Didacus: Difference between revisions

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It also has a simple extension to prime 11 - undecimal didacus, by tempering out [[176/175]], the valinorsma, so that (5/4)2 is equated to [[11/7]] and 9 generators stack to [[11/4]]; prime 13 can be found by tempering out [[640/637]], equating [[16/13]] to [[49/40]], and thereby putting the 13th harmonic 8 generators down. Beyond tridecimal didacus, further extensions to primes 17 and 19, known as roulette and mediantone, are also possible, sharing in common the interpretation of the generator as [[19/17]].

As for prime 3, while didacus has as a weak extension (among others) [[septimal meantone]], strong extensions that include 3 are rather complex. [[Hemithirds]] (25 & 31) tempers out [[1029/1024]] to find the fifth at [[3/2]] [[~]] ([[8/7]])3, and therefore the 3rd harmonic 15 generators down; and hemiwürschmidt (31 & 37) tempers out [[2401/2400]] so that ([[5/4]])8 is equated to [[6/1]], finding the 3rd harmonic 16 generators up (and as described for the page for 5-limit [[würschmidt]], there is also a free extension to find [[23/1]] at 28 generators). These two mappings intersect in 31edo, though the latter hews closer to the optimal range for undecimal didacus specifically.

As for prime 3, while didacus has as a weak extension (among others) [[septimal meantone]], strong extensions that include 3 are rather complex. Hemithirds (25 & 31) tempers out [[1029/1024]] to find the fifth at [[3/2]] [[~]] ([[8/7]])3, and therefore the 3rd harmonic 15 generators down; and hemiwürschmidt (31 & 37) tempers out [[2401/2400]] so that ([[5/4]])8 is equated to [[6/1]], finding the 3rd harmonic 16 generators up (and as described for the page for 5-limit [[würschmidt]], there is also a free extension to find [[23/1]] at 28 generators). These two mappings intersect in 31edo, though the latter hews closer to the optimal range for undecimal didacus specifically.

For technical data, see [[Hemimean clan #Didacus]].

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 It also has a simple extension to prime 11 - undecimal didacus, by tempering out [[176/175]], the valinorsma, so that (5/4)<sup>2</sup> is equated to [[11/7]] and 9 generators stack to [[11/4]]; prime 13 can be found by tempering out [[640/637]], equating [[16/13]] to [[49/40]], and thereby putting the 13th harmonic 8 generators down. Beyond tridecimal didacus, further extensions to primes 17 and 19, known as roulette and mediantone, are also possible, sharing in common the interpretation of the generator as [[19/17]].
-As for prime 3, while didacus has as a weak extension (among others) [[septimal meantone]], strong extensions that include 3 are rather complex. [[Hemithirds]] (25 & 31) tempers out [[1029/1024]] to find the fifth at [[3/2]] [[~]] ([[8/7]])<sup>3</sup>, and therefore the 3rd harmonic 15 generators down; and hemiwürschmidt (31 & 37) tempers out [[2401/2400]] so that ([[5/4]])<sup>8</sup> is equated to [[6/1]], finding the 3rd harmonic 16 generators up (and as described for the page for 5-limit [[würschmidt]], there is also a free extension to find [[23/1]] at 28 generators). These two mappings intersect in 31edo, though the latter hews closer to the optimal range for undecimal didacus specifically.
+As for prime 3, while didacus has as a weak extension (among others) [[septimal meantone]], strong extensions that include 3 are rather complex. Hemithirds (25 & 31) tempers out [[1029/1024]] to find the fifth at [[3/2]] [[~]] ([[8/7]])<sup>3</sup>, and therefore the 3rd harmonic 15 generators down; and hemiwürschmidt (31 & 37) tempers out [[2401/2400]] so that ([[5/4]])<sup>8</sup> is equated to [[6/1]], finding the 3rd harmonic 16 generators up (and as described for the page for 5-limit [[würschmidt]], there is also a free extension to find [[23/1]] at 28 generators). These two mappings intersect in 31edo, though the latter hews closer to the optimal range for undecimal didacus specifically.
 For technical data, see [[Hemimean clan #Didacus]].