54edo
| ← 53edo | 54edo | 55edo → |
54 equal divisions of the octave (abbreviated 54edo or 54ed2), also called 54-tone equal temperament (54tet) or 54 equal temperament (54et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 54 equal parts of about 22.2 ¢ each. Each step represents a frequency ratio of 21/54, or the 54th root of 2.
Theory
54edo is suitable for usage with dual-fifth tuning systems, or alternately, no-fifth tuning systems. 54edo has an ultrahard diatonic scale using the sharp fifth of 27edo and an ultrasoft diatonic using the flat fifth. The soft diatonic scale is so soft, with L/s = 8/7, that it stops sounding like meantone or even flattone, but just sounds like a circulating temperament of 7edo.
It's a rare temperament which adds better approximations of the 11th and 15th harmonics from 27edo, which it doubles. 54edo contains an alternate (flat) mapping of the fifth and an "extreme bayati" 6 6 10 10 2 10 10 diatonic scale.
Using the patent val, 54edo tempers out 2048/2025 in the 5-limit, making it a diaschismic system. It is the highest EDO in which the best mappings of the major 3rd (5/4) and harmonic 7th (7/4), 17\54 and 44\54, are exactly 600 cents apart, making them suitable for harmonies using tritone substitutions. In other words, this is the last EDO tempering out 50/49. This means it extends quite simply to the 7 and 11 limits using the pajarous mapping and to the 13-limit using the f val, falling neatly between the 7 and 13 limit minimax tunings. The 54cd val makes for an excellent tuning of 7-limit hexe temperament, while the bdf val does higher limit muggles about as well as it can be tuned. However, even these best temperament interpretations are quite high in badness compared to it's immediate neighbours 53 & 55edo, both of which are historically significant for different reasons, leaving it mostly unexplored so far.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +9.16 | -8.54 | +8.95 | -3.91 | +4.24 | +3.92 | +0.62 | +6.16 | -8.62 | -4.11 | -6.05 |
| relative (%) | +41 | -38 | +40 | -18 | +19 | +18 | +3 | +28 | -39 | -19 | -27 | |
| Steps (reduced) |
86 (32) |
125 (17) |
152 (44) |
171 (9) |
187 (25) |
200 (38) |
211 (49) |
221 (5) |
229 (13) |
237 (21) |
244 (28) | |
Intervals
