81edo
Theory
81edo is notable as a tuning for meantone and related temperaments and is the optimal patent val for a number of them. In particular it is the optimal patent val for 5-limit meantone, 7-limit meantone, 11-limit meanpop, 13-limit meanpop, and the rank three temperament erato. The electronic music pioneer Daphne Oram was interested in 81edo.
As a step in the Golden meantone series of EDOs, 81 EDO marks the point at which the series ceases to display audible changes to meantone temperament, and is also the EDO with the lowest average and most evenly spread Just-error across the scale (though 31 EDO does have the best harmonic 7th).
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.66 | -1.13 | -5.86 | +3.50 | -3.17 | +3.92 | -6.79 | -1.25 | -1.22 | +3.29 | -6.05 |
| Relative (%) | -38.2 | -7.6 | -39.6 | +23.6 | -21.4 | +26.4 | -45.8 | -8.4 | -8.2 | +22.2 | -40.9 | |
| Steps (reduced) |
128 (47) |
188 (26) |
227 (65) |
257 (14) |
280 (37) |
300 (57) |
316 (73) |
331 (7) |
344 (20) |
356 (32) |
366 (42) | |
Notation
81 EDO Accidentals created and used by Tom Winspear, based on those provided in Scala though with a logic correction. The innermost accidentals represent one EDOstep, followed by two, then the bracket representing three. Conventional sharp/doublesharp/flat/doubleflat accidentals are reached in steps of five and the pattern repeats itself on them. The chromatic scale can be notated utilizing only six accidentals in either direction - the rest are for enharmonics.
Regular temperament properties
Commas
- 5-limit commas: 81/80, [-48 1 20⟩
- 7-limit commas: 81/80, 126/125, [-24 1 0 8⟩
- 11-limit commas: 81/80, 126/125, 385/384, 12005/11979
- 13-limit commas: 81/80, 105/104, 144/143, 196/195, 6655/6591