Draft:13 equal temperament
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13 equal temperament, 13 equal divisions of the octave, 13et, 13edo, or 13tet is the tuning system which divides the octave in 13 equal parts. Each step represents a frequency ratio of 21/13, or the 13th root of 2.
Theory
[edit]13edo is notable for its lack of a (usable) perfect fifth, with its best approximation being 8 steps, or roughly 738.5 cents. For comparison, the just perfect fifth is roughly 702 cents. This makes 13edo's best fifth nearly 40% of a step sharp. Its second-best fifth, found at the 7th step, is roughly 646.2 cents, a whopping 55.8 cents flat.
On the other hand, one step of 13edo is very close to the major chroma (135:128) by direct approximation.
Harmonics
[edit]Here are the relative errors of all odd harmonics up to 21 in 13edo.
3: 39.5%
5: -18.5%
7: -49.6%
11: 2.7%
13: -10.6%
15: 21%
17: -13.7%
19: -22.3%
21: -10%