353edo

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353edo

354edo →

Prime factorization	353 (is prime)
Step size	3.39943 ¢
Fifth	206\353 (700.283 ¢)
Semitones (A1:m2)	30:29 (102 ¢ : 98.58 ¢)
Dual sharp fifth	207\353 (703.683 ¢)
Dual flat fifth	206\353 (700.283 ¢)
Dual major 2nd	60\353 (203.966 ¢)
Consistency limit	3
Distinct consistency limit	3

Theory

Approximation of odd harmonics in 353edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-1.67	+1.22	+0.01	+0.06	-0.61	-0.87	-0.45	+0.43	+1.64	-1.66	+0.62
	Relative (%)	-49.2	+35.9	+0.4	+1.6	-17.9	-25.5	-13.2	+12.6	+48.2	-48.8	+18.3
Steps (reduced)		559 (206)	820 (114)	991 (285)	1119 (60)	1221 (162)	1306 (247)	1379 (320)	1443 (31)	1500 (88)	1550 (138)	1597 (185)

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 (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, 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). 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, the generator. Temperance of 10985/10976 means that three 14/13s 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.

Other

It's possible to use superpyth fifth of Rectified Hebrew fifth, 209\353, as a generator. In this case, 76 & 353 temperament is obtained. In the 2.5.7.13 subgroup, this results in the fifth being equal to 98/65 and the comma basis of 10985/10976, [-103 0 -38 51 0 13⟩.

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	Note 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	196/195
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	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

Regular temperament properties

Assuming 353edo is treated as the 2.5.7.11.13.17 subgroup temperament.

Subgroup	Comma List	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.5	820 -353	[⟨353 820]]	-0.263	0.263	7.74
2.5.7	3136/3125, 209 -9 -67	[⟨353 820 991]]	-0.177	0.247	7.26
2.5.7.11	3136/3125, 5767168/5764801, -20 -6  1 9	[⟨353 820 991 1221]]	-0.089	0.263	7.73
2.5.7.11.13	3136/3125, 4394/4375, 6656/6655, 5767168/5764801	[⟨353 820 991 1221 1306]]	-0.024	0.268	7.89
2.5.7.11.13.17	3136/3125, 4394/4375, 7744/7735, 60112/60025, 64141/64000	[⟨353 820 991 1221 1306 1443]]	-0.037	0.247	7.26

Rank-2 temperaments

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	19\353	64.59	26/25	Rectified Hebrew
1	34\353	115.58	77/72	Apparatus
1	152\353	516.71	27/20	Marvo (353c) / zarvo (353cd)

Scales

• RectifiedHebrew[19] - 18L 1s
• RectifiedHebrew[130] - 93L 37s
• Austro-Hungarian Minor[9] - 57 38 38 38 38 38 38 38 30

See also

• 293edo
• Maximal evenness

Music

• Bottom Text by Mercury Amalgam

Links

• Rectified Hebrew Calendar