How Many PC Sets

This page catalogues the number of distinct pitch class sets (hereinafter referred to as PC sets) to be found amongst n-Tone Equal Temperament (nTET) systems, for n = 2 to 31.

In addition to bulk statistics for each n, we also show their distributions amongst their corresponding k-sets for k = 1 to n. For example, the number of PC sets in 12TET is 351, and they are distributed as 1 1-set, 6 2-set, 19 3-set, 43 4-set, 66 5-set, 80 6-set etc.

The first table shows those PC set frequencies

The second table enumerates the non-symmetric PC sets.

The third table enumerates symmetric PC sets.

The fourth table enumerates the even rarer sets exhibiting rotational repetitions.

Table 1a - Number of k-sets within nTET for 2 ≤ n ≤ 21

k\n23456789101112131415161718192021
111111111111111111111
21122334455667788991010
311245710121519222631354046515764
411351014223043557391116140172204245285
511371426426699143201273364476612776969
6114102242801322173355047281038142819442586
711412306613224642971511441768265238765538
811515439921742981014302438397863109690
91151955143335715143027044862839814000
101162273201504114424384862925216796
11116269127372817683978839816796
121173111636410382652631014000
1311735140476142838769690
141184017261219445538
15118462047762586
1611951245969
1711957285
18111064
191110
2011
211
k/n235713193559107187351631118121914115771114601275955248799879

Table 1b - Number of k-sets within nTET for 22 ≤ n ≤ 31

k\n22232425262728293031
11111111111
211111212131314141515
370778592100109117126136145
43353854465065786507358199171015
51197146317712126253029903510409547515481
6339943895620708488661096613468163801981123751
77752106591442119228253003289042288538206786084825
8145502131830667432636011582225111041148005195143254475
922610355305448481719120175173593246675345345476913650325
1029414497428175213075220434731245546875469069010016031430715
113206658786104006178296297160482885766935119301018209102731365
1229414587861127202080123715166438561086601178951528832894552275
1322610497421040062080124000247429001337220234013539919956653325
141455035530817521782963715167429001432860267444048476378554275
15775221318544841307522971606438561337220267444051706049694845
1633991065930667817192043474828851086601234013548476379694845
17119743891442143263120175312455766935178951539919958554275
18335146356201922860115173593468754119301028832896653325
197038517717084253008222524667569069018209104552275
201177446212688663289011104134534510016032731365
2111185506253010966422881480054769131430715
2211129257829901346853820195143650325
23111210065035101638067860254475
24111310973540951981184825
251113117819475123751
2611141269175481
2711141361015
281115145
291115
3011
311
k/n1907453647236992511342183258142749710679587579185127913579256769273667

Table 2a - Number of non-symmetric k-sets within nTET for 2 ≤ n ≤ 21

Table of frequencies of non-invertible PC k-sets. They always occur in mirror image pairs - as represented by k-gonal sub-polygons of regular n-gons. Each paired k-gon flips into the other over an axis passing through a particular one of its vertices or - if n is even - over one particular pair of vertices or of opposite edges, the particularity depending upon the set. Such non-symmetric k-sets constitute the vast majority of sets amongst the possible ones enumerated in the first table.

k\n23456789101112131415161718192021
1
2
3224681014162024283238424854
424812202840527088112136168200240
54820325684128180252336448576740924
66123258112180300444672948134418162466
78205611222639468010881712256837925418
810288418039473613602304385260889480
91440128300680136026324736827213788
101652180444108823044736898016544
11207025267217123852827216544
1224883369482568608813788
1328112448134437929480
143213657618165418
15381687402466
1642200924
1748240
1854
19
20
21
k/n2412286012425450498619363720720013804265725089297828

