Archytas–chromatic equivalence continuum: Difference between revisions
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{{ | {{Mathematical interest}} | ||
The '''Archytas–chromatic equivalence continuum''', or '''septimal–chromatic equivalence continuum''', is a [[equivalence continuum|continuum]] of [[7-limit]] temperaments which equate a number of [[64/63|Archytas commas (64/63)]] with the [[2187/2048|Pythagorean apotome (2187/2048)]]. | The '''Archytas–chromatic equivalence continuum''', or '''septimal–chromatic equivalence continuum''', is a [[equivalence continuum|continuum]] of [[7-limit]] temperaments which equate a number of [[64/63|Archytas commas (64/63)]] with the [[2187/2048|Pythagorean apotome (2187/2048)]]. | ||
All temperaments in the continuum satisfy {{nowrap|(64/63)<sup>''n''</sup> ~ 2187/2048}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. The just value of ''n'' is 4.169771, and temperaments near this tend to be the most accurate ones. | All temperaments in the continuum satisfy {{nowrap|(64/63)<sup>''n''</sup> ~ 2187/2048}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. The just value of ''n'' is 4.169771, and temperaments near this tend to be the most accurate ones. | ||
{| class="wikitable center-1 | {| class="wikitable center-1" | ||
|+ style="font-size: 105%;" | Temperaments in the continuum | |+ style="font-size: 105%;" | Temperaments in the continuum | ||
|- | |- | ||
| Line 13: | Line 13: | ||
! Ratio | ! Ratio | ||
! Monzo | ! Monzo | ||
|- | |||
| −2 | |||
| [[Dicot]] restriction | |||
| [[54/49]] | |||
| {{Monzo| 1 3 0 -2 }} | |||
|- | |||
| −1 | |||
| [[Armodue (temperament)|Armodue]] restriction | |||
| [[243/224]] | |||
| {{Monzo| -5 5 0 -1 }} | |||
|- | |- | ||
| 0 | | 0 | ||
| [[Whitewood]] | | [[Whitewood]] restriction | ||
| [[2187/2048]] | | [[2187/2048]] | ||
| {{ | | {{Monzo| -11 7 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| [[Flattone]] | | [[Flattone]] restriction | ||
| [[137781/131072]] | | [[137781/131072]] | ||
| {{ | | {{Monzo| -17 9 0 1 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| | | [[Mohajira]] restriction | ||
| [[8680203/8388608]] | | [[8680203/8388608]] | ||
| {{ | | {{Monzo| -23 11 0 2 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| | | [[Coendou]]/[[Squirrel]] restriction | ||
| [[546852789/536870912]] | | [[546852789/536870912]] | ||
| {{ | | {{Monzo| -29 13 0 3 }} | ||
|- | |- | ||
| 4 | | 4 | ||
| | | [[Sesquiquartififths]] restriction | ||
| [[34451725707/34359738368]] | | [[34451725707/34359738368]] | ||
| {{ | | {{Monzo| -35 15 0 4 }} | ||
|- | |- | ||
| 4{{frac|1|6}} | | 4{{frac|1|6}} | ||
| | | ({{nowrap|1848 & 3431}}) | ||
| <abbr title="105343182492594861947326056299830127783078740498060829800981944087/105312291668557186697918027683670432318895095400549111254310977536">[very long]</abbr> | | <abbr title="105343182492594861947326056299830127783078740498060829800981944087/105312291668557186697918027683670432318895095400549111254310977536">[very long]</abbr> | ||
| {{ | | {{Monzo| -216 92 0 25 }} | ||
|- | |- | ||
| … | | … | ||
| Line 50: | Line 60: | ||
|- | |- | ||
| ∞ | | ∞ | ||
| [[ | | [[Archy]] | ||
| [[64/63]] | | [[64/63]] | ||
| {{ | | {{Monzo| 6 -2 0 -1 }} | ||
|} | |} | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||