Superpyth

Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value).Archy is a temperament where the generator is 4/3, tuned flat so that stacking two of them gives the interval 7/4. This means that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, two 3/2 perfect fifths octave-reduced gives 8/7. Three fourths reach a subminor third that approximates 7/6, while four fifths reach a supermajor third that approximates 9/7. This means that the septimal comma (64/63) is tempered out.

Since the generator is a perfect fourth or perfect fifth, archy 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 and 16\27 are typical tunings of the generator.

If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out 245/243, so C–D♯ (an augmented second or limma-flat major third) is 5/4, leading to an extension often called superpyth. 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.

If intervals of 11 are desired, the canonical way is to map 11/8 to +16 generators, or a doubly augmented second (C–Dx), 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.

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.