Ultrapyth: Difference between revisions
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{{Infobox regtemp | |||
| Title = Ultrapyth | |||
| Subgroups = 2.3.5.7, 2.3.5.7.13 | |||
| Comma basis = [[64/63]], [[6860/6561]] (2.3.5.7)<br>[[64/63]], [[91/90]], [[6125/6084]] (2.3.5.7.13) | |||
| Edo join 1 = 5 | Edo join 2 = 32 | |||
| Mapping = 1; 1 14 -2 18 | |||
| Generators = 3/2 | |||
| Generators tuning = 713.6 | |||
| Optimization method = CWE | |||
| MOS scales = [[5L 7s]], [[5L 12s]], [[5L 17s]], [[5L 22s]] | |||
| Pergen = (P8, P5) | |||
| Odd limit 1 = 7 | Mistuning 1 = 11.4 | Complexity 1 = 17 | |||
| Odd limit 2 = 2.3.5.7.13 21 | Mistuning 2 = 22.8 | Complexity 2 = 22 | |||
}} | |||
'''Ultrapyth''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], tempering out [[64/63]]. The difference is that instead of extending to 2.3.5.7 by mapping 5 to +9 generators, it extends to the 2.3.7.13/5 subgroup (known as '''oceanfront''') by mapping the ultramajor third [[13/10]] to +4 generators (which is also the diatonic major third), tempering out [[91/90]]. This makes sense because the tunings of 2.3.7 archy that optimize for the simplest 2.3.7 intervals (8/7 and [[7/6]]) are sharp of the optimal tuning for 9/7, making that third more ultramajor than supermajor. If intervals of 5 and 13 independently are desired (i.e. [[5/4]], [[13/8]]), then oceanfront may be extended to ultrapyth by mapping 5 to +14 fifths (a double-augmented unison) and 13 to +18 fifths (a double-augmented third). The best tunings for ultrapyth are between 712 and 714 cents. | '''Ultrapyth''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], tempering out [[64/63]]. The difference is that instead of extending to 2.3.5.7 by mapping 5 to +9 generators, it extends to the 2.3.7.13/5 subgroup (known as '''oceanfront''') by mapping the ultramajor third [[13/10]] to +4 generators (which is also the diatonic major third), tempering out [[91/90]]. This makes sense because the tunings of 2.3.7 archy that optimize for the simplest 2.3.7 intervals (8/7 and [[7/6]]) are sharp of the optimal tuning for 9/7, making that third more ultramajor than supermajor. If intervals of 5 and 13 independently are desired (i.e. [[5/4]], [[13/8]]), then oceanfront may be extended to ultrapyth by mapping 5 to +14 fifths (a double-augmented unison) and 13 to +18 fifths (a double-augmented third). The best tunings for ultrapyth are between 712 and 714 cents. | ||
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| | | | ||
| 709.091 | | 709.091 | ||
| | | 22ccff val | ||
|- | |- | ||
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| | | | ||
| 711.111 | | 711.111 | ||
| | | 27cf val | ||
|- | |- | ||
| '''[[32edo|19\32]]''' | | '''[[32edo|19\32]]''' | ||
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| 7- and 9-odd-limit minimax | | 7- and 9-odd-limit minimax | ||
|- | |- | ||
| [[37edo|22\37]] | | | ||
|13/8 | |||
|713.363 | |||
| 2.3.5.7.13 13- to 21-odd-limit minimax | |||
|- | |||
| '''[[37edo|22\37]]''' | |||
| | | | ||
| '''713.514''' | |||
| '''Lower bound of 2.3.5.7.13 13-odd-limit diamond monotone<br>2.3.5.7.13 15- and 21-odd-limit diamond monotone (singleton) | |||
|- | |||
| | |||
|13/10 | |||
|713.553 | |||
| | |||
|- | |||
| | |||
|14/13 | |||
|713.585 | |||
| | |||
|- | |- | ||
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| 713.593 | | 713.593 | ||
| | | | ||
|- | |||
| | |||
|13/12 | |||
|714.034 | |||
| | |||
|- | |- | ||
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| 714.181 | | 714.181 | ||
| | | | ||
|- | |||
| | |||
|21/13 | |||
|714.197 | |||
| | |||
|- | |- | ||
| [[42edo|25\42]] | | [[42edo|25\42]] | ||
| | | | ||
| 714.286 | | 714.286 | ||
| | | 42f val | ||
|- | |- | ||
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| 714.369 | | 714.369 | ||
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|- | |||
| | |||
|13/9 | |||
|714.789 | |||
| | |||
|- | |- | ||
| [[47edo|28\47]] | | [[47edo|28\47]] | ||
| | | | ||
| 714.894 | | 714.894 | ||
| | | 47bcff val | ||
|- | |- | ||
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| Line 194: | Line 238: | ||
| 715.587 | | 715.587 | ||
| | | | ||
|- | |||
| | |||
|15/13 | |||
|717.420 | |||
| | |||
|- | |- | ||
| '''[[5edo|3\5]]''' | | '''[[5edo|3\5]]''' | ||
| | | | ||
| '''720.000''' | | '''720.000''' | ||
| '''Upper bound of 7- and 9-odd-limit diamond monotone''' | | '''Upper bound of 7- and 9-odd-limit,<br>2.3.5.7.13 13-odd-limit diamond monotone''' | ||
|- | |- | ||
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