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 out both commas and thus tempers out 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.

The 22-comma is the characteristic 3-limit comma tempered out in 22edo, and has many advantages as a target. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of harmonic 3 in the generator chain. For an n that is not coprime with 22, however, the corresponding temperament splits the octave into gcd(n, 22) parts, and splits the interval class of 3 into n/gcd(n, 22). For example:

Quasisuper (n = 1) is generated by a fifth with an unsplit octave;
Diaschismic (n = 2) splits the octave in two, as 2 divides 22;
Porcupine (n = 3) splits the fourth in three, as 3 is coprime with 22;
Etc.

Temperaments in the continuum
n	Temperament	Comma
n	Temperament	Ratio	Monzo
0	22 & 22c	(22 digits)	[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	(23 digits)	[25 -23 5⟩
…	…	…	…
∞	Superpyth	20480/19683	[12 -9 1⟩

We may also invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the quasisuper-22 equivalence continuum, which is essentially the same thing. The just value of m is 1.778495… The quasisuper comma is both larger and more complex than the superpyth comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.

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

Temperaments with fractional n and m
n	m	Temperament	Comma
11/5 = 2.2	11/6 = 1.83	Hendecatonic	[43 -11 -11⟩
9/4 = 2.25	9/5 = 1.8	Escapade	[32 -7 -9⟩
16/7 = 2.285714	16/9 = 1.8	Kwazy	[-53 10 16⟩
7/3 = 2.3	7/4 = 1.75	Orson	[-21 3 7⟩
5/2 = 2.5	5/3 = 1.6	Magic	[-10 -1 5⟩

Superpyth (5-limit)

For extensions, see Archytas clan #Superpyth and Jubilismic clan #Bipyth.

In the 5-limit, superpyth tempers out 20480/19683. It has a fifth generator of ~3/2 = ~710¢ and ~5/4 is found at +9 generator steps, as an augmented second (C–D#).

Subgroup: 2.3.5

Comma list: 20480/19683

Mapping: [⟨1 0 -12], ⟨0 1 9]]

mapping generators: ~2, ~3

Optimal tunings: