Carlos Alpha

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Carlos Alpha is a non-octave equal temperament with step size about 77.965 ¢ in the standard tuning[1]. It was invented by Wendy Carlos.

In this temperament, the interval of 9 steps approximates 3/2, that of 5 steps approximates 5/4, and that of 4 steps approximates 6/5.

Theory

Carlos optimized the tuning on 3/2, 5/4 and 6/5, such that the tuning divides the octave in [math]\frac{9^2 + 5^2 + 4^2}{9\log_2(3/2) + 5\log_2(5/4) + 4\log_2(6/5)}[/math] ≃ 15.391524 equal steps and the fifth in 9.003464 equal steps of 77.964990 cents each. It is thus very close to the equal division of the just perfect fifth into nine parts of 77.995 cents each (9ed3/2), corresponding to 15.3856edo.

Carlos Alpha is very closely related to valentine temperament. For a list of 11-limit dyadic chords of Carlos Alpha, see Chords of valentine.

It has a very good representation of the 7:8:10:11:12 chord from the harmonic series.

Sequence of EDOs with lower and lower relative error: 15, 31, 46, 77, 354

Intervals

Degrees Cents ~ Cents octave-reduced Approximate JI interval (11-limit)
0 0 1/1 (exact)
1 78 25/24, 22/21, 21/20
2 156 12/11, 35/32, 11/10
3 234 8/7, 55/48
4 312 6/5
5 390 5/4
6 468 72/55, 21/16
7 546 15/11, 48/35, 11/8
8 624 10/7, 63/44, 36/25
9 702 3/2 (exact)
10 780 25/16, 11/7, 63/40
11 858 18/11, 105/64, 33/20
12 936 12/7, 55/32
13 1014 9/5
14 1092 15/8
15 1170 108/55, 63/32
16 1248 ~ 48 45/22, 72/35, 33/16
17 1326 ~ 126 15/7, 189/88, 54/25
18 1404 ~ 204 9/4 (exact)
19 1482 ~ 282
20 1560 ~ 360
21 1638 ~ 438
22 1716 ~ 516
23 1794 ~ 594
24 1872 ~ 672
25 1950 ~ 750
26 2028 ~ 828
27 2106 ~ 906
28 2184 ~ 984
29 2262 ~ 1062
30 2340 ~ 1140
31 2418 ~ 18
32 2496 ~ 96
33 2574 ~ 174
34 2652 ~ 252
35 2730 ~ 330
36 2808 ~ 408

Music

See also: Category:Carlos Alpha tracks

See also

References

  1. Wendy Carlos, "Tuning: At the Crossroads", Computer Music Journal vol. 11 no. 1, 1987, pp. 29-43

External links