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, for example, A♯ is sharper than B♭ (in contrast to meantone where A♯ is flatter than B♭, or 12edo where they are identical). 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 the 5th harmonic is used at all, 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 has 3/2 tempered narrow so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 3/2 tempered wide so that intervals of 7 are simple while intervals of 5 are complex.
If intervals of 11 are desired, the canonical way is to map 11/8 to +16 generators, so 11/8 is a double augmented second (C-Dx), tempering out 100/99. Yet a simpler but reasonable way is to map it to -6 generators, so 11/8 is a diminished fifth (C-G♭), by tempering out 99/98. The latter is called supra, or suprapyth. The two mappings unite on 22edo.
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 | |
| 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 | |
| 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.