

vv

v vv

vf f

f

f[SSR+4]

0R, 8S, 3S

Figure 15: PSTP types with type R tetrachords.

18

v

v vvv v

vv

f ff

f[NWA+6]

8A, 3N, 5W

v

v vvv

v vv

ff

f

f[MAA+11]

8A, 1A, 11M

v

v vvv

v

v

v ff

ff

[TFA+11]

8A, 11T, 3F

v

v vvvv

vv

f

fff[WKA+11]

8A, 11W, 6K

v

v vvvvv

v ff

f f[LAA+9]

8A, 6L, 11A

v

v vvvv

vv ff

ff

[UAA+7]

8A, 11U, 9A

Figure 16: PSTP types with type A tetrachords.

19

PSTP Q N M T X V W S F L K D U R A

QQQ 0, 4, 8 - - - - - - - - - - - - - -QQX 0, 6 - - - 4 - - - - - - - - - -QQR 0, 1 - - - - - - - - - - - - 4 -

QMM 0 - 4, 6 - - - - - - - - - - - -QML 0 - 11 - - - - - - 1 - - - - -QVS 0 - - - - 1 - 3 - - - - - - -

QWW 0 - - - - - 4, 11 - - - - - - - -NNL - 8, 9 - - - - - - - 10 - - - - -NND - 3, 9 - - - - - - - - - 2 - - -

NTR - 9 - 11 - - - - - - - - - 1 -NVK - 9 - - - 1 - - - - 2 - - - -NKV - 9 - - - 8 - - - - 10 - - - -

NWA - 9 - - - - 11 - - - - - - - 2MMX - - 0, 6 - 2 - - - - - - - - - -MVU - - 0 - - 8 - - - - - - 2 - -

MUR - - 0 - - - - - - - - - 6 8 -MRU - - 0 - - - - - - - - - 10 1 -MAA - - 0 - - - - - - - - - - - 2, 9

TTL - - - 0, 9 - - - - - 10 - - - - -TTR - - - 0, 2 - - - - - - - - - 3 -TFA - - - 0 - - - - 4 - - - - - 9

TKR - - - 0 - - - - - - 1 - - 8 -XXX - - - - 0, 2, 4 - - - - - - - - - -XXD - - - - 0, 3 - - - - - - 2 - - -

XSS - - - - 0 - - 2, 8 - - - - - - -XFF - - - - 0 - - - 4, 5 - - - - - -XUU - - - - 0 - - - - - - - 3, 9 - -

Q N M T X V W S F L K D U R A

Table 2: Pitch-class relations of PSTP constituents.

20

PSTP Q N M T X V W S F L K D U R A

VVV - - - - - 2, 6, 10 - - - - - - - - -VVD - - - - - 2, 8 - - - - - 1 - - -VWW - - - - - 2 1, 4 - - - - - - - -

VLL - - - - - 2 - - - 1, 3 - - - - -VRR - - - - - 2 - - - - - - - 3, 6 -WWS - - - - - - 0, 1 8 - - - - - - -

WFR - - - - - - 0 - 1 - - - - 8 -WKA - - - - - - 0 - - - 7 - - - 9SSS - - - - - - - 3, 7, 11 - - - - - - -

SSR - - - - - - - 4, 11 - - - - - 8 -SLU - - - - - - - 11 - 3 - - 9 - -SUU - - - - - - - 11 - - - - 8, 10 - -

FFD - - - - - - - - 2, 5 - - 1 - - -LLD - - - - - - - - - 3, 9 - 2 - - -LKK - - - - - - - - - 9 2, 7 - - - -

LAA - - - - - - - - - 9 - - - - 2, 11KKD - - - - - - - - - - 2, 8 1 - - -DDD - - - - - - - - - - - 0, 1, 2 - - -

UAA - - - - - - - - - - - - 4 - 1, 2RRR - - - - - - - - - - - - - 0, 4, 8 -

Q N M T X V W S F L K D U R A

Table 3: Pitch-class relations of PSTP constituents (continued).

