353edo
| ← 352edo | 353edo | 354edo → |
The 353 equal divisions of the octave (353edo) divides the octave into parts of 3.3994 cents each.
Theory
Script error: No such module "primes_in_edo".
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.
Rectified Hebrew temperament is a 13-limit extension of the didacus. In the 13-limit, the it tempers out 3136/3125, 4394/4375, 10985/10976, and 1968512/1953125. 5 instances of 5/4 and two of 7/4 both amount to 30 generators (570 steps).
While the just 3/2 is on 206 in 353edo, the more rational way would be to use the perfect fifth that is provided by the 19-tone scale resulting from the generator. This produces the 353bbbbb val: [353 562 820 991 1306⟩, where the fifth is on 11*19 = 209 steps and measures about 710 cents. Such an usage of the rectified Hebrew temperament tempers out 91/90, 169/168, 196/195, 625/624, 686/675, 875/864, 2197/2160, and [45,-27,6,-7,1⟩.
Specific chords and intervals
353bbbbb val offers the following resolution sequence:13/8 D4/3 - D7 - T53, or in steps: 247-0-38-152 - 209-323-57-152 - 0-114-209, or 0-95-209. This has a very pleasant sound, with 13/8 acting as a "doubled resolvant" or "resolution into resolution". In the patent val, 169/168 amounts to 3 steps, which is the L step of the full 93L 37s rectified Hebrew scale.
18L 1s of Rectified Hebrew gives 19edo a unique stretch: 6\19 corresponds to 5/4, 13\19 corresponds to 13/8, and 15\19 corresponds to 7/4. When measured relative to the generator, the error is less than 1 in 5000. 7\19 corresponds to 13/10 when measured using the patent val (1306 - 820 - 353 = 133), however the direct approximation using the number is 134 steps. Since patent val is used to define if a comma is "tempered out", repeatedly stacking 7\19 3 times and reducing arrives at 46\353, an approximation for 35/32. The approach using 134 is inconistent by itself already, so therefore it can't be used.
Just as a large amount of 12edo music can be played consistently in 19edo, it can also be played consistently in the 18L 1s subset of Rectified Hebrew.
Table of intervals
| Step | Name
(diatonic Hebrew[19] version) |
Associated ratio
(2.5.7.13 subgroup) |
|---|---|---|
| 0 | C | 1/1 |
| 1 | C-C# | |
| 2 | C-Db | |
| 3 | C-D | 169/168 |
| 4 | C-D# | |
| 19 | C# | |
| 38 | Db | 14/13 |
| 41 | 13/12 | |
| 46 | Db-F | 35/32 |
| 57 | D | |
| 76 | D# | |
| 95 | Eb | |
| 114 | E | 5/4 |
| 133 | E# | 13/10 I (patent val approximation) |
| 134 | E#-C# | 13/10 II (direct approximation) |
| 152 | F | |
| 171 | F# | 7/5 |
| 190 | Gb | |
| 206 | Gb-Bb | |
| 209 | G | 98/65, |
| 228 | G# | |
| 247 | Ab | 13/8 |
| 266 | A | |
| 285 | A# | 7/4 |
| 304 | Bb | |
| 323 | B | |
| 342 | B#/Cb | |
| 353 | C | 2/1 |
Scales
- RectifiedHebrew[19] - 18L 1s
- 18-Glacial[19] - same as above
- RectifiedHebrew[130] - 93L 37s