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Carlos Gamma

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Carlos Gamma is a non-octave equal temperament whose step size is 35.0985422804 cents[1]. It was invented by Wendy Carlos.

English Wikipedia has an article on:

In this temperament, the interval of 20 steps approximates 3/2, that of 11 steps approximates 5/4, and that of 9 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]\displaystyle{ \frac{20^2 + 11^2 + 9^2}{20\log_2(3/2) + 11\log_2(5/4) + 9\log_2(6/5)} }[/math] ≃ 34.189454 equal steps and the fifth in 19.999549 equal steps of 35.098542 cents each. It is thus very close to the equal division of the just perfect fifth into twenty parts of 35.0978 cents each (20ed3/2), which has been oft-misquoted as actually being Wendy Carlos's gamma scale.

Carlos Gamma is closely related to the gammic temperament.

Lookalikes: 34edo, 54edt, 96ed7, 171edo

Intervals

The first steps up to two just perfect fifths should give a feeling of the granularity of this system…

Degrees ed233\420-5¢ ed31\54 ed121/81 (~ed11\19) ed696¢ (~69ed4!) ed32\55 ed700¢=r¢ ed3/2 Pyrite ed708¢ ed122/81 (~ed13\22) ed34\57 ed37\60+5¢
(~ed17\29) (~ed10\17)
1 33.04-33.29 34.44 34.74 34.8 34.91 35 35.1 35.18 35.29 35.4 35.45 35.79 37-37.25
2 66.07-66.57 68.89 69.48 69.6 69.82 70 70.2 70.36 70.59 70.8 70.91 71.58 74-74.5
3 99.11-99.86 103.33 104.22 104.4 104.73 105 105.29 105.545 105.88 106.2 106.36 107.37 111-111.75
4 132.14-133.14 137.78 138.96 139.2 139.64 140 140.39 140.72 141.17 141.6 141.81 143.16 148-149
5 165.18-166.43 172.22 173.7 174 174.545 175 175.49 175.91 176.46 177 177.27 178.95 185-186.25
6 198.21-199.71 206.67 208.445 208.8 209.455 210 210.59 211.09 211.76 212.4 212.72 214.74 222-223.5
7 231.25-233 241.11 243.19 243.6 244.36 245 245.68 246.27 247.05 247.8 248.17 250.53 259-260.75
8 264.29-266.29 275.56 277.93 278.4 279.27 280 280.78 281.45 282.34 283.2 283.63 286.32 296-298
9 297.32-299.57 310 312.67 313.2 314.18 315 315.88 316.64 317.635 318.6 319.08 322.105 333-335.25
10 330.35-332.86 344.44 347.41 348 349.09 350 350.98 351.82 352.93 354 354.53 357.895 370-372.5
11 363.39-366.14 378.89 382.15 382.8 384 385 386.075 387 388.22 389.4 389.99 393.68 407-409.75
12 396.43-399.43 413.33 416.89 417.6 418.91 420 421.17 422.18 423.51 424.8 425.44 429.47 444-447
13 429.46-432.71 447.78 451.63 452.4 453.82 455 456.27 457.36 458.81 460.2 460.89 465.26 481-484.25
14 462.5-466 482.22 486.37 487.2 488.73 490 491.37 492.545 494.1 495.6 496.345 501.05 518-521.5
15 495.54-499.29 516.67 521.11 522 523.64 525 526.47 527.73 529.39 531 531.8 536.84 555-558.75
16 528.57-532.57 551.11 555.85 556.8 558.545 560 561.56 562.91 564.68 566.4 567.25 572.63 592-596
17 561.61-565.86 585.56 590.59 591.6 593.455 595 596.66 598.09 599.98 601.8 602.705 608.42 629-633.25
18 594.64-599.14 620 625.33 626.4 628.36 630 631.76 633.27 635.27 637.2 638.16 644.21 666-670.5
19 627.68-632.43 654.44 660.075 661.2 663.27 665 666.86 668.45 670.56 672.6 673.61 680 703-707.75
20 660.71-665.71 688.89 694.82 696 698.18 700 701.955 703.64 705.86 708 709.065 715.79 740-745
21 693.75-699 723.33 729.56 732.8 733.09 735 737.05 738.82 741.15 743.4 744.52 751.58 777-782.25
22 726.79-732.29 757.78 764.3 765.6 768 770 772.15 774 776.44 778.8 779.97 787.37 814-819.5
23 759.82-765.57 792.22 799.04 800.4 802.91 805 807.25 809.18 811.73 813.2 815.425 823.16 851-856.75
24 792.86-798.86 826.67 833.78 835.2 837.82 840 842.35 844.36 847.03 848.6 850.88 858.95 888-894
25 825.89-832.14 861.11 868.52 870 872.73 875 877.44 879.545 882.32 885 886.33 894.74 925-931.25
26 858.93-865.43 895.6 903.26 904.8 907.64 910 912.54 914.73 917.61 920.4 921.78 930.53 962-968.5
27 891.96-898.71 930 938 939.6 942.545 945 947.64 949.91 952.905 955.8 957.24 966.32 999-1005.75
28 925-932 964.4 972.74 974.4 977.455 980 982.74 985.09 988.2 991.2 992.69 1002.105 1036-1043
29 958.04-965.29 998.89 1007.48 1009.2 1012.36 1015 1017.835 1020.27 1023.49 1026.6 1028.14 1039.895 1073-1081.25
30 991.07-998.57 1033.33 1042.22 1044 1047.27 1050 1052.93 1055.45 1058.78 1062 1063.6 1073.68 1110-1117.5
31 1024.11-1031.86 1067.78 1076.965 1078.8 1082.18 1085 1088.03 1090.64 1094.08 1097.4 1099.05 1109.47 1147-1154.75
32 1057.14-1065.14 1102.22 1111.705 1113.6 1117.09 1120 1123.13 1125.82 1129.37 1132.8 1134.5 1145.26 1184-1192
33 1090.18-1098.43 1136.67 1146.45 1148.4 1152 1155 1158.23 1161 1164.66 1168.2 1169.96 1181.05 1221-1229.25
34 1123.21-1131.71 1171.1 1181.19 1183.2 1186.91 1190 1193.32 1196.18 1199.95 1203.6 1205.41 1216.84 1258-1266.5
35 1156.25-1165 1205.56 1215.93 1218 1221.82 1225 1228.42 1231.36 1235.25 1239 1240.86 1252.63 1295-1303.75
36 1189.29-1198.29 1240 1250.67 1252.8 1256.73 1260 1263.52 1266.545 1270.54 1274.4 1276.32 1288.42 1332-1341
37 1222.32-1231.57 1274.44 1285.41 1287.6 1291.64 1295 1298.62 1301.73 1305.83 1309.8 1311.77 1324.42 1369-1377.25
38 1255.36-1263.86 1308.89 1320.15 1322.4 1326.545 1330 1333.715 1336.91 1341.125 1345.2 1347.22 1360 1406-1415.5
39 1288.39-1298.14 1343.33 1354.89 1357.2 1361.455 1365 1368.81 1372.09 1376.42 1380.6 1382.68 1395.79 1443-1452.75
40 1321.43-1331.43 1377.78 1389.6 1392 1396.36 1400 1403.91 1407.27 1411.71 1416 1418.13 1431.6 1480-1490

Compositions

See also

Reference

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

External links

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