Superpyth
Superpyth, also known as superpythagorean, is a temperament where ~3/2 is a generator, and the septimal comma (64/63) is tempered out, so that a stack of two perfect fifths octave-reduced gives a major whole tone that represents both 9/8 and 8/7 (likewise, two perfect fourths give a minor seventh that represents both 7/4 and 16/9, so intervals such as A–G and C–B♭ are harmonic sevenths). Since 3/2 is a generator we can use the same standard chain-of-fifths notation that is also used for meantone and 12edo, with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in Pythagorean tuning, in contrast to meantone where sharps are flatter than or equal to the corresponding flats. 10\17, 13\22, and 16\27 are typical tunings of the generator.
Such a temperament without the 5th harmonic is also called archy. If intervals of 5 are desired, it is mapped to +9 generators through tempering out 245/243, so C–D♯ is 5/4. So superpyth is the "opposite" of septimal meantone in several different ways: Meantone (including 12edo) has 3/2 tuned flat so that the 5th harmonic's intervals are simple and the 7th harmonic's intervals are complex, while superpyth has 3/2 tuned sharp so that the 7th harmonic's intervals are simple while the 5th harmonic's intervals are complex.
If intervals of 11 are desired, the canonical way is to map 11/8 to +16 generators, or a doubly augmented second (C–D𝄪), tempering out 100/99. Yet a simpler but reasonable way is to map it to −6 generators, or a diminished fifth (C–G♭), by tempering out 99/98. The latter is called supra, or suprapyth. The two mappings unite on 22edo.
If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪), by tempering out 31213/31104.
MOS scales of superpyth have cardinalities of 5, 7, 12, 17, or 22.
For more technical data, see Archytas clan#Superpyth.
Interval chains
In these tables, odd harmonics 1–11 are in bold.
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 709.6 | 3/2 |
| 2 | 219.2 | 8/7, 9/8 |
| 3 | 928.8 | 12/7 |
| 4 | 438.4 | 9/7 |
| 5 | 1148.0 | 27/14 |
| 6 | 657.6 | 72/49, 81/56 |
| 7 | 167.2 | 54/49 |
* In 2.3.7-subgroup CTE tuning
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 708.5 | 3/2 |
| 2 | 216.9 | 8/7, 9/8 |
| 3 | 925.4 | 12/7 |
| 4 | 433.8 | 9/7, 14/11 |
| 5 | 1142.3 | 21/11, 27/14, 64/33 |
| 6 | 650.7 | 16/11 |
| 7 | 159.2 | 12/11 |
* In 2.3.7.11-subgroup CTE tuning
| # | Cents* | Approximate Ratios | ||
|---|---|---|---|---|
| 7-limit | 11-limit Extension | |||
| Superpyth | Suprapyth | |||
| 0 | 0.0 | 1/1 | ||
| 1 | 709.6 | 3/2 | ||
| 2 | 219.2 | 8/7, 9/8 | ||
| 3 | 928.8 | 12/7 | ||
| 4 | 438.4 | 9/7 | 14/11 | |
| 5 | 1148.0 | 27/14, 35/18 | 88/45 | 21/11, 64/33 |
| 6 | 657.5 | 35/24, 40/27 | 22/15 | 16/11 |
| 7 | 167.1 | 10/9 | 11/10 | 12/11 |
| 8 | 876.7 | 5/3 | 33/20 | 18/11 |
| 9 | 386.3 | 5/4 | 27/22 | |
| 10 | 1095.9 | 15/8, 40/21 | ||
| 11 | 605.5 | 10/7 | ||
| 12 | 115.1 | 15/14 | ||
| 13 | 824.7 | 45/28 | 44/27 | |
| 14 | 334.3 | 60/49 | 11/9 | 40/33 |
| 15 | 1043.9 | 50/27 | 11/6 | 20/11 |
| 16 | 553.5 | 25/18 | 11/8 | 15/11 |
| 17 | 63.0 | 25/24 | 22/21, 33/32 | 45/44 |
* In 7-limit CTE tuning
Scales
- 5-note MOS (2L 3s, proper)
- Archy5 – archy in 472edo tuning
- 7-note MOS (5L 2s, improper)
In contrast to the meantone diatonic scale, the superpyth diatonic is improper.
