106edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 105edo106edo107edo →
Prime factorization 2 × 53
Step size 11.3208¢ 
Fifth 62\106 (701.887¢) (→31\53)
Semitones (A1:m2) 10:8 (113.2¢ : 90.57¢)
Consistency limit 5
Distinct consistency limit 5

106 equal divisions of the octave (abbreviated 106edo or 106ed2), also called 106-tone equal temperament (106tet) or 106 equal temperament (106et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 106 equal parts of about 11.3 ¢ each. Each step represents a frequency ratio of 21/106, or the 106th root of 2.

Theory

Since 106 = 2 × 53, 106edo is closely related to 53edo, and is contorted through the 7-limit, tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the 5-limit, 3125/3087, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports spectacle temperament and borwell temperament.

The division is notable for the fact that it is related to the turkish cent, or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the relative cent division for 106edo. Conversely, it makes the Pythagorean relative cent (or pion, symbol π¢, πr¢), which most closely approximates equally dividing an exact 3/2.

Prime harmonics

Approximation of prime harmonics in 106edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53
Error Absolute (¢) +0.00 -0.07 -1.41 +4.76 +3.40 -2.79 -3.07 -3.17 -5.63 +0.61 -1.64 -2.29 +1.13 -2.08 +2.42 -1.81
Relative (%) +0.0 -0.6 -12.4 +42.0 +30.0 -24.7 -27.1 -28.0 -49.8 +5.4 -14.5 -20.2 +9.9 -18.4 +21.4 -16.0
Steps
(reduced) 		106
(0) 		168
(62) 		246
(34) 		298
(86) 		367
(49) 		392
(74) 		433
(9) 		450
(26) 		479
(55) 		515
(91) 		525
(101) 		552
(22) 		568
(38) 		575
(45) 		589
(59) 		607
(77)

53edo for comparison:

Approximation of prime harmonics in 53edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53
Error Absolute (¢) +0.00 -0.07 -1.41 +4.76 -7.92 -2.79 +8.25 -3.17 +5.69 -10.71 +9.68 -2.29 +1.13 +9.24 -8.90 +9.51
Relative (%) +0.0 -0.3 -6.2 +21.0 -35.0 -12.3 +36.4 -14.0 +25.1 -47.3 +42.8 -10.1 +5.0 +40.8 -39.3 +42.0
Steps
(reduced) 		53
(0) 		84
(31) 		123
(17) 		149
(43) 		183
(24) 		196
(37) 		217
(5) 		225
(13) 		240
(28) 		257
(45) 		263
(51) 		276
(11) 		284
(19) 		288
(23) 		294
(29) 		304
(39)

