54edo
54-EDO is an equal temperament that divides the octave into 54 equal parts, each 22.2222 cents in size.
The immediate close presence of 53edo obscures 54edo and puts this temperament out of popular usage.
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Theory
54edo is suitable for usage with dual-fifth tuning systems, or alternately, no-fifth tuning systems.
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.
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. 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.
Using the patent val, 54edo tempers out 2048/2025 in the 5-limit.
| Degree | Name | Cents | Approximate Ratios |
|---|---|---|---|
| 0 | Natural Unison | 0.000 | |
| 1 | Ninth-tone | 22.222 | |
| 2 | Extreme bayati quarter-tone | 44.444 | |
| 3 | Third-tone | 66.666 | |
| 4 | 88.888 | ||
| 5 | 111.111 | ||
| 6 | Extreme bayati neutral second | 133.333 | |
| 7 | 155.555 | ||
| 8 | Minor whole tone | 177.777 | 10/9 |
| 9 | Symmetric whole tone | 200.000 | 9/8 |
| 10 | Extreme bayati whole tone | 222.222 | |
| 12 | Septimal submajor third | 266.666 | 7/6 |
| 17 | Classical major third | 377.777 | 5/4 |
| 18 | Symmetric major third | 400.000 | 29/23, 19/16 |
| 25 | Undecimal superfourth | 555.555 | 11/8 |
| 26 | Septimal minor tritone | 577.777 | 7/5 |
| 27 | Symmetric tritone | 600.000 | |
| 28 | Septimal major tritone | 633.333 | 10/7 |
| 36 | Symmetric augmented fifth | 800.000 | |
| 44 | Harmonic seventh | 977.777 | 7/4 |
| 54 | Octave | 1200.000 | Exact 2/1 |