Augmented–cloudy equivalence continuum: Difference between revisions
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The ''' | {{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 [[ | 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 | {| 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 & 15 | | 2 & 15 | ||
| [[16807/15625]] | | [[16807/15625]] | ||
| {{Monzo|0 0 -6 5}} | | {{Monzo| 0 0 -6 5 }} | ||
| | | −2 | ||
| 15 & 14c | | 15 & 14c | ||
| [[282475249/262144000]] | | [[282475249/262144000]] | ||
| {{Monzo|-21 0 -3 10}} | | {{Monzo| -21 0 -3 10 }} | ||
|- | |- | ||
| | | −1 | ||
| 4 & 15 | | 4 & 15 | ||
| [[16807/16000]] | | [[16807/16000]] | ||
| {{Monzo|-7 0 -3 5}} | | {{Monzo| -7 0 -3 5 }} | ||
| | | −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 | ||
| [[ | | [[Augment]] | ||
| [[128/125]] | | [[128/125]] | ||
| {{Monzo|7 0 -3}} | | {{Monzo| 7 0 -3 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| [[ | | [[Rainy]] | ||
| [[2100875/2097152]] | | [[2100875/2097152]] | ||
| {{Monzo|-21 0 3 5}} | | {{Monzo| -21 0 3 5 }} | ||
| 1 | | 1 | ||
| [[ | | [[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: | ||
|- | |- | ||
| ∞ | | ∞ | ||
| [[ | | [[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 | ||
Mapping: [{{Val| 1 3 2}}, {{Val| 0 -5 6}}] | |||
EDOs: {{EDOs| 7, 15, 22, 37, 44, 52, 49, 74, 89 }} | EDOs: {{EDOs| 7, 15, 22, 37, 44, 52, 49, 74, 89 }} | ||
| Line 113: | Line 113: | ||
== 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 | ||
Mapping: [{{Val| 1 5 2}}, {{Val| 0 -10 3}}] | |||
EDOs: {{EDOs| 11c, 15, 26, 41, 56, 67c, 71, 97 }} | EDOs: {{EDOs| 11c, 15, 26, 41, 56, 67c, 71, 97 }} | ||
| Line 125: | Line 124: | ||
== 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 | ||
Mapping: [{{Val| 1 0 7}}, {{Val| 0 5 -9}}] | |||
EDOs: {{EDOs| 13d, 15, 28, 43, 58, 71d, 73 }} | EDOs: {{EDOs| 13d, 15, 28, 43, 58, 71d, 73 }} | ||
| Line 137: | Line 135: | ||
== 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 | ||
Mapping: [{{Val| 3 9 8}}, {{Val| 0 -5 1}}] | |||
EDOs: {{EDOs| 15, 21c, 36c, 51, 66, 81, 96d, 117c }} | EDOs: {{EDOs| 15, 21c, 36c, 51, 66, 81, 96d, 117c }} | ||
| Line 149: | Line 146: | ||
== 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 | ||
Mapping: [{{Val| 1 1 2}}, {{Val| 0 5 3}}] | |||
EDOs: {{EDOs| 4, 11, 15, 19, 23d, 26c }} | EDOs: {{EDOs| 4, 11, 15, 19, 23d, 26c }} | ||
| Line 161: | Line 157: | ||
== 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 | ||
Mapping: [{{Val| 1 5 6}}, {{Val| 0 -5 -6}}] | |||
EDOs: {{EDOs| 2, 13, 15, 17c }} | EDOs: {{EDOs| 2, 13, 15, 17c }} | ||
| Line 173: | Line 168: | ||
== 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 | ||
Mapping: [{{Val| 1 3 3}}, {{Val| 0 -10 -3}}] | |||
EDOs: {{EDOs| 14c, 15, 29 }} | EDOs: {{EDOs| 14c, 15, 29 }} | ||
| Line 185: | Line 179: | ||
== 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 | ||
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 | ||
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 | ||
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 | ||
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 | ||
Mapping: [{{Val| 1 1 4}}, {{Val| 0 10 -9}}] | |||
EDOs: {{EDOs| 15, 38, 53, 68, 83, 106, 121, 136, 151 }} | EDOs: {{EDOs| 15, 38, 53, 68, 83, 106, 121, 136, 151 }} | ||
| Line 245: | Line 234: | ||
== 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 | ||
Mapping: [{{Val| 3 8 8}}, {{Val| 0 -5 2}}] | |||
EDOs: {{EDOs| 15, 42c, 57, 72, 87, 102, 129c, 144, 159 }} | EDOs: {{EDOs| 15, 42c, 57, 72, 87, 102, 129c, 144, 159 }} | ||
| Line 257: | Line 245: | ||
[[Category:15edo]] | [[Category:15edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||