Syntonic-31 equivalence continuum

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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 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

Examples of temperaments with fractional values of n:

Quadlayo (31 & 12c)

In fifths notation, 5/4 is mapped to the quadruple-diminished fifth.

Comma list: [-45 27 1 = 38127987424935/35184372088832

Mapping: [1 2 -9], 0 -1 27]]

POTE generator: ~4/3 = 503.050

Vals: 12c, 19c, 31, 43c, 50c

Badness: 2.993628

The temperament finder - 5-limit 31 & 12c

Quadlaleyo (31 & 70c)

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

Mapping: [1 3 0], 0 -11 18]]

POTE generator: ~32768/30375 = 154.597

Vals: 8c, 23c, 31, 39c, 132, 163

Badness: 2.067160

The temperament finder - 5-limit 31 & 70c

Ampersand (31 & 41)

See also: Gamelismic clan #Miracle

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

Mapping: [1 1 3], 0 6 -7]]

POTE generator: ~16/15 = 116.673

Vals: 10, 21, 31, 41, 72

Badness: 0.165755

Lalasepbigu (31 & 13c)

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

Mapping: [1 7 12], 0 -14 -25]]

POTE generator: ~25000/19683 = 464.423

Vals: 13c, 18bc, 31, 44c, 49bc, 75c, 80bc

Badness: 2.094918

The temperament finder - 5-limit 31 & 13c