93edo
← 92edo | 93edo | 94edo → |
93 equal divisions of the octave (abbreviated 93edo or 93ed2), also called 93-tone equal temperament (93tet) or 93 equal temperament (93et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 93 equal parts of about 12.9 ¢ each. Each step represents a frequency ratio of 21/93, or the 93rd root of 2.
Theory
Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | absolute (¢) | -5.18 | +0.78 | -1.08 | +2.54 | +3.52 | -1.82 | -4.40 | -1.73 | -0.74 | -6.26 | +3.98 |
relative (%) | -40 | +6 | -8 | +20 | +27 | -14 | -34 | -13 | -6 | -49 | +31 | |
Steps (reduced) |
147 (54) |
216 (30) |
261 (75) |
295 (16) |
322 (43) |
344 (65) |
363 (84) |
380 (8) |
395 (23) |
408 (36) |
421 (49) |
93 = 3 * 31, and 93 is a contorted 31 through the 7 limit. In the 11-limit the patent val tempers out 4000/3993 and in the 13-limit 144/143, 1188/1183 and 364/363. It provides the optimal patent val for the 11-limit prajapati and 13-limit kumhar temperaments, and the 11 and 13 limit 43&50 temperament. It is the 13th no-3s zeta peak edo.
Since 93edo has good approximations of 13th, 17th and 19th harmonics unlike 31edo (as 838.710 ¢, 103.226 ¢, and 296.774 ¢ respectively, octave-reduced), it also allows one to give a clearer harmonic identity to 31edo's excellent approximation of 13:17:19.
Intervals
Steps | Cents | Ups and downs notation (dual flat fifth 54\93) |
Ups and downs notation (dual sharp fifth 55\93) |
Approximate ratios |
---|---|---|---|---|
0 | 0 | D | D | 1/1 |
1 | 12.9032 | ↑D, ↓↓E♭♭ | ↑D, ↓3E♭ | |
2 | 25.8065 | ↑↑D, ↓E♭♭ | ↑↑D, ↓↓E♭ | 65/64, 66/65 |
3 | 38.7097 | ↑3D, E♭♭ | ↑3D, ↓E♭ | 49/48, 50/49 |
4 | 51.6129 | ↑4D, ↓5E♭ | ↑4D, E♭ | 33/32 |
5 | 64.5161 | ↑5D, ↓4E♭ | ↑5D, ↓12E | 26/25, 80/77 |
6 | 77.4194 | D♯, ↓3E♭ | ↑6D, ↓11E | |
7 | 90.3226 | ↑D♯, ↓↓E♭ | ↑7D, ↓10E | |
8 | 103.226 | ↑↑D♯, ↓E♭ | ↑8D, ↓9E | 35/33, 52/49 |
9 | 116.129 | ↑3D♯, E♭ | ↑9D, ↓8E | 15/14, 16/15 |
10 | 129.032 | ↑4D♯, ↓5E | ↑10D, ↓7E | 14/13 |
11 | 141.935 | ↑5D♯, ↓4E | ↑11D, ↓6E | 13/12 |
12 | 154.839 | D𝄪, ↓3E | ↑12D, ↓5E | 35/32 |
