63edo

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63edo

64edo →

Prime factorization	32 × 7
Step size	19.0476 ¢
Fifth	37\63 (704.762 ¢)
Semitones (A1:m2)	7:4 (133.3 ¢ : 76.19 ¢)
Consistency limit	7
Distinct consistency limit	7

63 equal divisions of the octave (abbreviated 63edo or 63ed2), also called 63-tone equal temperament (63tet) or 63 equal temperament (63et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 63 equal parts of about 19 ¢ each. Each step represents a frequency ratio of 2^1/63, or the 63rd root of 2.

Theory

63edo tempers out 3125/3072 in the 5-limit and 225/224, 245/243, and 875/864 in the 7-limit, so that it supports magic temperament. In the 11-limit it tempers out 100/99, supporting 11-limit magic, plus 385/384 and 540/539, 896/891. In the 13-limit it tempers out 169/168, 275/273, 640/637, 352/351, 364/363 and 676/675. It provides the optimal patent val for immune, the 29 & 34d temperament in the 7-, 11- and 13-limit.

63 is also a fascinating division to look at in the 47-limit. Although it does not deal as well with primes 5, 17, 19, 37 and 41, it excels in the 2.3.7.11.13.23.29.31.43.47 subgroup, and is a great candidate for a gentle tuning. Its regular augmented fourth (+6 fifths) is less than 0.3 cents sharp of 23/16, therefore tempering out 736/729. Its diesis (+12 fifths) can represent 33/32, 32/31, 30/29, 29/28, 28/27, as well as 91/88, and more, so it is very versatile, making chains of fifths of 12 tones or longer very useful in covering harmonic and melodic ground while providing a lot of different colour in different keys. We can take advantage of the representation of 27:28:29:30:31:32:33, which splits 11/9 into six "small dieses" as a result; here it can be seen more clearly why these are not regular quarter-tones so are best distinguished from such with the qualifier "large", as otherwise we would expect to see some flavour of minor third after six of them.

A 17-tone fifths chain looks on the surface a little similar to 17edo, but as −17 fifths gets us to 64/63, observing the comma becomes an essential part in progressions favouring prime 7. Furthermore, its prime 5 is far from unusable; although 25/16 is barely inconsistent, this affords the tuning supporting 7-limit magic, which may be considered interesting or desirable in of itself. And if this was not enough, if you really want to, it offers reasonable approximations to some yet higher primes too; namely 43/32, 47/32, and 53/32; see the tables below.

Prime harmonics

Approximation of prime harmonics in 63edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	+0.00	+2.81	-5.36	+2.60	+1.06	-2.43	+9.33	+7.25	+0.30	-1.01	-2.18	-3.72
	Relative (%)	+0.0	+14.7	-28.1	+13.7	+5.6	-12.8	+49.0	+38.1	+1.6	-5.3	-11.4	-19.6
Steps (reduced)		63 (0)	100 (37)	146 (20)	177 (51)	218 (29)	233 (44)	258 (6)	268 (16)	285 (33)	306 (54)	312 (60)	328 (13)

Approximation of prime harmonics in 63edo (continued)

Harmonic		41	43	47	53	59	61	67	71	73	79	83	89
Error	Absolute (¢)	+9.03	+2.77	+1.16	+2.69	+7.50	+6.92	-3.12	-8.27	+0.78	-2.63	+7.10	+0.55
	Relative (%)	+47.4	+14.5	+6.1	+14.1	+39.3	+36.4	-16.4	-43.4	+4.1	-13.8	+37.3	+2.9
Steps (reduced)		338 (23)	342 (27)	350 (35)	361 (46)	371 (56)	374 (59)	382 (4)	387 (9)	390 (12)	397 (19)	402 (24)	408 (30)

Subsets and supersets

Since 63 factors into 3² × 7, 63edo has subset edos 3, 7, 9, and 21.

Its representation of the 2.3.5.7.13 subgroup (no-11's 13-limit) can uniquely be described in terms of accurate approximations contained in its main subsets of 7edo and 9edo:

• 1\9 = ~14/13~13/12, implying (the much more accurate) 2\9 = ~7/6 (septiennealic)
• 2\7 = ~39/32~128/105, via 4096/4095 and the akjaysma (which are naturally paired)

If we avoid equating 14/13 and 13/12 (which is by far the highest damage equivalence) so that we achieve 7/6 = 2\9 directly, we get the 63 & 441 microtemperament in the same subgroup.

Intervals

The following table was created using Godtone's code with the command interpret_edo(63,ol=53,no=[5,17,19,25,27,37,41,51],add=[73,75,87,89,91,93,105],dec="''",wiki=23) (run in a Python 3 interactive console) plus manual correction of the order of some inconsistent intervals and removal of unsimplified intervals of 75.

