Augmented–cloudy equivalence continuum

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The augmented–cloudy equivalence continuum is a continuum of 2.5.7 subgroup temperaments which equate a number of lesser 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 augment 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	Augment	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⟩
…	…	…	…	…	…	…	…
∞	Augment	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

The temperament finder - 2.5.7 37 & 15

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

The temperament finder - 2.5.7 15 & 41

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

The temperament finder - 2.5.7 15 & 28

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

The temperament finder - 2.5.7 15 & 51

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

The temperament finder - 2.5.7 4 & 15

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

The temperament finder - 2.5.7 2 & 15

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

The temperament finder - 2.5.7 15 & 14c

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

The temperament finder - 2.5.7 379 & 4184

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

The temperament finder - 2.5.7 410 & 3675

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

The temperament finder - 2.5.7 441 & 1308

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

The temperament finder - 2.5.7 851 & 1687

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

The temperament finder - 2.5.7 68 & 15

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

The temperament finder - 2.5.7 15 & 72