Pitch-Class Sets

REPRESENTATION

pcset — Fortean pitch-class set representation

DESCRIPTION

The **pcset representation provides a means for indicating pitch-class sets. The representation is based on a simple extension of the system devised by Allen Forte (see REFERENCE). Fortean set names consist of two numbers separated by a dash, with an optional up-case letter `Z' preceding the second number. The first number indicates the pitch-class cardinality — that is, the number of unique pitch-classes in the set. For example, all major and minor chords have a cardinality of `3' since all such chords consist of three unique pitch-classes. (Inversionally-related pitch-class sets have the same set designation.)

The number following the dash simply distinguishes different pitch-class sets having the same cardinality. The letter `Z' is used to indicate that the set shares the same interval-class content as some other set. (said to be Z-related sets). For example, sets 4-Z15 and 4-Z29 are said to be Z-related since they both exhibit the same interval-class content. (See iv representation.)

Barlines are represented using the "common system" barlines — see barlines.

FILE TYPE

It is recommended that files containing predominantly pcs data should be given names with the distinguishing `.pcs' extension.

SIGNIFIERS

The following table summarizes the **pcset mappings of signifiers and signifieds.

0-9 pitch-class set labels; measure numbers . null token - dash; separates cardinality for set number = barline; == double barline —– ——————————————–

Summary of **pcset Signifiers

EXAMPLES

The following sample document shows a pitch-class representation for the opening measures of Schoenberg’s "Sommerm\o’u\(..'d" from Three Songs, Opus 48. The right-most spine shows a pcset representation identifying the pitch-class content of each sonority. (The pcset spine might be generated using the pcset command, after the pc pitch-class spines are filled-out using the fill command.)

!! Arnold Schoenberg, "Sommermued" (1933) **pc *Ipiano *M4/4 *MM72 =1 =1 r r (7 . 8') . r . r r (9 . 8') . r . =2 =2 r r 9 (7 (9` 8 11) . r . r . r . (9 [11 . . 8) =3 =3 *- *- ——————————————— ———- ———- ———– ———–
**pc **pc **text **pcset *Ipiano *Ivoice *Deutsch * *M4/4 *M4/4 *M4/4 * *MM72 *MM72 *MM72 * =1 =1 =1 r . . (11 . . 2-4 9\^) r . 2-2 1 Wenn 3-4 . . 2-4 . . 1-1 (11 2 du 3-7 7\^) . . 3-9 0 schon 3-4 . . 2-5 =2 =2 =2 (0 glaubst , 1-1 [6 | 3-2 . . 3-2 . . 3-7 . . 2-2 7') 6]) | 2-1 . . 1-1 3 es 2-6 3 ist 3-8 [5 e- 3-10 =3 =3 =3 *- *- *-

PERTINENT COMMANDS

The following humdrum commands accept **pcset encoded data as inputs:

iv determine interval vectors for successive vertical sonorities nf determine normal form for successive vertical sonorities pcset convert pitch and pitch-class information to set-theoretic representations pf prime form representation

The following Humdrum command produces **pcset data as output:

pcset convert pitch and pitch-class information to set-theoretic representations – ————————————- —————————————————————————-

TANDEM INTERPRETATIONS

The following tandem interpretations can be used in conjunction with **pcset:

meter signatures *M6/8 key signatures *k[ ] key *c#: tempo *MM96.3 timebase *tb32 —————— ———-

Tandem interpretations for pcset

REFERENCE

Forte, A. The Structure of Atonal Music. New Haven: Yale University Press, 1973.

APPENDIX

The following table provides an extended list of all possible Fortean-type pitch-class set names. The corresponding pitch-class set and interval vectors are also shown.

