9th-octave temperaments: Difference between revisions

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Therefore, one can consider it as interpreting [[9edo]] as a circle of [[7/6]]'s (corresponding to tempering the [[septimal ennealimma]]) and as a circle of [[27/25]]'s (corresponding to tempering the [[ennealimma]]), which is an equivalent description which implies tempering the [[landscape comma]] which makes [[63/50]] equal to exactly [[3edo|a third of an octave]].

An alternative 7-limit 9th-octave temperament supported by more edos is to preserve the mapping of 7/6 but not that of 27/25, resulting in [[septiennealimmal]], with many extensions possible. ~~Arguably the main~~ edo of interest is [[63edo]], a tuning doing very well in the no-17's no-19's no-41's [[47-limit]] if you forgive inconsistencies arising from its [[magic]]-tempered ~[[5/4]].

An alternative 7-limit 9th-octave temperament supported by more edos is to preserve the mapping of 7/6 but not that of 27/25, resulting in [[septiennealimmal]], with many extensions possible. An important edo of interest for this strategy is [[63edo]], a tuning doing very well in the no-17's no-19's no-41's [[47-limit]] if you forgive inconsistencies arising from its [[magic]]-tempered ~[[5/4]].

Some higher-limit interpretations of interest for both routes are [[14/13]]~[[13/12]] (tempering [[169/168|S13]]) for lower-complexity interpretations of 1\9 [[34/27]] for 1\3 (tempering [[19683/19652]] to give an interpretation to [[3edo]]) and the "rooted/harmonic wolf fifth" [[47/32]] for 5\9, by tempering ([[64/47]])/[[49/36|(7/6)<sup>2</sup>]] = [[2304/2303|S48]] = ([[48/47]])/([[49/48]]).

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 Therefore, one can consider it as interpreting [[9edo]] as a circle of [[7/6]]'s (corresponding to tempering the [[septimal ennealimma]]) and as a circle of [[27/25]]'s (corresponding to tempering the [[ennealimma]]), which is an equivalent description which implies tempering the [[landscape comma]] which makes [[63/50]] equal to exactly [[3edo|a third of an octave]].
-An alternative 7-limit 9th-octave temperament supported by more edos is to preserve the mapping of 7/6 but not that of 27/25, resulting in [[septiennealimmal]], with many extensions possible. Arguably the main edo of interest is [[63edo]], a tuning doing very well in the no-17's no-19's no-41's [[47-limit]] if you forgive inconsistencies arising from its [[magic]]-tempered ~[[5/4]].
+An alternative 7-limit 9th-octave temperament supported by more edos is to preserve the mapping of 7/6 but not that of 27/25, resulting in [[septiennealimmal]], with many extensions possible. An important edo of interest for this strategy is [[63edo]], a tuning doing very well in the no-17's no-19's no-41's [[47-limit]] if you forgive inconsistencies arising from its [[magic]]-tempered ~[[5/4]].
 Some higher-limit interpretations of interest for both routes are [[14/13]]~[[13/12]] (tempering [[169/168|S13]]) for lower-complexity interpretations of 1\9 [[34/27]] for 1\3 (tempering [[19683/19652]] to give an interpretation to [[3edo]]) and the "rooted/harmonic wolf fifth" [[47/32]] for 5\9, by tempering ([[64/47]])/[[49/36|(7/6)<sup>2</sup>]] = [[2304/2303|S48]] = ([[48/47]])/([[49/48]]).