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 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 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 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 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.

Temperament data

Main article: 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)

• Archy7 – archy in 472edo tuning
• Supra7 – supra in 56edo tuning

In contrast to the meantone diatonic scale, the superpyth diatonic is improper.

12-note MOS (5L 7s, borderline improper)

• Archy12 – archy in 472edo tuning
• Supra12 – supra in 56edo tuning
• 12-22a – superpyth in 22edo tuning

The boundary of propriety is 17edo.

Tuning spectrum

ET Generator	Eigenmonzo (Unchanged Interval)	Generator (¢)	Comments
	4/3	701.955	Pythagorean tuning
10\17		705.882	Lower bound of 7- and 9-odd-limit diamond monotone
	28/27	707.408	1/5 comma
23\39		707.692
	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
3\5		720.000	Upper bound of 7- and 9-odd-limit diamond monotone

Music

• 12of22studyPentUp4thsMstr^{[dead link]}
• 12of22gamelan1b^{[dead link]}
• 12of22study3 (children's story)^{[dead link]}
• 12of22study7^{[dead link]}

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

• Superpyth[12] chromatic riff
• Trio in Superpyth Temperament for Irish Whistle, Piano, and Cello

Both by Lillian Hearne in 22edo tuning

See also

• Ultrapyth