21

4 Musical Significance

In my opinion, the primary significance of any style, technique, or system of musicalcomposition subsists in the works of its practitioners. John L. Baker of Vancouver,Canada is a composer who has produced several works using this technique, includingthose listed in Table 4. This website includes a performance and score of his 2009Prelude, Toccata, and Fugue.

title date instrumentation duration

Passacaglia and Fugue 1993 sop. recorder, tenor crumhorn 4mPrelude in E 1994 piano solo 3m 30sSarabanda from Homage in B 1996 piano solo 3m 10sNocturne with Balletto 1997 treble, tenor, and bass viols 6mFantasy No. 1 2002 piano solo 5m 45sFantasy No. 2 2003 wind quintet (fl, ob, cl, hn, bn) 11mFantasy No. 3 2006 ob, cl, tbn, vln, cello 6m 30sSix Inventions 2007 flute, viola, and piano 8-9mThe Cantor-Schröder-Bernstein

Theorem, a cantata 2008 12 voices unaccompanied 8mPrelude, Toccata, and Fugue 2009 piano solo 4m

Table 4: PSTP compositions by John L. Baker.

Here are a few desultory observations on the properties of PS tetrachords andPSTPs.

• In case the constituent tetrachords of a PSTP share an axis of symmetry (as, forexample, those of type QML), their mutual symmetry can structure a harmonicprogression (at or below the surface), thus:

& wwwwwwww#n

wwww##bb œœ#n

wwwwnn

CM DL D≤Q

( )

( )

CM

• Say that a 12-tone row x realizes a PSTP y if the constituent tetrachord typesof y comprise the first, middle, and last four elements of x. Most PSTPs9 can

9All except those of types TTR, DDD, and RRR. See appendix A.

22

be realized by a fully combinatorial 12-tone row.10 For example, the the fullycombinatorial row

〈0, 2, 7, 9, 4, 5, 1, 8, 10, 6, 3, 11〉

realizes the PSTP

{{0, 7, 2, 9}, {11, 6, 3, 10}, {4, 1, 8, 5}} = {0Q, 11M, 1L}.

The PSTPs admitting such realizations are enumerated in Appendix A.

My interest in PS tetrachords was triggered by Elliott Antokoletz’s11 detailedanalysis of Bartók’s use of pitch-symmetry in general and of the S, W, and X 12 typetetrachords in particular. However, Bartók does not use these PS tetrachords aspart of a PSTP-like system or technique: in fact, the way in which he combines Sand W types is incompatible with completion by a third PS tetrachord to form aPSTP; likewise for his combinations of W and X types, which cannot occur togetherat all in a PSTP. (See Table 2.) Hence, for what it may be worth, I considerthat the PSTP technique described here, although informed by the growing use ofinversionally symmetrical pitch formations in European music from the latter partof the nineteenth century up to Bartók’s time,13 is nevertheless original with me.

10For definitions relating to combinatoriality, see “Pitch-Symmetric Tetrachords, etc” on thiswebsite, p 9.

11The Music of Béla Bartók (University of California Press, 1984).12Following George Perle (“Symmetrical Formations in the String Quartets of Béla Bartók” [Mu-

sic Review no. 16 (November 1955), pp. 300-312]) and Leo Treitler (“Harmonic Procedure in theFourth Quartet of Béla Bartók” [Journal of Music Theory v.3, no.2 (November 1959), pp. 292-297]),Antokoletz refers to PS tetrachords of types S, W, and X as cells X, Y, and Z respectively.

13Antokoletz, pp.4-25.

23

A PSTPs Realizable by Fully Combinatorial Rows

It is well known that the fully combinatorial 12-tone rows are precisely those whosefirst and second halves are both hexachords of one of the types P1, P2, P3, P4, P5,or P6 in George Perle’s enumeration,14 as shown in Figure 17. For a PTSP to be

v vvvv

v v vvv

vv

P1

v vvvv

v vvvv

v vP2

v v v

vvvvvv

vvv

P3

v vvvv

vv

vvv

v vP4

v vvvvv

v

vvvv

vP5

v v vvvvvvvvv

vP6

Figure 17: Hexachords (Circle-of-Fifths Order) for Fully Combinatorial Rows.

realizable by a fully combinatorial row in the sense given above (p 22), it is merelynecessary for two of its constituent tetrachords to be included individually in thetwo hexachords of one of these types. One can determine whether this is so for anyparticular PSTP by inspection, comparing Figures 2-16 with Figure 17. Considerfor example the PSTP QML as shown in Figure 2. There are three orientations ofthe diagram for P6 in Figure 17 in which its black hexachord covers the positionsof the black tetrachord in Figure 2, namely counter-clockwise rotations by 0, 1, or 2positions. Rotated counter-clockwise by one position, the red hexachord of Figure 17covers the positions of the M (red) tetrachord of QML in Figure 2; this shows thatthere are fully combinatorial tone rows beginning with the elements of 0Q and endingwith the elements of 11M (as in the example on p 23). On the other hand, rotatedcounter-clockwise by 0 or 2 positions, the red hexachord of Figure 17 does not cover

14Perle’s enumeration is shown in full in “Hauer’s Tropes and the Enumeration of Twelve-ToneHexachords” on this website. Note that the diagrams there are in circle-of-semitones order, so thatthe appearances of the diagrams for P4 and P6 are interchanged.

