74edo
| ← 73edo | 74edo | 75edo → |
74 equal divisions of the octave (abbreviated 74edo or 74ed2), also called 74-tone equal temperament (74tet) or 74 equal temperament (74et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 74 equal parts of about 16.2 ¢ each. Each step represents a frequency ratio of 21/74, or the 74th root of 2.
Theory
74edo is most notable as a meantone tuning, tempering out 81/80 in the 5-limit; 126/125 and 225/224 in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and 144/143 and 847/845 in the 13-limit. Discarding 847/845 from that gives the 13-limit meantone extension grosstone, for which 74edo gives the optimal patent val; and discarding 144/143 gives semimeantone, a 13-limit 62 & 74 temperament with half-octave period and two parallel tracks of meantone.
74edo tunes harmonic 11 only 1/30 of a cent sharp, and 13 2.7 cents sharp, making it a distinctly interesting choice for higher-limit meantone.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.66 | +2.88 | +4.15 | +6.90 | +0.03 | +2.72 | -1.78 | -7.66 | -5.62 | -0.51 | +4.16 |
| Relative (%) | -28.7 | +17.7 | +25.6 | +42.6 | +0.2 | +16.7 | -11.0 | -47.2 | -34.7 | -3.1 | +25.6 | |
| Steps (reduced) |
117 (43) |
172 (24) |
208 (60) |
235 (13) |
256 (34) |
274 (52) |
289 (67) |
302 (6) |
314 (18) |
325 (29) |
335 (39) | |
Subsets and supersets
Since 74 factors into 2 × 37, 74edo contains 2edo and 37edo as its subsets.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 16.2 | ^D, vE♭♭ | |
| 2 | 32.4 | ^^D, E♭♭ | |
| 3 | 48.6 | 34/33, 37/36, 38/37 | vvD♯, ^E♭♭ |
| 4 | 64.9 | 26/25 | vD♯, ^^E♭♭ |
| 5 | 81.1 | 21/20, 22/21 | D♯, vvE♭ |
| 6 | 97.3 | 18/17 | ^D♯, vE♭ |
| 7 | 113.5 | 16/15 | ^^D♯, E♭ |
| 8 | 129.7 | 14/13 | vvD𝄪, ^E♭ |
| 9 | 145.9 | 25/23, 37/34 | vD𝄪, ^^E♭ |
| 10 | 162.2 | 11/10 | D𝄪, vvE |
| 11 | 178.4 | 31/28 | ^D𝄪, vE |
| 12 | 194.6 | 19/17, 28/25, 37/33 | E |
| 13 | 210.8 | 26/23, 35/31 | ^E, vF♭ |
| 14 | 227 | 33/29 | ^^E, F♭ |
| 15 | 243.2 | 23/20, 38/33 | vvE♯, ^F♭ |
| 16 | 259.5 | vE♯, ^^F♭ | |
| 17 | 275.7 | 34/29 | E♯, vvF |
| 18 | 291.9 | 13/11 | ^E♯, vF |
| 19 | 308.1 | F | |
| 20 | 324.3 | 29/24 | ^F, vG♭♭ |
| 21 | 340.5 | 28/23 | ^^F, G♭♭ |
| 22 | 356.8 | 16/13 | vvF♯, ^G♭♭ |
| 23 | 373 | 26/21, 31/25, 36/29 | vF♯, ^^G♭♭ |
| 24 | 389.2 | 5/4 | F♯, vvG♭ |
| 25 | 405.4 | 24/19 | ^F♯, vG♭ |
| 26 | 421.6 | 37/29 | ^^F♯, G♭ |
| 27 | 437.8 | vvF𝄪, ^G♭ | |
| 28 | 454.1 | 13/10 | vF𝄪, ^^G♭ |
| 29 | 470.3 | 21/16, 38/29 | F𝄪, vvG |
| 30 | 486.5 | ^F𝄪, vG | |
| 31 | 502.7 | G | |
| 32 | 518.9 | 31/23 | ^G, vA♭♭ |
| 33 | 535.1 | 15/11 | ^^G, A♭♭ |
| 34 | 551.4 | 11/8 | vvG♯, ^A♭♭ |
| 35 | 567.6 | vG♯, ^^A♭♭ | |
| 36 | 583.8 | 7/5 | G♯, vvA♭ |
| 37 | 600 | 17/12, 24/17 | ^G♯, vA♭ |
| 38 | 616.2 | 10/7 | ^^G♯, A♭ |
| 39 | 632.4 | vvG𝄪, ^A♭ | |
| 40 | 648.6 | 16/11 | vG𝄪, ^^A♭ |
| 41 | 664.9 | 22/15 | G𝄪, vvA |
| 42 | 681.1 | ^G𝄪, vA | |
| 43 | 697.3 | A | |
| 44 | 713.5 | ^A, vB♭♭ | |
| 45 | 729.7 | 29/19, 32/21, 35/23 | ^^A, B♭♭ |
| 46 | 745.9 | 20/13, 37/24 | vvA♯, ^B♭♭ |
| 47 | 762.2 | 31/20 | vA♯, ^^B♭♭ |
| 48 | 778.4 | A♯, vvB♭ | |
| 49 | 794.6 | 19/12 | ^A♯, vB♭ |
| 50 | 810.8 | 8/5 | ^^A♯, B♭ |
| 51 | 827 | 21/13, 29/18 | vvA𝄪, ^B♭ |
| 52 | 843.2 | 13/8 | vA𝄪, ^^B♭ |
| 53 | 859.5 | 23/14 | A𝄪, vvB |
| 54 | 875.7 | ^A𝄪, vB | |
| 55 | 891.9 | B | |
| 56 | 908.1 | 22/13 | ^B, vC♭ |
| 57 | 924.3 | 29/17 | ^^B, C♭ |
| 58 | 940.5 | vvB♯, ^C♭ | |
| 59 | 956.8 | 33/19 | vB♯, ^^C♭ |
| 60 | 973 | B♯, vvC | |
| 61 | 989.2 | 23/13 | ^B♯, vC |
| 62 | 1005.4 | 25/14, 34/19 | C |
| 63 | 1021.6 | ^C, vD♭♭ | |
| 64 | 1037.8 | 20/11 | ^^C, D♭♭ |
| 65 | 1054.1 | vvC♯, ^D♭♭ | |
| 66 | 1070.3 | 13/7 | vC♯, ^^D♭♭ |
| 67 | 1086.5 | 15/8 | C♯, vvD♭ |
| 68 | 1102.7 | 17/9 | ^C♯, vD♭ |
| 69 | 1118.9 | 21/11 | ^^C♯, D♭ |
| 70 | 1135.1 | 25/13 | vvC𝄪, ^D♭ |
| 71 | 1151.4 | 33/17, 37/19 | vC𝄪, ^^D♭ |
| 72 | 1167.6 | C𝄪, vvD | |
| 73 | 1183.8 | ^C𝄪, vD | |
| 74 | 1200 | 2/1 | D |
Notation
Sagittal notation
Evo flavor
Revo flavor
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
Music
Modern renderings
- Maple Leaf Rag (1899) – rendered by Francium (2024)
- Maple Leaf Rag (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)