106edo

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

See also

Artists using 106 et: