Augmented–cloudy equivalence continuum: Difference between revisions

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Created page with "The '''augmented-cloudy equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of dieses (128/125) with the 16807/16384|cloudy com..."
 
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Line 108: Line 108:
Map: [{{Val| 1 3 2}}, {{Val| 0 -5 6}}]
Map: [{{Val| 1 3 2}}, {{Val| 0 -5 6}}]


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


[http://x31eq.com/cgi-bin/rt.cgi?ets=37_15&limit=2_5_7 The temperament finder - 2.5.7 37 & 15]
[http://x31eq.com/cgi-bin/rt.cgi?ets=37_15&limit=2_5_7 The temperament finder - 2.5.7 37 & 15]
Line 120: Line 120:
Map: [{{Val| 1 5 2}}, {{Val| 0 -10 3}}]
Map: [{{Val| 1 5 2}}, {{Val| 0 -10 3}}]


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


[http://x31eq.com/cgi-bin/rt.cgi?ets=15_41&limit=2_5_7 The temperament finder - 2.5.7 15 & 41]
[http://x31eq.com/cgi-bin/rt.cgi?ets=15_41&limit=2_5_7 The temperament finder - 2.5.7 15 & 41]
Line 132: Line 132:
Map: [{{Val| 1 0 7}}, {{Val| 0 5 -9}}]
Map: [{{Val| 1 0 7}}, {{Val| 0 5 -9}}]


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


[http://x31eq.com/cgi-bin/rt.cgi?ets=15_28&limit=2_5_7 The temperament finder - 2.5.7 15 & 28]
[http://x31eq.com/cgi-bin/rt.cgi?ets=15_28&limit=2_5_7 The temperament finder - 2.5.7 15 & 28]
Line 144: Line 144:
Map: [{{Val| 3 9 8}}, {{Val| 0 -5 1}}]
Map: [{{Val| 3 9 8}}, {{Val| 0 -5 1}}]


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


[http://x31eq.com/cgi-bin/rt.cgi?ets=15_51&limit=2_5_7 The temperament finder - 2.5.7 15 & 51]
[http://x31eq.com/cgi-bin/rt.cgi?ets=15_51&limit=2_5_7 The temperament finder - 2.5.7 15 & 51]
Line 156: Line 156:
Map: [{{Val| 1 1 2}}, {{Val| 0 5 3}}]
Map: [{{Val| 1 1 2}}, {{Val| 0 5 3}}]


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


[http://x31eq.com/cgi-bin/rt.cgi?ets=4_15&limit=2_5_7 The temperament finder - 2.5.7 4 & 15]
[http://x31eq.com/cgi-bin/rt.cgi?ets=4_15&limit=2_5_7 The temperament finder - 2.5.7 4 & 15]
Line 168: Line 168:
Map: [{{Val| 1 5 6}}, {{Val| 0 -5 -6}}]
Map: [{{Val| 1 5 6}}, {{Val| 0 -5 -6}}]


EDOs: {{EDOs| 2, 13, 15, 17c, 30 }}
EDOs: {{EDOs| 2, 13, 15, 17c }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=2_15&limit=2_5_7 The temperament finder - 2.5.7 2 & 15]
[http://x31eq.com/cgi-bin/rt.cgi?ets=2_15&limit=2_5_7 The temperament finder - 2.5.7 2 & 15]
Line 180: Line 180:
Map: [{{Val| 1 3 3}}, {{Val| 0 -10 -3}}]
Map: [{{Val| 1 3 3}}, {{Val| 0 -10 -3}}]


EDOs: {{EDOs| 14c, 15, 29, 30 }}
EDOs: {{EDOs| 14c, 15, 29 }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=15_14c&limit=2_5_7 The temperament finder - 2.5.7 15 & 14c]
[http://x31eq.com/cgi-bin/rt.cgi?ets=15_14c&limit=2_5_7 The temperament finder - 2.5.7 15 & 14c]
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Map: [{{Val| 1 1 4}}, {{Val| 0 10 -9}}]
Map: [{{Val| 1 1 4}}, {{Val| 0 10 -9}}]


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


[http://x31eq.com/cgi-bin/rt.cgi?ets=68_15&limit=2_5_7 The temperament finder - 2.5.7 68 & 15]
[http://x31eq.com/cgi-bin/rt.cgi?ets=68_15&limit=2_5_7 The temperament finder - 2.5.7 68 & 15]
Line 252: Line 252:
Map: [{{Val| 3 8 8}}, {{Val| 0 -5 2}}]
Map: [{{Val| 3 8 8}}, {{Val| 0 -5 2}}]


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


[http://x31eq.com/cgi-bin/rt.cgi?ets=15_72&limit=2_5_7 The temperament finder - 2.5.7 15 & 72]
[http://x31eq.com/cgi-bin/rt.cgi?ets=15_72&limit=2_5_7 The temperament finder - 2.5.7 15 & 72]

Revision as of 08:51, 16 March 2021

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, 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, 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

Map: [⟨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

Map: [⟨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

Map: [⟨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

Map: [⟨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

Map: [⟨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

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, 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, 42c, 57, 72, 87, 102, 129c, 144, 159

The temperament finder - 2.5.7 15 & 72

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