26edo: Difference between revisions
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{{problematic}} | {{problematic}} | ||
'''26edo''' is the equal | '''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== | ==Theory== | ||
===JI approximation=== | ===JI approximation=== | ||
26edo is characterized by a flat tuning of harmonics 3, 5, and 13 and | 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. 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]]. The flat 13 means that [[13/8]] is tempered to the same interval as [[8/5]], tempering out [[65/64]], and the combination of the accurate 11 and flat 3 implies that [[11/9]] is tempered to the same interval as 16/13 and 5/4, so [[144/143]] and [[45/44]] are also tempered out. | ||
26edo is an even EDO and contains the 600¢ half-octave tritone, | |||
{{Harmonics in ED|26|31|0}} | {{Harmonics in ED|26|31|0}} | ||
Revision as of 00:24, 2 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. 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. The flat 13 means that 13/8 is tempered to the same interval as 8/5, tempering out 65/64, and the combination of the accurate 11 and flat 3 implies that 11/9 is tempered to the same interval as 16/13 and 5/4, so 144/143 and 45/44 are also tempered out.
26edo is an even EDO and contains the 600¢ half-octave tritone,
| 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) | |
TODO:
- write about flattone
- write about usability in the 13-limit
