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]]

| Edo join 1 = 5 | Edo join 2 = 32

| Mapping = 1; 1 14 -2 18

| Generator = 3/2

| Generator 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.

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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]]
+| Edo join 1 = 5 | Edo join 2 = 32
+| Mapping = 1; 1 14 -2 18
+| Generator = 3/2
+| Generator 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.