24

the position of either the M (red) or the L (white) tetrachord of QML in Figure 2;this shows that 0QML is not realizable by any other fully combinatorial rows basedon P6 beginning with the elements of 0Q.

v v vvvv

vv vv ffvv ffQQQ

P6: 0Q – 8Q

v v vvvvv

vv

vf

fv

vf

fQQX(1)

P5: 0Q – 6Q

v v vvvvv

v vv ffv vf f

QQX(2)

P6: 0Q – 6Q

v v vvvvv

v vv

ff

vv

ff

QQX(3)

P6: 0Q – 6Q

v v vvvv

vv vv

ff

vv ffQQR

P6: 0Q – 1Q

v v vvvv

vv vv ffvvff QMM(1)

P6: 0Q – 6M

v v vvvv vvv

vf f

vv ff[QMM(1a)]

[P6: 0Q – 4M ]

v v vvv vv

v vv

ff

vv

ff QML(1)

P5: 0Q – 1L

v v vvv

vv

vv

vf

f

vv ffQML(2)

P6: 0Q – 11M

v v vvv vv

v vv

ff

vv ffQVS(1)

P5: 0Q – 3S

v v vvvv

vv v

vf

fv

vf

fQVS(2)

P6: 0Q – 1V

Figure 18: PTSPs realizable by rows with type Q incipit.

Figures 18-31 exhibit the results of all such comparisons, and therefore show allthe ways in which PSTPs can be realized by fully combinatorial tone rows. Notationfor these figures is generally as explained on p 3, with the following alterations.

25

The black dots in each circle, which are in the same positions as in the corre-sponding diagram in Figures 2-16, indicate the initial tetrachord (“incipit”, let ussay) of the (directly) realizing tone rows. The red dots, which correspond either tothe red or to the white dots in the corresponding diagram in Figures 2-16, indicatethe final tetrachord of the same tone rows. The other four dots, coloured grey orpink, combine with the black or red dots respectively to make up the initial and finalhexachords of those rows. Internal sub-captions (like “P6: 0Q – 11M” in Figure 18)give the hexachord type and name the initial and final tetrachords. Note that, if aPSTP in this listing has initial and final tetrachords of different types, then it willappear in the figure for each type as incipit.

Note also that there are no realizations with type D incipit.

The assertion above that “Figures 18-31 exhibit . . . all the ways in which PSTPscan be realized by fully combinatorial tone rows” requires some clarification. Therealizing tone rows directly represented in those figures, for example

〈0, 7, 2, 9, 4, 11, 8, 3, 6, 1, 10, 5〉, (A.1)

which realizes the PSTP 0QMM as shown in the diagram QMM(1) of Figure 18,is merely one member of the complex of 48 rows comprising all those obtainablefrom it by combinations of transposition, inversion, and reversal, and the diagramin question should be considered to express the fact that all the member rows of allthe complexes including rows thus directly represented realize some PTSP of typeQMM. It follows that each diagram indicates the realizability of a PTSP type in aparticular way by up to 4! · 2 · 2 · 4! · 48 = 110, 592 fully combinatorial 12-tone rows.15

As a consequence of this interpretation of the diagrams, some possible coveringsof tetrachords in Figures 2-16 by the hexachords of Figure 17 are redundant. Forexample, one might suppose that, in addition to the covering of 0QMM (Figure 2)by P6 indicated by QMM(1) (Figure 18), another covering [QMM(1a)] (Figure 18),obtained by rotating P6 two positions counter-clockwise, would also be relevant.However, because [QMM(1a)] is a reflection of QMM(1), the set of fully combinatorialrows realizing these two PTSPs is in fact identical. (A.1), for example is the I0 formof

P0 = 〈0, 5, 10, 3, 8, 1, 4, 9, 6, 11, 2, 7〉,

the P9 form of which (= 〈9, 2, 7, 0, 5, 10, 1, 6, 3, 8, 11, 4〉) realizes [QMM(1a)].15The factor 4! · 2 · 2 · 4! = 2, 304 corresponds to the (independent) permutations of{0, 7, 2, 9}, {4, 11}, {5, 8}, and {6, 1, 10, 5}—the sets represented by the four colours of dots.