Table 2b - Number of non-symmetric k-sets within nTET for 22 ≤ n ≤ 31

k\n22232425262728293031
1
2
3606674808896104112122130
4280330380440500572644728812910
51152140817162060246429123432400446605376
6322442245386686485641068013084160161933223296
77632104941425619008250803260442002534566749684370
8142002098830144427685936081510109988147004193708253110
922400352005415081224119680172872245960344344475904648960
102892049280809041299602029763111684666246886889984201427712
113181458324103544177504296368481598765648119100818189082728362
1228920583241117322070883696886421321083416178651228779724547270
1322400492801035442070883991007411841335504233713239889926648320
141420035200809041775043696887411841429202267100848408088547840
15763220988541501299602963686421321335504267100851671509688410
1632241049430144812242029764815981083416233713248408089688410
17115242241425642768119680311168765648178651239889928547840
18280140853861900859360172872466624119100828779726648320
196033017166864250808151024596068868818189084547270
206638020608564326041099883443449984202728362
2174440246410680420021470044759041427712
228050029121308453456193708648960
238857234321601667496253110
249664440041933284370
25104728466023296
261128125376
27122910
28130
29
30
31
k/n1875503606286928401333992256863649546569562014184800243574141869208132

Table 3a - Number of symmetric k-sets within nTET for 2 ≤ n ≤ 21

Symmetric sets invert into themselves. A set which - when represented by a cyclotomic polygon (a convex polygon constructed from subsets of the n vertices of regular n-gons) - is indistinguishable from the original when flipped over at least one axis of symmetry. These are relatively rare, and get proportionally rarer as n increases. Musically they correspond to scales which, when traversed downwards from some fixed pitch class within the scale (dependent upon the set) by the same interval pattern used in traversing up the scale, do not leave the scale. E.g. The Dorian mode of the diatonic scale from its tonic note. No other mode of the diatonic inverts on their tonic, but will invert on - variously - supertonic, mediant, dominant etc.

k\n23456789101112131415161718192021
111111111111111111111
211223344556677889910
3121333454666787999
41234681012151821242832364045
5133651010151520212828363545
61338101720323353567884116117
714410102019353556568484119
8145121532356470120126200210
9154151533357068126126207
1015618205356120126245252
1116621215656126126252
1216624287884200207
1317728288484210
141783236116119
151873635117
161894045
1719945
18199
19110
201
21
k/n12367141828396281126175246360510728102214852030

Table 3b - Number of symmetric k-sets within nTET for 22 ≤ n ≤ 31

k\n22232425262728293031
11111111111
210111112121313141415
310111012121213141315
4505560667278849198105
5455555656678789190105
6160165212220280282358364444455
7120165165220220286285364364455
8320330480495700715980100113441365
921033032749549571171510019971365
10452462782790127212871987200229803003
1125146246279279212871287200220023003
12452462900924169617102968300349625005
1321046246292492317161716300330035005
14320330782792169617163411343264006435
1512033032779079217101716343234256435
16160165480495127212872968300364006435
174516516549549512871287300330036435
1850552122207007111987200249625005
19105555220220715715200220025005
2010116065280286980100129803003
2111110666628228510019973003
2211112727835836413441365
231121278783643641365
24112128491444455
25113139190455
261131498105
2711413105
2811415
29115
301
31
k/n3007409460308184121591635224381327664884965534

Table 4a - Number of rotationally repeating k-sets within nTET for 2 ≤ n ≤ 21

Technically non-symmetric (these PC sets cannot be flipped over some axis and remain 'the same'). But they are cyclicly equivalent because - as polygons - they do not require a full rotation of 360 degrees to look the same), they occur as congruent clockwise and anti-clockwise pairs. They do not show up in the nTET universe until n is 12. The first bi-cyclic polygonal pattern (a yin yang) requires two consecutive segments of three contiguous edges of three distinct sizes. These smallest distinct sizes must be 1, 2 and 3 (forming a segment of 123 or 132) and it must repeat, as either 123123 or 132132. Clearly these are both mutually inverse and mutually complementary hexagons, relatively clockwise/anticlockwise, in 12TET. Larger order patterns are the triskelion (Manx Legs), the tetraskelion (fylfot, gammadion, svastika, etc). The 30TET space is the first capable of supporting a pentaskelion (with the aforementioned uninvertible segment 123 or 132, repeated five times).

k\n23456789101112131415161718192021
1
2
3
4
5
622468
7
824812
922
104820
11
126122
13
148
15
16
17
18
19
20
21
k/n241230604

Table 4b - Number of rotationally repeating k-sets within nTET for 22 ≤ n ≤ 31

k\n22232425262728293031
1
2
3
4
5
61014162024
7
82028405270
9468
10325684128180
11
123221128212
13
142056112226394
15482
161028842394
17
181440612812
19
201652180
218
222070
23
2424
25
26
27
28
29
30
31
k/n124206504286301378

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