Archytas–diatonic equivalence continuum: Difference between revisions

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The '''Archytas-diatonic equivalence continuum''' or the '''septimal-diatonic equivalence continuum''' is a 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)]].

All temperaments in the continuum satisfy (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 ~~[[Just intonation subgroup|~~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.

All temperaments in the continuum satisfy (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 [[5edo]]. In each case, we notice that ''n'' equals the order of harmonic 7 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale.

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 The '''Archytas-diatonic equivalence continuum''' or the '''septimal-diatonic equivalence continuum''' is a 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)]].
-All temperaments in the continuum satisfy (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 [[Just intonation subgroup|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.
+All temperaments in the continuum satisfy (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.
 /243 is the characteristic [[3-limit]] comma tempered out in [[5edo]]. In each case, we notice that ''n'' equals the order of harmonic 7 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale.