51edo
← 50edo | 51edo | 52edo → |
51 equal divisions of the octave (abbreviated 51edo or 51ed2), also called 51-tone equal temperament (51tet) or 51 equal temperament (51et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 51 equal parts of about 23.5 ¢ each. Each step represents a frequency ratio of 21/51, or the 51st root of 2.
Theory
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | absolute (¢) | +0.0 | +3.9 | -9.8 | -4.1 | -10.1 | +6.5 | -10.8 | +8.4 | +7.0 | +5.7 | +7.9 |
relative (%) | +0 | +17 | -42 | -18 | -43 | +28 | -46 | +36 | +30 | +24 | +34 | |
Steps (reduced) |
51 (0) |
81 (30) |
118 (16) |
143 (41) |
176 (23) |
189 (36) |
208 (4) |
217 (13) |
231 (27) |
248 (44) |
253 (49) |
51 EDO tempers out 250/243 in the 5-limit, 225/224 and 2401/2400 in the 7-limit, and 55/54 and 100/99 in the 11-limit. It is the optimal patent val for sonic, the rank three temperament tempering out 250/243, 55/54 and 100/99, and also for the rank four temperament tempering out 55/54. It provides an alternative tuning to 22edo for porcupine temperament, with a nice fifth but a rather flat major third, and the optimal patent val for 7 and 11-limit porky temperament, which is sonic plus 225/224. 51 contains an Archeotonic scale based on repetitions of 8\51, creating a scale with a whole tone-like drive towards the tonic through the 17edo semitone at the top.
51edo's step is the closest direct approximation to the Pythagorean comma by edo steps, though that comma itself is mapped to a different interval.
Intervals
Degrees | Cents | Ups and Downs Notation | ||
---|---|---|---|---|
0 | 0.000 | Perfect 1sn | P1 | D |
1 | 23.529 | Up 1sn | ^1 | ^D |
2 | 47.059 | Downminor 2nd | vm2 | vEb |
3 | 70.588 | Minor 2nd | m2 | Eb |
4 | 94.118 | Upminor 2nd | ^m2 | ^Eb |
5 | 117.647 | Downmid 2nd | v~2 | ^^Eb |
6 | 141.176 | Mid 2nd | ~2 | vvvE, ^^^Eb |
7 | 164.706 | Upmid 2nd | ^~2 | vvE |
8 | 188.235 | Downmajor 2nd | vM2 | vE |
9 | 211.765 | Major 2nd | M2 | E |
10 | 235.294 | Upmajor 2nd | ^M2 | ^E |
11 | 258.824 | Downminor 3rd | vm3 | vF |
12 | 282.353 | Minor 3rd | m3 | F |
13 | 305.882 | Upminor 3rd | ^m3 | ^F |
14 | 329.412 | Downmid 3rd | v~3 | ^^F |
15 | 352.941 | Mid 3rd | ~3 | ^^^F, vvvF# |
16 | 376.471 | Upmid 3rd | ^~3 | vvF# |
17 | 400.000 | Downmajor 3rd | vM3 | vF# |
18 | 423.529 | Major 3rd | M3 | F# |
19 | 447.509 | Upmajor 3rd | ^M3 | ^F# |
20 | 470.588 | Down 4th | v4 | vG |
21 | 494.118 | Perfect 4th | P4 | G |
22 | 517.647 | Up 4th | ^4 | ^G |
23 | 541.176 | Downdim 5th | vd5 | vAb |
24 | 564.706 | Dim 5th | d5 | Ab |
25 | 588.235 | Updim 5th | ^d5 | ^Ab |
26 | 611.765 | Downaug 4th | vA4 | vG# |
27 | 635.294 | Aug 4th | A4 | G# |
28 | 658.824 | Upaug 4th | ^A4 | ^G# |
29 | 682.353 | Down 5th | v5 | vA |
30 | 705.882 | Perfect 5th | P5 | A |
31 | 729.412 | Up 5th | ^5 | ^A |
32 | 752.941 | Downminor 6th | vm6 | vBb |
33 | 776.471 | Minor 6th | m6 | Bb |
34 | 800.000 | Upminor 6th | ^m6 | ^Bb |
35 | 823.529 | Downmid 6th | v~6 | ^^Bb |
36 | 847.059 | Mid 6th | ~6 | vvvB, ^^^Bb |
37 | 870.588 | Upmid 6th | ^~6 | vvB |
38 | 894.118 | Downmajor 6th | vM6 | vB |
39 | 917.647 | Major 6th | M6 | B |
40 | 941.176 | Upmajor 6th | ^M6 | ^B |
41 | 964.706 | Downminor 7th | vm7 | vC |
42 | 988.235 | Minor 7th | m7 | C |
43 | 1011.765 | Upminor 7th | ^m7 | ^C |
44 | 1035.294 | Downmid 7th | v~7 | ^^C |
45 | 1058.824 | Mid 7th | ~7 | ^^^C, vvvC# |
46 | 1082.353 | Upmid 7th | ^~7 | vvC# |
47 | 1105.882 | Downmajor 7th | vM7 | vC# |
48 | 1129.412 | Major 7th | M7 | C# |
49 | 1152.941 | Upmajor 7th | ^M7 | ^C# |
50 | 1176.471 | Down 8ve | v8 | vD |
51 | 1200.000 | Perfect 8ve | P8 | D |
Scales
Palace (subset of Porky[15])
7 7 7 9 7 7 7
UFO scale (inflected MOS of Teefs[19])
2 2 4 1 2 2 2 4 2 5 2 4 4 2 2 1 4 2 2
Cosmic scale (subset of UFO scale)
21 9 4 9 8
Instruments
Lumatone
See Lumatone mapping for 51edo
Music
- Whalectric (2022)
James Mulvale (FAST-fast)
- STARS (Thoughts and Prayers) (51 EDO for 51 States) (2020)