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
Multiples
104edo
104edo is a strong no-5 Parapyth tuning.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +1.9 | -5.5 | +0.4 | +2.5 | +1.8 | -1.1 | +2.5 | -5.2 | -2.7 | -2.7 | +2.5 | -2.1 | -3.8 | +3.7 |
| Relative (%) | 0.0 | +16.4 | -48.1 | +3.5 | +21.9 | +15.4 | -9.6 | +21.6 | -45.0 | -23.0 | -23.6 | +21.7 | -18.5 | -33.2 | +32.3 | |
| Steps
(reduced) |
104
(0) |
165
(61) |
241
(33) |
292
(84) |
360
(48) |
385
(73) |
425
(9) |
442
(26) |
470
(54) |
505
(89) |
515
(99) |
542
(22) |
557
(37) |
564
(44) |
578
(58) | |
