Quasisuper
| Quasisuper; quasisupra |
64/63, 99/98, 121/120 (11-limit)
11-limit 15-odd-limit: 14.9 ¢
11-limit 15-odd-limit: 17 notes
Quasisuper 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 this extension maps prime 5 to −13 generators, as a double-diminished fifth (C–G𝄫). This extension works in the range 17c-edo to 22-edo. In contrast, full 7-limit superpyth does not work in this range, as tunings with a flatter fifth than 22edo swap the sizes of 7/5 and 10/7. This extension may be preferred over superpyth due to having a softer diatonic scale, with a small step of around 60 cents compared to about 50 cents in regular 7-limit superpyth.
The best extension to the 11-limit, quasisupra, maps prime 11 to −6 generators as a diminished fifth (C–G♭), tempering out 99/98 as well as 121/120 and 540/539. Removing prime 5 from quasisupra results in a 2.3.7.11-subgroup restriction, called supra, which is notable for its simplicity. Finally, taking every other step of supra gives a 2.9.7.11-subgroup restriction, called machine.
For technical data see Archytas clan #Quasisuper and #Supra.
Interval chain
In the following tables, odd harmonics and subharmonics 1–11 are in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 707.5 | 3/2 |
| 2 | 215.0 | 8/7, 9/8 |
| 3 | 922.5 | 12/7 |
| 4 | 430.0 | 9/7, 14/11 |
| 5 | 1137.5 | 21/11, 27/14, 64/33 |
| 6 | 645.0 | 16/11 |
| 7 | 152.5 | 12/11 |
* In 2.3.7.11-subgroup CWE tuning,
octave reduced
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 708.3 | 3/2 |
| 2 | 216.6 | 8/7, 9/8 |
| 3 | 925.0 | 12/7 |
| 4 | 433.3 | 9/7, 14/11 |
| 5 | 1141.6 | 21/11, 27/14 |
| 6 | 649.9 | 16/11, 22/15 |
| 7 | 158.2 | 11/10, 12/11 |
| 8 | 866.6 | 18/11 |
| 9 | 374.9 | 27/22, 56/45 |
| 10 | 1083.2 | 28/15 |
| 11 | 591.5 | 7/5 |
| 12 | 99.8 | 16/15 |
| 13 | 808.2 | 8/5 |
| 14 | 316.5 | 6/5 |
| 15 | 1024.8 | 9/5 |
| 16 | 533.1 | 27/20 |
| 17 | 41.4 | 81/80, 56/55 |
* in 11-limit CWE tuning, octave reduced
Scales
Scala files
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 708.7690 ¢ | CWE: ~3/2 = 708.3716 ¢ | POTE: ~3/2 = 708.2385 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 708.7131 ¢ | CWE: ~3/2 = 708.3200 ¢ | POTE: ~3/2 = 708.2046 ¢ |
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 7\12 | 700.000 | 12cc val | |
| 3/2 | 701.955 | Pythagorean tuning | |
| 21/11 | 703.893 | ||
| 11/7 | 704.377 | ||
| 10\17 | 705.882 | 17c val, lower bound of 7-, 9-, and 11-odd-limit diamond monotone | |
| 11/9 | 706.574 | ||
| 21/20 | 707.039 | ||
| 33\56 | 707.143 | 56cd val | |
| 11/6 | 707.234 | ||
| 7/5 | 707.501 | ||
| 23\39 | 707.692 | 39d val | |
| 9/5 | 707.840 | ||
| 15/14 | 708.056 | ||
| 11/8 | 708.114 | ||
| 36\61 | 708.197 | 61d val | |
| 5/3 | 708.260 | ||
| 5/4 | 708.745 | ||
| 9/7 | 708.771 | ||
| 13\22 | 709.091 | Upper bound of 7-, 9-, and 11-odd-limit diamond monotone | |
| 11/10 | 709.286 | ||
| 15/8 | 709.311 | ||
| 15/11 | 710.508 | ||
| 7/6 | 711.043 | ||
| 16\27 | 711.111 | 27c val | |
| 7/4 | 715.587 | ||
| 3\5 | 720.000 | 5c val | |
| 21/16 | 729.219 |
* Besides the octave