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
		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	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⟩ = 38127987424935/35184372088832

Mapping: [⟨1 0 45], ⟨0 1 -27]]

Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 696.950

Optimal ET sequence: 12c, 19c, 31, 43c, 50c

Badness (Smith): 2.993628

The temperament finder - 5-limit 31 & 12c

Ampersand

For extensions, see Gamelismic clan #Miracle.

Subgroup: 2.3.5

Comma list: [-25 7 6⟩ = 34171875/33554432

Mapping: [⟨1 1 3], ⟨0 6 -7]]

Optimal tuning (CWE): ~2 = 1200.000, ~16/15 = 116.680

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

Badness:

• Smith: 0.165755
• Dirichlet: 3.89

Quadlaleyo (31 & 70c)

Subgroup: 2.3.5

Comma list: [-54 18 11⟩ = 18917016064453125/18014398509481984

Mapping: [⟨1 3 0], ⟨0 -11 18]]

Optimal tuning (POTE): ~2 = 1200.000, ~32768/30375 = 154.597

Optimal ET sequence: 8c, 23c, 31, 39c, 132, 163

Badness (Smith): 2.067160

The temperament finder - 5-limit 31 & 70c

Lalasepbigu (31 & 13c)

Subgroup: 2.3.5

Comma list: [-7 25 -14⟩ = 847288609443/781250000000

Mapping: [⟨1 7 12], ⟨0 -14 -25]]

Optimal tuning (POTE): ~2 = 1200.000, ~25000/19683 = 464.423

Optimal ET sequence: 13c, 18bc, 31, 44c, 49bc, 75c, 80bc

Badness (Smith): 2.094918

The temperament finder - 5-limit 31 & 13c

Counterwürschmidt

See also: Mirkwai clan § Grendel

Subgroup: 2.3.5

Comma list: [55 -1 -23⟩

Mapping: [⟨1 9 2], ⟨0 -23 1]]

mapping generators: ~2, ~5/4

Optimal tuning (CTE): ~2 = 1200.0000, ~5/4 = 386.8710

Optimal ET sequence: 28b, 31, 90, 121, 152, 335, 822, 1157c, 1492c

Badness (Smith): 0.420