Archytas–chromatic equivalence continuum: Difference between revisions
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+ another 7-limit temperament of 29&36, with 36edo patent val |
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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 12: | 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]] 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}} | |||
| ({{nowrap|1848 & 3431}}) | |||
| <abbr title="105343182492594861947326056299830127783078740498060829800981944087/105312291668557186697918027683670432318895095400549111254310977536">[very long]</abbr> | |||
| {{Monzo| -216 92 0 25 }} | |||
|- | |- | ||
| … | | … | ||
| Line 44: | Line 60: | ||
|- | |- | ||
| ∞ | | ∞ | ||
| [[ | | [[Archy]] | ||
| [[64/63]] | | [[64/63]] | ||
| {{ | | {{Monzo| 6 -2 0 -1 }} | ||
|} | |} | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Latest revision as of 08:24, 19 November 2025
|
This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
The Archytas–chromatic equivalence continuum, or septimal–chromatic equivalence continuum, is a continuum of 7-limit temperaments which equate a number of Archytas commas (64/63) with the Pythagorean apotome (2187/2048).
All temperaments in the continuum satisfy (64/63)n ~ 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.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| −2 | Dicot restriction | 54/49 | [1 3 0 -2⟩ |
| −1 | Armodue restriction | 243/224 | [-5 5 0 -1⟩ |
| 0 | Whitewood restriction | 2187/2048 | [-11 7⟩ |
| 1 | Flattone restriction | 137781/131072 | [-17 9 0 1⟩ |
| 2 | Mohajira restriction | 8680203/8388608 | [-23 11 0 2⟩ |
| 3 | Coendou/Squirrel restriction | 546852789/536870912 | [-29 13 0 3⟩ |
| 4 | Sesquiquartififths restriction | 34451725707/34359738368 | [-35 15 0 4⟩ |
| 41⁄6 | (1848 & 3431) | [very long] | [-216 92 0 25⟩ |
| … | … | … | |
| ∞ | Archy | 64/63 | [6 -2 0 -1⟩ |