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

External links

• Rectified Hebrew Calendar