Rectified Hebrew

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Rectified Hebrew is a 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.


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




( 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

External links