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\n	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
2	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8	8	9	9	10	10
3		1	1	2	4	5	7	10	12	15	19	22	26	31	35	40	46	51	57	64
4			1	1	3	5	10	14	22	30	43	55	73	91	116	140	172	204	245	285
5				1	1	3	7	14	26	42	66	99	143	201	273	364	476	612	776	969
6					1	1	4	10	22	42	80	132	217	335	504	728	1038	1428	1944	2586
7						1	1	4	12	30	66	132	246	429	715	1144	1768	2652	3876	5538
8							1	1	5	15	43	99	217	429	810	1430	2438	3978	6310	9690
9								1	1	5	19	55	143	335	715	1430	2704	4862	8398	14000
10									1	1	6	22	73	201	504	1144	2438	4862	9252	16796
11										1	1	6	26	91	273	728	1768	3978	8398	16796
12											1	1	7	31	116	364	1038	2652	6310	14000
13												1	1	7	35	140	476	1428	3876	9690
14													1	1	8	40	172	612	1944	5538
15														1	1	8	46	204	776	2586
16															1	1	9	51	245	969
17																1	1	9	57	285
18																	1	1	10	64
19																		1	1	10
20																			1	1
21																				1
k/n	2	3	5	7	13	19	35	59	107	187	351	631	1181	2191	4115	7711	14601	27595	52487	99879

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

k\n	22	23	24	25	26	27	28	29	30	31
1	1	1	1	1	1	1	1	1	1	1
2	11	11	12	12	13	13	14	14	15	15
3	70	77	85	92	100	109	117	126	136	145
4	335	385	446	506	578	650	735	819	917	1015
5	1197	1463	1771	2126	2530	2990	3510	4095	4751	5481
6	3399	4389	5620	7084	8866	10966	13468	16380	19811	23751
7	7752	10659	14421	19228	25300	32890	42288	53820	67860	84825
8	14550	21318	30667	43263	60115	82225	111041	148005	195143	254475
9	22610	35530	54484	81719	120175	173593	246675	345345	476913	650325
10	29414	49742	81752	130752	204347	312455	468754	690690	1001603	1430715
11	32066	58786	104006	178296	297160	482885	766935	1193010	1820910	2731365
12	29414	58786	112720	208012	371516	643856	1086601	1789515	2883289	4552275
13	22610	49742	104006	208012	400024	742900	1337220	2340135	3991995	6653325
14	14550	35530	81752	178296	371516	742900	1432860	2674440	4847637	8554275
15	7752	21318	54484	130752	297160	643856	1337220	2674440	5170604	9694845
16	3399	10659	30667	81719	204347	482885	1086601	2340135	4847637	9694845
17	1197	4389	14421	43263	120175	312455	766935	1789515	3991995	8554275
18	335	1463	5620	19228	60115	173593	468754	1193010	2883289	6653325
19	70	385	1771	7084	25300	82225	246675	690690	1820910	4552275
20	11	77	446	2126	8866	32890	111041	345345	1001603	2731365
21	1	11	85	506	2530	10966	42288	148005	476913	1430715
22	1	1	12	92	578	2990	13468	53820	195143	650325
23		1	1	12	100	650	3510	16380	67860	254475
24			1	1	13	109	735	4095	19811	84825
25				1	1	13	117	819	4751	23751
26					1	1	14	126	917	5481
27						1	1	14	136	1015
28							1	1	15	145
29								1	1	15
30									1	1
31										1
k/n	190745	364723	699251	1342183	2581427	4971067	9587579	18512791	35792567	69273667

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\n	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
1
2
3					2	2	4	6	8	10	14	16	20	24	28	32	38	42	48	54
4						2	4	8	12	20	28	40	52	70	88	112	136	168	200	240
5							4	8	20	32	56	84	128	180	252	336	448	576	740	924
6								6	12	32	58	112	180	300	444	672	948	1344	1816	2466
7									8	20	56	112	226	394	680	1088	1712	2568	3792	5418
8										10	28	84	180	394	736	1360	2304	3852	6088	9480
9											14	40	128	300	680	1360	2632	4736	8272	13788
10												16	52	180	444	1088	2304	4736	8980	16544
11													20	70	252	672	1712	3852	8272	16544
12														24	88	336	948	2568	6088	13788
13															28	112	448	1344	3792	9480
14																32	136	576	1816	5418
15																	38	168	740	2466
16																		42	200	924
17																			48	240
18																				54
19
20
21
k/n					2	4	12	28	60	124	254	504	986	1936	3720	7200	13804	26572	50892	97828

