Carlos Alpha: Difference between revisions

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→Intervals: Expand; reorder; formatting
Line 26: Line 26:
|-
|-
! Degrees
! Degrees
! Cents ~ Cents octave-reduced
! Cents<br>~(octave-reduced)
! Approximate JI interval (11-limit)
! Approximate JI interval (11-limit)
|-
|-
| 0
| 0
| 0
| 0
| | 1/1 (exact)
| 1/1
|-
|-
| | 1
| 1
| | 77.965
| 77.965
| |25/24, 22/21, 21/20
| 25/24, 21/20, 22/21
|-
|-
| | 2
| 2
| | 155.930
| 155.930
| | 12/11, 35/32, 11/10
| 11/10, 12/11, 35/32
|-
|-
| | 3
| 3
| | 233.895
| 233.895
| | 8/7, 55/48
| 8/7, 55/48
|-
|-
| | 4
| 4
| | 311.860
| 311.860
| | 6/5
| 6/5
|-
|-
| | 5
| 5
| | 389.825
| 389.825
| | 5/4
| 5/4
|-
|-
| | 6
| 6
| | 467.790
| 467.790
| | 72/55, 21/16
| 21/16, 72/55
|-
|-
| | 7
| 7
| | 545.755
| 545.755
| | 15/11, 48/35, 11/8
| 11/8, 15/11, 48/35
|-
|-
| | 8
| 8
| | 623.720
| 623.720
| | 10/7, 63/44, 36/25
| 10/7, 36/25, 63/44
|-
|-
| | 9
| 9
| | 701.685
| 701.685
| | 3/2
| 3/2
|-
|-
| | 10
| 10
| | 779.650
| 779.650
| | 25/16, 11/7, 63/40
| 11/7, 25/16, 63/40
|-
|-
| | 11
| 11
| | 857.615
| 857.615
| | 18/11, 105/64, 33/20
| 18/11, 33/20, 105/64
|-
|-
| | 12
| 12
| | 935.580
| 935.580
| | 12/7, 55/32
| 12/7, 55/32
|-
|-
| | 13
| 13
| | 1013.545
| 1013.545
| | 9/5
| 9/5
|-
|-
| | 14
| 14
| | 1091.510
| 1091.510
| | 15/8
| 15/8
|-
|-
| | 15
| 15
| | 1169.475
| 1169.475
| | 108/55, 63/32
| 63/32, 108/55
|-
|-
| | 16
| 16
| | 1247.440 ~ 47.440
| 1247.440 ~ 47.440
| | 45/22, 72/35, 33/16
| 72/35, 33/16, 45/22
|-
|-
| | 17
| 17
| | 1325.4045 ~ 125.405
| 1325.4045 ~ 125.405
| | 15/7, 189/88, 54/25
| 15/7, 54/25, 189/88
|-
|-
| | 18
| 18
| | 1403.370 ~ 203.370
| 1403.370 ~ 203.370
| | 9/4
| 9/4
|-
|-
| | 19
| 19
| | 1481.335 ~ 281.335
| 1481.335 ~ 281.335
|
| 33/14
|-
|-
| | 20
| 20
| | 1559.300 ~ 359.300
| 1559.300 ~ 359.300
|  
| 27/11 ([[Valentino]] 32/13)
|-
|-
| | 21
| 21
| | 1637.265 ~ 437.265
| 1637.265 ~ 437.265
|
| 18/7
|-
|-
| | 22
| 22
| | 1715.230 ~ 515.230
| 1715.230 ~ 515.230
|
| 27/20
|-
|-
| | 23
| 23
| | 1793.195 ~ 593.195
| 1793.195 ~ 593.195
|
| 45/32
|-
|-
| | 24
| 24
| | 1871.160 ~ 671.160
| 1871.160 ~ 671.160
|
|  
|-
|-
| | 25
| 25
| | 1949.125 ~ 749.125
| 1949.125 ~ 749.125
|
|
|-
|-
| | 26
| 26
| | 2027.090 ~ 827.090
| 2027.090 ~ 827.090
|
|
|-
|-
| | 27
| 27
| | 2105.055 ~ 905.055
| 2105.055 ~ 905.055
|
|
|-
|-
| | 28
| 28
| | 2183.020 ~ 983.020
| 2183.020 ~ 983.020
|
|
|-
|-
| | 29
| 29
| | 2260.985 ~ 1060.985
| 2260.985 ~ 1060.985
|
|
|-
|-
| | 30
| 30
| | 2330.950 ~ 1138.950
| 2330.950 ~ 1138.950
|
|
|-
|-
| | 31
| 31
| | 2416.915 ~ 16.915
| 2416.915 ~ 16.915
|
|
|-
|-
| | 32
| 32
| | 2494.880 ~ 94.880
| 2494.880 ~ 94.880
|
|
|-
|-
| | 33
| 33
| | 2572.845 ~ 172.845
| 2572.845 ~ 172.845
|
|
|-
|-
| | 34
| 34
| | 2650.810 ~ 250.810
| 2650.810 ~ 250.810
|
|
|-
|-
| | 35
| 35
| | 2728.775 ~ 328.775
| 2728.775 ~ 328.775
|
|
|-
|-
| | 36
| 36
| | 2806.740 ~ 406.740
| 2806.740 ~ 406.740
|
|
|}
|}

