Superpyth
Superpyth is a temperament of the archytas clan where ~3/2 is a generator, and the Archytas comma 64/63 is tempered out, so a stack of two generators octave-reduced represents 8/7 in addition to 9/8 (in other words, 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. 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.
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 the following tables, odd harmonics 1–11 are in bold.
- Archy (2.3.7)
| # | 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
- Supra (2.3.7.11)
| # | 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
- Full 7-limit superpyth
| # | Cents* | Approximate Ratios | ||
|---|---|---|---|---|
| 7-limit | 11-limit Extension (Superpyth) |
11-limit Extension (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
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¢ Eigenmonzo basis (unchanged-interval basis): 2.49/3 (2/5-comma tuning) |
| Skewed-equilateral | CSEE: ~3/2 = 711.9997¢ Eigenmonzo basis (unchanged-interval basis): 2.823543/243 (7/19-comma tuning) |
| Benedetti/Wilson | CBE: ~3/2 = 707.7286¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 -49 0 18⟩ (18/85-comma tuning) |
| Skewed-Benedetti/Wilson | CSBE: ~3/2 = 707.9869¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 -63 25⟩ (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¢ Eigenmonzo basis (unchanged-interval basis): 2.5859375/49 |
| Skewed-equilateral | CSEE: ~3/2 = 710.2428¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 3 -37 18⟩ |
| Benedetti/Wilson | CBE: ~3/2 = 709.4859¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 -1225 -3969 450⟩ |
| Skewed-Benedetti/Wilson | CSBE: ~3/2 = 710.0321¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 665 -15771 5160⟩ |
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 | ||
| 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 | |
| 15/14 | 709.954 | ||
| 25/24 | 710.040 | ||
| 29\49 | 710.204 | ||
| 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 | ||
| 49/48 | 712.861 | 2/5 comma, 2.3.7 subgroup CEE tuning | |
| 22\37 | 713.514 | ||
| 25\42 | 714.286 | ||
| 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 |
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.