- 12-note MOS (5L 7s, borderline improper)
The boundary of propriety is 17edo.
Tunings
The plastic number has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming an octave period, constitutes a variety of superpyth. This can be explained since superpyth equates 21/16 and 4/3, making the 9:12:16:21 chord evenly spaced by ~4/3, and when keeping ~9 + ~12 = ~21 the generator becomes the plastic number.
Prime-optimized tunings
| Weight-skew\Order | Euclidean |
|---|---|
| Tenney | CTE: ~3/2 = 709.5948¢ |
| Weil | CWE: ~3/2 = 709.3901¢ |
| Equilateral | CEE: ~3/2 = 712.8606¢ (2/5-comma tuning) |
| Skewed-equilateral | CSEE: ~3/2 = 711.9997¢ (7/19-comma tuning) |
| Benedetti/Wilson | CBE: ~3/2 = 707.7286¢ (18/85-comma tuning) |
| Skewed-Benedetti/Wilson | CSBE: ~3/2 = 707.9869¢ (25/113-comma tuning) |
| Weight-skew\Order | Euclidean |
|---|---|
| Tenney | CTE: ~3/2 = 709.5907¢ |
| Weil | CWE: ~3/2 = 710.1193¢ |
| Equilateral | CEE: ~3/2 = 709.7805¢ |
| Skewed-equilateral | CSEE: ~3/2 = 710.2428¢ |
| Benedetti/Wilson | CBE: ~3/2 = 709.4859¢ |
| Skewed-Benedetti/Wilson | CSBE: ~3/2 = 710.0321¢ |
Tuning spectrum
| Edo Generator |
Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 3/2 | 701.955 | Pythagorean tuning | |
| 10\17 | 705.882 | Lower bound of 7- and 9-odd-limit diamond monotone | |
| 81/56 | 706.499 | 1/6 comma | |
| 27/14 | 707.408 | 1/5 comma | |
| 23\39 | 707.692 | 39cd val | |
| 9/7 | 708.771 | 1/4 comma, {1, 3, 7, 9} minimax | |
| 15/8 | 708.807 | ||
| 13\22 | 709.091 | ||
| 5/4 | 709.590 | 9-odd-limit minimax | |
| 49/27 | 709.745 | 2/7 comma | |
| 42\71 | 709.859 | 71d val | |
| 15/14 | 709.954 | ||
| 25/24 | 710.040 | ||
| 29\49 | 710.204 | ||
| 45\76 | 710.526 | 76bcd val | |
| 5/3 | 710.545 | ||
| 7/5 | 710.681 | 7-odd-limit minimax | |
| 7/6 | 711.043 | 1/3 comma, {1, 3, 7} minimax | |
| 16\27 | 711.111 | ||
| 21/20 | 711.553 | ||
| 9/5 | 711.772 | ||
| 19\32 | 712.500 | 32c val | |
| 55/32 | 712.544 | Suprapyth mapping | |
| 49/48 | 712.861 | 2/5 comma, 2.3.7 subgroup CEE tuning | |
| 22\37 | 713.514 | 37cc val | |
| 25\42 | 714.286 | 42cc val | |
| 7/4 | 715.587 | 1/2 comma | |
| 3\5 | 720.000 | Upper bound of 7- and 9-odd-limit diamond monotone | |
| 21/16 | 729.219 | Full comma |
* Besides the octave
Other tunings
- DKW (2.3.5 Superpyth): ~2 = 1\1, ~3/2 = 709.758
- DKW (2.3.7 Archy): ~3/2 = 712.585
Music
- Superpyth[12] chromatic riff (2015)
- Trio in Superpyth Temperament for Irish Whistle, Piano, and Cello (2015)
Both in 22edo tuning
- 12of22studyPentUp4thsMstr [dead link]
- 12of22gamelan1b [dead link]
- 12of22study3 (children's story) [dead link]
- 12of22study7 [dead link]
All in superpyth[12] in 22edo tuning.