291edo: Difference between revisions
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291edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with three mappings possible for the 5-limit: {{val| 291 461 676 }} ([[patent val]]), {{val| 291 462 676 }} (291b), and {{val| 291 461 675 }} (291c). | |||
Using the 291b val, it tempers out 15625/15552 and |80 -46 -3 | Using the patent val, it [[tempering out|tempers out]] [[393216/390625]] and {{monzo| -47 37 -5 }} in the 5-limit; [[2401/2400]], [[3136/3125]], and 1162261467/1146880000 in the 7-limit; [[243/242]], [[441/440]], [[5632/5625]], and 58720256/58461513 in the 11-limit; [[351/350]], [[1001/1000]], [[1575/1573]], 3584/3575, and 43940/43923 in the 13-limit, so that it provides the [[optimal patent val]] for the 13-limit [[hemiwürschmidt]] temperament. | ||
Using the 291b val, it tempers out 15625/15552 and {{monzo| 80 -46 -3 }} in the 5-limit. | |||
Using the 291c val, it tempers out 390625000/387420489 and 1121008359375/1099511627776 in the 5-limit. | Using the 291c val, it tempers out 390625000/387420489 and 1121008359375/1099511627776 in the 5-limit. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|291}} | {{Harmonics in equal|291}} | ||
[[Category: | |||
=== Subsets and supersets === | |||
Since 291 factors into {{factorization|291}}, 291edo contains [[3edo]] and [[97edo]] as its subsets. | |||
[[Category:Hemiwürschmidt]] | |||
Latest revision as of 19:29, 20 February 2025
| ← 290edo | 291edo | 292edo → |
291 equal divisions of the octave (abbreviated 291edo or 291ed2), also called 291-tone equal temperament (291tet) or 291 equal temperament (291et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 291 equal parts of about 4.12 ¢ each. Each step represents a frequency ratio of 21/291, or the 291st root of 2.
291edo is inconsistent to the 5-odd-limit and higher limits, with three mappings possible for the 5-limit: ⟨291 461 676] (patent val), ⟨291 462 676] (291b), and ⟨291 461 675] (291c).
Using the patent val, it tempers out 393216/390625 and [-47 37 -5⟩ in the 5-limit; 2401/2400, 3136/3125, and 1162261467/1146880000 in the 7-limit; 243/242, 441/440, 5632/5625, and 58720256/58461513 in the 11-limit; 351/350, 1001/1000, 1575/1573, 3584/3575, and 43940/43923 in the 13-limit, so that it provides the optimal patent val for the 13-limit hemiwürschmidt temperament.
Using the 291b val, it tempers out 15625/15552 and [80 -46 -3⟩ in the 5-limit.
Using the 291c val, it tempers out 390625000/387420489 and 1121008359375/1099511627776 in the 5-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.92 | +1.32 | +0.25 | +1.26 | +0.71 | -1.86 | -0.61 | -1.47 | +1.35 | +1.36 |
| Relative (%) | +0.0 | -22.4 | +31.9 | +6.0 | +30.5 | +17.2 | -45.2 | -14.7 | -35.7 | +32.8 | +32.9 | |
| Steps (reduced) |
291 (0) |
461 (170) |
676 (94) |
817 (235) |
1007 (134) |
1077 (204) |
1189 (25) |
1236 (72) |
1316 (152) |
1414 (250) |
1442 (278) | |
Subsets and supersets
Since 291 factors into 3 × 97, 291edo contains 3edo and 97edo as its subsets.