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A 13-limit "pseudo-Pythagorean" scale

🔗Danny Wier <dawiertx@sbcglobal.net>

2/4/2007 1:03:29 PM

Good news. I'm finally writing music again.

One of the compositions I'm working on is something in F minor (and mostly 13/8 time). It's not at all hardcore microtonal and sounds just fine in 12-TET. I retuned it to traditional JI (with minor intervals of 16/15, 6/5, 8/5 and 9/5), and didn't like it. I got a "sad" minor but I wanted an "angry" minor. I really needed Pythagorean or something like it, or just leave it in equal.

I also tried a subset of the 15-limit square I've been using a lot lately, plus 21/20 for the minor second, which turned out to be a decent enough Pythagorean-type tuning for a fast-tempo piece. Given 1/1 is F:

0: 1/1 = 0.0000 cents
1: 21/20 = 84.4672
2: 9/8 = 203.9100
3: 13/11 = 289.2097
4: 5/4 = 386.3137
5: 4/3 = 498.0450
6: 7/5 = 582.5122
7: 3/2 = 701.9550
8: 11/7 = 782.4920
9: 5/3 = 884.3587
10: 16/9 = 996.0900
11: 15/8 = 1088.2687
12: 2/1 = 1200.0000

The 40/27 wolf is between the major second (here G) and the major sixth (D natural), and I don't have that fifth in the work yet.

Besides the normal 3/2 fifth, there are some other deviant fifths, all fairly acceptable:

B (Cb)-Gb: 112/75 = 694.2435 (a septimal kleisma flat of pure)
Db-Ab: 220/147 = 698.0248
Ab-Eb: 182/121 = 706.7177
Eb-Bb: 176/117 = 706.8803

I wouldn't necessarily recommend this tuning for a slow, harmonic piece, since the beats in some of these fifths might be a little rough.

~D.