Superpyth

Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value). Superpyth, sometimes called archy in the no-5 subgroup, is a temperament where the generator is a perfect fifth, tuned sharp such that a stack of two perfect fifths octave-reduced gives a whole tone that represents both 9/8 and 8/7, tempering out the septimal comma, 64/63. Likewise, two perfect fourths give a minor seventh that represents both 7/4 and 16/9, so that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, three fourths reach a minor third that approximates 7/6, while four fifths reach a major third that approximates 9/7.

Since the generator is a perfect fifth, superpyth can be notated using the same standard chain-of-fifths notation that is also used for meantone, 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. 13\22 (~1/4 septimal comma) and 16\27 (~1/3 septimal comma) are the most common tunings of the generator.

If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out 245/243, so 5/4 is an augmented second (e.g. C–D♯, a limma-flat major third). Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including 12edo) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of 7 are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.

Alternatively, for a sharper tuning, the 5th harmonic can be mapped to +14 generators, resulting in ultrapyth.

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. A simpler way to map it is 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 these 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
		7-limit	Superpyth	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

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.

Prime-optimized tunings

2.3.7-subgroup prime-optimized tunings
	Euclidean
	Unskewed	Skewed
Equilateral	CEE: ~3/2 = 712.8606¢ (2/5-comma)	CSEE: ~3/2 = 711.9997¢ (7/19-comma)
Tenney	CTE: ~3/2 = 709.5948¢	CWE: ~3/2 = 709.3901¢
Benedetti, Wilson	CBE: ~3/2 = 707.7286¢ (18/85-comma)	CSBE: ~3/2 = 707.9869¢ (25/113-comma)

7-limit prime-optimized tunings
	Euclidean
	Unskewed	Skewed
Equilateral	CEE: ~3/2 = 709.7805¢	CSEE: ~3/2 = 710.2428¢
Tenney	CTE: ~3/2 = 709.5907¢	CWE: ~3/2 = 710.1193¢
Benedetti, Wilson	CBE: ~3/2 = 709.4859¢	CSBE: ~3/2 = 710.0321¢

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	39cd val
	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
42\71		709.859	71d val
	15/14	709.954
	25/24	710.040
29\49		710.204
45\76		710.526	76bcd val
	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	32c val
	55/32	712.544	Suprapyth mapping
	49/48	712.861	2/5 comma, 2.3.7 subgroup CEE tuning
22\37		713.514	37cc val
25\42		714.286	42cc val
	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

* Besides the octave

Other tunings

DKW (2.3.5 superpyth): ~2 = 1200.000, ~3/2 = 709.758
DKW (2.3.7 archy): ~2 = 1200.000, ~3/2 = 712.585

Music

Lillian Hearne

Both in 22edo tuning

Joel Grant Taylor

All in Superpyth[12], 22edo tuning.