119edo
← 118edo | 119edo | 120edo → |
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
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) |
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