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 out both commas and thus tempers out 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-commatic		[-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	Mynic	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⟩

Examples of temperaments with fractional values of n:

Notable temperaments of fractional n
Temperament	n	Comma
Slender	13/2 = 6.5	[-46 10 13⟩
Eris	29/4 = 7.25	[-80 8 29⟩
Tertiaseptal	22/3 = 7.3	[-59 5 22⟩
Luna	15/2 = 7.5	[38 -2 -15⟩
Quasiorwell	38/5 = 7.6	[93 -3 -38⟩
Counterwürschmidt	23/3 = 7.6	[55 -1 -23⟩
Birds	31/4 = 7.75	[72 0 -31⟩
Countermiracle	25/3 = 8.3	[47 7 -25⟩
Casablanca	19/2 = 9.5	[22 14 -19⟩

Quadlayo (31 & 12c)

In the chain-of-fifths notation, 5/4 is mapped to the quadruple-diminished fifth (C-Gbbbb).

Subgroup: 2.3.5

Comma list: [-45 27 1⟩

Mapping: [⟨1 0 45], ⟨0 1 -27]]

mapping generators: ~2, ~3

Optimal tunings:

WE: ~2 = 1201.6167 ¢, ~3/2 = 697.8886 ¢

error map: ⟨+1.617 -2.450 -0.204]

CWE: ~2 = 1200.000 ¢, ~3/2 = 696.9075 ¢

error map: ⟨0.000 -5.048 -2.815]

Optimal ET sequence: 12c, 19c, 31, 136bc, 167bc, 198bc, 229bc

Badness (Sintel): 70.2

The temperament finder - 5-limit 31 & 12c

Ampersand

For extensions, see Gamelismic clan #Miracle.

Ampersand is the 5-limit version of miracle, tempering out the ampersand comma, which is the difference between a perfect fifth and a stack of six classical diatonic semitones. It can be described as the 31 & 41 temperament, corresponding to n = 6.

Subgroup: 2.3.5

Comma list: 34171875/33554432

Mapping: [⟨1 1 3], ⟨0 6 -7]]

mapping generators: ~2, ~16/15

Optimal tunings:

WE: ~2 = 1200.8367 ¢, ~16/15 = 116.7546 ¢

error map: ⟨+0.837 -0.591 -1.086]

CWE: ~2 = 1200.000 ¢, ~16/15 = 116.6802 ¢

error map: ⟨0.000 -1.874 -3.075]

Optimal ET sequence: 10, 21, 31, 41, 72

Badness (Sintel): 3.89

Valentine (5-limit)

For extensions, see Gamelismic clan #Valentine.

The 5-limit version of valentine tempers out the valentine comma, which is the difference between a perfect fifth and a stack of nine classical chromatic semitones. It can be described as the 31 & 46 temperament, corresponding to n = 9.

Subgroup: 2.3.5

Comma list: 1990656/1953125

Mapping: [⟨1 1 2], ⟨0 9 5]]

mapping generators: ~2, ~25/24

Optimal tunings: