Superpyth–22 equivalence continuum

The superpyth-22 equivalence continuum is a continuum of 5-limit temperaments which equate a number of superpyth commas, 20480/19683 = [12 -9 1⟩, with the 22-comma, [35 -22⟩. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 22edo.

All temperaments in the continuum satisfy (20480/19683)ⁿ ~ 250/243. Varying n results in different temperaments listed in the table below. It converges to 5-limit superpyth as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 22edo due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of n is approximately 2.284531…, and temperaments having n near this value tend to be the most accurate ones.

Temperaments in the continuum

n	Temperament	Comma
		Ratio	Monzo
0	22 & 22c		[35 -22⟩
1	Quasisuper	8388608/7971615	[23 -13 -1⟩
2	Diaschismic	2048/2025	[11 -4 -2⟩
3	Porcupine	250/243	[1 -5 3⟩
4	Comic	5120000/4782969	[13 -14 4⟩
5	22 & 3cc		[25 -23 5⟩
…	…	…	…
∞	Superpyth	20480/19683	[12 -9 1⟩

Examples of temperaments with fractional values of n:

• 22 & 39cc (n = 1/2 = 0.5)
• 22 & 29c (n = 3/2 = 1.5)
• Hendecatonic (n = 11/5 = 2.2)
• Escapade (n = 9/4 = 2.25)
• Kwazy (n = 16/7 = 2.285714...)
• Orson (n = 7/3 = 2.333...)
• Magic (n = 5/2 = 2.5)

We may also invert the continuum by setting m such that 1/m + 1/n = 1. The just value of m is 1.778495…

Temperaments in the continuum

m	Temperament	Comma
		Ratio	Monzo
0	22 & 22c		[35 -22⟩
1	Superpyth	20480/19683	[12 -9 1⟩
2	Diaschismic	2048/2025	[11 -4 -2⟩
3	22 & 29c		[34 -17 -3⟩
…	…	…	…
∞	Quasisuper	8388608/7971615	[23 -13 -1⟩