Cassandra triads

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The cassandra triads are triads in which the two step sizes are (tempered versions of) 13:11 and 14:11 and the total span is a tempered perfect fifth. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.

This tempers out the 364:363, which is one of the pentacircle commas. Cassandra triads appear in 17edo, and even more closely in 29edo. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad could be considered to function as both rootsubminor and as rootminor, and similarly the Cassandra major chord functions as both rootmajor and rootsupermajor.

Consonance of Cassandra triads

The 13:11 and 14:11 dyads both reside near local 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.

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.

The subminor triad is primarily an otonal 6:7:9 and secondarily an otonal 1 / (14:18:21). The supermajor triad is primarily an otonal 14:18:21 and secondarily a utonal 1 / (6:7:9).

The consonance of otonal chords comes primarily, I believe, from the fact that they share a common fundamental frequency; by the same token, utonal chords share a common overtone. Because these are two different mechanisms, it may be a good idea when calculating harmonic entropy to evaluate the two components separately.

This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 and as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone and fundamental.

This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive to me.

In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-patent val approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new comma pump options. Also, it means that 12edo melodies and chord progressions can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the patent minor and major triads.