20160edo
| ← 20159edo | 20160edo | 20161edo → |
20160 equal divisions of the octave (abbreviated 20160edo or 20160ed2), also called 20160-tone equal temperament (20160tet) or 20160 equal temperament (20160et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 20160 equal parts of about 0.0595 ¢ each. Each step represents a frequency ratio of 21/20160, or the 20160th root of 2.
20160edo is the 23rd highly composite edo, although it is not a member of the superabundant edos. It is also a highly factorable number edo, carrying the trait of being broken down into subsets into more ways than any number before it.
Aside from divisors, it is an excellent no-7s 29-limit system, having errors of 25% or less, and with the 28% error on the 7th harmonic it is overall a satisfactory 29-limit tuning, making it a candidate for an interval size measure.
Eliora proposes that one step of 20160edo be called pian, since piano manufacturers have a tendency to avoid the prominence of 7th harmonic in their sound. A semitone is therefore 1680 pians, a step of 224edo is 90 pians, and the Dröbisch angle, one step of 360edo, is 56 pians.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0093 | -0.0042 | -0.0164 | -0.0084 | +0.0081 | -0.0149 | -0.0130 | -0.0005 | +0.0061 | +0.0240 |
| Relative (%) | +0.0 | +15.6 | -7.0 | -27.5 | -14.1 | +13.5 | -25.1 | -21.9 | -0.9 | +10.3 | +40.2 | |
| Steps (reduced) |
20160 (0) |
31953 (11793) |
46810 (6490) |
56596 (16276) |
69742 (9262) |
74601 (14121) |
82403 (1763) |
85638 (4998) |
91195 (10555) |
97937 (17297) |
99877 (19237) | |