Fortean pitch-class interval Fortean pitch-class interval set-name set vector set-name set vector 0-0 (empty) <000000> 12-1 (0,1,2,3,4,5,6,7,8,9,10,11) <12,12,12,12,12,12> 1-1 (0) <000000> 11-1 (0,1,2,3,4,5,6,7,8,9,10) <10,10,10,10,10,10> 2-1 (0,1) <100000> 10-1 (0,1,2,3,4,5,6,7,8,9) <988888> 2-2 (0,2) <010000> 10-2 (0,1,2,3,4,5,6,7,8,A) <898888> 2-3 (0,3) <001000> 10-3 (0,1,2,3,4,5,6,7,9,A) <889888> 2-4 (0,4) <000100> 10-4 (0,1,2,3,4,5,6,8,9,A) <888988> 2-5 (0,5) <000010> 10-5 (0,1,2,3,4,5,7,8,9,A) <888898> 2-6 (0,6) <000001> 10-6 (0,1,2,3,4,6,7,8,9,A) <888889> 3-1 (0,1,2) <210000> 9-1 (0,1,2,3,4,5,6,7,8) <876663> 3-2 (0,1,3) <111000> 9-2 (0,1,2,3,4,5,6,7,9) <777663> 3-3 (0,1,4) <101100> 9-3 (0,1,2,3,4,5,6,8,9) <767763> 3-4 (0,1,5) <100110> 9-4 (0,1,2,3,4,5,7,8,9) <766773> 3-5 (0,1,6) <100011> 9-5 (0,1,2,3,4,6,7,8,9) <766674> 3-6 (0,2,4) <020100> 9-6 (0,1,2,3,4,5,6,8,10) <686763> 3-7 (0,2,5) <011010> 9-7 (0,1,2,3,4,5,7,8,10) <677673> 3-8 (0,2,6) <010101> 9-8 (0,1,2,3,4,6,7,8,10) <676764> 3-9 (0,2,7) <010020> 9-9 (0,1,2,3,5,6,7,8,10) <676683> 3-10 (0,3,6) <002001> 9-10 (0,1,2,3,4,6,7,9,10) <668664> 3-11 (0,3,7) <001110> 9-11 (0,1,2,3,5,6,7,9,10) <667773> 3-12 (0,4,8) <000300> 9-12 (0,1,2,4,5,6,8,9,10) <666963> 4-1 (0,1,2,3) <321000> 8-1 (0,1,2,3,4,5,6,7) <765442> 4-2 (0,1,2,4) <221100> 8-2 (0,1,2,3,4,5,6,8) <665542> 4-3 (0,1,3,4) <212100> 8-3 (0,1,2,3,4,5,6,9) <656542> 4-4 (0,1,2,5) <211110> 8-4 (0,1,2,3,4,5,7,8) <655552> 4-5 (0,1,2,6) <210111> 8-5 (0,1,2,3,4,6,7,8) <654553> 4-6 (0,1,2,7) <210021> 8-6 (0,1,2,3,5,6,7,8) <654463> 4-7 (0,1,4,5) <201210> 8-7 (0,1,2,3,4,5,8,9) <645652> 4-8 (0,1,5,6) <200121> 8-8 (0,1,2,3,4,7,8,9) <644563> 4-9 (0,1,6,7) <200022> 8-9 (0,1,2,3,6,7,8,9) <644464> 4-10 (0,2,3,5) <122010> 8-10 (0,2,3,4,5,6,7,9) <566452> 4-11 (0,1,3,5) <121110> 8-11 (0,1,2,3,4,5,7,9) <565552> 4-12 (0,2,3,6) <112101> 8-12 (0,1,3,4,5,6,7,9) <556543> 4-13 (0,1,3,6) <112011> 8-13 (0,1,2,3,4,6,7,9) <556453> 4-14 (0,2,3,7) <111120> 8-14 (0,1,2,4,5,6,7,9) <555562> 4-Z15 (0,1,4,6) <111111> 8-Z15 (0,1,2,3,4,6,8,9) <555553> 4-16 (0,1,5,7) <110121> 8-16 (0,1,2,3,5,7,8,9) <554563> 4-17 (0,3,4,7) <102210> 8-17 (0,1,3,4,5,6,8,9) <546652> 4-18 (0,1,4,7) <102111> 8-18 (0,1,2,3,5,6,8,9) <546553> 4-19 (0,1,4,8) <101310> 8-19 (0,1,2,4,5,6,8,9) <545752> 4-20 (0,1,5,8) <101220> 8-20 (0,1,2,4,5,7,8,9) <545662> 4-21 (0,2,4,6) <030201> 8-21 (0,1,2,3,4,6,8,10) <474643> 4-22 (0,2,4,7) <021120> 8-22 (0,1,2,3,5,6,8,10) <465562> 4-23 (0,2,5,7) <021030> 8-23 (0,1,2,3,5,7,8,10) <465472> 4-24 (0,2,4,8) <020301> 8-24 (0,1,2,4,5,6,8,10) <464743> 4-25 (0,2,6,8) <020202> 8-25 (0,1,2,4,6,7,8,10) <464644> 4-26 (0,3,5,8) <012120> 8-26 (0,1,2,4,5,7,9,10) <456562> 4-27 (0,2,5,8) <012111> 8-27 (0,1,2,4,5,7,8,10) <456553> 4-28 (0,3,6,9) <004002> 8-28 (0,1,3,4,6,7,9,10) <448444> 4-Z29 (0,1,3,7) <111111> 8-Z29 (0,1,2,3,5,6,7,9) <555553> 5-1 (0,1,2,3,4) <432100> 7-1 (0,1,2,3,4,5,6) <654321> 5-2 (0,1,2,3,5) <332110> 7-2 (0,1,2,3,4,5,7) <554331> 5-3 (0,1,2,4,5) <322210> 7-3 (0,1,2,3,4,5,8) <544431> 5-4 (0,1,2,3,6) <322111> 7-4 (0,1,2,3,4,6,7) <544332> 5-5 (0,1,2,3,7) <321121> 7-5 (0,1,2,3,5,6,7) <543342> 5-6 (0,1,2,5,6) <311221> 7-6 (0,1,2,3,4,7,8) <533442> 5-7 (0,1,2,6,7) <310132> 7-7 (0,1,2,3,6,7,8) <532353> 5-8 (0,2,3,4,6) <232201> 7-8 (0,2,3,4,5,6,8) <454422> 5-9 (0,1,2,4,6) <231211> 7-9 (0,1,2,3,4,6,8) <453432> 5-10 (0,1,3,4,6) <223111> 7-10 (0,1,2,3,4,6,9) <445332> 5-11 (0,2,3,4,7) <222220> 7-11 (0,1,3,4,5,6,8) <444441> 5-Z12 (0,1,3,5,6) <222121> 7-Z12 (0,1,2,3,4,7,9) <444342> 5-13 (0,1,2,4,8) <221311> 7-13 (0,1,2,4,5,6,8) <443532> 5-14 (0,1,2,5,7) <221131> 7-14 (0,1,2,3,5,7,8) <443352> 5-15 (0,1,2,6,8) <220222> 7-15 (0,1,2,4,6,7,8) <442443> 5-16 (0,1,3,4,7) <213211> 7-16 (0,1,2,3,5,6,9) <435432> 5-Z17 (0,1,3,4,8) <212320> 7-Z17 (0,1,2,4,5,6,9) <434541> 5-Z18 (0,1,4,5,7) <212221> 7-Z18 (0,1,2,3,5,8,9) <434442> 5-19 (0,1,3,6,7) <212122> 7-19 (0,1,2,3,6,7,9) <434343> 5-20 (0,1,3,7,8) <211231> 7-20 (0,1,2,4,7,8,9) <433452> 5-21 (0,1,4,5,8) <202420> 7-21 (0,1,2,4,5,8,9) <424641> 5-22 (0,1,4,7,8) <202321> 7-22 (0,1,2,5,6,8,9) <424542> 5-23 (0,2,3,5,7) <132130> 7-23 (0,2,3,4,5,7,9) <354351> 5-24 (0,1,3,5,7) <131221> 7-24 (0,1,2,3,5,7,9) <353442> 5-25 (0,2,3,5,8) <123121> 7-25 (0,2,3,4,6,7,9) <345342> 5-26 (0,2,4,5,8) <122311> 7-26 (0,1,3,4,5,7,9) <344532> 