353edo: Difference between revisions
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=== Relation to a calendar reform === | === 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. | 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. 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. | |||
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. | |||
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). | |||
=== Specific chords and intervals === | === 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". | 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. | 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. | ||
Revision as of 20:13, 26 March 2022
| ← 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. 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.
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.
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
- 18-Glacial[19] - same as above
- RectifiedHebrew[130] - 93L 37s