Rectified hebrew
Rectified Hebrew is a 2.5.7.13 subgroup temperament. Being a weak extension of didacus, it is notable due to its ability to reach several simple intervals in just a few generators.
Its name derives from a calendar layout by the same name.
Theory
353edo-specific theory
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. 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 19\353, the error is less than 1 in 5000. 5 instances of 5/4 and two of 7/4 both amount to 30 generators (570 steps). Tempering 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, the generator. Tempering of 10985/10976 means that a stack of three 14/13's are equated with 5/4.
The 11-step perfect fifth in this scale becomes 209\353, and it corresponds to 98/65, which is sharp of 3/2 by 196/195.
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.
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.
169/168 amounts to 3 steps, which is the L step of the full 93L 37s rectified Hebrew scale.
Specific chords and intervals
Rectified hebrew supports the tridecimal neutral seventh chords and a cadence invented by Eliora.
The tridecimal neutral seventh chord, noted as 13/8 N7, is represented in 353edo with steps 114 95 106, and its inversions respectively: 13/8 N65: 95 106 38, 13/8 N43: 106 38 114, 13/8 N42 (or 13/8 N2): 38 114 95. 114 steps is 6 generators, 95 steps is 5 generators, 38 steps is 2 generators, and 106 is closure of 13/8 against the octave, which consists of 5 generators with an octave residue to 19 generators.
The tridecimal neutral cadence is the following: 13/8 N43 - D7 - T53, or in 353edo 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 regular temperament theory of 353edo, one can think of it as the 353bbbbb val, where 209\353 fifth represents 3/2.
Miscellaneous properties
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.
Interval chain
| Generator
steps |
Interval
(2.5.7.13 subgroup) |
|---|---|
| 0 | 1/1 |
| 1 | 26/25 |
| 2 | 14/13 |
| 3 | 28/25 |
| 4 | 65/56 |
| 6 | 5/4 |
| 7 | 13/10 |
| 9 | 7/5 |
| 11 | 98/65 |
| 13 | 13/8 |
| 15 | 7/4 |