117edo: Difference between revisions
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It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 11-limit: <117 185 272 328 405| (patent val), <117 186 272 329 405| (117bd), <117 185 271 328 404| (117ce), and <117 185 272 329 405| (117d). | It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 11-limit: <117 185 272 328 405| (patent val), <117 186 272 329 405| (117bd), <117 185 271 328 404| (117ce), and <117 185 272 329 405| (117d). | ||
== Theory == | |||
=== Commas === | |||
Using the patent val, it tempers out the [[syntonic comma]] (81/80) and |69 -1 -29> in the 5-limit; 6144/6125, 31104/30625, and 403368/390625 in the 7-limit, supporting the 7-limit [[Mohajira]] temperament; 540/539, 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, supporting the [[Didymus rank three family|terpsichore rank-3 temperament]]; 144/143, 196/195, 364/363, 729/715, and 3146/3125 in the 13-limit. | Using the patent val, it tempers out the [[syntonic comma]] (81/80) and |69 -1 -29> in the 5-limit; 6144/6125, 31104/30625, and 403368/390625 in the 7-limit, supporting the 7-limit [[Mohajira]] temperament; 540/539, 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, supporting the [[Didymus rank three family|terpsichore rank-3 temperament]]; 144/143, 196/195, 364/363, 729/715, and 3146/3125 in the 13-limit. | ||
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Using the 117d val, it tempers out 126/125, 225/224, and 14495514624/13841287201 in the 7-limit; 99/98, 176/175, 441/440, and 12582912/12400927 in the 11-limit; 144/143, 640/637, 648/637, 1001/1000, and 43940/43923 in the 13-limit, supporting the 13-limit [[Grosstone]] temperament. | Using the 117d val, it tempers out 126/125, 225/224, and 14495514624/13841287201 in the 7-limit; 99/98, 176/175, 441/440, and 12582912/12400927 in the 11-limit; 144/143, 640/637, 648/637, 1001/1000, and 43940/43923 in the 13-limit, supporting the 13-limit [[Grosstone]] temperament. | ||
=== Harmonics === | |||
{{Harmonics in equal|117}} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
Revision as of 02:50, 24 June 2023
| ← 116edo | 117edo | 118edo → |
117edo is the equal division of the octave into 117 parts of 10.2564102564 cents each.
It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 11-limit: <117 185 272 328 405| (patent val), <117 186 272 329 405| (117bd), <117 185 271 328 404| (117ce), and <117 185 272 329 405| (117d).
Theory
Commas
Using the patent val, it tempers out the syntonic comma (81/80) and |69 -1 -29> in the 5-limit; 6144/6125, 31104/30625, and 403368/390625 in the 7-limit, supporting the 7-limit Mohajira temperament; 540/539, 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, supporting the terpsichore rank-3 temperament; 144/143, 196/195, 364/363, 729/715, and 3146/3125 in the 13-limit.
Using the 117bd val, it tempers out the kleisma, 15625/15552 and 17179869184/16142520375 in the 5-limit; 245/243, 3136/3125, and 51200/50421 in the 7-limit; 176/175, 1232/1215, 1375/1372, and 2560/2541 in the 11-limit; 169/168, 364/363, 640/637, 832/825, and 3200/3159 in the 13-limit.
Using the 117ce val, it tempers out the small diesis, 3125/3072 and 282429536481/268435456000 in the 5-limit; 2401/2400, 3645/3584, and 4375/4374 in the 7-limit; 243/242, 441/440, and 1815/1792 in the 11-limit; 105/104, 275/273, 1287/1280, and 2025/2002 in the 13-limit.
Using the 117d val, it tempers out 126/125, 225/224, and 14495514624/13841287201 in the 7-limit; 99/98, 176/175, 441/440, and 12582912/12400927 in the 11-limit; 144/143, 640/637, 648/637, 1001/1000, and 43940/43923 in the 13-limit, supporting the 13-limit Grosstone temperament.
Harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.52 | +3.43 | -4.72 | +1.22 | +2.53 | +0.50 | -1.09 | -2.39 | -0.08 | +1.01 | -2.63 |
| Relative (%) | -44.1 | +33.4 | -46.1 | +11.9 | +24.7 | +4.9 | -10.6 | -23.3 | -0.8 | +9.9 | -25.7 | |
| Steps (reduced) |
185 (68) |
272 (38) |
328 (94) |
371 (20) |
405 (54) |
433 (82) |
457 (106) |
478 (10) |
497 (29) |
514 (46) |
529 (61) | |