Ultrapyth
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