Augmented-cloudy equivalence continuum

The augmented-cloudy equivalence continuum is a continuum of 2.5.7 subgroup temperaments which equate a number of dieses (128/125) with the cloudy comma (16807/16384).

All temperaments in the continuum satisfy (128/125)n ~ 16807/16384. Varying n results in different temperaments listed in the table below. It converges to augmented as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 2.5.7 subgroup temperaments supported by 15edo (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 1.0747..., and temperaments having n near this value tend to be the most accurate ones.

Temperaments in the continuum
n Temperament Comma 1 / n Temperament Comma
Ratio Monzo Ratio Monzo
-2 2 & 15 16807/15625 [0 0 -6 5 -2 15 & 14c 282475249/262144000 [-21 0 -3 10
-1 4 & 15 16807/16000 [-7 0 -3 5 -1 4 & 15 16807/16000 [-7 0 -3 5
0 Cloudy 16807/16384 [-14 0 0 5 0 Augmented 128/125 [7 0 -3
1 Rainy 2100875/2097152 [-21 0 3 5 1 Rainy 2100875/2097152 [-21 0 3 5
2 37 & 15 268435456/262609375 [-28 0 6 5 2 15 & 41 35309406125/34359738368 [-35 0 3 10
3 15 & 28 [-35 0 9 5 3 15 & 51 [-49 0 3 15
Augmented 128/125 [7 0 -3 Cloudy 16807/16384 [-14 0 0 5

Examples of temperaments with fractional values of n not listed above:

• 15 & 72 (n = 2/3)
• 379 & 4184 (n = 13/12)
• 410 & 3675 (n = 14/13)
• 851 & 1687 (n = 29/27)
• 441 & 1308 (n = 15/14)
• 68 & 15 (n = 3/2)

37 & 15

Commas: [-28 0 6 5 = 268435456/262609375

POTE generator: 162.1073 cents

Mapping: [1 3 2], 0 -5 6]]

EDOs: 7, 15, 22, 37, 44, 52, 49, 74, 89

15 & 41

Commas: [-35 0 3 10 = 35309406125/34359738368

POTE generator: 321.5916 cents

Mapping: [1 5 2], 0 -10 3]]

EDOs: 11c, 15, 26, 41, 56, 67c, 71, 97

15 & 28

Commas: [-35 0 9 5 = 34359738368/32826171875

POTE generator: 558.3680 cents

Mapping: [1 0 7], 0 5 -9]]

EDOs: 13d, 15, 28, 43, 58, 71d, 73

15 & 51

Commas: [-49 0 3 15 = 593445188742875/562949953421312

POTE generator: 163.2989 cents

Mapping: [3 9 8], 0 -5 1]]

EDOs: 15, 21c, 36c, 51, 66, 81, 96d, 117c

4 & 15

Commas: [0 0 -6 5 = 16807/16800

POTE generator: 163.2989 cents

Mapping: [1 1 2], 0 5 3]]

EDOs: 4, 11, 15, 19, 23d, 26c

2 & 15

Commas: [-7 0 -3 5 = 16807/15625

POTE generator: 640.6490 cents

Mapping: [1 5 6], 0 -5 -6]]

EDOs: 2, 13, 15, 17c

15 & 14c

Commas: [-21 0 -3 10 = 282475249/262144000

POTE generator: 81.6979 cents

Mapping: [1 3 3], 0 -10 -3]]

EDOs: 14c, 15, 29

379 & 4184

Commas: [280 0 -42 -65

POTE generator: 319.7896 cents

Mapping: [1 -15 14], 0 65 -42]]

EDOs: 379, 758, 1137, 1516, 3426, 3805, 4184, 4563, 4942, 5321

410 & 3675

Commas: [-301 0 45 70

POTE generator: 79.0215 cents

Mapping: [5 7 17], 0 14 -9]]

EDOs: 410, 820, 1230, 1640, 2050, 2855, 3265, 3675, 4085, 4495

441 & 1308

Commas: [-322 0 48 75

POTE generator: 160.5487 cents

Mapping: [3 17 2], 0 -25 16]]

EDOs: 426, 441, 867, 882, 1308, 1323, 1749, 2190, 2631, 3057

851 & 1687

Commas: [-623 0 93 145

POTE generator: 320.0945 cents

Mapping: [1 41 -22], 0 -145 93]]

EDOs: 15, 836, 851, 1687, 1702, 2538, 3374, 3389, 4225, 5076

68 & 15

Commas: [-49 0 9 10 = 562949953421312/551709470703125

POTE generator: 158.7877 cents

Mapping: [1 1 4], 0 10 -9]]

EDOs: 15, 38, 53, 68, 83, 106, 121, 136, 151

15 & 72

Commas: [-56 0 6 15 = 74180648592859375/72057594037927936

POTE generator: 83.0570 cents

Mapping: [3 8 8], 0 -5 2]]

EDOs: 15, 42c, 57, 72, 87, 102, 129c, 144, 159