As the command and description indicates, it is a(n accurate) "no-5's"* no-17's no-19's no-25's no-27's no-37's no-41's 49-odd-limit add-53 add-63 add-73 add-87 add-89 add-91 add-93 add-105 interpretation, tuned to the strengths of 63edo. * Note that because of the cancellation of factors, some odd harmonics of 5 (the simpler/more relevant ones) are present, EG 75/3 = 25, 45/3 = 15, 105/75 = 7/5, 75/35/2 = 15/14, and 45/9 = 5, so it isn't really "no-5's", just has a de-emphasized focus.

Intervals are listed in order of size, so that one can know their relative order at a glance and deem the value of the interpretation for a harmonic context, and 23-limit intervals are highlighted for navigability as 13-limit intervals are more likely to already have pages, and as we are excluding primes 17 and 19, we are only adding prime 23 to the 13-limit.

Inconsistent intervals are in italics.

Degree	Cents	Approximate ratios
0	0.0	1/1
1	19.05	106/105, 105/104, 94/93, 93/92, 92/91, 91/90, 90/89, 89/88, 88/87, 87/86, 73/72, 65/64, 64/63
2	38.1	66/65, 53/52, 49/48, 48/47, 47/46, 93/91, 46/45, 91/89, 45/44, 89/87, 44/43, 43/42
3	57.14	36/35, 33/32, 32/31, 94/91, 31/30, 92/89, 91/88, 30/29, 89/86, 29/28, 25/24
4	76.19	26/25, 49/47, 73/70, 24/23, 47/45, 93/89, 23/22, 91/87, 45/43, 22/21
5	95.24	98/93, 96/91, 94/89, 56/53, 93/88, 92/87, 91/86, 89/84, 35/33, 52/49
6	114.29	33/31, 49/46, 16/15, 47/44, 78/73, 31/29, 46/43, 15/14
7	133.33	14/13, 96/89, 94/87, 93/86, 53/49, 13/12
8	152.38	49/45, 12/11, 47/43, 35/32, 58/53, 23/21
9	171.43	11/10, 98/89, 43/39, 32/29, 53/48, 116/105, 73/66, 52/47, 31/28
10	190.48	49/44, 39/35, 29/26, 48/43, 105/94, 104/93, 47/42
11	209.52	28/25, 9/8, 98/87, 53/47, 44/39, 35/31, 26/23, 60/53, 25/22
12	228.57	33/29, 49/43, 106/93, 73/64, 89/78, 105/92, 8/7
13	247.62	84/73, 53/46, 15/13, 52/45
14	266.67	29/25, 36/31, 106/91, 7/6, 104/89, 62/53
15	285.71	73/62, 53/45, 86/73, 33/28, 46/39, 105/89, 124/105, 13/11, 58/49
16	304.76	106/89, 56/47, 87/73, 31/26, 105/88, 43/36, 104/87
17	323.81	6/5, 112/93, 53/44, 47/39, 88/73, 35/29, 64/53, 29/24, 52/43
18	342.86	73/60, 28/23, 106/87, 39/32, 128/105, 89/73, 105/86, 11/9, 60/49
19	361.9	43/35, 16/13, 53/43, 90/73, 58/47, 89/72, 26/21
20	380.95	31/25, 36/29, 87/70, 56/45, 66/53, 91/73, 116/93, 5/4
21	400.0	49/39, 44/35, 39/31, 112/89, 73/58, 92/73, 29/23, 53/42, 91/72, 62/49
22	419.05	33/26, 89/70, 14/11, 93/73, 116/91, 60/47, 23/18
23	438.1	32/25, 9/7, 112/87, 94/73, 58/45, 40/31, 31/24
24	457.14	13/10, 56/43, 43/33, 116/89, 73/56, 30/23, 47/36, 64/49
25	476.19	21/16, 46/35, 96/73, 29/22, 120/91, 62/47, 70/53
26	495.24	93/70, 4/3
27	514.29	98/73, 47/35, 43/32, 39/29, 35/26, 66/49, 31/23, 120/89, 89/66, 58/43
28	533.33	42/31, 72/53, 53/39, 87/64, 49/36, 64/47, 124/91, 15/11
29	552.38	48/35, 11/8, 128/93, 73/53, 62/45, 91/66, 40/29, 29/21
30	571.43	18/13, 43/31, 146/105, 89/64, 32/23, 39/28, 124/89, 46/33, 60/43
31	590.48	7/5, 87/62, 73/52, 66/47, 45/32, 128/91, 31/22
