Augmented–cloudy equivalence continuum: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
15,213 edits
m ArrowHead294 moved page Augmented-cloudy equivalence continuum to Augmented–cloudy equivalence continuum: The dash in titles like these should be an en dash, not a hyphen-minus, since "Augmented" does not modify "cloudy"
Flirora (talk | contribs)
autopatrolled, emailconfirmed
290 edits
m fix typo
 
(5 intermediate revisions by 3 users not shown)
Line 1: Line 1:
The '''augmented-cloudy equivalence continuum''' is a continuum of 2.5.7 subgroup temperaments which equate a number of [[128/125|dieses (128/125)]] with the [[16807/16384|cloudy comma (16807/16384)]].
{{Technical data page}}
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 (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"
|+ Temperaments in the continuum
|+ style="font-size: 105%;" | Temperaments in the continuum
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
! rowspan="2" | 1 / ''n''
! rowspan="2" | 1/''n''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
Line 18: Line 19:
! Monzo
! Monzo
|-
|-
| -2
| −2
| 2 & 15
| 2 & 15
| [[16807/15625]]
| [[16807/15625]]
| {{Monzo|0 0 -6 5}}
| {{Monzo| 0 0 -6 5 }}
| -2
| −2
| 15 & 14c
| 15 & 14c
| [[282475249/262144000]]
| [[282475249/262144000]]
| {{Monzo|-21 0 -3 10}}
| {{Monzo| -21 0 -3 10 }}
|-
|-
| -1
| −1
| 4 & 15
| 4 & 15
| [[16807/16000]]
| [[16807/16000]]
| {{Monzo|-7 0 -3 5}}
| {{Monzo| -7 0 -3 5 }}
| -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 82: Line 83:
|-
|-
| ∞
| ∞
| [[Augmented family|Augmented]]
| [[Augment]]
| [[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 }}
|}
|}


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


* 15 & 72 (''n'' = 2/3)
* 15 & 72 ({{nowrap|''n'' {{=}} 2/3}})
* 379 & 4184 (''n'' = 13/12)
* 379 & 4184 ({{nowrap|''n'' {{=}} 13/12}})
* 410 & 3675 (''n'' = 14/13)
* 410 & 3675 ({{nowrap|''n'' {{=}} 14/13}})
* 851 & 1687 (''n'' = 29/27)
* 851 & 1687 ({{nowrap|''n'' {{=}} 29/27}})
* 441 & 1308 (''n'' = 15/14)
* 441 & 1308 ({{nowrap|''n'' {{=}} 15/14}})
* 68 & 15 (''n'' = 3/2)
* 68 & 15 ({{nowrap|''n'' {{=}} 3/2}})


== 37 & 15 ==
== 37 & 15 ==
Commas: {{Monzo|-28 0 6 5}} = 268435456/262609375
Commas: {{Monzo|-28 0 6 5}} = 268435456/262609375


Line 113: Line 113:


== 15 & 41 ==
== 15 & 41 ==
Commas: {{Monzo|-35 0 3 10}} = 35309406125/34359738368
Commas: {{Monzo|-35 0 3 10}} = 35309406125/34359738368


Line 125: Line 124:


== 15 & 28 ==
== 15 & 28 ==
Commas: {{Monzo|-35 0 9 5}} = 34359738368/32826171875
Commas: {{Monzo|-35 0 9 5}} = 34359738368/32826171875


Line 137: Line 135:


== 15 & 51 ==
== 15 & 51 ==
Commas: {{Monzo|-49 0 3 15}} = 593445188742875/562949953421312
Commas: {{Monzo|-49 0 3 15}} = 593445188742875/562949953421312


Line 149: Line 146:


== 4 & 15 ==
== 4 & 15 ==
Commas: {{Monzo|0 0 -6 5}} = 16807/16800
Commas: {{Monzo|0 0 -6 5}} = 16807/16800


Line 161: Line 157:


== 2 & 15 ==
== 2 & 15 ==
Commas: {{Monzo|-7 0 -3 5}} = 16807/15625
Commas: {{Monzo|-7 0 -3 5}} = 16807/15625


Line 173: Line 168:


== 15 & 14c ==
== 15 & 14c ==
Commas: {{Monzo|-21 0 -3 10}} = 282475249/262144000
Commas: {{Monzo|-21 0 -3 10}} = 282475249/262144000


Line 185: Line 179:


== 379 & 4184 ==
== 379 & 4184 ==
Commas: {{Monzo|280 0 -42 -65}}
Commas: {{Monzo|280 0 -42 -65}}


Line 197: Line 190:


== 410 & 3675 ==
== 410 & 3675 ==
Commas: {{Monzo|-301 0 45 70}}
Commas: {{Monzo|-301 0 45 70}}


Line 209: Line 201:


== 441 & 1308 ==
== 441 & 1308 ==
Commas: {{Monzo|-322 0 48 75}}
Commas: {{Monzo|-322 0 48 75}}


Line 221: Line 212:


== 851 & 1687 ==
== 851 & 1687 ==
Commas: {{Monzo|-623 0 93 145}}
Commas: {{Monzo|-623 0 93 145}}


Line 233: Line 223:


== 68 & 15 ==
== 68 & 15 ==
Commas: {{Monzo|-49 0 9 10}} = 562949953421312/551709470703125
Commas: {{Monzo|-49 0 9 10}} = 562949953421312/551709470703125


Line 245: Line 234:


== 15 & 72 ==
== 15 & 72 ==
Commas: {{Monzo|-56 0 6 15}} = 74180648592859375/72057594037927936
Commas: {{Monzo|-56 0 6 15}} = 74180648592859375/72057594037927936


Latest revision as of 20:52, 15 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⟩
… … … … … … … …
∞ 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

Retrieved from "https://en.xen.wiki/index.php?title=Augmented–cloudy_equivalence_continuum&oldid=209963"