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¢, 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 -1 -12 +42 +30 -25 -27 -28 -50 +5 -14 -20 +10 -18 +21 -16
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 -6 +21 -35 -12 +36 -14 +25 -47 +43 -10 +5 +41 -39 +42
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 Ups and downs notation Approximate ratios
0 0 D 1/1
1 11.3208 ↑D, ↓7E♭
2 22.6415 ↑↑D, ↓6E♭ 65/64, 81/80
3 33.9623 ↑3D, ↓5E♭ 55/54, 56/55
4 45.283 ↑4D, ↓4E♭ 36/35, 40/39
5 56.6038 ↑5D, ↓3E♭ 33/32
6 67.9245 ↑6D, ↓↓E♭ 25/24, 26/25, 27/26
7 79.2453 ↑7D, ↓E♭ 22/21
8 90.566 ↑8D, E♭
9 101.887 ↑9D, ↓9E 35/33
10 113.208 D♯, ↓8E 16/15
11 124.528 ↑D♯, ↓7E
12 135.849 ↑↑D♯, ↓6E 13/12, 27/25
13 147.17 ↑3D♯, ↓5E 12/11
14 158.491 ↑4D♯, ↓4E 35/32
15 169.811 ↑5D♯, ↓3E
16 181.132 ↑6D♯, ↓↓E 10/9, 72/65
17 192.453 ↑7D♯, ↓E
18 203.774 E 9/8
19 215.094 ↑E, ↓7F
20 226.415 ↑↑E, ↓6F
21 237.736 ↑3E, ↓5F 55/48, 63/55
22 249.057 ↑4E, ↓4F 15/13, 52/45, 81/70
23 260.377 ↑5E, ↓3F 64/55
24 271.698 ↑6E, ↓↓F 75/64
25 283.019 ↑7E, ↓F 33/28
26 294.34 F 32/27
27 305.66 ↑F, ↓7G♭
28 316.981 ↑↑F, ↓6G♭ 6/5, 65/54
29 328.302 ↑3F, ↓5G♭
30 339.623 ↑4F, ↓4G♭ 39/32
31 350.943 ↑5F, ↓3G♭ 11/9, 27/22
32 362.264 ↑6F, ↓↓G♭ 16/13
33 373.585 ↑7F, ↓G♭
34 384.906 ↑8F, G♭ 5/4, 81/65
35 396.226 ↑9F, ↓9G 44/35
36 407.547 F♯, ↓8G 81/64
37 418.868 ↑F♯, ↓7G 14/11
38 430.189 ↑↑F♯, ↓6G 32/25, 50/39
39 441.509 ↑3F♯, ↓5G
40 452.83 ↑4F♯, ↓4G 13/10, 35/27
41 464.151 ↑5F♯, ↓3G 55/42, 72/55
42 475.472 ↑6F♯, ↓↓G
43 486.792 ↑7F♯, ↓G
44 498.113 G 4/3
45 509.434 ↑G, ↓7A♭
46 520.755 ↑↑G, ↓6A♭ 27/20, 65/48
47 532.075 ↑3G, ↓5A♭
48 543.396 ↑4G, ↓4A♭ 48/35
49 554.717 ↑5G, ↓3A♭ 11/8
50 566.038 ↑6G, ↓↓A♭ 18/13, 25/18
51 577.358 ↑7G, ↓A♭
52 588.679 ↑8G, A♭ 45/32
53 600 ↑9G, ↓9A
54 611.321 G♯, ↓8A 64/45
55 622.642 ↑G♯, ↓7A 63/44
56 633.962 ↑↑G♯, ↓6A 13/9, 36/25, 75/52
57 645.283 ↑3G♯, ↓5A 16/11
58 656.604 ↑4G♯, ↓4A 35/24
59 667.925 ↑5G♯, ↓3A 81/55
60 679.245 ↑6G♯, ↓↓A 40/27
61 690.566 ↑7G♯, ↓A
62 701.887 A 3/2
63 713.208 ↑A, ↓7B♭
64 724.528 ↑↑A, ↓6B♭
65 735.849 ↑3A, ↓5B♭ 55/36
66 747.17 ↑4A, ↓4B♭ 20/13, 54/35
67 758.491 ↑5A, ↓3B♭
68 769.811 ↑6A, ↓↓B♭ 25/16, 39/25, 81/52
69 781.132 ↑7A, ↓B♭ 11/7
70 792.453 ↑8A, B♭
71 803.774 ↑9A, ↓9B 35/22
72 815.094 A♯, ↓8B 8/5
73 826.415 ↑A♯, ↓7B
74 837.736 ↑↑A♯, ↓6B 13/8, 81/50
75 849.057 ↑3A♯, ↓5B 18/11, 44/27
76 860.377 ↑4A♯, ↓4B 64/39
77 871.698 ↑5A♯, ↓3B
78 883.019 ↑6A♯, ↓↓B 5/3
79 894.34 ↑7A♯, ↓B
80 905.66 B 27/16
81 916.981 ↑B, ↓7C 56/33
82 928.302 ↑↑B, ↓6C
83 939.623 ↑3B, ↓5C 55/32
84 950.943 ↑4B, ↓4C 26/15, 45/26
85 962.264 ↑5B, ↓3C
86 973.585 ↑6B, ↓↓C
87 984.906 ↑7B, ↓C
88 996.226 C 16/9
89 1007.55 ↑C, ↓7D♭
90 1018.87 ↑↑C, ↓6D♭ 9/5, 65/36
91 1030.19 ↑3C, ↓5D♭
92 1041.51 ↑4C, ↓4D♭ 64/35
93 1052.83 ↑5C, ↓3D♭ 11/6, 81/44
94 1064.15 ↑6C, ↓↓D♭ 24/13, 50/27
95 1075.47 ↑7C, ↓D♭
96 1086.79 ↑8C, D♭ 15/8
97 1098.11 ↑9C, ↓9D 66/35
98 1109.43 C♯, ↓8D
99 1120.75 ↑C♯, ↓7D 21/11
100 1132.08 ↑↑C♯, ↓6D 25/13, 48/25, 52/27
101 1143.4 ↑3C♯, ↓5D 64/33
102 1154.72 ↑4C♯, ↓4D 35/18, 39/20
103 1166.04 ↑5C♯, ↓3D 55/28
104 1177.36 ↑6C♯, ↓↓D
105 1188.68 ↑7C♯, ↓D
106 1200 D 2/1

See also

Artists using 106 et:

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