Xenharmonic Wiki

Augmented–cloudy equivalence continuum: Difference between revisions

VisualWikitext
Flirora (talk | contribs)
autopatrolled, emailconfirmed
290 edits
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..."
 
Flirora (talk | contribs)
autopatrolled, emailconfirmed
290 edits
m fix typo
 
(12 intermediate revisions by 7 users not shown)
Line 1: Line 1:
The '''augmented-cloudy equivalence continuum''' is a continuum of 5-limit 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
| [[Subgroup temperaments#Rainy|Rainy]]
| [[Rainy]]
| [[2100875/2097152]]
| [[2100875/2097152]]
| {{Monzo|-21 0 3 5}}
| {{Monzo| -21 0 3 5 }}
| 1
| 1
| [[Subgroup temperaments#Rainy|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


POTE generator: 162.1073 cents
POTE generator: 162.1073 cents


Map: [{{Val| 1 3 2}}, {{Val| 0 -5 6}}]
Mapping: [{{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]


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


POTE generator: 321.5916 cents
POTE generator: 321.5916 cents


Map: [{{Val| 1 5 2}}, {{Val| 0 -10 3}}]
Mapping: [{{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]


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


POTE generator: 558.3680 cents
POTE generator: 558.3680 cents


Map: [{{Val| 1 0 7}}, {{Val| 0 5 -9}}]
Mapping: [{{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]


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


POTE generator: 163.2989 cents
POTE generator: 163.2989 cents


Map: [{{Val| 3 9 8}}, {{Val| 0 -5 1}}]
Mapping: [{{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]


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


POTE generator: 163.2989 cents
POTE generator: 163.2989 cents


Map: [{{Val| 1 1 2}}, {{Val| 0 5 3}}]
Mapping: [{{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]


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


POTE generator: 640.6490 cents
POTE generator: 640.6490 cents


Map: [{{Val| 1 5 6}}, {{Val| 0 -5 -6}}]
Mapping: [{{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]


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


POTE generator: 81.6979 cents
POTE generator: 81.6979 cents


Map: [{{Val| 1 3 3}}, {{Val| 0 -10 -3}}]
Mapping: [{{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]


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


POTE generator: 319.7896 cents
POTE generator: 319.7896 cents


Map: [{{Val| 1 -15 14}}, {{Val| 0 65 -42}}]
Mapping: [{{Val| 1 -15 14}}, {{Val| 0 65 -42}}]


EDOs: {{EDOs| 379, 758, 1137, 1516, 3426, 3805, 4184, 4563, 4942, 5321 }}
EDOs: {{EDOs| 379, 758, 1137, 1516, 3426, 3805, 4184, 4563, 4942, 5321 }}
Line 197: Line 190:


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


POTE generator: 79.0215 cents
POTE generator: 79.0215 cents


Map: [{{Val| 5 7 17}}, {{Val| 0 14 -9}}]
Mapping: [{{Val| 5 7 17}}, {{Val| 0 14 -9}}]


EDOs: {{EDOs| 410, 820, 1230, 1640, 2050, 2855, 3265, 3675, 4085, 4495 }}
EDOs: {{EDOs| 410, 820, 1230, 1640, 2050, 2855, 3265, 3675, 4085, 4495 }}
Line 209: Line 201:


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


POTE generator: 160.5487 cents
POTE generator: 160.5487 cents


Map: [{{Val| 3 17 2}}, {{Val| 0 -25 16}}]
Mapping: [{{Val| 3 17 2}}, {{Val| 0 -25 16}}]


EDOs: {{EDOs| 426, 441, 867, 882, 1308, 1323, 1749, 2190, 2631, 3057 }}
EDOs: {{EDOs| 426, 441, 867, 882, 1308, 1323, 1749, 2190, 2631, 3057 }}
Line 221: Line 212:


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


POTE generator: 320.0945 cents
POTE generator: 320.0945 cents


Map: [{{Val| 1 41 -22}}, {{Val| 0 -145 93}}]
Mapping: [{{Val| 1 41 -22}}, {{Val| 0 -145 93}}]


EDOs: {{EDOs| 15, 836, 851, 1687, 1702, 2538, 3374, 3389, 4225, 5076 }}
EDOs: {{EDOs| 15, 836, 851, 1687, 1702, 2538, 3374, 3389, 4225, 5076 }}
Line 233: Line 223:


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


POTE generator: 158.7877 cents
POTE generator: 158.7877 cents


Map: [{{Val| 1 1 4}}, {{Val| 0 10 -9}}]
Mapping: [{{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]


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


POTE generator: 83.0570 cents
POTE generator: 83.0570 cents


Map: [{{Val| 3 8 8}}, {{Val| 0 -5 2}}]
Mapping: [{{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]


[[Category:15edo]]
[[Category:15edo]]
[[Category:Theory]]
[[Category:Temperament]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
Retrieved from "https://en.xen.wiki/w/Augmented–cloudy_equivalence_continuum"