Syntonic–31 equivalence continuum

The syntonic-31 equivalence continuum is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with a 31-comma ([-49 31⟩). This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 31edo.

All temperaments in the continuum satisfy (81/80)ⁿ ~ [-49 31⟩. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 31edo (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 7.46781..., and temperaments having n near this value tend to be the most accurate ones.

Temperaments in the continuum
n	Temperament	Comma
n	Temperament	Ratio	Monzo
0	31 & 31c		[-49 31⟩
1	31 & 12c		[-45 27 1⟩
2	Quasimoha	2353579470675/2199023255552	[-41 23 2⟩
3	Oncle	145282683375/137438953472	[-37 19 3⟩
4	Sentinel	8968066875/8589934592	[-33 15 4⟩
5	Tritonic	553584375/536870912	[-29 11 5⟩
6	Ampersand	34171875/33554432	[-25 7 6⟩
7	Orson	2109375/2097152	[-21 3 7⟩
8	Würschmidt	393216/390625	[-17 -1 8⟩
9	Valentine	1990656/1953125	[-13 -5 9⟩
10	Myna	10077696/9765625	[9 9 -10⟩
11	Nusecond	51018336/48828125	[5 13 -11⟩
12	Cypress	258280326/244140625	[1 17 -12⟩
13	Diesic	10460353203/9765625000	[-3 21 -13⟩
14	31 & 13c	847288609443/781250000000	[-7 25 -14⟩
…	…	…	…
∞	Meantone	81/80	[-4 4 -1⟩