Superpyth
Superpyth, a member of the Archytas clan, has 4/3 as a generator, and the Archytas comma 64/63 is tempered out, so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for meantone and 12edo, with the understanding that, for example, A# is sharper than Bb (in contrast to meantone where A# is flatter than Bb, or 12edo where they are identical). An interesting coincidence is that the plastic numberhas a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth.
If the 5th harmonic is used at all, it is mapped to -9 generators, so C-D# is 5/4. So superpyth is "the opposite of" septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex.
If intervals of 11 are desired the simplest reasonable way is to map 11/8 to 6 generators (so 11/8 is a "diminished fifth"), by tempering out 99/98.
This temperament is called "supra", or "suprapyth" if you include 5 as well.
MOSes include 5, 7, 12, 17, and 22.
Superpyth[edit]
Commas: 64/63, 245/243
POTE generator: ~3/2 = 710.291
Map: [<1 0 -12 6|, <0 1 9 -2|]
Wedgie: <<1 9 -2 12 -6 -30||
EDOs: 5, 17, 22, 27, 49
Badness: 0.0323
11-limit[edit]
Commas: 64/63, 100/99, 245/243
POTE generator: ~3/2 = 710.175
Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]
EDOs: 22, 27e, 49
Badness: 0.0250
13-limit[edit]
Commas: 64/63, 78/77, 91/90, 100/99
POTE generator: ~3/2 = 710.479
Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|]
EDOs: 22, 27e, 49, 76bcde
Badness: 0.0247
Suprapyth[edit]
Commas: 55/54, 64/63, 99/98
POTE generator: ~3/2 = 709.495
Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]
EDOs: 5, 17, 22
Badness: 0.0328
Interval chains[edit]
Basic superpyth (2.3.7)[edit]
| 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[edit]
| 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)[edit]
| 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[edit]
| 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 |
MOSes[edit]
5-note (LsLss, proper)[edit]
See 2L 3s.
7-note (LLLsLLs, improper)[edit]
See 5L 2s. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.
12-note (LsLsLssLsLss, borderline improper)[edit]
See 5L 7s. The boundary of propriety is 17edo.
Music[edit]
12of22study3 (children's story)
By Joel Grant Taylor, all in Superpyth[12] in 22edo tuning.