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Superpyth | Suprapyth | Music

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

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

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

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

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

Basic superpyth (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

MOSes

5-note (LsLss, proper)

See 2L 3s.

7-note (LLLsLLs, improper)

See 5L 2s. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.

12-note (LsLsLssLsLss, borderline improper)

See 5L 7s. The boundary of propriety is 17edo.

Music

12of22studyPentUp4thsMstr
12of22gamelan1b
12of22study3 (children's story)
12of22study7
By Joel Grant Taylor, all in Superpyth[12] in 22edo tuning.