13 | 167.742 | ↑D𝄪, ↓↓E | D♯, ↓4E | 11/10 |
14 | 180.645 | ↑↑D𝄪, ↓E | ↑D♯, ↓3E | |
15 | 193.548 | E | ↑↑D♯, ↓↓E | 28/25 |
16 | 206.452 | ↑E, ↓↓F♭ | ↑3D♯, ↓E | |
17 | 219.355 | ↑↑E, ↓F♭ | E | 25/22 |
18 | 232.258 | ↑3E, F♭ | ↑E, ↓3F | 8/7 |
19 | 245.161 | ↑4E, ↓5F | ↑↑E, ↓↓F | 15/13 |
20 | 258.065 | ↑5E, ↓4F | ↑3E, ↓F | 64/55, 65/56 |
21 | 270.968 | E♯, ↓3F | F | 7/6, 75/64 |
22 | 283.871 | ↑E♯, ↓↓F | ↑F, ↓3G♭ | 33/28 |
23 | 296.774 | ↑↑E♯, ↓F | ↑↑F, ↓↓G♭ | 77/65 |
24 | 309.677 | F | ↑3F, ↓G♭ | |
25 | 322.581 | ↑F, ↓↓G♭♭ | ↑4F, G♭ | 77/64 |
26 | 335.484 | ↑↑F, ↓G♭♭ | ↑5F, ↓12G | 40/33 |
27 | 348.387 | ↑3F, G♭♭ | ↑6F, ↓11G | 49/40, 60/49 |
28 | 361.29 | ↑4F, ↓5G♭ | ↑7F, ↓10G | 16/13 |
29 | 374.194 | ↑5F, ↓4G♭ | ↑8F, ↓9G | 26/21 |
30 | 387.097 | F♯, ↓3G♭ | ↑9F, ↓8G | 5/4 |
31 | 400 | ↑F♯, ↓↓G♭ | ↑10F, ↓7G | 44/35, 49/39 |
32 | 412.903 | ↑↑F♯, ↓G♭ | ↑11F, ↓6G | 14/11, 33/26 |
33 | 425.806 | ↑3F♯, G♭ | ↑12F, ↓5G | 32/25 |
34 | 438.71 | ↑4F♯, ↓5G | F♯, ↓4G | |
35 | 451.613 | ↑5F♯, ↓4G | ↑F♯, ↓3G | 13/10 |
36 | 464.516 | F𝄪, ↓3G | ↑↑F♯, ↓↓G | 64/49 |
37 | 477.419 | ↑F𝄪, ↓↓G | ↑3F♯, ↓G | 33/25 |
38 | 490.323 | ↑↑F𝄪, ↓G | G | 65/49 |
39 | 503.226 | G | ↑G, ↓3A♭ | 75/56 |
40 | 516.129 | ↑G, ↓↓A♭♭ | ↑↑G, ↓↓A♭ | 35/26, 66/49 |
41 | 529.032 | ↑↑G, ↓A♭♭ | ↑3G, ↓A♭ | 65/48 |
42 | 541.935 | ↑3G, A♭♭ | ↑4G, A♭ | 48/35 |
43 | 554.839 | ↑4G, ↓5A♭ | ↑5G, ↓12A | 11/8 |
44 | 567.742 | ↑5G, ↓4A♭ | ↑6G, ↓11A | |
45 | 580.645 | G♯, ↓3A♭ | ↑7G, ↓10A | 7/5 |
46 | 593.548 | ↑G♯, ↓↓A♭ | ↑8G, ↓9A | |
47 | 606.452 | ↑↑G♯, ↓A♭ | ↑9G, ↓8A | |
48 | 619.355 | ↑3G♯, A♭ | ↑10G, ↓7A | 10/7 |
49 | 632.258 | ↑4G♯, ↓5A | ↑11G, ↓6A | 75/52 |
50 | 645.161 | ↑5G♯, ↓4A | ↑12G, ↓5A | 16/11 |
51 | 658.065 | G𝄪, ↓3A | G♯, ↓4A | 35/24 |
52 | 670.968 | ↑G𝄪, ↓↓A | ↑G♯, ↓3A | 65/44 |
53 | 683.871 | ↑↑G𝄪, ↓A | ↑↑G♯, ↓↓A | 49/33, 52/35, 77/52 |
54 | 696.774 | A | ↑3G♯, ↓A | |
55 | 709.677 | ↑A, ↓↓B♭♭ | A | |
56 | 722.581 | ↑↑A, ↓B♭♭ | ↑A, ↓3B♭ | 50/33 |
57 | 735.484 | ↑3A, B♭♭ | ↑↑A, ↓↓B♭ | 49/32, 75/49 |
58 | 748.387 | ↑4A, ↓5B♭ | ↑3A, ↓B♭ | 20/13, 77/50 |
