353edo

The 353 equal divisions of the octave (353edo) divides the octave into parts of 3.3994 cents each.

Theory

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.

Such a temperament gives 19edo a unique stretch: 6\19 corresponds to 5/4, 13\19 corresponds to 13/8, and 15\19 corresponds to 7/4. When measured relative to the generator, the error is less than 1 in 5000. In the 13-limit, the it tempers out 3136/3125, 4394/4375, 10985/10976, and 1968512/1953125. This gives it a few more unique intervals.

Table of intervals

Step	Name (diatonic Hebrew[19] version)	Associated ratio (2.5.7.13 subgroup)
0	C	1/1
19	C#
38	Db	14/13
57	D
76	D#
95	Eb
114	E	6/5
133	E#/Fb	13/10 minor (best approximation is 134)
152	F
171	F#	7/5
190	Gb
209	G	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

See also

• 293edo
• Maximal evenness

Links

• Rectified Hebrew Calendar