5-27 (0,1,3,5,8) <122230> 7-27 (0,1,2,4,5,7,9) <344451> 5-28 (0,2,3,6,8) <122212> 7-28 (0,1,3,5,6,7,9) <344433> 5-29 (0,1,3,6,8) <122131> 7-29 (0,1,2,4,6,7,9) <344352> 5-30 (0,1,4,6,8) <121321> 7-30 (0,1,2,4,6,8,9) <343542> 5-31 (0,1,3,6,9) <114112> 7-31 (0,1,3,4,6,7,9) <336333> 5-32 (0,1,4,6,9) <113221> 7-32 (0,1,3,4,6,8,9) <335442> 5-33 (0,2,4,6,8) <040402> 7-33 (0,1,2,4,6,8,10) <262623> 5-34 (0,2,4,6,9) <032221> 7-34 (0,1,3,4,6,8,10) <254442> 5-35 (0,2,4,7,9) <032140> 7-35 (0,1,3,5,6,8,10) <254361> 5-Z36 (0,1,2,4,7) <222121> 7-Z36 (0,1,2,3,5,6,8) <444342> 5-Z37 (0,3,4,5,8) <212320> 7-Z37 (0,1,3,4,5,7,8) <434541> 5-Z38 (0,1,2,5,8) <212221> 7-Z38 (0,1,2,4,5,7,8) <434442> 6-1 (0,1,2,3,4,5) <543210> 6-Z26 (0,1,3,5,7,8) <232341> 6-2 (0,1,2,3,4,6) <443211> 6-27 (0,1,3,4,6,9) <225222> 6-Z3 (0,1,2,3,5,6) <433221> 6-Z28 (0,1,3,5,6,9) <224322> 6-Z4 (0,1,2,4,5,6) <432321> 6-Z29 (0,1,3,6,8,9) <224232> 6-5 (0,1,2,3,6,7) <422232> 6-30 (0,1,3,6,7,9) <224223> 6-Z6 (0,1,2,5,6,7) <421242> 6-31 (0,1,3,5,8,9) <223431> 6-7 (0,1,2,6,7,8) <420243> 6-32 (0,2,4,5,7,9) <143250> 6-8 (0,2,3,4,5,7) <343230> 6-33 (0,2,3,5,7,9) <143241> 6-9 (0,1,2,3,5,7) <342231> 6-34 (0,1,3,5,7,9) <142422> 6-Z10 (0,1,3,4,5,7) <333321> 6-35 (0,2,4,6,8,10) <060603> 6-Z11 (0,1,2,4,5,7) <333231> 6-Z36 (0,1,2,3,4,7) <433221> 6-Z12 (0,1,2,4,6,7) <332232> 6-Z37 (0,1,2,3,4,8) <432321> 6-Z13 (0,1,3,4,6,7) <324222> 6-Z38 (0,1,2,3,7,8) <421242> 6-14 (0,1,3,4,5,8) <323430> 6-Z39 (0,2,3,4,5,8) <333321> 6-15 (0,1,2,4,5,8) <323421> 6-Z40 (0,1,2,3,5,8) <333231> 6-16 (0,1,4,5,6,8) <322431> 6-Z41 (0,1,2,3,6,8) <332232> 6-Z17 (0,1,2,4,7,8) <322332> 6-Z42 (0,1,2,3,6,9) <324222> 6-18 (0,1,2,5,7,8) <322242> 6-Z43 (0,1,2,5,6,8) <322332> 6-Z19 (0,1,3,4,7,8) <313431> 6-Z44 (0,1,2,5,6,9) <313431> 6-20 (0,1,4,5,8,9) <303630> 6-Z45 (0,2,3,4,6,9) <234222> 6-21 (0,2,3,4,6,8) <242412> 6-Z46 (0,1,2,4,6,9) <233331> 6-22 (0,1,2,4,6,8) <241422> 6-Z47 (0,1,2,4,7,9) <233241> 6-Z23 (0,2,3,5,6,8) <234222> 6-Z48 (0,1,2,5,7,9) <232341> 6-Z24 (0,1,3,4,6,8) <233331> 6-Z49 (0,1,3,4,7,9) <224322> 6-Z25 (0,1,3,5,6,8) <233241> 6-Z50 (0,1,4,6,7,9) <224232> ———- ————— ———— – – – ———- —————————– ———————–