Intervals

Steps Cents Approximate Ratios Ups and Downs Notation
0 0 1/1 D
1 11.321 ^D, v7E♭
2 22.642 ^^D, v6E♭
3 33.962 ^3D, v5E♭
4 45.283 37/36, 38/37, 39/38, 40/39 ^4D, v4E♭
5 56.604 30/29, 31/30, 32/31 ^5D, v3E♭
6 67.925 26/25 ^6D, vvE♭
7 79.245 22/21, 45/43 ^7D, vE♭
8 90.566 20/19, 39/37 ^8D, E♭
9 101.887 35/33 ^9D, v9E
10 113.208 16/15, 31/29 D♯, v8E
11 124.528 29/27, 43/40, 44/41 ^D♯, v7E
12 135.849 40/37 ^^D♯, v6E
13 147.17 37/34 ^3D♯, v5E
14 158.491 34/31 ^4D♯, v4E
15 169.811 32/29, 43/39 ^5D♯, v3E
16 181.132 10/9 ^6D♯, vvE
17 192.453 19/17 ^7D♯, vE
18 203.774 9/8 E
19 215.094 17/15, 43/38 ^E, v7F
20 226.415 41/36 ^^E, v6F
21 237.736 31/27, 39/34 ^3E, v5F
22 249.057 15/13, 37/32 ^4E, v4F
23 260.377 36/31, 43/37 ^5E, v3F
24 271.698 41/35 ^6E, vvF
25 283.019 20/17, 33/28 ^7E, vF
26 294.34 32/27, 45/38 F
27 305.66 31/26, 37/31, 43/36 ^F, v7G♭
28 316.981 6/5 ^^F, v6G♭
29 328.302 29/24 ^3F, v5G♭
30 339.623 45/37 ^4F, v4G♭
31 350.943 38/31 ^5F, v3G♭
32 362.264 37/30 ^6F, vvG♭
33 373.585 31/25, 36/29, 41/33 ^7F, vG♭
34 384.906 5/4 ^8F, G♭
35 396.226 39/31, 44/35 ^9F, v9G
36 407.547 19/15, 43/34 F♯, v8G
37 418.868 14/11 ^F♯, v7G
38 430.189 41/32 ^^F♯, v6G
39 441.509 31/24, 40/31 ^3F♯, v5G
40 452.83 13/10 ^4F♯, v4G
41 464.151 17/13 ^5F♯, v3G
42 475.472 25/19 ^6F♯, vvG
43 486.792 45/34 ^7F♯, vG
44 498.113 4/3 G
45 509.434 43/32 ^G, v7A♭
46 520.755 27/20 ^^G, v6A♭
47 532.075 34/25 ^3G, v5A♭
48 543.396 26/19, 37/27 ^4G, v4A♭
49 554.717 40/29 ^5G, v3A♭
50 566.038 43/31 ^6G, vvA♭
51 577.358 ^7G, vA♭
52 588.679 45/32 ^8G, A♭
53 600 41/29 ^9G, v9A
54 611.321 37/26 G♯, v8A
55 622.642 43/30 ^G♯, v7A
56 633.962 ^^G♯, v6A
57 645.283 29/20, 45/31 ^3G♯, v5A
58 656.604 19/13 ^4G♯, v4A
59 667.925 25/17 ^5G♯, v3A
60 679.245 37/25, 40/27 ^6G♯, vvA
61 690.566 ^7G♯, vA
62 701.887 3/2 A
63 713.208 ^A, v7B♭
64 724.528 38/25, 41/27 ^^A, v6B♭
65 735.849 26/17 ^3A, v5B♭
66 747.17 20/13, 37/24 ^4A, v4B♭
67 758.491 31/20, 45/29 ^5A, v3B♭
68 769.811 39/25 ^6A, vvB♭
69 781.132 11/7 ^7A, vB♭
70 792.453 30/19 ^8A, B♭
71 803.774 35/22, 43/27 ^9A, v9B
72 815.094 8/5 A♯, v8B
73 826.415 29/18 ^A♯, v7B
74 837.736 ^^A♯, v6B
75 849.057 31/19 ^3A♯, v5B
76 860.377 ^4A♯, v4B
77 871.698 43/26 ^5A♯, v3B
78 883.019 5/3 ^6A♯, vvB
79 894.34 ^7A♯, vB
80 905.66 27/16 B
81 916.981 17/10 ^B, v7C
82 928.302 41/24 ^^B, v6C
83 939.623 31/18, 43/25 ^3B, v5C
84 950.943 26/15, 45/26 ^4B, v4C
85 962.264 ^5B, v3C
86 973.585 ^6B, vvC
87 984.906 30/17 ^7B, vC
88 996.226 16/9 C
89 1007.547 34/19, 43/24 ^C, v7D♭
90 1018.868 9/5 ^^C, v6D♭
91 1030.189 29/16 ^3C, v5D♭
92 1041.509 31/17 ^4C, v4D♭
93 1052.83 ^5C, v3D♭
94 1064.151 37/20 ^6C, vvD♭
95 1075.472 41/22 ^7C, vD♭
96 1086.792 15/8 ^8C, D♭
97 1098.113 ^9C, v9D
98 1109.434 19/10 C♯, v8D
99 1120.755 21/11 ^C♯, v7D
100 1132.075 25/13 ^^C♯, v6D
101 1143.396 29/15, 31/16 ^3C♯, v5D
102 1154.717 37/19, 39/20 ^4C♯, v4D
103 1166.038 ^5C♯, v3D
104 1177.358 ^6C♯, vvD
105 1188.679 ^7C♯, vD
106 1200 2/1 D

See also

Artists using 106 et:

Retrieved from "https://en.xen.wiki/index.php?title=106edo&oldid=132896"