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 number has 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".

MOSes include 5, 7, 12, 17, and 22.

Temperament data

Main article: Archytas clan § Superpyth

Technical data

Subgroup: 2.3.5.7

Comma list: 64/63, 245/243

Mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]

Wedgie: ⟨⟨1 9 -2 12 -6 -30]]

Badness: 0.0323

11-limit superpyth

Technical data

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 245/243

Mapping: [⟨1 0 -12 6 -22], ⟨0 1 9 -2 16]]

Badness: 0.0250

13-limit superpyth

Technical data

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 100/99

Mapping: [⟨1 0 -12 6 -22 -17], ⟨0 1 9 -2 16 13]]

Badness: 0.0247

11-limit suprapyth

Technical data

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 99/98

Mapping: [⟨1 0 -12 6 13], ⟨0 1 9 -2 -6]]

Badness: 0.0328

13-limit suprapyth

Technical data

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 65/63, 99/98

Mapping: [⟨1 0 -12 6 13 18], ⟨0 1 9 -2 -6 -9]]

Badness: 0.0363

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

MOS scales

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.

Spectrum of superpyth tunings

Eigenmonzo	Generator	Comment
4/3	701.955
(10\17)	705.882
28/27	707.408	1/5 comma
9/7	708.771	1/4 comma, 1.3.7.9 minimax
16/15	708.807
(13\22)	709.091
5/4	709.590	9-odd-limit minimax
54/49	709.745	2/7 comma
25/24	710.040
(29\49)	710.204
6/5	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
10/9	711.772
49/48	712.861	2/5 comma
8/7	715.587	1/2 comma

Music

• 12of22studyPentUp4thsMstr
• 12of22gamelan1b
• 12of22study3 (children's story)
• 12of22study7

By Joel Grant Taylor, all in Superpyth[12] in 22edo tuning.