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]]''' | ||
Latest revision as of 10:08, 9 February 2026
| Ultrapyth |
64/63, 91/90, 6125/6084 (2.3.5.7.13)
2.3.5.7.13 21-odd-limit: 22.8 ¢
2.3.5.7.13 21-odd-limit: 22 notes
Ultrapyth is an alternative extension of the archy chain of fifths to superpyth. Like superpyth, it is a 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.
If intervals of 11 are desired, 14/11 may be mapped to +9 generators, implying 16/11 is (fittingly) mapped to +11 generators and 11/9 is tempered together with 6/5 (a feature common to many systems with sharp fifths).
The oceanfront mos scales take the form of 5L (5n + 2)s, for n up to 7. Most of these scales resemble 5edo. 37edo makes a good tuning of oceanfront or ultrapyth.
Both oceanfront and ultrapyth were named by Mike Battaglia in 2011[1].
For technical data, see The Biosphere #Oceanfront and Archytas clan #Ultrapyth.
Interval chain
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 711.7 | 3/2 |
| 2 | 223.5 | 8/7, 9/8 |
| 3 | 935.2 | 12/7, 26/15 |
| 4 | 447.0 | 9/7, 13/10 |
| 5 | 1158.7 | 27/14, 39/20 |
| 6 | 670.4 | 52/35, 72/49 |
| 7 | 182.2 | 39/35, 54/49 |
* In 2.3.7.13/5-subgroup CWE tuning,
octave reduced
| # | Cents* | Approximate ratios | ||
|---|---|---|---|---|
| 2.3.5.7.13 subgroup | Full 13-limit extensions | |||
| Ultrapyth | Ultramarine | |||
| 0 | 0.0 | 1/1 | ||
| 1 | 713.6 | 3/2 | ||
| 2 | 227.3 | 8/7, 9/8 | ||
| 3 | 940.9 | 12/7, 26/15 | ||
| 4 | 454.5 | 9/7, 13/10 | ||
| 5 | 1168.1 | 27/14, 39/20 | ||
| 6 | 681.8 | 52/35, 72/49 | ||
| 7 | 195.4 | 39/35, 54/49 | ||
| 8 | 909.0 | 81/49, 117/70 | 56/33 | 22/13 |
| 9 | 422.7 | 35/27 | 14/11 | 33/26 |
| 10 | 1136.3 | 25/13, 35/18 | 21/11, 64/33 | 88/45 |
| 11 | 649.9 | 35/24, 40/27 | 16/11 | 22/15 |
| 12 | 163.5 | 10/9 | 12/11 | 11/10 |
| 13 | 877.2 | 5/3 | 18/11 | 33/20 |
| 14 | 390.8 | 5/4 | ||
| 15 | 1104.4 | 15/8, 40/21, 52/27 | ||
| 16 | 618.1 | 10/7, 13/9 | ||
| 17 | 131.7 | 13/12, 15/14 | ||
| 18 | 845.3 | 13/8 | ||
* In 2.3.5.7.13-subgroup CWE tuning, octave reduced
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 713.2179 ¢ | CWE: ~3/2 = 713.5430 ¢ | POTE: ~3/2 = 713.6509 ¢ |
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 3/2 | 701.955 | Pythagorean tuning | |
| 9/7 | 708.771 | ||
| 13\22 | 709.091 | 22ccff val | |
| 7/6 | 711.043 | ||
| 16\27 | 711.111 | 27cf val | |
| 19\32 | 712.500 | Lower bound of 7- and 9-odd-limit diamond monotone | |
| 15/8 | 712.551 | ||
| 15/14 | 712.908 | ||
| 5/4 | 713.308 | 7- and 9-odd-limit minimax | |
| 13/8 | 713.363 | 2.3.5.7.13 13- to 21-odd-limit minimax | |
| 22\37 | 713.514 | Lower bound of 2.3.5.7.13 13-odd-limit diamond monotone 2.3.5.7.13 15- and 21-odd-limit diamond monotone (singleton) | |
| 13/10 | 713.553 | ||
| 14/13 | 713.585 | ||
| 7/5 | 713.593 | ||
| 13/12 | 714.034 | ||
| 5/3 | 714.181 | ||
| 21/13 | 714.197 | ||
| 25\42 | 714.286 | 42f val | |
| 21/20 | 714.369 | ||
| 13/9 | 714.789 | ||
| 28\47 | 714.894 | 47bcff val | |
| 9/5 | 715.200 | ||
| 7/4 | 715.587 | ||
| 15/13 | 717.420 | ||
| 3\5 | 720.000 | Upper bound of 7- and 9-odd-limit, 2.3.5.7.13 13-odd-limit diamond monotone | |
| 21/16 | 729.219 |
* Besides the octave