Archytas–diatonic equivalence continuum: Difference between revisions
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The '''Archytas–diatonic equivalence continuum''', or '''septimal–diatonic equivalence continuum''', is a [[equivalence continuum|continuum]] of [[2.3.7 subgroup]] temperaments which equate a number of [[64/63|Archytas commas (64/63)]] with the [[256/243|limma (256/243)]]. | The '''Archytas–diatonic equivalence continuum''', or '''septimal–diatonic equivalence continuum''', is a [[equivalence continuum|continuum]] of [[2.3.7 subgroup]] temperaments which equate a number of [[64/63|Archytas' commas (64/63)]] with the [[256/243|Pythagorean limma (256/243)]]. This continuum is theoretically interesting in that these are all [[2.3.7 subgroup|2.3.7-subgroup]] temperaments [[support]]ed by [[5edo]]. | ||
All temperaments in the continuum satisfy {{nowrap|(64/63)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by | All temperaments in the continuum satisfy {{nowrap|(64/63)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by 5edo 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 3.3093…, and temperaments near this tend to be the most accurate ones. | ||
256/243 is the characteristic [[3-limit]] comma tempered out in | 256/243 is the characteristic [[3-limit]] comma tempered out in 5edo, and has many advantages as a target. In each case, ''n'' equals the order of harmonic 7 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain. | ||
{| 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 | ||
! | ! S. monzo | ||
|- | |- | ||
| 0 | | 0 | ||
| [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=off&weights=weil&target=&edos=5+%26+5d&submit_edo=submit&commas= 2.3.7 blackwood] | | [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=off&weights=weil&target=&edos=5+%26+5d&submit_edo=submit&commas= 2.3.7 blackwood] | ||
| [[256/243]] | | [[256/243]] | ||
| {{ | | {{Monzo| 8 -5 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| [[ | | [[Trienstonian]] | ||
| [[28/27]] | | [[28/27]] | ||
| {{ | | {{Monzo| 2 -3 1 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| [[Semaphore]] | | [[Semaphore]] | ||
| [[49/48]] | | [[49/48]] | ||
| {{ | | {{Monzo| -4 -1 2 }} | ||
|- | |- | ||
| 2.5 | | 2.5 | ||
| [[Cloudy]] | | [[Cloudy]] retraction | ||
| [[16807/16384]] | | [[16807/16384]] | ||
| {{ | | {{Monzo| -14 0 5 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| [[Slendric]] | | [[Slendric]] | ||
| [[1029/1024]] | | [[1029/1024]] | ||
| {{ | | {{Monzo| -10 1 3 }} | ||
|- | |- | ||
| 3.3 | | 3.3 | ||
| 5 & 436 | | 5 & 436 | ||
| | | (72 digits) | ||
| {{ | | {{Monzo| 118 -16 -33 }} | ||
|- | |- | ||
| 10/3 | | 10/3 | ||
| [[ | | [[Slendroschismic]] | ||
| 68719476736/68641485507 | | 68719476736/68641485507 | ||
| {{ | | {{Monzo| 36 -5 -10 }} | ||
|- | |- | ||
| 3.5 | | 3.5 | ||
| [[Septiness]] | | [[Septiness]] restriction | ||
| 67108864/66706983 | | 67108864/66706983 | ||
| {{ | | {{Monzo| 26 -4 -7 }} | ||
|- | |- | ||
| 4 | | 4 | ||
| [[Buzzard]] | | [[Buzzard]] | ||
| [[65536/64827]] | | [[65536/64827]] | ||
| {{ | | {{Monzo| 16 -3 -4 }} | ||
|- | |- | ||
| 5 | | 5 | ||
| [[ | | [[Obscenity]] | ||
| 4194304/4084101 | | [[4194304/4084101]] | ||
| {{ | | {{Monzo| 22 -5 -5 }} | ||
|- | |- | ||
| … | | … | ||
| Line 71: | Line 71: | ||
|- | |- | ||
| ∞ | | ∞ | ||
| [[ | | [[Archy]] | ||
| [[64/63]] | | [[64/63]] | ||
| {{ | | {{Monzo| 6 -2 -1 }} | ||
|} | |} | ||
[[Category:5edo]] | [[Category:5edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Revision as of 12:40, 1 May 2026
The Archytas–diatonic equivalence continuum, or septimal–diatonic equivalence continuum, is a continuum of 2.3.7 subgroup temperaments which equate a number of Archytas' commas (64/63) with the Pythagorean limma (256/243). This continuum is theoretically interesting in that these are all 2.3.7-subgroup temperaments supported by 5edo.
All temperaments in the continuum satisfy (64/63)n ~ 256/243. Varying n results in different temperaments listed in the table below. It converges to archy as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 2.3.7 subgroup temperaments supported by 5edo 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 3.3093…, and temperaments near this tend to be the most accurate ones.
256/243 is the characteristic 3-limit comma tempered out in 5edo, and has many advantages as a target. In each case, n equals the order of harmonic 7 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | S. monzo | ||
| 0 | 2.3.7 blackwood | 256/243 | [8 -5⟩ |
| 1 | Trienstonian | 28/27 | [2 -3 1⟩ |
| 2 | Semaphore | 49/48 | [-4 -1 2⟩ |
| 2.5 | Cloudy retraction | 16807/16384 | [-14 0 5⟩ |
| 3 | Slendric | 1029/1024 | [-10 1 3⟩ |
| 3.3 | 5 & 436 | (72 digits) | [118 -16 -33⟩ |
| 10/3 | Slendroschismic | 68719476736/68641485507 | [36 -5 -10⟩ |
| 3.5 | Septiness restriction | 67108864/66706983 | [26 -4 -7⟩ |
| 4 | Buzzard | 65536/64827 | [16 -3 -4⟩ |
| 5 | Obscenity | 4194304/4084101 | [22 -5 -5⟩ |
| … | … | … | |
| ∞ | Archy | 64/63 | [6 -2 -1⟩ |