Superpyth

Superpyth is a temperament of the archytas clan where ~3/2 is a generator, and the Archytas comma 64/63 is tempered out, so a stack of two generators octave-reduced represents 8/7 in addition to 9/8 (in other words, intervals such as A–G and C–B♭ are harmonic sevenths). Since 3/2 is a generator we can use the same standard chain-of-fifths notation that is also used for meantone and 12edo, with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in Pythagorean tuning, in contrast to meantone where sharps are flatter than or equal to the corresponding flats. The plastic number has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming an octave period, constitutes a variety of superpyth. This can be explained since superpyth equates 21/16 and 4/3, making the 9:12:16:21 chord evenly spaced by ~4/3, and when keeping ~9 + ~12 = ~21 the generator becomes the plastic number.

Such a temperament without the 5th harmonic is also called archy. If intervals of 5 are desired, 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 (including 12edo) has 3/2 tuned flat so that the 5th harmonic's intervals are simple and the 7th harmonic's intervals are complex, while superpyth has 3/2 tuned sharp so that the 7th harmonic's intervals are simple while the 5th harmonic's intervals are complex.

If intervals of 11 are desired, the canonical way is to map 11/8 to +16 generators, or a doubly-augmented second (C–D𝄪), tempering out 100/99. Yet a simpler but reasonable way is to map it to −6 generators, or a diminished fifth (C–G♭), by tempering out 99/98. The latter is called supra, or suprapyth. The two mappings unite on 22edo.

If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly-augmented fourth (C–F𝄪), by tempering out 31213/31104.

Mos scales of superpyth have cardinalities of 5, 7, 12, 17, or 22.

For more technical data, see Archytas clan #Superpyth.

Interval chains

In the following tables, odd harmonics 1–11 are in bold.

Archy (2.3.7)

#	Cents*	Approximate Ratios
0	0.0	1/1
1	709.6	3/2
2	219.2	8/7, 9/8
3	928.8	12/7
4	438.4	9/7
5	1148.0	27/14
6	657.6	72/49, 81/56
7	167.2	54/49

* in 2.3.7-subgroup CTE tuning

Supra (2.3.7.11)

#	Cents*	Approximate Ratios
0	0.0	1/1
1	708.5	3/2
2	216.9	8/7, 9/8
3	925.4	12/7
4	433.8	9/7, 14/11
5	1142.3	21/11, 27/14, 64/33
6	650.7	16/11
7	159.2	12/11

* in 2.3.7.11-subgroup CTE tuning

Full 7-limit superpyth

#	Cents*	Approximate Ratios
		7-limit	11-limit Extension (Superpyth)	11-limit Extension (Suprapyth)
0	0.0	1/1
1	709.6	3/2
2	219.2	8/7, 9/8
3	928.8	12/7
4	438.4	9/7		14/11
5	1148.0	27/14, 35/18	88/45	21/11, 64/33
6	657.5	35/24, 40/27	22/15	16/11
7	167.1	10/9	11/10	12/11
8	876.7	5/3	33/20	18/11
9	386.3	5/4		27/22
10	1095.9	15/8, 40/21
11	605.5	10/7
12	115.1	15/14
13	824.7	45/28	44/27
14	334.3	60/49	11/9	40/33
15	1043.9	50/27	11/6	20/11
16	553.5	25/18	11/8	15/11
17	63.0	25/24	22/21, 33/32	45/44

* in 7-limit CTE tuning

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 Subgroup Prime-Optimized Tunings

Weight-skew\Order	Euclidean
Tenney	CTE: ~3/2 = 709.5948¢
Weil	CWE: ~3/2 = 709.3901¢
Equilateral	CEE: ~3/2 = 712.8606¢ Eigenmonzo basis (unchanged-interval basis): 2.49/3 (2/5-comma tuning)
Skewed-equilateral	CSEE: ~3/2 = 711.9997¢ Eigenmonzo basis (unchanged-interval basis): 2.823543/243 (7/19-comma tuning)
Benedetti/Wilson	CBE: ~3/2 = 707.7286¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 -49 0 18⟩ (18/85-comma tuning)
Skewed-Benedetti/Wilson	CSBE: ~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¢
Weil	CWE: ~3/2 = 710.1193¢
Equilateral	CEE: ~3/2 = 709.7805¢ Eigenmonzo basis (unchanged-interval basis): 2.5859375/49
Skewed-equilateral	CSEE: ~3/2 = 710.2428¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 3 -37 18⟩
Benedetti/Wilson	CBE: ~3/2 = 709.4859¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 -1225 -3969 450⟩
Skewed-Benedetti/Wilson	CSBE: ~3/2 = 710.0321¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 665 -15771 5160⟩

Tuning spectrum

Edo Generator	Eigenmonzo (Unchanged-interval)	Generator (¢)	Comments
	3/2	701.955	Pythagorean tuning
10\17		705.882	Lower bound of 7- and 9-odd-limit diamond monotone
	81/56	706.499	1/6 comma
	27/14	707.408	1/5 comma
23\39		707.692
	9/7	708.771	1/4 comma, {1, 3, 7, 9} minimax
	15/8	708.807
13\22		709.091
	5/4	709.590	9-odd-limit minimax
	49/27	709.745	2/7 comma
	15/14	709.954
	25/24	710.040
29\49		710.204
	5/3	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
	21/20	711.553
	9/5	711.772
19\32		712.500
	49/48	712.861	2/5 comma, 2.3.7 subgroup CEE tuning
22\37		713.514
25\42		714.286
	7/4	715.587	1/2 comma
3\5		720.000	Upper bound of 7- and 9-odd-limit diamond monotone
	21/16	729.219	Full comma

Music

Lillian Hearne

• Superpyth[12] chromatic riff (2015)
• Trio in Superpyth Temperament for Irish Whistle, Piano, and Cello (2015)

Both in 22edo tuning

Joel Grant Taylor

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

All in superpyth[12] in 22edo tuning.

See also

• Ultrapyth