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 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 {{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 [[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.

{| class="wikitable center-1 center-2"

|+Temperaments in the continuum

|+ style="font-size: 105%;" | Temperaments in the continuum

|-

! rowspan="2" | ''n''

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| {{monzo| 6 -2 -1 }}

|}

<nowiki>*~~</nowiki>~~ The name "slendrismic" may be changed to minimise confusion with confusingly named temperaments, or those temperaments may be changed instead, either way consensus would first need to be formed.

<nowiki />* The name "slendrismic" may be changed to minimise confusion with confusingly named temperaments, or those temperaments may be changed instead, either way consensus would first need to be formed.

[[Category:5edo]]

[[Category:Equivalence continua]]

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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 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 {{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.
 /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.
 {| class="wikitable center-1 center-2"
-|+Temperaments in the continuum
+|+ style="font-size: 105%;" | Temperaments in the continuum
 |-
 ! rowspan="2" | ''n''
@@ Line 70: / Line 70: @@
 | {{monzo| 6 -2 -1 }}
 |}
-<nowiki>*</nowiki> The name "slendrismic" may be changed to minimise confusion with confusingly named temperaments, or those temperaments may be changed instead, either way consensus would first need to be formed.
+<nowiki />* The name "slendrismic" may be changed to minimise confusion with confusingly named temperaments, or those temperaments may be changed instead, either way consensus would first need to be formed.
 [[Category:5edo]]
 [[Category:Equivalence continua]]