54edo
| ← 53edo | 54edo | 55edo → |
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. In other words, this is the last EDO tempering out 50/49. 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.
The immediate close presence of 53edo obscures 54edo and puts this temperament out of popular usage.
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) | |
Intervals
| 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 | 19/18, 20/19 | |
| 5 | 111.111 | 16/15 | |
| 6 | Extreme bayati neutral second | 133.333 | 13/12 |
| 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 | 8/7, 17/15 |
| 11 | 244.444 | 15/13, 23/20 | |
| 12 | Septimal submajor third | 266.666 | 7/6 |
| 13 | Gothic minor third | 288.888 | 13/11, 20/17 |
| 14 | Classical minor third | 311.111 | 6/5, 19/16 |
| 15 | 333.333 | 17/14 | |
| 16 | 355.555 | 11/9, 16/13 | |
| 17 | Classical major third | 377.777 | 5/4 |
| 18 | Symmetric major third | 400.000 | 29/23 |
| 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 |