89edo
Theory
89et tempers out the commas 126/125, 1728/1715, 32805/32768, 2401/2400, 176/175, 243/242, 441/440 and 540/539. It is an especially good tuning for the myna temperament, both in the 7-limit, tempering out 126/125 and 1728/1715, and in the 11-limit, where 176/175 is tempered out also. It is likewise a good tuning for the rank-3 temperament thrush, tempering out 126/125 and 176/175.
89 has a fifth less than a cent flat and a major third less than five cents sharp, with a 7 two cents sharp and an 11 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas. On a related note, a notable characteristic of this edo is that it is the lowest in a series of four consecutive edos to temper out quartisma.
89edo is the 24th prime edo, and the 11th in the Fibonacci sequence, which means its 55th step approximates logarithmic φ (i.e. (φ - 1)×1200 cents) within a fraction of a cent.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.83 | +4.70 | +1.96 | +1.49 | -4.57 | +2.91 | -0.88 | +5.43 | -4.86 | +1.03 |
| Relative (%) | +0.0 | -6.2 | +34.8 | +14.5 | +11.1 | -33.9 | +21.6 | -6.6 | +40.3 | -36.0 | +7.7 | |
| Steps (reduced) |
89 (0) |
141 (52) |
207 (29) |
250 (72) |
308 (41) |
329 (62) |
364 (8) |
378 (22) |
403 (47) |
432 (76) |
441 (85) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-141 89⟩ | [⟨89 141]] | +0.262 | 0.262 | 1.95 |
| 2.3.5 | 32805/32768, 10077696/9765625 | [⟨89 141 207]] | -0.500 | 1.098 | 8.15 |
| 2.3.5.7 | 126/125, 1728/1715, 32805/32768 | [⟨89 141 207 250]] | -0.550 | 0.955 | 7.08 |
| 2.3.5.7.11 | 126/125, 176/175, 243/242, 16384/16335 | [⟨89 141 207 250 308]] | -0.526 | 0.855 | 6.35 |