26edo
26edo, or 26 equal divisions of the octave (sometimes called 26-TET or 26-tone equal temperament), is the equal tuning featuring steps of (1200/26) ~= 46.15 cents, 26 of which stack to the perfect octave 2/1.
26edo has a perfect fifth, 692.3¢, which is tuned even flatter than that of 19edo. Its diatonic scale is thus very soft (homoioheptatonic). Its thirds can still be taken, if inaccurately, to approximate 6/5 and 5/4, supporting Meantone. In terms of 7-limit properties, 26edo is notably the smallest EDO to distinguish all of 9/8, 8/7, 7/6, 6/5, and 5/4 (although "9/8" in particular is far closer to 10/9), and does so consistently.
Where 26edo truly shines, however, is in higher limits. We can observe it closely approximates both the 7th and 11th harmonics. Structurally, 26edo's fifth spans 15 edosteps, which means that it can be split into 3 parts and into 5. Both splits result in almost perfectly just intervals: 8/7 serves as 1/3 (5 edosteps), resulting in Slendric temperament, and 13/12 as 1/5 of the fifth (3 edosteps). These intervals are tuned particularly well as a result of 26edo approximating the natural fifth of e2/5. Anchored by 8/7, 11/8, and 13/12, 26edo consistently represents the 13-odd-limit, beating 22edo's consistency record of 11. Prime 17 can also be included in the mix, as it is tuned similarly to 13 and 3, although intervals of 15 are inconsistently mapped.
26edo, despite ostensibly supporting familiar harmonic organization in the form of Meantone, is overall very xenharmonic in structure due to its flat tuning. Furthermore, as a claimant for the smallest EDO to merit consideration as a 17-limit system, and with primes 7 and 11 tuned far more accurately than 3 and 5, 26edo is extremely capable of supporting harmonic systems that rely at most minimally on the diatonic scale or 5-limit harmony at all.
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.
| Edostep | Cents | Notation | 13-limit JI approximation | ADIN interval category |
|---|---|---|---|---|
| 0 | 0 | C | 1/1 | unison |
| 1 | 46.2 | C# | 25/24, 33/32, 36/35, 49/48 | superunison |
| 2 | 92.3 | Cx, Dbb | 21/20, 22/21, 26/25 | farminor second |
| 3 | 138.5 | Db | 12/11, 13/12, 14/13 | supraminor second |
| 4 | 184.6 | D | 9/8, 10/9, 11/10 | submajor second |
| 5 | 230.8 | D# | 8/7 | supermajor second |
| 6 | 276.9 | Dx, Ebb | 7/6, 13/11 | farminor third |
| 7 | 323.1 | Eb | 6/5 | supraminor third |
| 8 | 369.2 | E | 5/4, 16/13, 11/9 | submajor third |
| 9 | 415.4 | E#, Fbb | 14/11, 9/7 | farmajor third |
| 10 | 461.5 | Ex, Fb | 21/16, 13/10 | subfourth |
| 11 | 507.7 | F | 4/3 | perfect fourth |
| 12 | 553.8 | F# | 11/8 | subaugmented fourth |
| 13 | 600 | Fx, Gbb | 7/5, 10/7 | tritone |
| 14 | 646.2 | Gb | 16/11 | supradiminished fifth |
| 15 | 692.3 | G | 3/2 | perfect fifth |
| 16 | 738.5 | G# | 32/21, 20/13 | superfifth |
| 17 | 784.6 | Gx, Abb | 11/7, 14/9 | farminor sixth |
| 18 | 830.8 | Ab | 8/5, 13/8, 18/11 | supraminor sixth |
| 19 | 876.9 | A | 5/3 | submajor sixth |
| 20 | 923.1 | A# | 12/7, 22/13 | farmajor sixth |
| 21 | 969.2 | Ax, Bbb | 7/4 | subminor seventh |
| 22 | 1015.4 | Bb | 9/5, 16/9, 20/11 | supraminor seventh |
| 23 | 1061.5 | B | 11/6, 13/7, 24/13 | submajor seventh |
| 24 | 1107.7 | B#, Cbb | 21/11, 25/13, 40/21 | farmajor seventh |
| 25 | 1153.8 | Bx, Cb | 64/33, 96/49, 35/18, 48/25 | suboctave |
| 26 | 1200 | C | 2/1 | octave |
Compositional theory
| Quality | Farminor | Supraminor | Submajor | Farmajor |
|---|---|---|---|---|
| Cents | 276.9 | 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) | |
130edo
130edo adds 26edo's accurate 7/4 and 10edo's accurate 13/8 to 65edo, resulting in a strong 2.3.5.7.11.13.19.23.31.47 system. It is a good Hemiwurschmidt tuning. It is also useful as an example for interval categorization.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -0.4 | +1.4 | +0.4 | +2.5 | -0.5 | -3.4 | -2.1 | -0.6 | +4.3 | -0.4 | -2.1 | -4.4 | -3.8 | -0.9 |
| Relative (%) | 0.0 | -4.5 | +14.9 | +4.4 | +27.4 | -5.7 | -37.0 | -23.1 | -6.3 | +46.2 | -4.6 | -22.9 | -48.2 | -41.4 | -9.7 | |
| Steps
(reduced) |
130
(0) |
206
(76) |
302
(42) |
365
(105) |
450
(60) |
481
(91) |
531
(11) |
552
(32) |
588
(68) |
632
(112) |
644
(124) |
677
(27) |
696
(46) |
705
(55) |
722
(72) | |
