119edo

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← 118edo 119edo 120edo →
Prime factorization 7 × 17
Step size 10.084¢ 
Fifth 70\119 (705.882¢) (→10\17)
Semitones (A1:m2) 14:7 (141.2¢ : 70.59¢)
Dual sharp fifth 70\119 (705.882¢) (→10\17)
Dual flat fifth 69\119 (695.798¢)
Dual major 2nd 20\119 (201.681¢)
Consistency limit 3
Distinct consistency limit 3

119 equal divisions of the octave (abbreviated 119edo or 119ed2), also called 119-tone equal temperament (119tet) or 119 equal temperament (119et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 119 equal parts of about 10.1 ¢ each. Each step represents a frequency ratio of 21/119, or the 119th root of 2.

Theory

119edo is inconsistent in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the optimal patent val for the 11-limit androboh and quasitemp temperaments. The patent val also tunes the 11-limit quadrawell temperament. 119c val tunes treecreeper, sensus, and senator as high as the 17-limit, while the 119b val is an extremely good approximation to 2/7-comma meantone in addition to supporting chlorine (by equating 25/24 very accurately to one step of 17edo).

Odd harmonics

Approximation of odd harmonics in 119edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23 25
Error Absolute (¢) +3.93 -3.12 -0.76 -2.23 +3.30 -3.55 +0.81 -4.12 +5.01 +3.17 -3.06 +3.84
Relative (%) +38.9 -30.9 -7.5 -22.1 +32.8 -35.2 +8.0 -40.8 +49.7 +31.4 -30.4 +38.1
Steps
(reduced) 		189
(70) 		276
(38) 		334
(96) 		377
(20) 		412
(55) 		440
(83) 		465
(108) 		486
(10) 		506
(30) 		523
(47) 		538
(62) 		553
(77)
Approximation of odd harmonics in 119edo (continued)
Harmonic 27 29 31 33 35 37 39 41 43 45 47 49
Error Absolute (¢) +1.70 -1.01 +4.54 -2.85 -3.88 +0.76 +0.37 +4.55 +2.77 +4.73 +0.04 -1.52
Relative (%) +16.8 -10.0 +45.1 -28.3 -38.5 +7.5 +3.7 +45.1 +27.4 +46.9 +0.4 -15.0
Steps
(reduced) 		566
(90) 		578
(102) 		590
(114) 		600
(5) 		610
(15) 		620
(25) 		629
(34) 		638
(43) 		646
(51) 		654
(59) 		661
(66) 		668
(73)

Subsets and supersets

Since 119edo factors as 7 × 17, it contains 7edo and 17edo as a subset. Hence it supports circles of fifths of those respective equal temperaments.

