Carlos Alpha

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Alpha scale

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	77.964990	25/24, 22/21, 21/20
2	155.929979	12/11, 35/32, 11/10
3	233.894969	8/7, 55/48
4	311.859958	6/5
5	389.824948	5/4
6	467.789937	72/55, 21/16
7	545.754927	15/11, 48/35, 11/8
8	623.719916	10/7, 63/44, 36/25
9	701.684906	3/2
10	779.6498955	25/16, 11/7, 63/40
11	857.614885	18/11, 105/64, 33/20
12	935.579875	12/7, 55/32
13	1013.544864	9/5
14	1091.509854	15/8
15	1169.474843	108/55, 63/32
16	1247.439833 ~ 47.439833	45/22, 72/35, 33/16
17	1325.404822 ~ 125.404822	15/7, 189/88, 54/25
18	1403.369812 ~ 203.369812	9/4
19	1481.3348015 ~ 281.3348015
20	1559.299791 ~ 359.299791
21	1637.264781 ~ 437.264781
22	1715.229770 ~ 515.229770
23	1793.194760 ~ 593.194769
24	1871.159749 ~ 671.159749
25	1949.124739 ~ 749.124739
26	2027.089728 ~ 827.089728
27	2105.054718 ~ 905.054718
28	2183.019707 ~ 983.019707
29	2260.984697 ~ 1060.984697
30	2330.949687 ~ 1138.949687
31	2416.914676 ~ 16.914676
32	2494.879666 ~ 94.879666
33	2572.844655 ~ 172.844655
34	2650.809645 ~ 250.809645
35	2728.774635 ~ 328.774634
36	2806.739624 ~ 406.739624