63edo
| ← 62edo | 63edo | 64edo → |
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
The equal temperament tempers out 3125/3072 in the 5-limit and 225/224, 245/243, 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
| 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) | |
| 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* |
|---|---|---|
| 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 |
* 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 with accurate or low-complexity intervals involving 5 added.
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
- Improvisation in 12-tone fifths chain (2015)
- those early dreams (2016)
- early dreams 2 (2016)
- Seconds and Otonal Shifts (2016)