Superpyth

Superpyth, a member of the archytas clan (and thus a version of archy with a mapping for prime 5), 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.

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)

• 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.

Tunings

Prime-optimized tunings

2.3.7 Prime-Optimized Tunings

Weight-skew\Order	Euclidean
Tenney	CTE ~3/2 = 709.5948¢
Tenney-Weil	CTWE ~3/2 = 709.3901¢
Equilateral	CEE ~3/2 = 712.8606¢ Eigenmonzo basis (unchanged-interval basis): 2.49/3 (2/5-comma tuning)
Equilateral-Weil	CEWE ~3/2 = 711.9997¢ Eigenmonzo basis (unchanged-interval basis): 2.823543/243 (7/19-comma tuning)
Benedetti	CBE ~3/2 = 707.7286¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 -49 0 18⟩ (18/85-comma tuning)
Benedetti-Weil	CBWE ~3/2 = 707.9869¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 -63 25⟩ (25/113-comma tuning)

7-limit Prime-Optimized Tunings

Weight-skew\Order	Euclidean
Tenney	CTE ~3/2 = 709.5907¢
Tenney-Weil	CTWE ~3/2 = 710.1193¢
Equilateral	CEE ~3/2 = 709.7805¢ Eigenmonzo basis (unchanged-interval basis): 2.5859375/49
Equilateral-Weil	CEWE ~3/2 = 710.2428¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 3 -37 18⟩
Benedetti	CBE ~3/2 = 709.4859¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 -1225 -3969 450⟩
Benedetti-Weil	CBWE ~3/2 = 710.0321¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 665 -15771 5160⟩

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, 2.3.7 CEE tuning
	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