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Keenan, this is an interesting idea, and I'm curious to see you expand on it.
I'm confused about your statement, For example, the list of amity "thirds" includes ...6/5 (339.5) (363.2) 5/4... where the intervals given in cents are not representable as simple JI intervals (243/200 and 100/81 are about as simple as you can get).
But in the 13-limit, 339.5 is 11/9 and 363.2 is 16/13, which are relatively simple. So I wonder if it would be fair to call amity a cluster temperament in the 13-limit, but not in a lower one? I understand it's a fuzzy definition right now, so maybe there are other reasons to not count amity that I don't understand.
- Andrew_Heathwaite June 20, 2012, 07:37:53 AM UTC-0700
Oh, I guess I missed this back in July. Right you are, Andrew. I guess you'd call that "184.108.40.206.13 hitchcock" temperament. (It doesn't have the same mapping of 11 as 11-limit amity.)
- keenanpepper November 24, 2012, 09:13:41 AM UTC-0800
Hey Keenan, just saw this page, very interesting! Would you say that the schismatic temperament is a cluster temperament? If so, would 10/9 and 9/8 be a cluster, also 32/27 and 6/5, etc? This seems like an example of "clustering" that's been done subconsciously in the West for centuries.
Some thoughts on firming things up mathematically: a cluster temperament can be found by taking two commas and tempering out the difference between them. In your slendric example, it's 49/48 and 64/63. In general, it's possible to combine these two commas to get a 3-limit comma via hermite reduction. This gives us 256/243, which implies quasi-5-edo, hence 5 clusters per octave, and pentatonic scales. The 3-limit comma is the sum of 3 7-limit commas, so there are 3 chromas from 9/8 to 32/27.
I think you could take any comma of the form 2^a·3^b·P^c, with c = ±1, like 81/80 or 64/63 or 33/32, and equate some number of them to any 3-limit comma, and get a cluster temperament. Equating 64/63 and the pythagorean comma implies 12-note scales, and makes 9/8 cluster with 8/7. The amity comma equates the apotome to 5 syntonic commas, hence 7 clusters = heptatonic. Each cluster contains 5 steps between 3-limit ratios, hence 6 ratios: 32/27 - 6/5 - 11/9 - 16/13 - 5/4 - 81/64.
- TallKite January 24, 2018, 06:34:40 PM UTC-0800