26

Apart from [QMM(1a)], included just to illustrate the foregoing point, there aretwo exceptions to the avoidance of this sort of redundancy in Figures 18-31:

• As mentioned above, if a PSTP in this listing has initial and final tetrachordsof different types, then it will appear in the figure for each type as incipit. Inthis case the second appearance are marked by “[ ] ” around the name in thecentre of the circle.

• Because NVK and NKV (Figure 3) are related by inversion as PSTP types,it follows that corresponding digrams in Figure 19, etc indicate the same setsof realizing tone rows. Likewise for MUR and MRU (Figure 4). In this caseall appearances of NKV and MRU are marked by (possibly additional) “[ ] ”around the name in the centre of the circle.

vv

vv

v vv

v

vv ff

vv

ff

NNL

P6: 9N – 8N

vv

vv

vv

vv

vv ffv vf f

NND

P6: 9N – 3N

vv

vv

vvv

vvv ff

vv

ff NVK(1)

P6: 9N – 1V

vv

vv

vvv

v

vv

ffv

vf

f[NKV(1)]

P6: 9N – 8V

Figure 19: PTSPs realizable by rows with type N incipit.

To summarize, this appendix answers three types of question:

• Which PSTPs are realizable by fully combinatorial 12-tone rows? Answer: Allexcept those of types TTR, DDD, or RRR.

• In what ways can a PSTP be realized by a fully combinatorial 12-tone row?That is, to which of the patterns of Figure 17 must a realizing row conform,and how can the constituent tetrachords of the PSTP be distributed withinthe row?

• Given a fully combinatorial row comprising three pitch-symmetric tetrachordsin sequence, which PSTP does it realize? Note that this question always has aunique answer except for the alternation between NVK and NKV (respectivelyMUR and MRU ).

27

v

vv

vv

vv

vvvff

vvffMMX(1)

P2: 0M – 6M

v

vv

vv

vv

v vvffvvff MMX(2)

P6: 0M – 6M

v

vv

vvvv

vvvff

v

v

f

fMVU(1)

P2: 0M – 2U

v

vv

vvv

v

v

vvffvv

ffMVU(2)

P6: 0M – 8V

v

vv

vvvvv

vvff vv ff MUR

P2: 0M – 6U

v

vv

vv v

vvvvff

vv

ff[MRU]

P2: 0M – 10U

v

vv

vv vv vvvff vv ffQMM(2)

P2: 0M – 10M

v

vv

vvvv

v vvffvv ff

[QMM(1)]

P6: 0M – 8Q

v

vv

vvv

vvvvff v

vf

fQML(3)

P2: 0M – 2L

v

vv

vvvvv vvff

v

v

f

f[QML(2)]

P6: 0M – 1Q

Figure 20: PTSPs realizable by rows with type M incipit.

28

v

v

v

vvv vv v

v

f

f vv ffTTL

P3: 0T – 9T

v

v

v

vv v

vv v

v

f

fvv ff TFA(1)

P3: 0T – 4F

v

v

v

vvv vv

v

v

f

f vvffTKR

P3: 0T – 8R

v

v

v

vvvv vv

v

f

fvvff

NTR

P3: 0T – 2R

Figure 21: PTSPs realizable by rows with type T incipit.

v

v

v

vv

v vv v

v

f

fv

vf

fXXX

P3: 0X – 2X

v

v

v

vv vv vv

v

f

f vv

ff XXD

P3: 0X – 3X

v

v

v

v vvv v

v

v

f

fvv ff XFF(1)

P3: 0X – 4F

Figure 22: PTSPs realizable by rows with type X incipit.