32	609.52	44/31, 91/64, 64/45, 47/33, 104/73, 124/87, 10/7
33	628.57	43/30, 33/23, 89/62, 56/39, 23/16, 128/89, 105/73, 62/43, 13/9
34	647.62	42/29, 29/20, 132/91, 45/31, 106/73, 93/64, 16/11, 35/24
35	666.67	22/15, 91/62, 47/32, 72/49, 128/87, 78/53, 53/36, 31/21
36	685.71	43/29, 132/89, 89/60, 46/31, 49/33, 52/35, 58/39, 64/43, 70/47, 73/49
37	704.76	3/2, 140/93
38	723.81	53/35, 47/31, 91/60, 44/29, 73/48, 35/23, 32/21
39	742.86	49/32, 72/47, 23/15, 112/73, 89/58, 66/43, 43/28, 20/13
40	761.9	48/31, 31/20, 45/29, 73/47, 87/56, 14/9, 25/16
41	780.95	36/23, 47/30, 91/58, 146/93, 11/7, 140/89, 52/33
42	800.0	49/31, 144/91, 84/53, 46/29, 73/46, 116/73, 89/56, 62/39, 35/22, 78/49
43	819.05	8/5, 93/58, 146/91, 53/33, 45/28, 140/87, 29/18, 50/31
44	838.1	21/13, 144/89, 47/29, 73/45, 86/53, 13/8, 70/43
45	857.14	49/30, 18/11, 172/105, 146/89, 105/64, 64/39, 87/53, 23/14, 120/73, 150/91
46	876.19	43/26, 48/29, 53/32, 58/35, 73/44, 78/47, 88/53, 93/56, 5/3
47	895.24	87/52, 72/43, 176/105, 52/31, 146/87, 47/28, 89/53, 150/89
48	914.29	49/29, 22/13, 105/62, 178/105, 39/23, 56/33, 73/43, 90/53, 124/73
49	933.33	53/31, 89/52, 12/7, 91/53, 31/18, 50/29
50	952.38	45/26, 26/15, 92/53, 73/42
51	971.43	7/4, 184/105, 156/89, 128/73, 93/53, 86/49, 58/33
52	990.48	44/25, 53/30, 23/13, 62/35, 39/22, 94/53, 87/49, 16/9, 25/14
53	1009.52	84/47, 93/52, 188/105, 43/24, 52/29, 70/39, 88/49
54	1028.57	56/31, 47/26, 132/73, 105/58, 96/53, 29/16, 78/43, 89/49, 20/11
55	1047.62	42/23, 53/29, 64/35, 86/47, 11/6, 90/49
56	1066.67	24/13, 98/53, 172/93, 87/47, 89/48, 13/7
57	1085.71	28/15, 43/23, 58/31, 73/39, 88/47, 15/8, 92/49, 62/33
58	1104.76	49/26, 66/35, 168/89, 172/91, 87/46, 176/93, 53/28, 89/47, 91/48, 93/49
59	1123.81	21/11, 86/45, 174/91, 44/23, 178/93, 90/47, 23/12, 140/73, 94/49, 25/13
60	1142.86	48/25, 56/29, 172/89, 29/15, 176/91, 89/46, 60/31, 91/47, 31/16, 64/33, 35/18
61	1161.9	84/43, 43/22, 174/89, 88/45, 178/91, 45/23, 182/93, 92/47, 47/24, 96/49, 104/53
62	1180.95	63/32, 144/73, 172/87, 87/44, 176/89, 89/45, 180/91, 91/46, 184/93, 93/47, 208/105, 105/53
63	1200.0	2/1

Notation

Sagittal notation

This notation uses the same sagittal sequence as 56-EDO.

Evo flavor

Revo flavor

Ups and downs notation

Using Helmholtz–Ellis accidentals, 63edo can be notated using ups and downs notation:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
Sharp symbol
Flat symbol

Approximation to JI

Zeta peak index

Tuning			Strength			Closest edo		Integer limit
ZPI	Steps per octave	Step size (cents)	Height	Integral	Gap	Edo	Octave (cents)	Consistent	Distinct
321zpi	63.0192885705350	19.0417890652143	6.768662	1.049023	15.412920	63edo	1199.63271110850	8	8

Scales

• Approximation of Pelog lima: 6 9 21 6 21
• Timeywimey (original/default tuning): 16 10 7 4 11 5 10
• Sandcastle (original/default tuning): 8 10 8 11 8 8 10

Music

Cam Taylor

• Improvisation in 12-tone fifths chain (2015)
• those early dreams (2016)
• early dreams 2 (2016)
• Seconds and Otonal Shifts (2016)