Augmented–cloudy equivalence continuum: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
m Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"
m Cleanup
Line 2: Line 2:
The '''augmented–cloudy equivalence continuum''' is a continuum of 2.5.7 subgroup temperaments which equate a number of [[128/125|lesser dieses (128/125)]] with the [[16807/16384|cloudy comma (16807/16384)]].
The '''augmented–cloudy equivalence continuum''' is a continuum of 2.5.7 subgroup temperaments which equate a number of [[128/125|lesser dieses (128/125)]] with the [[16807/16384|cloudy comma (16807/16384)]].


All temperaments in the continuum satisfy {{nowrap|(128/125)<sup>''n''</sup> ~ 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 [[Just intonation subgroup|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.
All temperaments in the continuum satisfy {{nowrap| (128/125)<sup>''n''</sup> ~ 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|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.


{| class="wikitable center-1 center-2 center-4 center-5"
{| class="wikitable center-1 center-5"
|+ style="font-size: 105%;" | Temperaments in the continuum
|+ style="font-size: 105%;" | Temperaments in the continuum
|-
|-
Line 19: Line 19:
! Monzo
! Monzo
|-
|-
| &minus;2
| −2
| 2 & 15
| 2 & 15
| [[16807/15625]]
| [[16807/15625]]
| {{Monzo|0 0 -6 5}}
| {{Monzo| 0 0 -6 5 }}
| &minus;2
| −2
| 15 & 14c
| 15 & 14c
| [[282475249/262144000]]
| [[282475249/262144000]]
| {{Monzo|-21 0 -3 10}}
| {{Monzo| -21 0 -3 10 }}
|-
|-
| &minus;1
| −1
| 4 & 15
| 4 & 15
| [[16807/16000]]
| [[16807/16000]]
| {{Monzo|-7 0 -3 5}}
| {{Monzo| -7 0 -3 5 }}
| &minus;1
| −1
| 4 & 15
| 4 & 15
| [[16807/16000]]
| [[16807/16000]]
| {{Monzo|-7 0 -3 5}}
| {{Monzo| -7 0 -3 5 }}
|-
|-
| 0
| 0
| Cloudy
| Cloudy
| [[16807/16384]]
| [[16807/16384]]
| {{Monzo|-14 0 0 5}}
| {{Monzo| -14 0 0 5 }}
| 0
| 0
| [[Augmented family|Augmented]]
| [[Augment]]
| [[128/125]]
| [[128/125]]
| {{Monzo|7 0 -3}}
| {{Monzo| 7 0 -3 }}
|-
|-
| 1
| 1
| [[Rainy]]
| [[Rainy]]
| [[2100875/2097152]]
| [[2100875/2097152]]
| {{Monzo|-21 0 3 5}}
| {{Monzo| -21 0 3 5 }}
| 1
| 1
| [[Rainy]]
| [[Rainy]]
| [[2100875/2097152]]
| [[2100875/2097152]]
| {{Monzo|-21 0 3 5}}
| {{Monzo| -21 0 3 5 }}
|-
|-
| 2
| 2
| 37 & 15
| 37 & 15
| [[268435456/262609375]]
| [[268435456/262609375]]
| {{Monzo|-28 0 6 5}}
| {{Monzo| -28 0 6 5 }}
| 2
| 2
| 15 & 41
| 15 & 41
| [[35309406125/34359738368]]
| [[35309406125/34359738368]]
| {{Monzo|-35 0 3 10}}
| {{Monzo| -35 0 3 10 }}
|-
|-
| 3
| 3
| 15 & 28
| 15 & 28
|  
|  
| {{Monzo|-35 0 9 5}}
| {{Monzo| -35 0 9 5 }}
| 3
| 3
| 15 & 51
| 15 & 51
|  
|  
| {{Monzo|-49 0 3 15}}
| {{Monzo| -49 0 3 15 }}
|-
|-
| …
| …
Line 83: Line 83:
|-
|-
| ∞
| ∞
| [[Augmented family|Augmented]]
| [[AAugment]]
| [[128/125]]
| [[128/125]]
| {{Monzo|7 0 -3}}
| {{Monzo| 7 0 -3 }}
| ∞
| ∞
| Cloudy
| Cloudy
| [[16807/16384]]
| [[16807/16384]]
| {{Monzo|-14 0 0 5}}
| {{Monzo| -14 0 0 5 }}
|}
|}


Revision as of 05:53, 11 September 2025

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⟩
… … … … … … … …
∞ AAugment 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

Retrieved from "https://en.xen.wiki/index.php?title=Augmented–cloudy_equivalence_continuum&oldid=209740"