Revision as of 19:31, 20 March 2026

English Wikipedia has an article on:

Carlos Alpha is a non-octave equal temperament invented by Wendy Carlos, with step size about 77.965 ¢. 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. The scale can be used with (i.e. 15L 1s) or without octaves.

Theory

Carlos provided the tuning of 78.0 ¢[1][2]. Based on her work, Dave Benson optimized the temperament for 3/2, 5/4 and 6/5, such that the tuning divides the octave in

[math]\displaystyle{ \displaystyle \frac{9^2 + 5^2 + 4^2}{9\log_2(3/2) + 5\log_2(5/4) + 4\log_2(6/5)} ≃ 15.391524 }[/math]

equal steps and the fifth in 9.003464 equal steps of 77.964990 ¢ each[3]. It is thus very close to the equal division of the just perfect fifth into nine parts of 77.995 ¢ each (9edf), 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
~(octave-reduced) 		Approximate JI interval (11-limit)
0 0 1/1
1 77.965 25/24, 21/20, 22/21
2 155.930 11/10, 12/11, 35/32
3 233.895 8/7, 55/48
4 311.860 6/5
5 389.825 5/4
6 467.790 21/16, 72/55
7 545.755 11/8, 15/11, 48/35
8 623.720 10/7, 36/25, 63/44
9 701.685 3/2
10 779.650 11/7, 25/16, 63/40
11 857.615 18/11, 33/20, 105/64
12 935.580 12/7, 55/32
13 1013.545 9/5
14 1091.510 15/8
15 1169.475 63/32, 108/55
16 1247.440 ~ 47.440 72/35, 33/16, 45/22
17 1325.4045 ~ 125.405 15/7, 54/25, 189/88
18 1403.370 ~ 203.370 9/4
19 1481.335 ~ 281.335 33/14
20 1559.300 ~ 359.300 27/11 (Valentino 32/13)
21 1637.265 ~ 437.265 18/7
22 1715.230 ~ 515.230 27/20
23 1793.195 ~ 593.195 45/32
24 1871.160 ~ 671.160
25 1949.125 ~ 749.125
26 2027.090 ~ 827.090
27 2105.055 ~ 905.055
28 2183.020 ~ 983.020
29 2260.985 ~ 1060.985
30 2330.950 ~ 1138.950
31 2416.915 ~ 16.915
32 2494.880 ~ 94.880
33 2572.845 ~ 172.845
34 2650.810 ~ 250.810
35 2728.775 ~ 328.775
36 2806.740 ~ 406.740

Music

See also: Category:Carlos Alpha tracks
Bazil Müzik
Bryan Deister
Mandrake
  • HUH? (2022) – in 9edf tuning
Omega9
Jean-Pierre Poulin
Carlo Serafini
Joel Grant Taylor
MagentaFlavouredMusic

See also

References

  1. ↑ Wendy Carlos, "Tuning: At the Crossroads", Computer Music Journal, vol. 11 no. 1, 1987, pp. 29-43
  2. ↑ Wendy Carlos, Three Asymmetric Divisions of the Octave.
  3. ↑ Dave Benson, Music: A Mathematical Offering, pp. 232-233.

External links

Retrieved from "https://en.xen.wiki/index.php?title=Carlos_Alpha&oldid=226532"