59 | 761.29 | ↑5A, ↓4B♭ | ↑4A, B♭ | |
60 | 774.194 | A♯, ↓3B♭ | ↑5A, ↓12B | 25/16 |
61 | 787.097 | ↑A♯, ↓↓B♭ | ↑6A, ↓11B | 11/7, 52/33 |
62 | 800 | ↑↑A♯, ↓B♭ | ↑7A, ↓10B | 35/22, 78/49 |
63 | 812.903 | ↑3A♯, B♭ | ↑8A, ↓9B | 8/5 |
64 | 825.806 | ↑4A♯, ↓5B | ↑9A, ↓8B | 21/13 |
65 | 838.71 | ↑5A♯, ↓4B | ↑10A, ↓7B | 13/8 |
66 | 851.613 | A𝄪, ↓3B | ↑11A, ↓6B | 49/30, 80/49 |
67 | 864.516 | ↑A𝄪, ↓↓B | ↑12A, ↓5B | 33/20 |
68 | 877.419 | ↑↑A𝄪, ↓B | A♯, ↓4B | |
69 | 890.323 | B | ↑A♯, ↓3B | |
70 | 903.226 | ↑B, ↓↓C♭ | ↑↑A♯, ↓↓B | |
71 | 916.129 | ↑↑B, ↓C♭ | ↑3A♯, ↓B | 56/33 |
72 | 929.032 | ↑3B, C♭ | B | 12/7 |
73 | 941.935 | ↑4B, ↓5C | ↑B, ↓3C | 55/32 |
74 | 954.839 | ↑5B, ↓4C | ↑↑B, ↓↓C | 26/15 |
75 | 967.742 | B♯, ↓3C | ↑3B, ↓C | 7/4 |
76 | 980.645 | ↑B♯, ↓↓C | C | 44/25 |
77 | 993.548 | ↑↑B♯, ↓C | ↑C, ↓3D♭ | |
78 | 1006.45 | C | ↑↑C, ↓↓D♭ | 25/14 |
79 | 1019.35 | ↑C, ↓↓D♭♭ | ↑3C, ↓D♭ | |
80 | 1032.26 | ↑↑C, ↓D♭♭ | ↑4C, D♭ | 20/11 |
81 | 1045.16 | ↑3C, D♭♭ | ↑5C, ↓12D | 64/35 |
82 | 1058.06 | ↑4C, ↓5D♭ | ↑6C, ↓11D | 24/13 |
83 | 1070.97 | ↑5C, ↓4D♭ | ↑7C, ↓10D | 13/7 |
84 | 1083.87 | C♯, ↓3D♭ | ↑8C, ↓9D | 15/8, 28/15 |
85 | 1096.77 | ↑C♯, ↓↓D♭ | ↑9C, ↓8D | 49/26, 66/35 |
86 | 1109.68 | ↑↑C♯, ↓D♭ | ↑10C, ↓7D | |
87 | 1122.58 | ↑3C♯, D♭ | ↑11C, ↓6D | |
88 | 1135.48 | ↑4C♯, ↓5D | ↑12C, ↓5D | 25/13, 77/40 |
89 | 1148.39 | ↑5C♯, ↓4D | C♯, ↓4D | 64/33 |
90 | 1161.29 | C𝄪, ↓3D | ↑C♯, ↓3D | 49/25 |
91 | 1174.19 | ↑C𝄪, ↓↓D | ↑↑C♯, ↓↓D | 65/33 |
92 | 1187.1 | ↑↑C𝄪, ↓D | ↑3C♯, ↓D | |
93 | 1200 | D | D | 2/1 |
Scales
Meantone Chromatic
- 116.129
- 193.548
- 309.677
- 387.097
- 503.226
- 580.645
- 696.774
- 812.903
- 890.323
- 1006.452
- 1083.871
- 1200.000
Superpyth Chromatic
- 51.613
- 219.355
- 270.968
- 438.710
- 490.323
- 658.065
- 709.677
- 761.290
- 929.032
- 980.645
- 1148.387
- 1200.000
Superpyth Shailaja
- 270.968
- 709.677
- 761.290
- 980.645
- 1200.000
Superpyth Subminor Hexatonic
- 219.355
- 270.968
- 490.323
- 709.677
- 980.645
- 1200.000
Superpyth Subminor Pentatonic
- 270.968
- 490.323
- 709.677
- 980.645
- 1200.000