# Syntonic-31 equivalence continuum

(Redirected from Counterwürschmidt)

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

In the circle-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 = 1\1, ~3/2 = 696.950

Subgroup: 2.3.5

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

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

Optimal tuning (POTE): ~2 = 1\1, ~32768/30375 = 154.597

## Ampersand

Subgroup: 2.3.5

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

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

Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 116.673

## Counterwürschmidt

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 = 1\1, ~5/4 = 386.8710