Archytas–chromatic equivalence continuum: Difference between revisions
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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 15: | Line 15: | ||
|- | |- | ||
| −2 | | −2 | ||
| [[Dicot]] | | [[Dicot]] restriction | ||
| [[54/49]] | | [[54/49]] | ||
| {{ | | {{Monzo| 1 3 0 -2 }} | ||
|- | |- | ||
| −1 | | −1 | ||
| [[Armodue]] | | [[Armodue (temperament)|Armodue]] restriction | ||
| [[243/224]] | | [[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]] | | [[Mohajira]] restriction | ||
| [[8680203/8388608]] | | [[8680203/8388608]] | ||
| {{ | | {{Monzo| -23 11 0 2 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| (29 & | | (29 & 36) | ||
| [[546852789/536870912]] | | [[546852789/536870912]] | ||
| {{ | | {{Monzo| -29 13 0 3 }} | ||
|- | |- | ||
| 4 | | 4 | ||
| [[Sesquiquartififths]] | | [[Sesquiquartififths]] restriction | ||
| [[34451725707/34359738368]] | | [[34451725707/34359738368]] | ||
| {{ | | {{Monzo| -35 15 0 4 }} | ||
|- | |- | ||
| 4{{frac|1|6}} | | 4{{frac|1|6}} | ||
| ({{nowrap|1848 & | | ({{nowrap|1848 & 3431}}) | ||
| <abbr title="105343182492594861947326056299830127783078740498060829800981944087/105312291668557186697918027683670432318895095400549111254310977536">[very long]</abbr> | | <abbr title="105343182492594861947326056299830127783078740498060829800981944087/105312291668557186697918027683670432318895095400549111254310977536">[very long]</abbr> | ||
| {{ | | {{Monzo| -216 92 0 25 }} | ||
|- | |- | ||
| … | | … | ||
| Line 60: | Line 60: | ||
|- | |- | ||
| ∞ | | ∞ | ||
| [[ | | [[Archy]] | ||
| [[64/63]] | | [[64/63]] | ||
| {{ | | {{Monzo| 6 -2 0 -1 }} | ||
|} | |} | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Revision as of 14:17, 6 September 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 | (29 & 36) | 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⟩ |