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
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
| 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 primes as 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
| Degree | Cents | Approximate ratios* |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 19.0 | 50/49, 55/54, 64/63, 65/64, 91/90, 105/104 |
| 2 | 38.1 | 45/44, 46/45, 49/48, 56/55, 66/65, 81/80 |
| 3 | 57.1 | 25/24, 28/27, 29/28, 30/29, 31/30, 32/31, 33/32, 36/35 |
| 4 | 76.2 | 22/21, 23/22, 24/23, 26/25, 27/26 |
| 5 | 95.2 | 21/20, 35/33 |
| 6 | 114.3 | 15/14, 16/15 |
| 7 | 133.3 | 13/12, 14/13 |
| 8 | 152.4 | 12/11 |
| 9 | 171.4 | 10/9, 11/10, 31/28, 32/29 |
| 10 | 190.5 | 29/26, 39/35, 49/44 |
| 11 | 209.5 | 9/8 |
| 12 | 228.6 | 8/7 |
| 13 | 247.6 | 15/13 |
| 14 | 266.7 | 7/6 |
| 15 | 285.7 | 13/11 |
| 16 | 304.8 | 31/26 |
| 17 | 323.8 | 6/5 |
| 18 | 342.9 | 11/9, 28/23, 39/32 |
| 19 | 361.9 | 16/13, 26/21, 27/22 |
| 20 | 381.0 | 5/4 |
| 21 | 400.0 | 29/23, 44/35, 49/39 |
| 22 | 419.0 | 14/11 |
| 23 | 438.1 | 9/7 |
| 24 | 457.1 | 13/10 |
| 25 | 476.2 | 21/16 |
| 26 | 495.2 | 4/3 |
| 27 | 514.3 | 35/26 |
| 28 | 533.3 | 15/11, 27/20 |
| 29 | 552.4 | 11/8 |
| 30 | 571.4 | 18/13, 32/23 |
| 31 | 590.5 | 7/5 |
| 32 | 609.5 | 10/7 |
| 33 | 628.6 | 13/9, 23/16 |
| 34 | 647.6 | 16/11 |
| 35 | 666.7 | 22/15 |
| 36 | 685.7 | 52/35 |
| 37 | 704.8 | 3/2 |
| 38 | 723.8 | 32/21 |
| 39 | 742.9 | 20/13 |
| 40 | 761.9 | 14/9 |
| 41 | 781.0 | 11/7 |
| 42 | 800.0 | 35/22, 46/29 |
| 43 | 819.0 | 8/5 |
| 44 | 838.1 | 13/8, 21/13, 44/27 |
| 45 | 857.1 | 18/11, 23/14, 64/39 |
| 46 | 876.2 | 5/3 |
| 47 | 895.2 | 52/31 |
| 48 | 914.3 | 22/13 |
| 49 | 933.3 | 12/7 |
| 50 | 952.4 | 26/15 |
| 51 | 971.4 | 7/4 |
| 52 | 990.5 | 16/9 |
| 53 | 1009.5 | 52/29, 70/39, 88/49 |
| 54 | 1028.6 | 9/5, 20/11, 29/16, 56/31 |
| 55 | 1047.6 | 11/6 |
| 56 | 1066.7 | 13/7, 24/13 |
| 57 | 1085.7 | 15/8, 28/15 |
| 58 | 1104.8 | 40/21, 66/35 |
| 59 | 1123.8 | 21/11, 23/12, 25/13, 44/23, 52/27 |
| 60 | 1142.9 | 27/14, 29/15, 31/16, 35/18, 48/25, 56/29, 60/31, 64/33 |
| 61 | 1161.9 | 45/23, 55/28, 88/45, 96/49, 160/81 |
| 62 | 1181.0 | 49/25, 63/32, 65/33, 108/55, 180/91, 208/105 |
| 63 | 1200.0 | 2/1 |
* As a 2.3.5.7.11.13.23.29.31-subgroup (no-17 no-19 31-limit) temperament, inconsistent intervals in italics
See also 63edo/Godtone's approach for some higher-limit ratios.
Notation
Sagittal notation
This notation uses the same sagittal sequence as 56edo.
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 |
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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
- Improvisation in 12-tone fifths chain (2015)
- those early dreams (2016)
- early dreams 2 (2016)
- Seconds and Otonal Shifts (2016)