353edo
The 353 equal divisions of the octave (353edo) divides the octave into parts of 3.3994 cents each.
Theory
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From the prime number standpoint, 353edo is suitable for use with 2.7.11.17.23.29.31.37 subgroup. This makes 353edo an "upside-down" EDO – poor approximation of the low harmonics, but an improvement over the high ones. Nonetheless, it provides the optimal patent val for didacus, the 2.5.7 subgroup temperament tempering out 3136/3125.
353edo is the 71st prime EDO.
Relation to a calendar reform
In the original Hebrew calendar, years number 3, 6, 8, 11, 14, 17, and 19 within a 19-year pattern (makhzor, plural:makhzorim) are leap. When converted to 19edo, this results in 5L 2s mode, and simply the diatonic major scale.
Following this logic, a temperament can be constructed for the Rectified Hebrew calendar (see below), containing 130 notes of the 353edo scale. Hebrew[130] scale has 334\353 as its generator, which is a supermajor seventh, or alternately, 19\353, about a third-tone, since inverting the generator has no effect on the scale. Using such small of a generator helps explore the 353edo's "upside down" side. In addition, every sub-pattern in a 19-note generator is actually a Hebrew makhzor, that is a mini-19edo on its own, until it is truncated to an 11-note pattern. Just as the original calendar reform consists of 18 makhzorim with 1 hendecaeteris, Hebrew[130] scale consists of a stack of naively 18 "major scales" finished with one 11-edo tetratonic.
Such a temperament stretches 19edo so that 15\19 corresponds to 7/4. When measured relative to the generator, the error is less than 1 in 5000. In the 13-limit, the it tempers out 3136/3125, 4394/4375, 10985/10976, and 1968512/1953125.
Scales
- RectifiedHebrew[19] - 18L 1s
- 18-Glacial[19] - same as above
- RectifiedHebrew[130] - 93L 37s