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

k\n	22	23	24	25	26	27	28	29	30	31
1
2
3	60	66	74	80	88	96	104	112	122	130
4	280	330	380	440	500	572	644	728	812	910
5	1152	1408	1716	2060	2464	2912	3432	4004	4660	5376
6	3224	4224	5386	6864	8564	10680	13084	16016	19332	23296
7	7632	10494	14256	19008	25080	32604	42002	53456	67496	84370
8	14200	20988	30144	42768	59360	81510	109988	147004	193708	253110
9	22400	35200	54150	81224	119680	172872	245960	344344	475904	648960
10	28920	49280	80904	129960	202976	311168	466624	688688	998420	1427712
11	31814	58324	103544	177504	296368	481598	765648	1191008	1818908	2728362
12	28920	58324	111732	207088	369688	642132	1083416	1786512	2877972	4547270
13	22400	49280	103544	207088	399100	741184	1335504	2337132	3988992	6648320
14	14200	35200	80904	177504	369688	741184	1429202	2671008	4840808	8547840
15	7632	20988	54150	129960	296368	642132	1335504	2671008	5167150	9688410
16	3224	10494	30144	81224	202976	481598	1083416	2337132	4840808	9688410
17	1152	4224	14256	42768	119680	311168	765648	1786512	3988992	8547840
18	280	1408	5386	19008	59360	172872	466624	1191008	2877972	6648320
19	60	330	1716	6864	25080	81510	245960	688688	1818908	4547270
20		66	380	2060	8564	32604	109988	344344	998420	2728362
21			74	440	2464	10680	42002	147004	475904	1427712
22				80	500	2912	13084	53456	193708	648960
23					88	572	3432	16016	67496	253110
24						96	644	4004	19332	84370
25							104	728	4660	23296
26								112	812	5376
27									122	910
28										130
29
30
31
k/n	187550	360628	692840	1333992	2568636	4954656	9562014	18480024	35741418	69208132

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\n	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
2		1	1	2	2	3	3	4	4	5	5	6	6	7	7	8	8	9	9	10
3			1	2	1	3	3	3	4	5	4	6	6	6	7	8	7	9	9	9
4				1	2	3	4	6	8	10	12	15	18	21	24	28	32	36	40	45
5					1	3	3	6	5	10	10	15	15	20	21	28	28	36	35	45
6						1	3	3	8	10	17	20	32	33	53	56	78	84	116	117
7							1	4	4	10	10	20	19	35	35	56	56	84	84	119
8								1	4	5	12	15	32	35	64	70	120	126	200	210
9									1	5	4	15	15	33	35	70	68	126	126	207
10										1	5	6	18	20	53	56	120	126	245	252
11											1	6	6	21	21	56	56	126	126	252
12												1	6	6	24	28	78	84	200	207
13													1	7	7	28	28	84	84	210
14														1	7	8	32	36	116	119
15															1	8	7	36	35	117
16																1	8	9	40	45
17																	1	9	9	45
18																		1	9	9
19																			1	10
20																				1
21
k/n	1	2	3	6	7	14	18	28	39	62	81	126	175	246	360	510	728	1022	1485	2030

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

k\n	22	23	24	25	26	27	28	29	30	31
1	1	1	1	1	1	1	1	1	1	1
2	10	11	11	12	12	13	13	14	14	15
3	10	11	10	12	12	12	13	14	13	15
4	50	55	60	66	72	78	84	91	98	105
5	45	55	55	65	66	78	78	91	90	105
6	160	165	212	220	280	282	358	364	444	455
7	120	165	165	220	220	286	285	364	364	455
8	320	330	480	495	700	715	980	1001	1344	1365
9	210	330	327	495	495	711	715	1001	997	1365
10	452	462	782	790	1272	1287	1987	2002	2980	3003
11	251	462	462	792	792	1287	1287	2002	2002	3003
12	452	462	900	924	1696	1710	2968	3003	4962	5005
13	210	462	462	924	923	1716	1716	3003	3003	5005
14	320	330	782	792	1696	1716	3411	3432	6400	6435
15	120	330	327	790	792	1710	1716	3432	3425	6435
16	160	165	480	495	1272	1287	2968	3003	6400	6435
17	45	165	165	495	495	1287	1287	3003	3003	6435
18	50	55	212	220	700	711	1987	2002	4962	5005
19	10	55	55	220	220	715	715	2002	2002	5005
20	10	11	60	65	280	286	980	1001	2980	3003
21	1	11	10	66	66	282	285	1001	997	3003
22		1	11	12	72	78	358	364	1344	1365
23			1	12	12	78	78	364	364	1365
24				1	12	12	84	91	444	455
25					1	13	13	91	90	455
26						1	13	14	98	105
27							1	14	13	105
28								1	14	15
29									1	15
30										1
31
k/n	3007	4094	6030	8184	12159	16352	24381	32766	48849	65534

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\n	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
1
2
3
4
5
6											2		2		4		6		8
7
8													2		4		8		12
9																	2			2
10															4		8		20
11
12																	6		12	2
13
14																			8
15
16
17
18
19
20
21
k/n											2		4		12		30		60	4

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

k\n	22	23	24	25	26	27	28	29	30	31
1
2
3
4
5
6	10		14		16		20		24
7
8	20		28		40		52		70
9			4			6			8
10	32		56		84		128		180
11
12	32		2		112	8	2		12
13
14	20		56		112		226		394
15			4			8			2
16	10		28		84		2		394
17
18			14		40	6	128		12
19
20					16		52		180
21									8
22							20		70
23
24									24
25
26
27
28
29
30
31
k/n	124		206		504	28	630		1378