29

vvv

v

vvv v

vv ff vv

ff VVV

P5: 2V – 6V

vvv

v

vvv

vvv

ff

vv

ff

VVD(1)

P4: 2V – 8V

vvv

v

vvv

v

vv ffv vf fVVD(2)

P5: 2V – 8V

vvv

v

vvv

v

vv

ffv

vf

fVVD(3)

P6: 2V – 8V

vvv

v

v vv

vvv ffv

vf

fVLL(1)

P5: 2V – 1L

vvv

v

vv

v

v

vv

ffvvff QVS(3)

P4: 2V – 4S

vvv

v

vvvv

vv

ff

v

v

f

f[QVS(2)]

P6: 2V – 1Q

vvv

vv vv

v

vv

ffv

vf

fNVK(2)

P4: 2V – 3K

vvv

vvv

vv

vv

ffv

vf

f [NVK(1)]

P6: 2V – 10N

vvv

v

vv v

vv

vf

fv

vf

f[NKV(2)]

P4: 2V – 4K

vvv

vv

v

vv

vv

ff

vv

ff[[NKV(1)]]

P6: 2V – 3N

vvv

v

vv vv

vv

ff

v

v

f

fMVU(3)

P4: 2V – 8U

vvv

v

v

vv

vv

vf

fvvff [ MVU(2)]

P6: 2V – 6M

Figure 23: PTSPs realizable by rows with type V incipit.

30

v vvvv

vv v

vvff v

vf

fWWS

P1: 0W – 1W

v vvv

v

vv

vvvff vv ff WFR(1)

P1: 0W – 1F

v vvvv

vv

vvvff v

v

f

fWKA

P1: 0W – 9A

v vvv

v vv

vvvff

vv

ff QWW

P1: 0W – 5W

v vvv

vv

vvv

vff

v

v

f

fNWA

P1: 0W – 3A

v vvv

vv v

vvvff

vv ffVWW

P1: 0W – 3W

Figure 24: PTSPs realizable by rows with type W incipit.

31

v

v

vvv

v vvv vf f

v

v

f

fSSS

P4: 11S – 7S

v

v

vvv

v

v

vvvff vv ff SSR

P4: 11S – 4S

v

v

vvvv

vvvvff

v

v

f

fSLU(1)

P4: 11S – 9U

v

v

vvv

v

vv vvffv

vf

f SLU(2)

P5: 11S – 3L

v

v

vv

vv vv

vv ffv

v

f

fSUU(1)

P4: 11S – 8U

v

v

vvvvv

vvvffv

vf

f[QVS(3)]

P4: 11S – 9V

v

v

vv

vvv v vvffv

v

f

f[QVS(1)]

P5: 11S – 8Q

v

v

vvv

v

vvvvff vvffXSS(1)

P4: 11S – 5S

v

v

vvv

v

vv vvffvvff XSS(2)

P5: 11S – 5S

Figure 25: PTSPs realizable by rows with type S incipit.

32

v

vv

vvv

vv vv

ff v

v

f

fFFD(1)

P1: 2F – 5F

v

vv

vvv

vv

v

v

f

f vv

ff

FFD(2)

P3: 2F – 5F

v

vv

v

v

v vvv

vf

fv

vf

f TFA(2)

P1: 2F – 1A

v

vv

v

vv

vv

v

v

f

f

vv

ff[TFA(1)]

P3: 2F – 4T

v

vv

v

v

v v

v vv

ff vv ff XFF(2)

P1: 2F – 1F

v

vv

v

vv v

v

v

v

f

fvv ff [XFF(1)]

P3: 2F – 4X

v

vv

v vv

vvv

vf

fvv

ff

[WFR(1)]

P1: 2F – 7W

v

vv

vvv v

v v

v

f

fv

vf

fWFR(2)

P3: 2F – 3R

Figure 26: PTSPs realizable by rows with type F incipit.

33

vv

v vv

v

vv vv

ff

vv

ff

LLD(1)

P2: 9L – 3L

vv

v vv

v

vv vv

ff

vv

ff LLD(2)

P5: 9L – 3L

vv

v vv

vv

vvv

ff

vvff[QML(3)]

P2: 9L – 7M

vv

v vvvv

v

vv

ffv

v

f

f[QML(1)]

P5: 9L – 8Q

vv

v vvvv

vv

vf

fv

vf

fVLL(2)

P2: 9L – 7L

vv

v vvv v v

vv

ff

vv

ff [VLL(1)]

P5: 9L – 10V

vv

v vvv

vv

vv

ff

v

v

f

fSLU(3)

P2: 9L – 3U

vv

v vv

v

vv v

vf

fvvff [SLU(2)]

P5: 9L – 5S

Figure 27: PTSPs realizable by rows with type L incipit.

v vv

v vvv

v

vv ffv vf fKKD

P4: 8K – 2K

v vv

v v vv

vvv

ff

vv

ff

[NVK(2)]

P4: 8K – 7V

v vv

v

vvv v

vv ff vv ff[[NKV(2)]]

P4: 8K – 6V

v vv

v

vvv

v vv ffvv

ff

LKK

P4: 8K – 1K

Figure 28: PTSPs realizable by rows with type K incipit.

