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Here is a temperament; I haven't mastered the notation; how should it be denoted

🔗tomhchappell <tomhchappell@yahoo.com>

10/22/2009 12:34:36 PM

Here is an even temperament, fairly close to just.
I haven't mastered the notation; how should this temperament be denoted?
2^840 is about 13^227.
7^153 is about 3^271.
2^128 is about 11^37.
5^295 is about 11^198.
5^157 is about 3^230.
3^49 is about 17^19.
2^485 is about 3^306.
5^146 is about 2^339.
7^109 is about 2^306.
7^287 is about 5^347.
13^242 is about 3^565.
17^217 is about 5^382.
2^327 is about 17^80.
11^219 is about 3^478.
7^29 is about 13^22.
5^51 is about 13^32.
11^261 is about 13^244.
7^182 is about 17^125.
7^313 is about 11^254.
17^153 is about 13^169.
11^13 is about 17^11.

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Here is another even temperament, somewhat less just.
How should I denote it?
13^10 is about 2^37.
13^3 is about 3^7.
11^13 is about 17^11.
11^11 is about 3^24.
3^12 is about 2^19.
2^45 is about 11^13.
2^7 is about 5^3.
7^5 is about 2^14.
2^45 is about 17^11.
3^22 is about 5^15.
7^13 is about 3^23.
17^11 is about 7^16.
5^3 is about 11^2.
11^13 is about 7^16.
5^8 is about 13^5.
17^7 is about 3^18.

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The first temperament requires the octave to be divided into
LCM(37, 80, 109, 146, 227, 306) or (2^4)*(3^2)*5*17*37*73*109*227 (that is, 818,009,518,320) increments,
and/or requires the twelfth to be divided into
LCM(19, 153, 157, 219, 242, 485) or 2*(3^2)*5*(11^2)*17*19*73*97*157 (that is, 3,910,431,195,990) increments.

The second temperament requires the octave to be divided into
LCM(3, 5, 10, 11, 12, 13) or (2^2)*3*5*11*13 (that is, 8,580) increments,
and/or requires the twelfth to be divided into
LCM(3, 7, 11, 13, 15, 19) or 3*5*7*11*13*19 (that is, 285,285) increments.