Ultrapyth: Difference between revisions
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{{Infobox | {{Infobox regtemp | ||
| Title = Ultrapyth | | Title = Ultrapyth | ||
| Subgroups = 2.3.5.7, 2.3.5.7.13 | | 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]] | | 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 | | Edo join 1 = 5 | Edo join 2 = 32 | ||
| Mapping = 1; 1 14 -2 18 | | Mapping = 1; 1 14 -2 18 | ||
| | | Generators = 3/2 | ||
| | | Generators tuning = 713.6 | ||
| Optimization method = CWE | | Optimization method = CWE | ||
| MOS scales = [[5L 7s]], [[5L 12s]], [[5L 17s]], [[5L 22s]] | | MOS scales = [[5L 7s]], [[5L 12s]], [[5L 17s]], [[5L 22s]] | ||
| Pergen = (P8, P5) | | Pergen = (P8, P5) | ||
| Odd limit 1 = 7 | Mistuning 1 = 11.4 | Complexity 1 = 17 | | Odd limit 1 = 7 | Mistuning 1 = 11.4 | Complexity 1 = 17 | ||
| Odd limit 2 = | | 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. | ||
| Line 172: | Line 172: | ||
|13/8 | |13/8 | ||
|713.363 | |713.363 | ||
| | | 2.3.5.7.13 13- to 21-odd-limit minimax | ||
|- | |- | ||
| '''[[37edo|22\37]]''' | | '''[[37edo|22\37]]''' | ||