26edo
26edo, or 26 equal divisions of the octave, is the equal tuning featuring steps of (1200/26) ~= 46.15 cents, 26 of which stack to the perfect octave 2/1.
Theory
JI approximation
26edo is characterized by a flat tuning of harmonics 3, 5, and 13 and slightly sharp but accurate tunings of 7 and 11. Although its primes 3, 5, and 13 are damaged, 26edo can be used as a 13-limit temperament as it is consistent to the 13-odd-limit. Additionally, the fact that primes 3 and 13 are flat by about the same amount and 5 is flat by about double that means that intervals such as 13/12 and 10/9 are approximated well. The accurate 7 combined with the flat 5 means that 7/5 and 10/7 are both mapped to the 600¢ half octave tritone, tempering out 50/49. 16/13 and 11/9 are mapped to the same interval as 5/4, tempering out 65/64, 144/143, and 45/44.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -9.6 | -17.1 | +0.4 | +2.5 | -9.8 | -12.6 | -20.6 | +17.9 | -14.2 | +8.8 |
| Relative (%) | 0.0 | -20.9 | -37.0 | +0.9 | +5.5 | -21.1 | -27.4 | -44.6 | +38.7 | -30.8 | +19.1 | |
| Steps
(reduced) |
26
(0) |
41
(15) |
60
(8) |
73
(21) |
90
(12) |
96
(18) |
106
(2) |
110
(6) |
118
(14) |
126
(22) |
129
(25) | |
Edostep interpretations
26edo's edostep has the following 13-limit interpretations:
- 25/24 (the difference between 5/4 and 6/5)
- 33/32 (the difference between 4/3 and 11/8)
- 36/35 (the difference between 5/4 and 9/7)
- 49/48 (the difference between 8/7 and 7/6)
Intervals and notation
Similar to 19edo, 26edo can be notated entirely with standard diatonic notation, with #/b = 1\26 and x/bb = 2\26. The equivalences are Cx = Dbb, E# = Fbb, and Ex = Fb.
Compositional theory
| Quality | Farminor | Supraminor | Submajor | Farmajor |
|---|---|---|---|---|
| Cents | 277 | 323.1 | 369.2 | 415.4 |
| Just interpretation | 7/6 | 6/5 | 5/4, 16/13 | 14/11, 9/7 |
| Steps | 6 | 7 | 8 | 9 |
Chords
TODO:
- write about flattone
Scales
| View • Talk • EditEqual temperaments | |
|---|---|
| EDOs | |
| Macrotonal | 5 • 7 • 8 • 9 • 10 • 11 |
| 12-23 | 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 |
| 24-35 | 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35 |
| 36-47 | 36 • 37 • 39 • 40 • 41 • 43 • 44 • 45 • 46 • 47 |
| 48-59 | 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58 |
| 60-71 | 60 • 63 • 64 • 65 • 67 • 68 • 70 |
| 72-83 | 72 • 77 • 80 • 81 |
| 84-95 | 84 • 87 • 89 • 90 • 93 • 94 |
| Large EDOs | 99 • 104 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 217 • 224 • 239 • 270 • 306 • 311 • 612 • 665 |
| Nonoctave equal temperaments | |
| Tritave | 4 • 9 • 13 • 17 • 26 • 39 |
| Fifth | 8 • 9 • 11 • 20 |
| Other | |
