63edo

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← 62edo 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 21/63, 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.

Intervals

The following table was created using Godtone's code with the command interpret_edo(63,ol=47,no=[5,17,19,25,27,37,41],add=[63,73,75,87,89,91,93],dec="''",wiki=23) (run in a Python 3 interactive console) plus manual correction of the order of some inconsistent intervals.

As the command 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 47-odd-limit add-63 add-73 add-75 add-87 add-89 add-91 add-93 interpretation, tuned to the strengths of 63edo. Note that because of the cancellation of factors, some odd harmonics of 5 (the more relevant ones) are present, specifically 75/3 = 25, 45/3 = 15, and 45/9 = 5.

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[note 1]
0 0.0 1/1
1 19.05 94/93, 93/92, 92/91, 91/90, 90/89, 89/88, 88/87, 87/86, 73/72, 64/63
2 38.1 63/62, 48/47, 47/46, 93/91, 46/45, 91/89, 45/44, 89/87, 44/43, 43/42, 75/73
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, 73/70, 24/23, 47/45, 93/89, 23/22, 91/87, 45/43, 22/21
5 95.24 21/20, 96/91, 94/89, 93/88, 92/87, 91/86, 89/84, 35/33
6 114.29 33/31, 16/15, 47/44, 78/73, 31/29, 46/43, 15/14
7 133.33 14/13, 96/89, 94/87, 93/86, 13/12
8 152.38 63/58, 12/11, 47/43, 35/32, 23/21
9 171.43 11/10, 43/39, 32/29, 73/66, 52/47, 31/28, 10/9
10 190.48 39/35, 29/26, 48/43, 104/93, 47/42
11 209.52 28/25, 9/8, 44/39, 35/31, 26/23, 25/22
12 228.57 33/29, 73/64, 89/78, 8/7
13 247.62 86/75, 84/73, 15/13, 52/45, 73/63
14 266.67 29/25, 36/31, 7/6, 104/89, 75/64
15 285.71 88/75, 73/62, 86/73, 33/28, 46/39, 13/11, 25/21
16 304.76 89/75, 56/47, 87/73, 31/26, 43/36, 104/87
17 323.81 6/5, 112/93, 47/39, 88/73, 35/29, 29/24, 52/43, 75/62
18 342.86 63/52, 91/75, 73/60, 28/23, 39/32, 89/73, 11/9
19 361.9 92/75, 43/35, 16/13, 90/73, 58/47, 89/72, 26/21
20 380.95 31/25, 36/29, 87/70, 56/45, 91/73, 116/93, 5/4
21 400.0 94/75, 44/35, 39/31, 112/89, 73/58, 92/73, 29/23, 91/72
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, 75/58
24 457.14 13/10, 56/43, 43/33, 116/89, 73/56, 30/23, 47/36
25 476.19 21/16, 46/35, 96/73, 29/22, 120/91, 62/47
26 495.24 93/70, 4/3, 75/56
27 514.29 63/47, 47/35, 43/32, 39/29, 35/26, 31/23, 120/89, 89/66, 58/43
28 533.33 42/31, 87/64, 64/47, 124/91, 15/11, 86/63
29 552.38 63/46, 48/35, 11/8, 128/93, 62/45, 91/66, 40/29, 29/21
30 571.43 18/13, 104/75, 43/31, 89/64, 32/23, 39/28, 124/89, 46/33, 60/43, 88/63
31 590.48 7/5, 87/62, 73/52, 66/47, 45/32, 128/91, 31/22, 89/63
32 609.52 126/89, 44/31, 91/64, 64/45, 47/33, 104/73, 124/87, 10/7
33 628.57 63/44, 43/30, 33/23, 89/62, 56/39, 23/16, 128/89, 62/43, 75/52, 13/9
34 647.62 42/29, 29/20, 132/91, 45/31, 93/64, 16/11, 35/24, 92/63
35 666.67 63/43, 22/15, 91/62, 47/32, 128/87, 31/21
36 685.71 43/29, 132/89, 89/60, 46/31, 52/35, 58/39, 64/43, 70/47, 94/63
37 704.76 112/75, 3/2, 140/93
38 723.81 47/31, 91/60, 44/29, 73/48, 35/23, 32/21
39 742.86 72/47, 23/15, 112/73, 89/58, 66/43, 43/28, 20/13
40 761.9 116/75, 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
42 800.0 144/91, 46/29, 73/46, 116/73, 89/56, 62/39, 35/22, 75/47
43 819.05 8/5, 93/58, 146/91, 45/28, 140/87, 29/18, 50/31
44 838.1 21/13, 144/89, 47/29, 73/45, 13/8, 70/43, 75/46
45 857.14 18/11, 146/89, 64/39, 23/14, 120/73, 150/91, 104/63
46 876.19 124/75, 43/26, 48/29, 58/35, 73/44, 78/47, 93/56, 5/3
47 895.24 87/52, 72/43, 52/31, 146/87, 47/28, 150/89
48 914.29 42/25, 22/13, 39/23, 56/33, 73/43, 124/73, 75/44
49 933.33 128/75, 89/52, 12/7, 31/18, 50/29
50 952.38 126/73, 45/26, 26/15, 73/42, 75/43
51 971.43 7/4, 156/89, 128/73, 58/33
52 990.48 44/25, 23/13, 62/35, 39/22, 16/9, 25/14
53 1009.52 84/47, 93/52, 43/24, 52/29, 70/39
54 1028.57 9/5, 56/31, 47/26, 132/73, 29/16, 78/43, 20/11
55 1047.62 42/23, 64/35, 86/47, 11/6, 116/63
56 1066.67 24/13, 172/93, 87/47, 89/48, 13/7
57 1085.71 28/15, 43/23, 58/31, 73/39, 88/47, 15/8, 62/33
58 1104.76 66/35, 168/89, 172/91, 87/46, 176/93, 89/47, 91/48, 40/21
59 1123.81 21/11, 86/45, 174/91, 44/23, 178/93, 90/47, 23/12, 140/73, 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 1161.9 146/75, 84/43, 43/22, 174/89, 88/45, 178/91, 45/23, 182/93, 92/47, 47/24, 124/63
62 1180.95 63/32, 144/73, 172/87, 87/44, 176/89, 89/45, 180/91, 91/46, 184/93, 93/47
63 1200.0 2/1

Notation

Sagittal notation

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

Evo flavor

63-EDO Evo Sagittal.svgSagittal notationPeriodic table of EDOs with sagittal notation64/6381/8033/32

Revo flavor

63-EDO Revo Sagittal.svgSagittal notationPeriodic table of EDOs with sagittal notation64/6381/8033/32

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
Heji18.svg
Heji19.svg
Heji20.svg
Heji21.svg
Heji22.svg
Heji23.svg
Heji24.svg
Heji25.svg
Heji26.svg
Heji27.svg
Heji28.svg
Heji29.svg
Heji30.svg
Heji31.svg
Heji32.svg
Heji33.svg
Heji34.svg
Heji35.svg
Flat Symbol
Heji17.svg
Heji16.svg
Heji15.svg
Heji14.svg
Heji13.svg
Heji12.svg
Heji11.svg
Heji10.svg
Heji9.svg
Heji8.svg
Heji7.svg
Heji6.svg
Heji5.svg
Heji4.svg
Heji3.svg
Heji2.svg
Heji1.svg

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

Notes

  1. Based on treating 63edo as a 2.3.5.7.11.13.23.29.31.43.47.73-subgroup (no-17's no-19's no-37's no-41's 47-limit add-73 add-89) temperament; other approaches are also possible. Accurate or low-complexity intervals involving 5 are also included here.