Intervals

Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 69\119) 		Ups and downs notation
(Dual sharp fifth 70\119)
0 0 1/1 D D
1 10.1 ^D, ^3E♭♭♭ ^D, v6E♭
2 20.2 ^^D, v3E♭♭ ^^D, v5E♭
3 30.3 ^3D, vvE♭♭ ^3D, v4E♭
4 40.3 42/41, 43/42, 44/43, 45/44 v3D♯, vE♭♭ ^4D, v3E♭
5 50.4 35/34 vvD♯, E♭♭ ^5D, vvE♭
6 60.5 29/28, 30/29 vD♯, ^E♭♭ ^6D, vE♭
7 70.6 49/47 D♯, ^^E♭♭ ^7D, E♭
8 80.7 22/21, 43/41, 45/43 ^D♯, ^3E♭♭ v6D♯, ^E♭
9 90.8 39/37 ^^D♯, v3E♭ v5D♯, ^^E♭
10 100.8 ^3D♯, vvE♭ v4D♯, ^3E♭
11 110.9 16/15, 49/46 v3D𝄪, vE♭ v3D♯, ^4E♭
12 121 15/14, 44/41 vvD𝄪, E♭ vvD♯, ^5E♭
13 131.1 41/38 vD𝄪, ^E♭ vD♯, ^6E♭
14 141.2 D𝄪, ^^E♭ D♯, v7E
15 151.3 12/11 ^D𝄪, ^3E♭ ^D♯, v6E
16 161.3 45/41 ^^D𝄪, v3E ^^D♯, v5E
17 171.4 21/19, 32/29 ^3D𝄪, vvE ^3D♯, v4E
18 181.5 v3D♯𝄪, vE ^4D♯, v3E
19 191.6 48/43 E ^5D♯, vvE
20 201.7 ^E, ^3F♭♭ ^6D♯, vE
21 211.8 26/23 ^^E, v3F♭ E
22 221.8 ^3E, vvF♭ ^E, v6F
23 231.9 8/7 v3E♯, vF♭ ^^E, v5F
24 242 23/20 vvE♯, F♭ ^3E, v4F
25 252.1 22/19, 37/32 vE♯, ^F♭ ^4E, v3F
26 262.2 43/37 E♯, ^^F♭ ^5E, vvF
27 272.3 48/41 ^E♯, ^3F♭ ^6E, vF
28 282.4 20/17 ^^E♯, v3F F
29 292.4 45/38 ^3E♯, vvF ^F, v6G♭
30 302.5 v3E𝄪, vF ^^F, v5G♭
31 312.6 F ^3F, v4G♭
32 322.7 47/39 ^F, ^3G♭♭♭ ^4F, v3G♭
33 332.8 ^^F, v3G♭♭ ^5F, vvG♭
34 342.9 39/32 ^3F, vvG♭♭ ^6F, vG♭
35 352.9 38/31, 49/40 v3F♯, vG♭♭ ^7F, G♭
36 363 37/30 vvF♯, G♭♭ v6F♯, ^G♭
37 373.1 vF♯, ^G♭♭ v5F♯, ^^G♭
38 383.2 F♯, ^^G♭♭ v4F♯, ^3G♭
39 393.3 49/39 ^F♯, ^3G♭♭ v3F♯, ^4G♭
40 403.4 24/19 ^^F♯, v3G♭ vvF♯, ^5G♭
41 413.4 47/37 ^3F♯, vvG♭ vF♯, ^6G♭
42 423.5 37/29 v3F𝄪, vG♭ F♯, v7G
43 433.6 vvF𝄪, G♭ ^F♯, v6G
44 443.7 31/24 vF𝄪, ^G♭ ^^F♯, v5G
45 453.8 13/10 F𝄪, ^^G♭ ^3F♯, v4G
46 463.9 17/13 ^F𝄪, ^3G♭ ^4F♯, v3G
47 473.9 46/35 ^^F𝄪, v3G ^5F♯, vvG
48 484 37/28, 41/31 ^3F𝄪, vvG ^6F♯, vG
49 494.1 v3F♯𝄪, vG G
50 504.2 G ^G, v6A♭
51 514.3 35/26, 39/29 ^G, ^3A♭♭♭ ^^G, v5A♭
52 524.4 23/17, 42/31 ^^G, v3A♭♭ ^3G, v4A♭
53 534.5 ^3G, vvA♭♭ ^4G, v3A♭
54 544.5 v3G♯, vA♭♭ ^5G, vvA♭
55 554.6 vvG♯, A♭♭ ^6G, vA♭
56 564.7 43/31 vG♯, ^A♭♭ ^7G, A♭
57 574.8 39/28 G♯, ^^A♭♭ v6G♯, ^A♭
58 584.9 ^G♯, ^3A♭♭ v5G♯, ^^A♭
59 595 31/22 ^^G♯, v3A♭ v4G♯, ^3A♭
