63edo: Difference between revisions
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The equal temperament [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], [[875/864]] in the 7-limit, so that it [[support]]s [[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. | The equal temperament [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], [[875/864]] in the 7-limit, so that it [[support]]s [[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, | 63 is also a fascinating division to look at in the [[31-limit]]. Although it does not deal as well with primes 5, 17, and 19, it excels in the 2.3.7.11.13.23.29.31 [[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 === | === Prime harmonics === | ||
{{Harmonics in equal|63|columns= | {{Harmonics in equal|63|columns=12}} | ||
{{Harmonics in equal|63|start=13|columns=12|collapsed=1|title=Approximation of prime harmonics in 63edo (continued)}} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 63 factors into {{factorization|63}}, 63edo has subset edos {{EDOs| 3, 7, 9, and 21 }}. | |||
== Intervals == | == Intervals == | ||
Revision as of 08:32, 7 February 2024
| ← 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. Although it does not deal as well with primes 5, 17, and 19, it excels in the 2.3.7.11.13.23.29.31 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
| 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 |