Quasisuper

Quasisuper is an alternative extension of the archy chain of fifths to superpyth. Like superpyth, it is a temperament generated by a perfect fifth, where stacking two of them reaches the interval of 8/7~9/8, tempering out 64/63. The difference is that this extension maps prime 5 to -13 generators, as a double-diminished fifth (C–G𝄫). This extension works in the range 17c-edo to 22-edo. In contrast, full 7-limit superpyth does not work in this range, as tunings with a flatter fifth than 22edo swap the sizes of 7/5 and 10/7. This extension may be preferred over superpyth due to having a softer diatonic scale, with a small step of around 60 cents compared to about 50 cents in regular 7-limit superpyth.

The best extension to the 11-limit, quasisupra, uses the supra mapping of prime 11 to -6 generators, as a diminished fifth (C–G♭). This tempers out 99/98 as in supra, as well as 121/120 and 540/539.

For technical data see Archytas clan #Quasisuper.

Interval chain

In the following table, odd harmonics and subharmonics 1–11 are in bold.

#	Cents*	Approximate ratios
0	0.0	1/1
1	708.3	3/2
2	216.6	8/7, 9/8
3	925.0	12/7
4	433.3	9/7, 14/11
5	1141.6	21/11, 27/14
6	649.9	16/11, 22/15
7	158.2	11/10, 12/11
8	866.6	18/11
9	374.9	27/22, 56/45
10	1083.2	28/15
11	591.5	7/5
12	99.8	16/15
13	808.2	8/5
14	316.5	6/5
15	1024.8	9/5
16	533.1	27/20
17	41.4	81/80, 56/55

* in 11-limit CWE tuning, octave reduced

Tunings

7-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 708.7690 ¢	CWE: ~3/2 = 708.3716 ¢	POTE: ~3/2 = 708.2385 ¢

11-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 708.7131 ¢	CWE: ~3/2 = 708.3200 ¢	POTE: ~3/2 = 708.2046 ¢

Todo: complete section