Canopus: Difference between revisions

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Canopus is the rank two 3.5.7 temperament tempering out 16875/16807. Having a generator of ~7:5, it possesses non-trivial MOS of the families 1L 2s (triad), 3L 1s (tetrad), 3L 4s ("neutral" diatonic) and 3L 7s (augmented neutral decatonic). On either side the greater region where it appears, there appear the most important, though as yet unnamed, tritave-equivalent temperaments which retain twos, they being important for using a (smeary) ~4:3 or 3:2 as a generator.

Canopus is the [[Rank-2 temperament|rank two]] [[7-limit|3.5.7]] temperament [[Temper out|tempering]] out [[16875/16807]]. Having a generator of ~[[7/5|7:5]], it possesses non-trivial [[MOS]] of the families [[1L 2s]] (triad), [[3L 1s]] (tetrad), [[3L 4s]] ("neutral" [[diatonic]]) and [[3L 7s]] (augmented neutral decatonic). On either side the greater region where it appears, there appear the most important, though as yet unnamed, [[tritave]]-equivalent temperaments which retain twos, they being important for using a (smeary) ~4:3 or 3:2 as a generator.

The Sigma and Anti-Sigma (Mu) MOS families of 8L+3s and 3L+8s (unfair) or 4L+7s and 7L+4s (fair), but especially the unfair families which by definition include an interval for the function of an "ordinary" ~2:1, are good scales to know for the conceptualizations they provide of how an "ordinary" diatonic or anti-diatonic scale extends into a tritave equivalence (8L+3s being in fact the [[wikipedia:Obikhod|Obikhod]] pitch set used in Russian Orthodox chants and in fiction, most obviously speaking, the secondary traditional tuning of cylindrical wind instrument humans e. g. clarinets). These scales are neighbors of the 7&3 region where the 3L+7s Canopus decatonic scale appears. Below is a list of equal temperaments which contain these scales using ~~generators~~ between or 475.5 and 713.2 cents:

The Sigma and Anti-Sigma (Mu) MOS families of [[8L 3s|8L+3s]] and [[3L 8s|3L+8s]] (unfair) or [[4L 7s|4L+7s]] and [[7L 4s|7L+4s]] (fair), but especially the unfair families which by definition include an interval for the function of an "ordinary" ~[[2/1|2:1]], are good scales to know for the conceptualizations they provide of how an "ordinary" diatonic or anti-diatonic scale extends into a tritave equivalence (8L+3s being in fact the [[wikipedia:Obikhod|Obikhod]] pitch set used in Russian Orthodox chants and in fiction, most obviously speaking, the secondary traditional tuning of cylindrical wind instrument humans e. g. clarinets). These scales are neighbors of the 7&3 region where the 3L+7s Canopus decatonic scale appears. Below is a list of equal temperaments which contain these scales using [[generator]]s between or 475.5 and 713.2 [[cents]]:

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-Canopus is the rank two 3.5.7 temperament tempering out 16875/16807. Having a generator of ~7:5, it possesses non-trivial MOS of the families 1L 2s (triad), 3L 1s (tetrad), 3L 4s ("neutral" diatonic) and 3L 7s (augmented neutral decatonic). On either side the greater region where it appears, there appear the most important, though as yet unnamed, tritave-equivalent temperaments which retain twos, they being important for using a (smeary) ~4:3 or 3:2 as a generator.
+Canopus is the [[Rank-2 temperament|rank two]] [[7-limit|3.5.7]] temperament [[Temper out|tempering]] out [[16875/16807]]. Having a generator of ~[[7/5|7:5]], it possesses non-trivial [[MOS]] of the families [[1L 2s]] (triad), [[3L 1s]] (tetrad), [[3L 4s]] ("neutral" [[diatonic]]) and [[3L 7s]] (augmented neutral decatonic). On either side the greater region where it appears, there appear the most important, though as yet unnamed, [[tritave]]-equivalent temperaments which retain twos, they being important for using a (smeary) ~4:3 or 3:2 as a generator.
-The Sigma and Anti-Sigma (Mu) MOS families of 8L+3s and 3L+8s (unfair) or 4L+7s and 7L+4s (fair), but especially the unfair families which by definition include an interval for the function of an "ordinary" ~2:1, are good scales to know for the conceptualizations they provide of how an "ordinary" diatonic or anti-diatonic scale extends into a tritave equivalence (8L+3s being in fact the [[wikipedia:Obikhod|Obikhod]] pitch set used in Russian Orthodox chants and in fiction, most obviously speaking, the secondary traditional tuning of cylindrical wind instrument humans e. g. clarinets). These scales are neighbors of the 7&amp;3 region where the 3L+7s Canopus decatonic scale appears. Below is a list of equal temperaments which contain these scales using generators between or 475.5 and 713.2 cents:
+The Sigma and Anti-Sigma (Mu) MOS families of [[8L 3s|8L+3s]] and [[3L 8s|3L+8s]] (unfair) or [[4L 7s|4L+7s]] and [[7L 4s|7L+4s]] (fair), but especially the unfair families which by definition include an interval for the function of an "ordinary" ~[[2/1|2:1]], are good scales to know for the conceptualizations they provide of how an "ordinary" diatonic or anti-diatonic scale extends into a tritave equivalence (8L+3s being in fact the [[wikipedia:Obikhod|Obikhod]] pitch set used in Russian Orthodox chants and in fiction, most obviously speaking, the secondary traditional tuning of cylindrical wind instrument humans e. g. clarinets). These scales are neighbors of the 7&amp;3 region where the 3L+7s Canopus decatonic scale appears. Below is a list of equal temperaments which contain these scales using [[generator]]s between or 475.5 and 713.2 [[cents]]:
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