353edo
Prime factorization | 353 (is prime) |
Step size | 3.3994 |
Fifth | 206\353 (700.28¢) |
The 353 equal divisions of the octave (353edo), or the 353-tone equal temperament (353tet), 353 equal temperament (353et) when viewed from a regular temperament perspective, divides the octave into 353 equal parts of about 3.4 cents each.
Theory
Prime number | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | absolute (¢) | +0.00 | -1.67 | +1.22 | +0.01 | -0.61 | -0.87 | +0.43 | +1.64 | +0.62 | +0.45 | +0.57 | +0.21 |
relative (%) | +0 | -49 | +36 | +0 | -18 | -26 | +13 | +48 | +18 | +13 | +17 | +6 | |
Steps (reduced) | 353 (0) | 559 (206) | 820 (114) | 991 (285) | 1221 (162) | 1306 (247) | 1443 (31) | 1500 (88) | 1597 (185) | 1715 (303) | 1749 (337) | 1839 (74) |
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 (מחזור), 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, which represents 353 years of the cycle. 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.
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). Temperance of 4394/4375 means that a stack of three 13/10s (7 generators) is equated with 35/32, octave-reduced, and also splits 14/13 (2 generators) into two parts each corresponding to 26/25. Temperance of 10985/10976 means that three 14/13s are equated with 5/4.
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. 18L 1s of Rectified Hebrew gives 19edo a unique stretch: 6 generators correspond to 5/4, 13 correspond to 13/8, and 15 correspond to 7/4. When measured relative to the generator, the error is less than 1 in 5000.
The number 353 in this version of the Hebrew calendar must not be confused with the number of days in shanah chaserah (שנה חסרה), the deficient year.
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". 169/168 amounts to 3 steps, which is the L step of the full 93L 37s rectified Hebrew scale.
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# | 26/25 |
38 | Db | 14/13 |
41 | Db-D | 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 | 3/2 |
209 | G | 169/112, 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
- RectifiedHebrew[130] - 93L 37s
- Austro-Hungarian Minor[9] - 57 38 38 38 38 38 38 38 30