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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 353 (is prime)
{{ED intro}}  
| Step size = 3.3994
| Sharp fifth = 207\353 (703.68¢)
| Flat fifth = 206\353 (700.28¢)
| Major 2nd = 60\353 (203.97¢)
}}
{{EDO intro|353}}  


== Theory ==
== Theory ==
{{Harmonics in equal|353}}
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]].  


353edo is the 71st [[prime EDO]].
Using the [[patent val]] nonetheless, 353edo supports [[apparatus]], [[marvo]] and [[zarvo]].


=== Relation to a calendar reform ===
=== Odd harmonics ===
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.
{{Harmonics in equal|353}}


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.  
=== Subsets and supersets ===
353edo is the 71st [[prime edo]].


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.  
=== 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.   
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.  
 
=== 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, {{Monzo|-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.
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}}.


== Table of intervals ==
== Table of intervals ==
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+
!Step
!Note 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**
|-
|-
|1
| 0
|C-C#
| C
|
| 1/1
|-
|-
|2
| 1
|C-Db
| C-C#
|
|  
|-
|-
|3
| 2
|C-D
| C-Db
|[[196/195]]
|  
|-
|-
|4
| 3
|C-D#
| C-D
|
| [[196/195]]
|-
|-
|19
| 4
|C#
| C-D#
|[[26/25]]
|  
|-
|-
|38
| 19
|Db
| C#
|[[14/13]]
| [[26/25]]
|-
|-
|41
| 38
|Db-D
| Db
|[[13/12]]
| [[14/13]]
|-
|-
|46
| 41
|Db-F
| Db-D
|[[35/32]]
| [[13/12]]
|-
|-
|57
| 46
|D
| Db-F
|
| [[35/32]]
|-
|-
|76
| 57
|D#
| D
|
|  
|-
|-
|95
| 76
|Eb
| D#
|
|  
|-
|-
|114
| 95
|E
| Eb
|[[5/4]]
|  
|-
|-
|133
| 114
|E#
| E
|[[13/10]] I (patent val approximation)
| [[5/4]]
|-
|-
|134
| 133
|E#-C#
| E#
|13/10 II (direct approximation)
| [[13/10]] I (patent val approximation)
|-
|-
|152
| 134
|F
| E#-C#
|
| 13/10 II (direct approximation)
|-
|-
|171
| 152
|F#
| F
|[[7/5]]
|  
|-
|-
|190
| 171
|Gb
| F#
|
| [[7/5]]
|-
|-
|206
| 190
|Gb-Bb
| Gb
|3/2
|  
|-
|-
|209
| 206
|G
| Gb-Bb
|[[98/65]]
| 3/2
|-
|-
|228
| 209
|G#
| G
|
| [[98/65]]
|-
|-
|247
| 228
|Ab
| G#
|[[13/8]]
|  
|-
|-
|266
| 247
|A
| Ab
|
| [[13/8]]
|-
|-
|285
| 266
|A#
| A
|[[7/4]]
|  
|-
|-
|304
| 285
|Bb
| A#
|
| [[7/4]]
|-
|-
|323
| 304
|B
| Bb
|
|  
|-
|-
|342
| 323
|B#/Cb
| B
|
|  
|-
|-
|353
| 342
|C
| B#/Cb
|2/1
|
|-
| 353
| C
| 2/1
|}
|}
<nowiki />* Diatonic Hebrew[19] version
<nowiki />** 2.5.7.13 subgroup


== Regular temperament properties ==
== Regular temperament properties ==
Assuming 353edo is treated as the 2.5.7.11.13.17 subgroup temperament.
Assuming 353edo is treated as the 2.5.7.11.13.17 subgroup temperament.
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal
8ve stretch (¢)
! colspan="2" |Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.5
| 2.5
|820 -353
| {{monzo| 820 -353 }}
|[{{val|353 820}}]
| {{mapping| 353 820 }}
| -0.263
| −0.263
|0.263
| 0.263
|7.74
| 7.74
|-
|-
|2.5.7
| 2.5.7
|3136/3125, 209 -9 -67
| 3136/3125, {{monzo| 209 -9 -67 }}
|[{{val|353 820 991}}]
| {{mapping| 353 820 991 }}
| -0.177
| −0.177
|0.247
| 0.247
|7.26
| 7.26
|-
|-
|2.5.7.11
| 2.5.7.11
|3136/3125, 5767168/5764801, -20 -6  1 9
| 3136/3125, 5767168/5764801, {{monzo| -20 -6  1 9 }}
|[{{val|353 820 991 1221}}]
| {{mapping| 353 820 991 1221 }}
| -0.089
| −0.089
|0.263
| 0.263
|7.73
| 7.73
|-
|-
|2.5.7.11.13
| 2.5.7.11.13
|3136/3125, 4394/4375, 6656/6655, 5767168/5764801
| 3136/3125, 4394/4375, 6656/6655, 5767168/5764801
|[{{val|353 820 991 1221 1306}}]
| {{mapping| 353 820 991 1221 1306 }}
| -0.024
| −0.024
|0.268
| 0.268
|7.89
| 7.89
|-
|-
|2.5.7.11.13.17
| 2.5.7.11.13.17
|3136/3125, 4394/4375, 7744/7735, 60112/60025, 64141/64000
| 3136/3125, 4394/4375, 7744/7735, 60112/60025, 64141/64000
|[{{val|353 820 991 1221 1306 1443}}]
| {{mapping| 353 820 991 1221 1306 1443 }}
| -0.037
| −0.037
|0.247
| 0.247
|7.26
| 7.26
|}
|}
=== Rank-2 temperaments ===
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Periods
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
per octave
|-
!Generator
! Periods<br />per 8ve
(reduced)
! Generator*
!Cents
! Cents*
(reduced)
! Associated<br />ratio*
!Associated
! Temperament
ratio
!Temperaments
|-
|-
|1
| 1
|19\353
| 19\353
|64.59
| 64.59
|26/25
| 26/25
|[[Hemimean clan#Rectified Hebrew|Rectified Hebrew]]
| [[Rectified hebrew]]
|-
|-
|1
| 1
|34\353
| 34\353
|115.58
| 115.58
|77/72
| 77/72
|[[Subgroup temperaments#Apparatus|Apparatus]]
| [[Subgroup temperaments#Apparatus|Apparatus]]
|-
|-
|1
| 1
|152\353
| 152\353
|516.71
| 516.71
|27/20
| 27/20
|[[Marvo]] (353c) / [[zarvo]] (353cd)
| [[Marvo]] (353c) / [[zarvo]] (353cd)
|}
|}
=== Scales ===
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
* RectifiedHebrew[19] - 18L 1s
 
* RectifiedHebrew[130] - 93L 37s
== Scales ==
* Austro-Hungarian Minor[9] - 57 38 38 38 38 38 38 38 30
* RectifiedHebrew[19] – 18L 1s
* RectifiedHebrew[130] – 93L 37s
* Austro-Hungarian Minor[9] – 57 38 38 38 38 38 38 38 30


== See also ==
== See also ==
Line 243: Line 234:


== Music ==
== 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


* [https://www.youtube.com/watch?v=z-SxvrnkTzU Bottom Text] by Mercury Amalgam
== External links ==
 
== 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|###]] <!-- 3-digit number -->
[[Category:Didacus]]
[[Category:Didacus]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Jewish]]
 
{{Todo| review }}
Retrieved from "https://en.xen.wiki/w/353edo"