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Can't understand this temperament
Okay, I've been trying to figure this out because it sounds weird and interesting, but it's not making sense.
I'm going off of this:
"...the 8:9:10:(12) chord as exactly analogous to the 4:5:6:(8) chord in meantone..."
Now, in meantone, the "third" 5/4 and the "fifth" 3/2 are related in this way: (3/2)^4/((5/4)*(2/1)^2) = 81/80 -> 1/1.
If you're telling me that 8:9:10 chords should do the "exactly analogous" thing, that makes me think that (5/4)^4/((9/8)*(3/2)^2) should be the comma that is tempered out. That's 625/648, the same comma as diminished.
This is all well and good and makes sense as a 5-limit temperament with 3/2 as the period (and either 6/5 or 5/4 as a generator). However, the MOS series of this temperament goes:
2edf, 3edf, 5edf, 7edf, 12edf, 19edf, 26edf, 33edf...
So your 9+11=20 scale doesn't pop out of the temperament I found at all. There must be some mistake somewhere?
What is the comma tempered out of your scale?
- keenanpepper December 26, 2011, 09:10:33 PM UTC-0800
Just going by the MOS numbers (9,11,20) it seems like you might be talking about the temperament with the comma 15625/15552 (period 3/2, generator 6/5 or 5/4), but I have no idea how to interpret the "exactly analogous" statement.
- keenanpepper December 26, 2011, 10:35:42 PM UTC-0800
Ha! My fault, it isn't "exactly". I wasn't thinking of how many 5/4 tempered flat end up with 9/8 just that they do. Sorry!
Good stuff with the edf/edo correspondences.
- Kosmorsky December 27, 2011, 01:14:37 AM UTC-0800
Oh, ok. So you meant that 5/4 is the generator and a bunch of them (not necessarily 4), reduced by 3/2, gives you 9/8.
So this is equivalent to the 15625/15552 temperament then, right?
(And the 648/625 temperament is something different I just accidentally discovered...)
- keenanpepper December 27, 2011, 06:58:19 AM UTC-0800
15625/15552 it is, indeed. Should be 6 of them instead of 4.
- Kosmorsky December 27, 2011, 11:49:02 AM UTC-0800
That's cool man. Totally different musical perspective on what's mathematically "the same" as hanson temperament.
- keenanpepper December 27, 2011, 09:14:44 PM UTC-0800