Augmented–cloudy equivalence continuum

The augmented-cloudy equivalence continuum is a continuum of 5-limit 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

Map: [⟨1 3 2], ⟨0 -5 6]]

EDOs: 7, 15, 22, 30, 37, 44, 52, 49, 74, 89

The temperament finder - 2.5.7 37 & 15

15 & 41

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

POTE generator: 321.5916 cents

Map: [⟨1 5 2], ⟨0 -10 3]]

EDOs: 11c, 15, 26, 30, 41, 56, 67c, 71, 82, 97

The temperament finder - 2.5.7 15 & 41

15 & 28

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

POTE generator: 558.3680 cents

Map: [⟨1 0 7], ⟨0 5 -9]]

EDOs: 13d, 15, 28, 30, 43, 45c, 58, 71d, 73

The temperament finder - 2.5.7 15 & 28

15 & 51

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

POTE generator: 163.2989 cents

Map: [⟨3 9 8], ⟨0 -5 1]]

EDOs: 15, 21c, 30, 36c, 45c, 51, 66, 81, 96d, 117c

The temperament finder - 2.5.7 15 & 51

4 & 15

Commas: [0 0 -6 5⟩ = 16807/16800

POTE generator: 163.2989 cents

Map: [⟨1 1 2], ⟨0 5 3]]

EDOs: 4, 11, 15, 19, 23d, 26c, 30

The temperament finder - 2.5.7 4 & 15

2 & 15

Commas: [-7 0 -3 5⟩ = 16807/15625

POTE generator: 640.6490 cents

Map: [⟨1 5 6], ⟨0 -5 -6]]

EDOs: 2, 13, 15, 17c, 30

The temperament finder - 2.5.7 2 & 15

15 & 14c

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

POTE generator: 81.6979 cents

Map: [⟨1 3 3], ⟨0 -10 -3]]

EDOs: 14c, 15, 29, 30

The temperament finder - 2.5.7 15 & 14c

379 & 4184

Commas: [280 0 -42 -65⟩

POTE generator: 319.7896 cents

Map: [⟨1 -15 14], ⟨0 65 -42]]

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

The temperament finder - 2.5.7 379 & 4184

410 & 3675

Commas: [-301 0 45 70⟩

POTE generator: 79.0215 cents

Map: [⟨5 7 17], ⟨0 14 -9]]

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

The temperament finder - 2.5.7 410 & 3675

441 & 1308

Commas: [-322 0 48 75⟩

POTE generator: 160.5487 cents

Map: [⟨3 17 2], ⟨0 -25 16]]

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

The temperament finder - 2.5.7 441 & 1308

851 & 1687

Commas: [-623 0 93 145⟩

POTE generator: 320.0945 cents

Map: [⟨1 41 -22], ⟨0 -145 93]]

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

The temperament finder - 2.5.7 851 & 1687

68 & 15

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

POTE generator: 158.7877 cents

Map: [⟨1 1 4], ⟨0 10 -9]]

EDOs: 15, 30, 38, 53, 68, 83, 106, 121, 136, 151

The temperament finder - 2.5.7 68 & 15

15 & 72

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

POTE generator: 83.0570 cents

Map: [⟨3 8 8], ⟨0 -5 2]]

EDOs: 15, 30, 42c, 57, 72, 87, 102, 129c, 144, 159

The temperament finder - 2.5.7 15 & 72