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. 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. An interesting coincidence is that 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.
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
- Archy (2.3.7)
| 1146.61 | 437.29 | 927.97 | 218.64 | 709.32 | 0 | 490.68 | 981.36 | 272.03 | 762.71 | 53.39 |
| 27/14 | 9/7 | 12/7 | 9/8~8/7 | 3/2 | 1/1 | 4/3 | 7/4~16/9 | 7/6 | 14/9 | 28/27 |
- Full 7-limit superpyth
| 613.20 | 1102.91 | 392.62 | 882.33 | 172.04 | 661.75 | 1151.46 | 441.16 | 930.87 | 220.58 | 710.29 | 0 | 489.71 | 979.42 | 269.13 | 758.84 | 48.54 | 538.25 | 1027.96 | 317.67 | 807.38 | 97.09 | 586.80 |
| 10/7 | 15/8 | 5/4 | 5/3 | 10/9 | 27/14 | 9/7 | 12/7 | 9/8~8/7 | 3/2 | 1/1 | 4/3 | 7/4~16/9 | 7/6 | 14/9 | 28/27 | 9/5 | 6/5 | 8/5 | 16/15 | 7/5 |
- Supra (2.3.7.11)
| 857.54 | 150.35 | 643.15 | 1135.96 | 428.77 | 921.58 | 214.38 | 707.19 | 0 | 492.81 | 985.62 | 278.42 | 771.23 | 64.04 | 556.85 | 1049.65 | 342.46 |
| 18/11 | 12/11 | 16/11 | 27/14 | 14/11~9/7 | 12/7 | 9/8~8/7 | 3/2 | 1/1 | 4/3 | 7/4~16/9 | 7/6 | 14/9~11/7 | 33/32~28/27 | 11/8 | 11/6 | 11/9 |
- Full 11-limit suprapyth
| 604.44 | 1094.94 | 385.45 | 875.96 | 166.46 | 656.97 | 1147.47 | 437.98 | 928.48 | 218.99 | 709.49 | 0 | 490.51 | 981.01 | 271.52 | 762.02 | 52.53 | 543.03 | 1033.54 | 324.04 | 814.55 | 105.06 | 595.56 |
| 10/7 | 15/8 | 5/4 | 18/11~5/3 | 12/11~10/9 | 16/11 | 27/14 | 14/11~9/7 | 12/7 | 9/8~8/7 | 3/2 | 1/1 | 4/3 | 7/4~16/9 | 7/6 | 14/9~11/7 | 33/32~28/27 | 11/8 | 9/5~11/6 | 6/5~11/9 | 8/5 | 16/15 | 7/5 |
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.