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 as a dual-fifth tuning system, or alternatively, a no-fifth tuning system. Using the sharp fifth, it can be viewed as two rings of 27edo, which adds better approximations of the 11th and 15th harmonics. Using the flat fifth, it generates an ultrasoft diatonic scale. This 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.
The patent val of this edo takes the same fifth as 27edo, but the mapping for harmonic 5 is different. It 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 ¢ 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-limit using the pajarous mapping and to the 13-limit using the 54f 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 54bdf val does higher limit muggles about as well as it can be tuned. However, even these best temperament interpretations of 54edo are quite high in badness compared to its immediate neighbours 53- and 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 (%) | +41.2 | -38.4 | +40.3 | -17.6 | +19.1 | +17.6 | +2.8 | +27.7 | -38.8 | -18.5 | -27.2 | |
| Steps (reduced) |
86 (32) |
125 (17) |
152 (44) |
171 (9) |
187 (25) |
200 (38) |
211 (49) |
221 (5) |
229 (13) |
237 (21) |
244 (28) | |
Octave stretch
54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by APS4/5méride, a stretched-octave version of 54edo. The trade-off is a slightly worse 2/1.
If one prefers a compressed-octave tuning instead, 86edt, 126ed5 and 152ed7 are possible choices. They improve upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1.
54ed255/128 is another compressed octave option. It improves upon 54edo’s 3/1, 5/1, 11/1, 13/1, 17/1 and 19/1, at slight cost to the 2/1 and 7/1. Its 2/1 is the least accurate of all the tunings mentioned in this section, though still accurate enough that it has low harmonic entropy.
There are also some nearby Zeta peak index (ZPI) tunings which can be used to improve 54edo’s approximation of JI: 262zpi, 263zpi, 264zpi and 265zpi. The main Zeta peak index page details all four tunings.
Subsets and supersets
Since 54 factors into 2 × 33, 54edo has subset edos 2, 3, 6, 9, 18, and 27.
Intervals
Using the sharp fifth as a generator, 54edo requires up to quadruple ups and downs to notate. But using the flat fifth as a generator, it requires up to septuple sharps and flats. 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) | 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, ^E♭ |
| 4 | 88.889 | D𝄪𝄪, E♭♭♭♭ | ^4D, ^^E♭ |
| 5 | 111.111 | D♯𝄪𝄪, E♭♭♭ | v3D♯, ^3E♭ |
| 6 | 133.333 | D𝄪𝄪𝄪, E♭♭ | vvD♯, v4E |
| 7 | 155.556 | D♯𝄪𝄪𝄪, E♭ | vD♯, 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, ^G♭ |
| 16 | 355.556 | F♯, G♭♭♭♭♭♭♭ | ^4F, ^^G♭ |
| 17 | 377.778 | F𝄪, G♭♭♭♭♭♭ | v3F♯, ^3G♭ |
| 18 | 400.000 | F♯𝄪, G♭♭♭♭♭ | vvF♯, v4G |
| 19 | 422.222 | F𝄪𝄪, G♭♭♭♭ | vF♯, 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, ^A♭ |
| 26 | 577.778 | G♯𝄪, A♭♭♭♭♭ | ^4G, ^^A♭ |
| 27 | 600.000 | G𝄪𝄪, A♭♭♭♭ | v3G♯, ^3A♭ |
| 28 | 622.222 | G♯𝄪𝄪, A♭♭♭ | vvG♯, v4A |
| 29 | 644.444 | G𝄪𝄪𝄪, A♭♭ | vG♯, 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, ^B♭ |
| 36 | 800.000 | A♯𝄪𝄪, B♭♭♭ | ^4A, ^^B♭ |
| 37 | 822.222 | A𝄪𝄪𝄪, B♭♭ | v3A♯, ^3B♭ |
| 38 | 844.444 | A♯𝄪𝄪𝄪, B♭ | vvA♯, v4B |
| 39 | 866.667 | B | vA♯, 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, ^D♭ |
| 48 | 1066.667 | C𝄪, D♭♭♭♭♭♭ | ^4C, ^^D♭ |
| 49 | 1088.889 | C♯𝄪, D♭♭♭♭♭ | v3C♯, ^3D♭ |
| 50 | 1111.111 | C𝄪𝄪, D♭♭♭♭ | vvC♯, v4D |
| 51 | 1133.333 | C♯𝄪𝄪, D♭♭♭ | vC♯, v3D |
| 52 | 1155.556 | C𝄪𝄪𝄪, D♭♭ | C♯, vvD |
| 53 | 1177.778 | C♯𝄪𝄪𝄪, D♭ | ^C♯, vD |
| 54 | 1200.000 | D | D |
Notation
Ups and downs notation
Using Helmholtz–Ellis accidentals, 54edo can also be notated using ups and downs notation:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
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| Flat symbol |  |
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Here, a sharp raises by eight steps, and a flat lowers by eight steps, so single, double, and triple arrows along with Stein–Zimmerman quarter-tone accidentals can be used to fill in the gap.
Sagittal notation
This notation uses the same sagittal sequence as 61-EDO, and is a superset of the notation for 27-EDO.
Evo flavor
Revo flavor
Evo-SZ 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.
Instruments
- Lumatone