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13-limit meanpop

🔗Gene Ward Smith <gwsmith@svpal.org>

8/20/2004 3:49:58 PM

This can be described as 13-limit 50&81:

13-limit meanpop

name: meanpop, meantone, 50&81
wedgie: <<1 4 10 -13 15 4 13 -24 20 12 -44 20 -71 5 100||
mapping: [<1 2 4 7 -2 10|, <0 -1 -4 -10 13 -15|]
13-limit poptimal generator: 105/181
TOP period: 1201.851 generator: 504.734
TM basis: {81/80, 105/104, 126/125, 144/143}
comma sequence: 81/80, 126/125, 385/384, 105/104
MOS: 5, 7, 12, 19, 31, 50, 81, 131, 181

This is the version of 13-limit meantone with a ratwolf fifth of
20/13; if we look at the 13-limit tonality diamond, we get something
for everything in the range +-11 except for +-7, and going to the
15-limit doesn't help there.

-11 {20/13}
-10 {8/7}
-9 {12/7}
-8 {9/7}
-7 {}
-6 {10/7}
-5 {14/13}
-4 {8/5}
-3 {6/5}
-2 {9/5, 16/9}
-1 {4/3}
0 {1}
1 {3/2}
2 {9/8, 10/9}
3 {5/3}
4 {5/4}
5 {13/7}
6 {7/5}
7 {}
8 {14/9}
9 {7/6}
10 {7/4}
11 {13/10}

While 20/13 may not seem like much of a consonance, 13/10 and 13/5
might be more inspiring. 181-equal gives a 20/13 flat by 3.245 cents;
of course getting nearer to the ratwolf fifth will improve on this;
the 50-equal 20/13 is 1.786 cents flat.