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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 353 (is prime)
{{ED intro}}  
| Step size = 3.3994
| Fifth = 206\353 (700.28¢)
}}
{{EDO intro|353}}  


== Theory ==
== Theory ==
{{primes in edo|353|columns=12}}
353edo is in[[consistent]] in the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. It is suitable for use with the 2.9.15.7.11.13.17.23.29.31.37 [[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]], and serves as a very close approximation of its just-[[7/4]] tuning.


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]].  
Using the [[patent val]] nonetheless, 353edo supports [[apparatus]], [[marvo]] and [[zarvo]].


353edo is the 71st [[prime EDO]].
=== Odd harmonics ===
{{Harmonics in equal|353}}


=== Relation to a calendar reform ===
=== Subsets and supersets ===
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.  
353edo is the 71st [[prime edo]].


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]]. 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. Temperance of 10985/10976 means that three 14/13s are equated with 5/4.  
=== Miscellaneous properties ===
[[Eliora]] associates 353edo with a reformed Hebrew calendar. 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 (→ [[rectified hebrew]]) can be constructed for the Rectified Hebrew calendar. 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.  


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. 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.
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.  
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.  


=== Specific chords and intervals ===
It is possible to use a superpyth-ish fifth of Rectified Hebrew fifth, 209\353, as a generator. In this case, {{nowrap|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, {{Monzo|-103 0 -38 51 0 13}}.
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 ==
== Table of intervals ==
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+
!Step
!Name
<small>(diatonic Hebrew[19] version</small>)
!Associated ratio
<small>(2.5.7.13 subgroup)</small>
|-
|-
|0
! Step
|C
! Note name*
|1/1
! Associated ratio**
|-
| 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
|
|-
|-
|1
| 76
|C-C#
| D#
|
|  
|-
|-
|2
| 95
|C-Db
| Eb
|
|  
|-
|-
|3
| 114
|C-D
| E
|[[169/168]]
| [[5/4]]
|-
|-
|4
| 133
|C-D#
| E#
|
| [[13/10]] I (patent val approximation)
|-
|-
|19
| 134
|C#
| E#-C#
|[[26/25]]
| 13/10 II (direct approximation)
|-
|-
|38
| 152
|Db
| F
|[[14/13]]
|  
|-
|-
|41
| 171
|Db-D
| F#
|[[13/12]]
| [[7/5]]
|-
|-
|46
| 190
|Db-F
| Gb
|[[35/32]]
|  
|-
|-
|57
| 206
|D
| Gb-Bb
|
| 3/2
|-
|-
|76
| 209
|D#
| G
|
| [[98/65]]
|-
|-
|95
| 228
|Eb
| G#
|
|  
|-
|-
|114
| 247
|E
| Ab
|[[5/4]]
| [[13/8]]
|-
|-
|133
| 266
|E#
| A
|[[13/10]] I (patent val approximation)
|  
|-
|-
|134
| 285
|E#-C#
| A#
|13/10 II (direct approximation)
| [[7/4]]
|-
|-
|152
| 304
|F
| Bb
|
|  
|-
|-
|171
| 323
|F#
| B
|[[7/5]]
|
|-
| 342
| B#/Cb
|
|-
| 353
| C
| 2/1
|}
<nowiki />* Diatonic Hebrew[19] version
 
