63edo
| ← 62edo | 63edo | 64edo → |
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 31-limit, as its regular augmented fourth (+6 fifths) is less than 0.3c sharp of 23/16, therefore tempering out 736/729, and its "large quarter-tone", or diesis, is only 2.2c off of 32/31 which is equated with 31/30 (even more accurate) and 30/29 on the other side, hence tempering S31 and S30, but also completing a streak of large quartertones/small dieses of superparticular intervals in the harmonic series by continuing to equate them on the large side with 29/28 and 28/27 (tempering S29 and S28) and on the small side with 33/32 (tempering S31). Although it doesn't deal as well with primes 5, 17, and 19, it excels in the 2.3.7.11.13.23.29.31 group, and is a great candidate for a rank-1 or rank-2 gentle tuning. As a fifths-system, the diesis after 12 fifths can represent 32:33, 27:28, any superparticular interval imbetween those, 88:91, and more, so it is very versatile, making chains of fifths 12 or longer very useful in covering harmonic and melodic ground while providing a lot of different colour in different keys. 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. Alternatively, using the quarter-tone interval 3\63 = 1\21, 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. 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 wasn't 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 table below.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | +9.03 | +2.77 | +1.16 | +2.69 |
| 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 | +47.4 | +14.5 | +6.1 | +14.1 | |
| 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) |
338 (23) |
342 (27) |
350 (35) |
361 (46) | |
Subsets and supersets
It is divisible by 3, 7, 9 and 21.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 19 | ^D, v3E♭ | |
| 2 | 38.1 | ^^D, vvE♭ | |
| 3 | 57.1 | 29/28, 30/29, 31/30, 32/31, 33/32 | ^3D, vE♭ |
| 4 | 76.2 | 23/22, 24/23 | v3D♯, E♭ |
| 5 | 95.2 | 18/17, 19/18 | vvD♯, ^E♭ |
| 6 | 114.3 | 16/15, 31/29 | vD♯, ^^E♭ |
| 7 | 133.3 | D♯, ^3E♭ | |
| 8 | 152.4 | 12/11, 35/32 | ^D♯, v3E |
| 9 | 171.4 | 21/19, 32/29 | ^^D♯, vvE |
| 10 | 190.5 | 19/17, 29/26 | ^3D♯, vE |
| 11 | 209.5 | 26/23, 35/31 | E |
| 12 | 228.6 | 8/7 | ^E, v3F |
| 13 | 247.6 | 15/13 | ^^E, vvF |
| 14 | 266.7 | 7/6 | ^3E, vF |
| 15 | 285.7 | 13/11, 33/28 | F |
| 16 | 304.8 | 31/26 | ^F, v3G♭ |
| 17 | 323.8 | 29/24, 35/29 | ^^F, vvG♭ |
| 18 | 342.9 | 28/23 | ^3F, vG♭ |
| 19 | 361.9 | 16/13, 21/17 | v3F♯, G♭ |
| 20 | 381 | vvF♯, ^G♭ | |
| 21 | 400 | 29/23, 34/27 | vF♯, ^^G♭ |
| 22 | 419 | 14/11 | F♯, ^3G♭ |
| 23 | 438.1 | 9/7 | ^F♯, v3G |
| 24 | 457.1 | 13/10, 30/23 | ^^F♯, vvG |
| 25 | 476.2 | 29/22 | ^3F♯, vG |
| 26 | 495.2 | 4/3 | G |
| 27 | 514.3 | 31/23, 35/26 | ^G, v3A♭ |
| 28 | 533.3 | 15/11 | ^^G, vvA♭ |
| 29 | 552.4 | 11/8 | ^3G, vA♭ |
| 30 | 571.4 | 32/23 | v3G♯, A♭ |
| 31 | 590.5 | 31/22 | vvG♯, ^A♭ |
| 32 | 609.5 | 27/19 | vG♯, ^^A♭ |
| 33 | 628.6 | 23/16, 33/23 | G♯, ^3A♭ |
| 34 | 647.6 | 16/11 | ^G♯, v3A |
| 35 | 666.7 | 22/15 | ^^G♯, vvA |
| 36 | 685.7 | ^3G♯, vA | |
| 37 | 704.8 | 3/2 | A |
| 38 | 723.8 | 35/23 | ^A, v3B♭ |
| 39 | 742.9 | 20/13, 23/15 | ^^A, vvB♭ |
| 40 | 761.9 | 14/9, 31/20 | ^3A, vB♭ |
| 41 | 781 | 11/7 | v3A♯, B♭ |
| 42 | 800 | 27/17, 35/22 | vvA♯, ^B♭ |
| 43 | 819 | vA♯, ^^B♭ | |
| 44 | 838.1 | 13/8, 34/21 | A♯, ^3B♭ |
| 45 | 857.1 | 23/14 | ^A♯, v3B |
| 46 | 876.2 | ^^A♯, vvB | |
| 47 | 895.2 | ^3A♯, vB | |
| 48 | 914.3 | 22/13 | B |
| 49 | 933.3 | 12/7 | ^B, v3C |
| 50 | 952.4 | 26/15, 33/19 | ^^B, vvC |
| 51 | 971.4 | 7/4 | ^3B, vC |
| 52 | 990.5 | 23/13 | C |
| 53 | 1009.5 | 34/19 | ^C, v3D♭ |
| 54 | 1028.6 | 29/16 | ^^C, vvD♭ |
| 55 | 1047.6 | 11/6 | ^3C, vD♭ |
| 56 | 1066.7 | v3C♯, D♭ | |
| 57 | 1085.7 | 15/8 | vvC♯, ^D♭ |
| 58 | 1104.8 | 17/9 | vC♯, ^^D♭ |
| 59 | 1123.8 | 23/12 | C♯, ^3D♭ |
| 60 | 1142.9 | 29/15, 31/16 | ^C♯, v3D |
| 61 | 1161.9 | ^^C♯, vvD | |
| 62 | 1181 | ^3C♯, vD | |
| 63 | 1200 | 2/1 | D |