60 605 44/31 ^3G♯, vvA♭ v3G♯, ^4A♭
61 615.1 v3G𝄪, vA♭ vvG♯, ^5A♭
62 625.2 43/30 vvG𝄪, A♭ vG♯, ^6A♭
63 635.3 vG𝄪, ^A♭ G♯, v7A
64 645.4 45/31 G𝄪, ^^A♭ ^G♯, v6A
65 655.5 ^G𝄪, ^3A♭ ^^G♯, v5A
66 665.5 47/32 ^^G𝄪, v3A ^3G♯, v4A
67 675.6 31/21, 34/23 ^3G𝄪, vvA ^4G♯, v3A
68 685.7 v3G♯𝄪, vA ^5G♯, vvA
69 695.8 A ^6G♯, vA
70 705.9 ^A, ^3B♭♭♭ A
71 716 ^^A, v3B♭♭ ^A, v6B♭
72 726.1 35/23 ^3A, vvB♭♭ ^^A, v5B♭
73 736.1 26/17, 49/32 v3A♯, vB♭♭ ^3A, v4B♭
74 746.2 20/13 vvA♯, B♭♭ ^4A, v3B♭
75 756.3 48/31 vA♯, ^B♭♭ ^5A, vvB♭
76 766.4 A♯, ^^B♭♭ ^6A, vB♭
77 776.5 47/30 ^A♯, ^3B♭♭ ^7A, B♭
78 786.6 ^^A♯, v3B♭ v6A♯, ^B♭
79 796.6 19/12 ^3A♯, vvB♭ v5A♯, ^^B♭
80 806.7 v3A𝄪, vB♭ v4A♯, ^3B♭
81 816.8 vvA𝄪, B♭ v3A♯, ^4B♭
82 826.9 vA𝄪, ^B♭ vvA♯, ^5B♭
83 837 47/29 A𝄪, ^^B♭ vA♯, ^6B♭
84 847.1 31/19 ^A𝄪, ^3B♭ A♯, v7B
85 857.1 ^^A𝄪, v3B ^A♯, v6B
86 867.2 ^3A𝄪, vvB ^^A♯, v5B
87 877.3 v3A♯𝄪, vB ^3A♯, v4B
88 887.4 B ^4A♯, v3B
89 897.5 47/28 ^B, ^3C♭♭ ^5A♯, vvB
90 907.6 49/29 ^^B, v3C♭ ^6A♯, vB
91 917.6 17/10 ^3B, vvC♭ B
92 927.7 41/24 v3B♯, vC♭ ^B, v6C
93 937.8 vvB♯, C♭ ^^B, v5C
94 947.9 19/11 vB♯, ^C♭ ^3B, v4C
95 958 40/23 B♯, ^^C♭ ^4B, v3C
96 968.1 7/4 ^B♯, ^3C♭ ^5B, vvC
97 978.2 ^^B♯, v3C ^6B, vC
98 988.2 23/13 ^3B♯, vvC C
99 998.3 v3B𝄪, vC ^C, v6D♭
100 1008.4 43/24 C ^^C, v5D♭
101 1018.5 ^C, ^3D♭♭♭ ^3C, v4D♭
102 1028.6 29/16, 38/21 ^^C, v3D♭♭ ^4C, v3D♭
103 1038.7 ^3C, vvD♭♭ ^5C, vvD♭
104 1048.7 11/6 v3C♯, vD♭♭ ^6C, vD♭
105 1058.8 vvC♯, D♭♭ ^7C, D♭
106 1068.9 vC♯, ^D♭♭ v6C♯, ^D♭
107 1079 28/15, 41/22 C♯, ^^D♭♭ v5C♯, ^^D♭
108 1089.1 15/8 ^C♯, ^3D♭♭ v4C♯, ^3D♭
109 1099.2 ^^C♯, v3D♭ v3C♯, ^4D♭
110 1109.2 ^3C♯, vvD♭ vvC♯, ^5D♭
111 1119.3 21/11 v3C𝄪, vD♭ vC♯, ^6D♭
112 1129.4 vvC𝄪, D♭ C♯, v7D
113 1139.5 29/15 vC𝄪, ^D♭ ^C♯, v6D
114 1149.6 C𝄪, ^^D♭ ^^C♯, v5D
115 1159.7 41/21, 43/22 ^C𝄪, ^3D♭ ^3C♯, v4D
116 1169.7 ^^C𝄪, v3D ^4C♯, v3D
117 1179.8 ^3C𝄪, vvD ^5C♯, vvD
118 1189.9 v3C♯𝄪, vD ^6C♯, vD
119 1200 2/1 D D

Scales

  • Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
  • Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2
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