119edo

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← 118edo119edo120edo →
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.

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 10.084 ^D, v4E♭♭ ^D, v6E♭
2 20.168 ^^D, v3E♭♭ ^^D, v5E♭
3 30.252 ^3D, vvE♭♭ ^3D, v4E♭
4 40.336 42/41, 43/42, 44/43, 45/44 ^4D, vE♭♭ ^4D, v3E♭
5 50.42 35/34 ^5D, E♭♭ ^5D, vvE♭
6 60.504 29/28, 30/29 ^6D, v6E♭ ^6D, vE♭
7 70.588 49/47 D♯, v5E♭ ^7D, E♭
8 80.672 22/21, 43/41, 45/43 ^D♯, v4E♭ ^8D, v13E
9 90.756 39/37 ^^D♯, v3E♭ ^9D, v12E
10 100.84 ^3D♯, vvE♭ ^10D, v11E
11 110.924 16/15, 49/46 ^4D♯, vE♭ ^11D, v10E
12 121.008 15/14, 44/41 ^5D♯, E♭ ^12D, v9E
13 131.092 41/38 ^6D♯, v6E ^13D, v8E
14 141.176 D𝄪, v5E D♯, v7E
15 151.261 12/11 ^D𝄪, v4E ^D♯, v6E
16 161.345 45/41 ^^D𝄪, v3E ^^D♯, v5E
17 171.429 21/19, 32/29 ^3D𝄪, vvE ^3D♯, v4E
18 181.513 ^4D𝄪, vE ^4D♯, v3E
19 191.597 48/43 E ^5D♯, vvE
20 201.681 ^E, v4F♭ ^6D♯, vE
21 211.765 26/23 ^^E, v3F♭ E
22 221.849 ^3E, vvF♭ ^E, v6F
23 231.933 8/7 ^4E, vF♭ ^^E, v5F
24 242.017 23/20 ^5E, F♭ ^3E, v4F
25 252.101 22/19, 37/32 ^6E, v6F ^4E, v3F
26 262.185 43/37 E♯, v5F ^5E, vvF
27 272.269 48/41 ^E♯, v4F ^6E, vF
28 282.353 20/17 ^^E♯, v3F F
29 292.437 45/38 ^3E♯, vvF ^F, v6G♭
30 302.521 ^4E♯, vF ^^F, v5G♭
31 312.605 F ^3F, v4G♭
32 322.689 47/39 ^F, v4G♭♭ ^4F, v3G♭
33 332.773 ^^F, v3G♭♭ ^5F, vvG♭
34 342.857 39/32 ^3F, vvG♭♭ ^6F, vG♭
35 352.941 38/31, 49/40 ^4F, vG♭♭ ^7F, G♭
36 363.025 37/30 ^5F, G♭♭ ^8F, v13G
37 373.109 ^6F, v6G♭ ^9F, v12G
38 383.193 F♯, v5G♭ ^10F, v11G
39 393.277 49/39 ^F♯, v4G♭ ^11F, v10G
40 403.361 24/19 ^^F♯, v3G♭ ^12F, v9G
41 413.445 47/37 ^3F♯, vvG♭ ^13F, v8G
42 423.529 37/29 ^4F♯, vG♭ F♯, v7G
43 433.613 ^5F♯, G♭ ^F♯, v6G
44 443.697 31/24 ^6F♯, v6G ^^F♯, v5G
45 453.782 13/10 F𝄪, v5G ^3F♯, v4G
46 463.866 17/13 ^F𝄪, v4G ^4F♯, v3G
47 473.95 46/35 ^^F𝄪, v3G ^5F♯, vvG
48 484.034 37/28, 41/31 ^3F𝄪, vvG ^6F♯, vG
49 494.118 ^4F𝄪, vG G
50 504.202 G ^G, v6A♭
51 514.286 35/26, 39/29 ^G, v4A♭♭ ^^G, v5A♭
52 524.37 23/17, 42/31 ^^G, v3A♭♭ ^3G, v4A♭
53 534.454 ^3G, vvA♭♭ ^4G, v3A♭
54 544.538 ^4G, vA♭♭ ^5G, vvA♭
55 554.622 ^5G, A♭♭ ^6G, vA♭
56 564.706 43/31 ^6G, v6A♭ ^7G, A♭
57 574.79 39/28 G♯, v5A♭ ^8G, v13A
58 584.874 ^G♯, v4A♭ ^9G, v12A
59 594.958 31/22 ^^G♯, v3A♭ ^10G, v11A
60 605.042 44/31 ^3G♯, vvA♭ ^11G, v10A
61 615.126 ^4G♯, vA♭ ^12G, v9A
62 625.21 43/30 ^5G♯, A♭ ^13G, v8A
63 635.294 ^6G♯, v6A G♯, v7A
64 645.378 45/31 G𝄪, v5A ^G♯, v6A
65 655.462 ^G𝄪, v4A ^^G♯, v5A
66 665.546 47/32 ^^G𝄪, v3A ^3G♯, v4A
67 675.63 31/21, 34/23 ^3G𝄪, vvA ^4G♯, v3A
68 685.714 ^4G𝄪, vA ^5G♯, vvA
69 695.798 A ^6G♯, vA
70 705.882 ^A, v4B♭♭ A
71 715.966 ^^A, v3B♭♭ ^A, v6B♭
72 726.05 35/23 ^3A, vvB♭♭ ^^A, v5B♭
73 736.134 26/17, 49/32 ^4A, vB♭♭ ^3A, v4B♭
74 746.218 20/13 ^5A, B♭♭ ^4A, v3B♭
75 756.303 48/31 ^6A, v6B♭ ^5A, vvB♭
76 766.387 A♯, v5B♭ ^6A, vB♭
77 776.471 47/30 ^A♯, v4B♭ ^7A, B♭
78 786.555 ^^A♯, v3B♭ ^8A, v13B
79 796.639 19/12 ^3A♯, vvB♭ ^9A, v12B
80 806.723 ^4A♯, vB♭ ^10A, v11B
81 816.807 ^5A♯, B♭ ^11A, v10B
82 826.891 ^6A♯, v6B ^12A, v9B
83 836.975 47/29 A𝄪, v5B ^13A, v8B
84 847.059 31/19 ^A𝄪, v4B A♯, v7B
85 857.143 ^^A𝄪, v3B ^A♯, v6B
86 867.227 ^3A𝄪, vvB ^^A♯, v5B
87 877.311 ^4A𝄪, vB ^3A♯, v4B
88 887.395 B ^4A♯, v3B
89 897.479 47/28 ^B, v4C♭ ^5A♯, vvB
90 907.563 49/29 ^^B, v3C♭ ^6A♯, vB
91 917.647 17/10 ^3B, vvC♭ B
92 927.731 41/24 ^4B, vC♭ ^B, v6C
93 937.815 ^5B, C♭ ^^B, v5C
94 947.899 19/11 ^6B, v6C ^3B, v4C
95 957.983 40/23 B♯, v5C ^4B, v3C
96 968.067 7/4 ^B♯, v4C ^5B, vvC
97 978.151 ^^B♯, v3C ^6B, vC
98 988.235 23/13 ^3B♯, vvC C
99 998.319 ^4B♯, vC ^C, v6D♭
100 1008.403 43/24 C ^^C, v5D♭
101 1018.487 ^C, v4D♭♭ ^3C, v4D♭
102 1028.571 29/16, 38/21 ^^C, v3D♭♭ ^4C, v3D♭
103 1038.655 ^3C, vvD♭♭ ^5C, vvD♭
104 1048.739 11/6 ^4C, vD♭♭ ^6C, vD♭
105 1058.824 ^5C, D♭♭ ^7C, D♭
106 1068.908 ^6C, v6D♭ ^8C, v13D
107 1078.992 28/15, 41/22 C♯, v5D♭ ^9C, v12D
108 1089.076 15/8 ^C♯, v4D♭ ^10C, v11D
109 1099.16 ^^C♯, v3D♭ ^11C, v10D
110 1109.244 ^3C♯, vvD♭ ^12C, v9D
111 1119.328 21/11 ^4C♯, vD♭ ^13C, v8D
112 1129.412 ^5C♯, D♭ C♯, v7D
113 1139.496 29/15 ^6C♯, v6D ^C♯, v6D
114 1149.58 C𝄪, v5D ^^C♯, v5D
115 1159.664 41/21, 43/22 ^C𝄪, v4D ^3C♯, v4D
116 1169.748 ^^C𝄪, v3D ^4C♯, v3D
117 1179.832 ^3C𝄪, vvD ^5C♯, vvD
118 1189.916 ^4C𝄪, 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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