21edo
21edo is an equal division of the octave into 21 steps of 1200c/21 ~= 57.1c each.
21edo is unusual 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) approximations. On the one hand, one could say 21edo has a 5 since its 15/8 is accurate due to error cancellation of the 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; the reverse is true of certain triad inversions, since 0-514-914 is close to being +1+1 DR (approximately 23:31:39).
Notable scales:
- Archylino diatonic (2L3m2s): 3423432 or 4323432
- Interseptimal diatonic (4L1m2s): 4143414
- 21edo is the first edo with a diasem scale: 323132313 (RH) or 313231323 (LH). Diasem provides basic 2.3.7 harmony, though 7/6, 28/27, and 9/7 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. This allows all intervals to have a "minor", "neutral/perfect", and "major" variant.
| Edostep | Cents | Notation (Ups and downs) | Interval name |
|---|---|---|---|
| 0 | 0 | C | Perfect unison |
| 1 | 57.1 | ^C | Up unison |
| 2 | 114.3 | vD | Down second |
| 3 | 171.4 | D | Perfect second |
| 4 | 228.6 | ^D | Up second |
| 5 | 285.7 | vE | Down third |
| 6 | 342.9 | E | Perfect third |
| 7 | 400 | ^E | Up third |
| 8 | 457.1 | vF | Down fourth |
| 9 | 514.3 | F | Perfect fourth |
| 10 | 571.4 | ^F | Up fourth |
| 11 | 628.6 | vG | Down fifth |
| 12 | 685.7 | G | Perfect fifth |
| 13 | 742.9 | ^G | Up fifth |
| 14 | 800 | vA | Down sixth |
| 15 | 857.1 | A | Perfect sixth |
| 16 | 914.3 | ^A | Up sixth |
| 17 | 971.4 | vB | Down seventh |
| 18 | 1028.6 | B | Perfect seventh |
| 19 | 1085.7 | ^B | Up seventh |
| 20 | 1142.9 | vC | Down octave |
| 21 | 1200 | C | Octave |
Prime harmonic approximations
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -16.2 | +13.7 | +2.6 | +20.1 | +16.6 | +9.3 | -11.8 | +0.3 | -1.0 |
| Relative (%) | 0.0 | -28.4 | +24.0 | +4.6 | +35.2 | +29.1 | +16.3 | -20.6 | +0.5 | -1.8 | |
| Steps
(reduced) |
21
(0) |
33
(12) |
49
(7) |
59
(17) |
73
(10) |
78
(15) |
86
(2) |
89
(5) |
95
(11) |
102
(18) | |
Erac group
As a temperament, 21edo may be described using eracs: 2.x>3.x<5.7.x<11.x<13.23.29. These specific eracs indicate that the primes are about 1/3 of an edostep off, and that 63edo is an accurate system.
Edostep interpretations
21edo's edostep has the following interpretations in the 2.3.5.7.23.29 subgroup:
- 24/23
- 30/29
- 29/28
- 49/48
- 50/49
- 46/45
- 64/63
Multiples
63edo
63edo provides a good representation of 2.3.5.7.11.13.23.29.31.
- The 3 is somewhat sharp, thus supporting Parapyth temperament, a rank-3 temperament 32/27 is tuned close to 13/11 and 81/64 is tempered together with 14/11, and where the "spacer" 28/27 is identified with 33/32.
- The 5 is quite flat, thus supporting Magic temperament where the stack of five 5/4 major thirds becomes one 3/1.
