Cassandra triads: Difference between revisions

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The 13:11 and 14:11 dyads both reside near local [[Harmonic_Entropy|harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.

However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound ''great'' harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.

However, appearances can be deceiving. I ([[Mason Green]]) find that Cassandra triads sound ''great'' harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.

There is an ambiguity in how we compute harmonic entropy for triads, depending on whether we use utonal or otonal approximations. The common major triad is otonally a 4:5:6 and utonally a 1 / (10:12:15). Because the otonal form is simpler, it dominates over the utonal form. For the minor triad, the situation is reversed; the minor triad is primarily a utonal 1 / (6:5:4) and secondarily an otonal 10:12:15.

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 The 13:11 and 14:11 dyads both reside near local [[Harmonic_Entropy|harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.
-However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound ''great'' harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.
+However, appearances can be deceiving. I ([[Mason Green]]) find that Cassandra triads sound ''great'' harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.
 There is an ambiguity in how we compute harmonic entropy for triads, depending on whether we use utonal or otonal approximations. The common major triad is otonally a 4:5:6 and utonally a 1 / (10:12:15). Because the otonal form is simpler, it dominates over the utonal form. For the minor triad, the situation is reversed; the minor triad is primarily a utonal 1 / (6:5:4) and secondarily an otonal 10:12:15.