34

vv

vv v

vv

v

vv

ffv vf f[MVU(1)]

P2: 4U – 2M

vv

vvv

v vv

vv

ff v

vf

f[MVU(3)]

P4: 4U – 10V

vv

vvv v

vv

vv

ff vv ff

[MUR]

P2: 4U – 10M

vv

vv

v

vv

v vv

ff

vvff[[MRU]]

P2: 4U – 6M

vv

vvv

v

vv

vv

ff v

vf

fXUU(1)

P2: 4U – 10U

vv

vvv

v

vvv

vf

f

vv

ff XUU(2)

P4: 4U – 10U

vv

vvv

vvv v

vf

f vv

ff[SLU(3)]

P2: 4U – 10L

vv

vv

v

vv vv

vf

fv vf f[SLU(1)]

P4: 4U – 6S

vv

vv

vvvv v

vf

f vv ff SUU(2)

P2: 4U – 6U

vv

vv

vv vv

vv

ff

v

v

f

f[SUU(1)]

P4: 4U – 7S

Figure 29: PTSPs realizable by rows with type U incipit.

35

v v

vvvv

vv

vv ff vv

ff [NTR]

P3: 0R – 10T

v v

vvv

vv

vvv ff

vv

ff[TKR]

P3: 0R – 4T

v v

vvvv v

v

vv ffv

vf

fVRR

P3: 0R – 3R

v v

vv

vv

vv

vv ffv

vf

f[WFR(2)]

P3: 0R – 5F

Figure 30: PTSPs realizable by rows with type R incipit.

v

v vv

v vv

vv

v

f

fv

vf

f [NWA]

P1: 8A – 5W

v

v vvv

v vv

v

v

f

fv

vf

f MAA

P1: 8A – 1A

v

v vvvv

vv

v

v

f

fv

v

f

f[TFA(2)]

P1: 8A – 3F

v

v vvvv

vv

v

v

f

f v vf

f[WKA]

P1: 8A – 11W

v

v vvv

vv v

v

v

f

f vv

ffLAA

P1: 8A – 11A

v

v vvvv

vv

v

v

f

f v

v

f

fUAA

P1: 8A – 9A

Figure 31: PTSPs realizable by rows with type A incipit.

36

B Enumerating the PSTPs

The Pascal16 program below produces something closely analogous to the listing inTables 2 and 3.17 It first generates a linked list of (PS) Tetrachord records in theorder Q, N, . . . , A; then with each in turn in the 0 position,18 tries to fit the sametetrachord (then each of the following ones) into the remaining positions. When itfinds a fit for two tetrachords, it then checks if the remaining four positions are alsoPS (i.e. on the list); if so, it reports.

Notice that the Tetrachord record contains all that is needed to generate andreport and also (in the member array content) the sets of positions occupied whenthe tetrachord is based at 0, 1, . . . , 11 respectively. This makes for quick testing:if T1^.content[i1]

procedure NewSymmTchord_A (R: PC; N: string; X: PC;

first, second: interval; var where: link);

{On return, where points to the new record, the .next field of which

points where where used to.}

var

i0, i1, i2, i3: PC;

oldwhere: link;

begin

oldwhere := where;

new(where);

with where^ do

begin

next := oldwhere;

root := R;

name := N;

xlimit := X;

for i0 := 0 to X do

{X is limit due to internal symmetry}

begin

i1 := (i0 + first) mod 12;

i2 := (i1 + second) mod 12;

i3 := (i2 + first) mod 12;

content[i0] := [i0, i1, i2, i3]

{symmetric 4-chord transposed by i0}

end

end

end;

38

procedure NewSymmTchord_B (R: PC; N: string; X: PC;

first: interval; var where: link);

