26edo: Difference between revisions
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104edo is a strong no-5 [[Parapyth]] tuning. | 104edo is a strong no-5 [[Parapyth]] tuning. | ||
{{Harmonics in ED|104|47|0}} | {{Harmonics in ED|104|47|0}} | ||
=== 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. | |||
{{Harmonics in ED|130|47|0}} | |||
{{navbox EDO}} | {{navbox EDO}} | ||
{{Cat|edos}} | {{Cat|edos}} | ||
Revision as of 03:00, 9 March 2026
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) | |
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.
| 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) | |
