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 (to within half a cent for the former and 3 cents for the latter). 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 overall, despite ostensibly supporting familiar harmonic organization in the form of Meantone, presents that organization vastly differently from 12edo due to its flat tuning. Furthermore, 26edo includes 13edo as a subset, and with it, the oneirotonic scale. And 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 quite 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, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5¢ JND, and therefore 26edo does so as well.
With the very flat tuning of 5, the 4:5:6 triad has more of a submajor quality, and 25/24, the distinction between the main 5-limit triads, is reduced to the size of a quartertone. Similarly to 22edo, this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary chthonic medials), and, as while 7 is accurate, 5 is flat enough in 26edo that 7/5 and 10/7 are both mapped to the 600¢ half-octave tritone.
26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also 11/10; this is characteristic of Flattone, a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that 11/9 is mapped to the same interval as 5/4, and to this we can add the successive mediants, 16/13 as well as 21/17 and 26/21. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.
| 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 interpretations in the 11-limit:
- 25/24 (the difference between 6/5 and 5/4)
- 33/32 (the difference between 4/3 and 11/8)
- 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
- 49/48 (the difference between 8/7 and 7/6)
- 64/63 (the difference between 9/8 and 8/7)
- 80/77 (the difference between 11/10 and 8/7)
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. JI approximations are given in the 17-limit 21-odd-limit, aside from the single edostep and its complement. Approximations within 3¢ are given in [brackets]. Harmonics 3-21 are bolded; inconsistent intervals (involving 15) are italicized.
| Edostep | Cents | Notation | 17-limit JI approximation | ADIN interval category |
|---|---|---|---|---|
| 0 | 0 | C | 1/1 | unison |
| 1 | 46.2 | C# | 25/24, 33/32, [36/35], 49/48, 64/63 | superunison |
| 2 | 92.3 | Cx, Dbb | 15/14, 17/16, 18/17, 21/20, 22/21 | farminor second |
| 3 | 138.5 | Db | 12/11, [13/12], 14/13, 16/15 | (supra)minor second |
| 4 | 184.6 | D | 9/8, [10/9], 11/10 | (sub)major second |
| 5 | 230.8 | D# | [8/7], 15/13, 17/15 | farmajor second |
| 6 | 276.9 | Dx, Ebb | 7/6, 13/11, 20/17 | farminor third |
| 7 | 323.1 | Eb | 6/5, 17/14 | (supra)minor third |
| 8 | 369.2 | E | 5/4, 11/9, 16/13, 21/17, [26/21] | (sub)major third |
| 9 | 415.4 | E#, Fbb | 9/7, [14/11] | farmajor third |
| 10 | 461.5 | Ex, Fb | 13/10, [17/13], 21/16, 22/17 | subfourth |
| 11 | 507.7 | F | 4/3, 15/11 | perfect fourth |
| 12 | 553.8 | F# | [11/8], 18/13 | (sub)augmented fourth |
| 13 | 600 | Fx, Gbb | 7/5, 10/7, 17/12, 24/17 | tritone |
| 14 | 646.2 | Gb | 13/9, [16/11] | (supra)diminished fifth |
| 15 | 692.3 | G | 3/2, 22/15 | perfect fifth |
| 16 | 738.5 | G# | 17/11, 20/13, [26/17], 32/21 | superfifth |
| 17 | 784.6 | Gx, Abb | [11/7], 14/9 | farminor sixth |
| 18 | 830.8 | Ab | 8/5, 13/8, 18/11, [21/13], 34/21 | (supra)minor sixth |
| 19 | 876.9 | A | 5/3, 28/17 | (sub)major sixth |
| 20 | 923.1 | A# | 12/7, 17/10, 22/13 | farmajor sixth |
| 21 | 969.2 | Ax, Bbb | [7/4], 26/15, 30/17 | farminor seventh |
| 22 | 1015.4 | Bb | [9/5], 16/9, 20/11 | (supra)minor seventh |
| 23 | 1061.5 | B | 11/6, 13/7, 15/8, [24/13] | (sub)major seventh |
| 24 | 1107.7 | B#, Cbb | 17/9, 21/11, 28/15, 32/17, 40/21 | farmajor seventh |
| 25 | 1153.8 | Bx, Cb | [35/18], 48/25, 63/32, 64/33, 96/49 | suboctave |
| 26 | 1200 | C | 2/1 | octave |
Compositional theory
Tertian structure
| Quality | Farminor | Supraminor | Submajor | Farmajor |
|---|---|---|---|---|
| Cents | 276.9 | 323.1 | 369.2 | 415.4 |
| Just interpretation | 7/6 (+10.0¢) | 6/5 (+7.4¢) | 11/9 (+22.2¢) 16/13 (+9.8¢) 5/4 (-17.1¢) |
14/11 (-2.1¢) 9/7 (-19.7¢) |
| Steps | 6 | 7 | 8 | 9 |
Diatonic thirds are bolded.
Chords
TODO:
- write about flattone
Scales
Organization of MOSes
26edo has six distinct intervals that define octave-periodic generator structures, not counting those of 13edo. These generator structures and consequent MOS scales organize themselves into two loops of three, each linked by the operation of tripling.
| Loop 1 | |||
|---|---|---|---|
| Temperament | Slendric | Flattone | Superkleismic |
| Scale (albitonic) | 5-5-5-5-6 | 4-4-3-4-4-4-3 | 2-5-2-5-2-5-5 |
| Generator | 5\, 21\ | 11\, 15\ | 7\, 19\ |
| Loop 2 | |||
| Temperament | Quartonic | Bleu | Roman |
| Scale (albitonic) | - (11-note scale has >10 steps interval) |
3-3-3-3-3-3-3-3-2 | 7-1-7-1-1-7-1-1 |
| Generator | 1\, 25\ | 3\, 23\ | 9\, 17\ |
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) | |