{On return, where points to the new record, the .next field of which

points where where used to.}

var

second, i0, i1, i2, i3: PC;

oldwhere: link;

begin

second := 6 - first;

oldwhere := where;

new(where);

with where^ do

begin

next := oldwhere;

root := R;

name := N;

xlimit := X;

for i0 := 0 to X do

{X is limit due to internal symmetry}

begin

i1 := (i0 + first) mod 12;

i2 := (i1 + first) mod 12;

i3 := (i2 + second) mod 12;

content[i0] := [i0, i1, i2, i3]

{symmetric 4-chord transposed by i0}

end

end

end;

39

procedure writePCset (X: PCset);

var

i: PC;

first: Boolean;

begin

first := true;

write(’[’);

for i := 0 to 11 do

if i in X then

if first then

begin

first := false;

write(i : 1)

end

else

write(i : 3);

write(’] ’);

end;

procedure ReportPartition (S0, S1, S2: link; j0, j1, j2: PC);

begin

with S0^ do

write((root + 7*j0) mod 12 : 2, name, ’ ’);

with S1^ do

write((root + 7*j1) mod 12 : 2, name, ’ ’);

with S2^ do

write((root + 7*j2) mod 12 : 2, name, ’ ’);

write(’ : ’);

writePCSet(S0^.content[j0]);

writePCSet(S1^.content[j1]);

writePCSet(S2^.content[j2]);

writeln;

end;

var

i1, i2, starti2: PC;

U1, U2: PCset;

Tlist, T0, T1, T2: link;

40

begin

{Build list of pitch-symmetric tetrachords,

in reverse of output canonical order: }

Tlist := nil;

{sets in circle-of-fifths order circ-of-st order}

NewSymmTchord_B( 8, ’A’, 11, 2, Tlist);

{Aug+Tt : [0, 2, 4, 8] [8, 0, 2, 4] }

NewSymmTchord_B( 0, ’R’, 11, 1, Tlist);

{Quintal+m9(Tt) : [0, 1, 2, 7] [0, 7, 2, 1] }

NewSymmTchord_A( 4, ’U’, 11, 4, 1, Tlist);

{4 PM7+m6 : [0, 4, 5, 9] [4, 11, 0, 3] }

NewSymmTchord_A( 0, ’D’, 2, 3, 3, Tlist);

{0 dd7 : [0, 3, 6, 9] [0, 3, 6, 9]; 4-fold symmetry}

NewSymmTchord_A( 8, ’K’, 11, 3, 2, Tlist);

{mixed cluster : [0, 3, 5, 8] [8, 9, 11, 0] }

NewSymmTchord_A( 9, ’L’, 11, 3, 1, Tlist);

{9 M+m3 : [0, 3, 4, 7] [9, 0, 1, 4] }

NewSymmTchord_A( 2, ’F’, 5, 2, 4, Tlist);

{Fr x6 : [0, 2, 6, 8] [2, 6, 8, 0]; 2-fold symmetry}

NewSymmTchord_A(11, ’S’, 11, 2, 3, Tlist);

{semitone cluster : [0, 2, 5, 7] [11, 0, 1, 2] }

NewSymmTchord_A( 0, ’W’, 11, 2, 2, Tlist);

{whole-tone cluster : [0, 2, 4, 6] [0, 2, 4, 6] }

NewSymmTchord_A( 2, ’V’, 11, 2, 1, Tlist);

{2 Pm7+M6 : [0, 2, 3, 5] [2, 9, 11, 0] }

NewSymmTchord_A( 0, ’X’, 5, 1, 5, Tlist);

{crossed tritones : [0, 1, 6, 7] [0, 1, 6, 7]; 2-fold symm}

NewSymmTchord_A( 0, ’T’, 11, 1, 4, Tlist);

{0 PM7+Tt : [0, 1, 5, 6] [0, 6, 7, 11] }

NewSymmTchord_A( 0, ’M’, 11, 1, 3, Tlist);

{0 MM7 : [0, 1, 4, 5] [0, 4, 7, 11] }

NewSymmTchord_A( 9, ’N’, 11, 1, 2, Tlist);

{9 mm7 : [0, 1, 3, 4] [9, 0, 4, 7] }

NewSymmTchord_A( 0, ’Q’, 11, 1, 1, Tlist);

{Quintal-Quartal set : [0, 1, 2, 3] [0, 7, 2, 9] }

41

{Report partitions, searching in direct canonical order:}

T0 := Tlist;

while T0 nil do

begin

U1 := All - T0^.content[0];

T1 := T0;

while T1 nil do

begin

for i1 := 1 to T1^.xlimit do

if T1^.content[i1]