21edo
21edo is an equal division of the octave into 21 steps of 1200c/21 ~= 57.1c each.
21edo is somewhat strange from a regular temperament perspective due to its mixture of very accurate (e.g. 23/16, 16/15) and very inaccurate (3/2, 6/5, 7/6, 5/4) approximations. On the one hand, one could say 21edo has a 5 since its 15/8 is accurate due to error cancellation of extremely sharp 5 with the flat 3. In root-position major triads, though, the 400c major third sounds much less isodifferential than even in 12edo due to the 3/2 being much flatter, and root-position major and minor triads sound somewhat neogothic as a result and also because of the neominor third; something of the reverse is true of triad inversions, since 0-514-914 is close to being isodifferential (approximately 26:35:44).
Notable scales:
- Archylino scale: 3423432 or 4323432
- 21edo is the first edo with a diasem scale: 323132313 (RH) or 313231323 (LH). Diasem provides basic 2.3.7 harmony, though 7/6 and 28/27 are not accurate at all in 21edo.
- Slentonic (5L6s, sLsLsLsLsLs), interpreted as Slendric[11], generated by stacking the ~8/7 (4\21)
- Oneirotonic (5L3s, LLsLLsLs), generated by stacking 8\21
Basic theory
Intervals and notation
Since 21edo's best fifth is from 7edo, 21edo is notated with CDEFGAB representing 7edo and up/down representing 1\21.
Prime harmonic approximations
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -16.2 | +13.7 | +2.6 | +20.1 | +16.6 | +9.3 | -11.8 | +0.3 | -1.0 | -2.2 |
| Relative (%) | 0.0 | -28.4 | +24.0 | +4.6 | +35.2 | +29.1 | +16.3 | -20.6 | +0.5 | -1.8 | -3.8 | |
| Steps
(reduced) |
21
(0) |
33
(12) |
49
(7) |
59
(17) |
73
(10) |
78
(15) |
86
(2) |
89
(5) |
95
(11) |
102
(18) |
104
(20) | |
Edostep interpretations
21edo's edostep has the following interpretations in the 2.3.5.7.23.29.31 subgroup:
- 32/31
- 31/30
- 30/29
- 29/28
- 49/48
- 46/45
- 64/63
