64edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|64}} | |||
== Theory == | == Theory == | ||
64edo has two options of fifth equally far from just. The sharp fifth is inherited from 32edo and produces a hard superpythagorean scale, while the flat fifth is within the meantone/flattone range, supporting flattone temperament. | |||
Still, the patent val tempers out [[648/625]] in the 5-limit and [[225/224]] in the 7-limit, plus [[66/65]], [[121/120]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&64 temperament, which would perhaps be of more interest if it was lower in badness. | |||
64be val is a tuning for the beatles temperament and for the rank-3 temperaments heimlaug and vili in the 17-limit. 64bccc tunes dichotic, although that is an exotemperament. 64cdf is a tuning for vibhu. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|64}} | {{Harmonics in equal|64}} | ||
=== Subsets and supersets === | |||
64edo is the 6th power of two edo, and it has subset edos 1, 2, 4, 8, 16, 32. | |||
== Intervals == | == Intervals == | ||
{| class="wikitable center-all right-2" | {| class="wikitable mw-collapsible mw-collapsed center-all right-2" | ||
|- | |- | ||
![[Degree|#]] | ![[Degree|#]] | ||
Revision as of 14:24, 3 July 2023
| ← 63edo | 64edo | 65edo → |
Theory
64edo has two options of fifth equally far from just. The sharp fifth is inherited from 32edo and produces a hard superpythagorean scale, while the flat fifth is within the meantone/flattone range, supporting flattone temperament.
Still, the patent val tempers out 648/625 in the 5-limit and 225/224 in the 7-limit, plus 66/65, 121/120 and 441/440 in the 11-limit and 144/143 in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&64 temperament, which would perhaps be of more interest if it was lower in badness.
64be val is a tuning for the beatles temperament and for the rank-3 temperaments heimlaug and vili in the 17-limit. 64bccc tunes dichotic, although that is an exotemperament. 64cdf is a tuning for vibhu.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -8.21 | +7.44 | +6.17 | +2.34 | -7.57 | +3.22 | -0.77 | +7.54 | +2.49 | -2.03 | +9.23 |
| Relative (%) | -43.8 | +39.7 | +32.9 | +12.5 | -40.4 | +17.2 | -4.1 | +40.2 | +13.3 | -10.8 | +49.2 | |
| Steps (reduced) |
101 (37) |
149 (21) |
180 (52) |
203 (11) |
221 (29) |
237 (45) |
250 (58) |
262 (6) |
272 (16) |
281 (25) |
290 (34) | |
Subsets and supersets
64edo is the 6th power of two edo, and it has subset edos 1, 2, 4, 8, 16, 32.
Intervals
| # | Cents |
|---|---|
| 0 | 0.00 |
| 1 | 18.75 |
| 2 | 37.50 |
| 3 | 56.25 |
| 4 | 75.00 |
| 5 | 93.75 |
| 6 | 112.50 |
| 7 | 131.25 |
| 8 | 150.00 |
| 9 | 168.75 |
| 10 | 187.50 |
| 11 | 206.25 |
| 12 | 225.00 |
| 13 | 243.75 |
| 14 | 262.50 |
| 15 | 281.25 |
| 16 | 300.00 |
| 17 | 318.75 |
| 18 | 337.50 |
| 19 | 356.25 |
| 20 | 375.00 |
| 21 | 393.75 |
| 22 | 412.50 |
| 23 | 431.25 |
| 24 | 450.00 |
| 25 | 468.75 |
| 26 | 487.50 |
| 27 | 506.25 |
| 28 | 525.00 |
| 29 | 543.75 |
| 30 | 562.50 |
| 31 | 581.25 |
| 32 | 600.00 |
| 33 | 618.75 |
| 34 | 637.50 |
| 35 | 656.25 |
| 36 | 675.00 |
| 37 | 693.75 |
| 38 | 712.50 |
| 39 | 731.25 |
| 40 | 750.00 |
| 41 | 768.75 |
| 42 | 787.50 |
| 43 | 806.25 |
| 44 | 825.00 |
| 45 | 843.75 |
| 46 | 862.50 |
| 47 | 881.25 |
| 48 | 900.00 |
| 49 | 918.75 |
| 50 | 937.50 |
| 51 | 956.25 |
| 52 | 975.00 |
| 53 | 993.75 |
| 54 | 1012.50 |
| 55 | 1031.25 |
| 56 | 1050.00 |
| 57 | 1068.75 |
| 58 | 1087.50 |
| 59 | 1106.25 |
| 60 | 1125.00 |
| 61 | 1143.75 |
| 62 | 1162.50 |
| 63 | 1181.25 |
| 64 | 1200.00 |