Using the sharp fifth as a generator, 54edo require an incredibly large amount of ups and downs to notate, and using the flat fifth as a generator, 54edo requires an incredibly large amount of sharps and flats to notate. Because the flat fifth generates a diatonic scale with a chroma of 1 step, ups and downs are not needed in notation if the flat fifth is used.
| Degree | Cents | Ups and downs notation (flat fifth 31\54) | Ups and downs notation (sharp fifth 16\27) |
|---|---|---|---|
| 0 | 0.000 | D | D |
| 1 | 22.222 | D♯, E♭♭♭♭♭♭♭ | ^D, vE♭ |
| 2 | 44.444 | D𝄪, E♭♭♭♭♭♭ | ^^D, E♭ |
| 3 | 66.667 | D♯𝄪, E♭♭♭♭♭ | ^3D, v7E |
| 4 | 88.889 | D𝄪𝄪, E♭♭♭♭ | ^4D, v6E |
| 5 | 111.111 | D♯𝄪𝄪, E♭♭♭ | ^5D, v5E |
| 6 | 133.333 | D𝄪𝄪𝄪, E♭♭ | ^6D, v4E |
| 7 | 155.556 | D♯𝄪𝄪𝄪, E♭ | ^7D, v3E |
| 8 | 177.778 | E | D♯, vvE |
| 9 | 200.000 | E♯, F♭♭♭♭♭♭ | ^D♯, vE |
| 10 | 222.222 | E𝄪, F♭♭♭♭♭ | E |
| 11 | 244.444 | E♯𝄪, F♭♭♭♭ | ^E, vF |
| 12 | 266.667 | E𝄪𝄪, F♭♭♭ | F |
| 13 | 288.889 | E♯𝄪𝄪, F♭♭ | ^F, vG♭ |
| 14 | 311.111 | E𝄪𝄪𝄪, F♭ | ^^F, G♭ |
| 15 | 333.333 | F | ^3F, v7G |
| 16 | 355.556 | F♯, G♭♭♭♭♭♭♭ | ^4F, v6G |
| 17 | 377.778 | F𝄪, G♭♭♭♭♭♭ | ^5F, v5G |
| 18 | 400.000 | F♯𝄪, G♭♭♭♭♭ | ^6F, v4G |
| 19 | 422.222 | F𝄪𝄪, G♭♭♭♭ | ^7F, v3G |
| 20 | 444.444 | F♯𝄪𝄪, G♭♭♭ | F♯, vvG |
| 21 | 466.667 | F𝄪𝄪𝄪, G♭♭ | ^F♯, vG |
| 22 | 488.889 | F♯𝄪𝄪𝄪, G♭ | G |
| 23 | 511.111 | G | ^G, vA♭ |
| 24 | 533.333 | G♯, A♭♭♭♭♭♭♭ | ^^G, A♭ |
| 25 | 555.556 | G𝄪, A♭♭♭♭♭♭ | ^3G, v7A |
| 26 | 577.778 | G♯𝄪, A♭♭♭♭♭ | ^4G, v6A |
| 27 | 600.000 | G𝄪𝄪, A♭♭♭♭ | ^5G, v5A |
| 28 | 622.222 | G♯𝄪𝄪, A♭♭♭ | ^6G, v4A |
| 29 | 644.444 | G𝄪𝄪𝄪, A♭♭ | ^7G, v3A |
| 30 | 666.667 | G♯𝄪𝄪𝄪, A♭ | G♯, vvA |
| 31 | 688.889 | A | ^G♯, vA |
| 32 | 711.111 | A♯, B♭♭♭♭♭♭♭ | A |
| 33 | 733.333 | A𝄪, B♭♭♭♭♭♭ | ^A, vB♭ |
| 34 | 755.556 | A♯𝄪, B♭♭♭♭♭ | ^^A, B♭ |
| 35 | 777.778 | A𝄪𝄪, B♭♭♭♭ | ^3A, v7B |
| 36 | 800.000 | A♯𝄪𝄪, B♭♭♭ | ^4A, v6B |
| 37 | 822.222 | A𝄪𝄪𝄪, B♭♭ | ^5A, v5B |
| 38 | 844.444 | A♯𝄪𝄪𝄪, B♭ | ^6A, v4B |
| 39 | 866.667 | B | ^7A, v3B |
| 40 | 888.889 | B♯, C♭♭♭♭♭♭ | A♯, vvB |
| 41 | 911.111 | B𝄪, C♭♭♭♭♭ | ^A♯, vB |
| 42 | 933.333 | B♯𝄪, C♭♭♭♭ | B |
| 43 | 955.556 | B𝄪𝄪, C♭♭♭ | ^B, vC |
| 44 | 977.778 | B♯𝄪𝄪, C♭♭ | C |
| 45 | 1000.000 | B𝄪𝄪𝄪, C♭ | ^C, vD♭ |
| 46 | 1022.222 | C | ^^C, D♭ |
| 47 | 1044.444 | C♯, D♭♭♭♭♭♭♭ | ^3C, v7D |
| 48 | 1066.667 | C𝄪, D♭♭♭♭♭♭ | ^4C, v6D |
| 49 | 1088.889 | C♯𝄪, D♭♭♭♭♭ | ^5C, v5D |
| 50 | 1111.111 | C𝄪𝄪, D♭♭♭♭ | ^6C, v4D |
| 51 | 1133.333 | C♯𝄪𝄪, D♭♭♭ | ^7C, v3D |
| 52 | 1155.556 | C𝄪𝄪𝄪, D♭♭ | C♯, vvD |
| 53 | 1177.778 | C♯𝄪𝄪𝄪, D♭ | ^C♯, vD |
| 54 | 1200.000 | D | D |
Instruments
Lumatone