<nowiki />** 2.5.7.13 subgroup
 
== Regular temperament properties ==
Assuming 353edo is treated as the 2.5.7.11.13.17 subgroup temperament.
{| class="wikitable center-4 center-5 center-6"
|-
|-
|190
! rowspan="2" | [[Subgroup]]
|Gb
! rowspan="2" | [[Comma list]]
|
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|206
! [[TE error|Absolute]] (¢)
|Gb-Bb
! [[TE simple badness|Relative]] (%)
|3/2
|-
|-
|209
| 2.5
|G
| {{monzo| 820 -353 }}
|169/112, 98/65
| {{mapping| 353 820 }}
| −0.263
| 0.263
| 7.74
|-
|-
|228
| 2.5.7
|G#
| 3136/3125, {{monzo| 209 -9 -67 }}
|
| {{mapping| 353 820 991 }}
| −0.177
| 0.247
| 7.26
|-
|-
|247
| 2.5.7.11
|Ab
| 3136/3125, 5767168/5764801, {{monzo| -20 -6  1 9 }}
|[[13/8]]
| {{mapping| 353 820 991 1221 }}
| −0.089
| 0.263
| 7.73
|-
|-
|266
| 2.5.7.11.13
|A
| 3136/3125, 4394/4375, 6656/6655, 5767168/5764801
|
| {{mapping| 353 820 991 1221 1306 }}
| −0.024
| 0.268
| 7.89
|-
|-
|285
| 2.5.7.11.13.17
|A#
| 3136/3125, 4394/4375, 7744/7735, 60112/60025, 64141/64000
|[[7/4]]
| {{mapping| 353 820 991 1221 1306 1443 }}
| −0.037
| 0.247
| 7.26
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
|304
! Periods<br />per 8ve
|Bb
! Generator*
|
! Cents*
! Associated<br />ratio*
! Temperament
|-
|-
|323
| 1
|B
| 19\353
|
| 64.59
| 26/25
| [[Rectified hebrew]]
|-
|-
|342
| 1
|B#/Cb
| 34\353
|
| 115.58
| 77/72
| [[Subgroup temperaments#Apparatus|Apparatus]]
|-
|-
|353
| 1
|C
| 152\353
|2/1
| 516.71
| 27/20
| [[Marvo]] (353c) / [[zarvo]] (353cd)
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
* RectifiedHebrew[19] - 18L 1s
* RectifiedHebrew[19] – 18L 1s
* RectifiedHebrew[130] - 93L 37s
* RectifiedHebrew[130] – 93L 37s
* Austro-Hungarian Minor[9] - 57 38 38 38 38 38 38 38 30
* Austro-Hungarian Minor[9] – 57 38 38 38 38 38 38 38 30


== See also ==
== See also ==
Line 158: Line 233:
* [[Maximal evenness]]
* [[Maximal evenness]]


== Links ==
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=JrSEGE6_oys ''Snow On My City''] (2022) – cover of [[wikipedia:Naomi Shemer|Naomi Shemer]] in Rectified Hebrew and apparatus
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=z-SxvrnkTzU ''Bottom Text''] (2022) in Rectified Hebrew
 
== External links ==
* [http://individual.utoronto.ca/kalendis/hebrew/rect.htm Rectified Hebrew Calendar]
* [http://individual.utoronto.ca/kalendis/hebrew/rect.htm Rectified Hebrew Calendar]


[[Category:Equal divisions of the octave]]
[[Category:Didacus]]
[[Category:Didacus]]
[[Category:Listen]]
{{Todo| review }}

Latest revision as of 05:06, 2 March 2025

← 352edo 353edo 354edo →
Prime factorization 353 (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

353 equal divisions of the octave (abbreviated 353edo or 353ed2), also called 353-tone equal temperament (353tet) or 353 equal temperament (353et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 353 equal parts of about 3.4 ¢ each. Each step represents a frequency ratio of 21/353, or the 353rd root of 2.

Theory

353edo is inconsistent in the 5-odd-limit and harmonic 3 is about halfway between its steps. It is suitable for use with the 2.9.15.7.11.13.17.23.29.31.37 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, and serves as a very close approximation of its just-7/4 tuning.

Using the patent val nonetheless, 353edo supports apparatus, marvo and zarvo.

Odd harmonics

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)

Subsets and supersets

353edo is the 71st prime edo.

Miscellaneous properties

Eliora associates 353edo with a reformed Hebrew calendar. 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 (→ rectified hebrew) can be constructed for the Rectified Hebrew calendar. 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.

It is possible to use a superpyth-ish 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⟩.

Table of intervals

Step Note name* Associated ratio**
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

* Diatonic Hebrew[19] version

** 2.5.7.13 subgroup

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

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
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)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

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

See also

Music

Eliora
Mercury Amalgam

External links

Retrieved from "https://en.xen.wiki/index.php?title=353edo&oldid=183884"