23rd-octave temperaments
Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.
Here are some EDOs with low relative error on simple harmonics, which are multiples of 23 - perhaps fertile ground for 23rd-octave temperaments: 46, 161, 207, 230, 414, 690, 805, 851, 920, 966, and 1012.
46 and 1012 are also members of zeta edo lists.
Icositritonic
The main 23rd-octave temperament of interest is icositritonic (temperament data given there), notable for being a 23-limit 23rd-octave temperament supported by 115edo, 161edo and 207edo, with 46edo serving as a trivial tuning. It is named after the recurrence of the number 23. It can be thought of as leveraging the most accurate approximations of 23-limit harmony supplied by the 23edo subset of 46edo while correcting others, by mapping:
- 33/32 (-1.1 ¢), 34/33 (+0.5 ¢) and 35/34 (+2.0 ¢) to 1\23
- (therefore) 17/16 (-0.7 ¢) and 35/33 (+2.5 ¢) to 2\23
- (and) 35/32 (+1.4 ¢) to 3\23, as well as 23/21 (-1.0 ¢)
- 44/39 to 4\23 (-0.1 ¢)
- 6/5 to 6\23 (-2.6 ¢) (among more complex 23-limit interpretations also mapped here, namely 55/46 and 115/96, both slightly more accurate)
- 21/17 to 7\23 (-0.6 ¢)
- 14/11 to 8\23 (-0.1 ¢)
- 21/16 to 9\23 (-1.2 ¢)
- 23/17 to 10\23 (-1.6 ¢)
- 32/23 (2.2 ¢), 46/33 (-1.1 ¢) and 39/28 (+0.3 ¢) to 11\23
This serves to define icositritonic in the 23-limit. It is quite accurate so that considering other 23rd-octave temperaments might in time prove unnecessary.
Arguably the 69-note MOS is the place to start, corresponding to 46L 23s, essentially a version of 46edo with half of its notes duplicated to allow for better tunings, as (if we pay attention to its mapping) that allows all primes except 13 and 19 to be found relative to the same root note, with ample opportunity for more complex harmony by combining 3 and 5 (found at 1 gen) with other primes which are often found at -1 gen, so that most of the complex harmony is actually accessible directly in 23edo, an impressive feat.
However, if we want to guarantee we can find all primes relative to the same root note, we require a 5*23=115-note MOS, corresponding to 46L 69s, that is, to 2L 3s (pentic) with a 1\23 period, a fascinating "microtonal minification" of the familiar pentatonic scale, using the small step for very subtle commatic variations on notes of 46edo, though few dare to use scales this elaborate, and the 23L 46s scale is likely sufficient for exploring most of the harmony offered (and likely more interesting to play with), so that it might also serve in some strange way as a well temperament of 69edo (not just 46edo), which is a flat tuning of meantone supporting many but not all of the equivalences of icositritonic.
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Todo: expand, add examples |