Augmented–cloudy equivalence continuum: Difference between revisions
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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)]]. | 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)]]. | ||
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 [[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. | ||
{| class="wikitable center-1 center-2 center-4 center-5" | {| class="wikitable center-1 center-2 center-4 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 18: | ||
! 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]] | ||
| Line 93: | Line 93: | ||
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 112: | ||
== 15 & 41 == | == 15 & 41 == | ||
Commas: {{Monzo|-35 0 3 10}} = 35309406125/34359738368 | Commas: {{Monzo|-35 0 3 10}} = 35309406125/34359738368 | ||
| Line 125: | Line 123: | ||
== 15 & 28 == | == 15 & 28 == | ||
Commas: {{Monzo|-35 0 9 5}} = 34359738368/32826171875 | Commas: {{Monzo|-35 0 9 5}} = 34359738368/32826171875 | ||
| Line 137: | Line 134: | ||
== 15 & 51 == | == 15 & 51 == | ||
Commas: {{Monzo|-49 0 3 15}} = 593445188742875/562949953421312 | Commas: {{Monzo|-49 0 3 15}} = 593445188742875/562949953421312 | ||
| Line 149: | Line 145: | ||
== 4 & 15 == | == 4 & 15 == | ||
Commas: {{Monzo|0 0 -6 5}} = 16807/16800 | Commas: {{Monzo|0 0 -6 5}} = 16807/16800 | ||
| Line 161: | Line 156: | ||
== 2 & 15 == | == 2 & 15 == | ||
Commas: {{Monzo|-7 0 -3 5}} = 16807/15625 | Commas: {{Monzo|-7 0 -3 5}} = 16807/15625 | ||
| Line 173: | Line 167: | ||
== 15 & 14c == | == 15 & 14c == | ||
Commas: {{Monzo|-21 0 -3 10}} = 282475249/262144000 | Commas: {{Monzo|-21 0 -3 10}} = 282475249/262144000 | ||
| Line 185: | Line 178: | ||
== 379 & 4184 == | == 379 & 4184 == | ||
Commas: {{Monzo|280 0 -42 -65}} | Commas: {{Monzo|280 0 -42 -65}} | ||
| Line 197: | Line 189: | ||
== 410 & 3675 == | == 410 & 3675 == | ||
Commas: {{Monzo|-301 0 45 70}} | Commas: {{Monzo|-301 0 45 70}} | ||
| Line 209: | Line 200: | ||
== 441 & 1308 == | == 441 & 1308 == | ||
Commas: {{Monzo|-322 0 48 75}} | Commas: {{Monzo|-322 0 48 75}} | ||
| Line 221: | Line 211: | ||
== 851 & 1687 == | == 851 & 1687 == | ||
Commas: {{Monzo|-623 0 93 145}} | Commas: {{Monzo|-623 0 93 145}} | ||
| Line 233: | Line 222: | ||
== 68 & 15 == | == 68 & 15 == | ||
Commas: {{Monzo|-49 0 9 10}} = 562949953421312/551709470703125 | Commas: {{Monzo|-49 0 9 10}} = 562949953421312/551709470703125 | ||
| Line 245: | Line 233: | ||
== 15 & 72 == | == 15 & 72 == | ||
Commas: {{Monzo|-56 0 6 15}} = 74180648592859375/72057594037927936 | Commas: {{Monzo|-56 0 6 15}} = 74180648592859375/72057594037927936 | ||