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The '''353 equal divisions of the octave''' ('''353edo''') divides the [[octave]] into parts of 3.3994 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== 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, plural:makhzorim) are leap. When converted to [[19edo]], this results in [[5L 2s]] mode, and simply the diatonic major scale.  
353edo is the 71st [[prime edo]].


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.  
=== 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.  


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]].  
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.


=== Specific chords and intervals ===
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.  
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.  


If 209\353 is loosely assumed to equal 3/2, it offers a 710c stretched fifth, and the following resolution sequence: 7/4 D7 - 13/8 D2 - D53 - T53. This has a very pleasant sound, with 13/8 acting as a "doubled resolvant" or "resolution into resolution". Such an arrangement also "tempers out" [[169/168]].
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
!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**
|-
|-
|19
| 0
|C#
| C
|
| 1/1
|-
|-
|38
| 1
|Db
| C-C#
|[[14/13]]
|  
|-
|-
|57
| 2
|D
| C-Db
|
|  
|-
|-
|76
| 3
|D#
| C-D
|
| [[196/195]]
|-
|-
|95
| 4
|Eb
| C-D#
|
|  
|-
|-
|114
| 19
|E
| C#
|[[5/4]]
| [[26/25]]
|-
|-
|133
| 38
|E#/Fb
| Db
|[[13/10]] minor (best approximation is 134)
| [[14/13]]
|-
|-
|152
| 41
|F
| Db-D
|
| [[13/12]]
|-
|-
|171
| 46
|F#
| Db-F
|[[7/5]]
| [[35/32]]
|-
|-
|190
| 57
|Gb
| D
|
|  
|-
|-
|209
| 76
|G
| D#
|98/65
|  
|-
|-
|228
| 95
|G#
| Eb
|
|  
|-
|-
|247
| 114
|Ab
| E
|[[13/8]]
| [[5/4]]
|-
|-
|266
| 133
|A
| E#
|
| [[13/10]] I (patent val approximation)
|-
|-
|285
| 134
|A#
| E#-C#
|[[7/4]]
| 13/10 II (direct approximation)
|-
|-
|304
| 152
|Bb
| F
|
|  
|-
|-
|323
| 171
|B
| F#
|
| [[7/5]]
|-
|-
|342
| 190
|B#/Cb
| Gb
|
|  
|-
|-
|353
| 206
|C
| Gb-Bb
|2/1
| 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
|}
|}
<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"
|-
! rowspan="2" | [[Subgroup]]
! 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
| {{monzo| 820 -353 }}
| {{mapping| 353 820 }}
| −0.263
| 0.263
| 7.74
|-
| 2.5.7
| 3136/3125, {{monzo| 209 -9 -67 }}
| {{mapping| 353 820 991 }}
| −0.177
| 0.247
| 7.26
|-
| 2.5.7.11
| 3136/3125, 5767168/5764801, {{monzo| -20 -6  1 9 }}
| {{mapping| 353 820 991 1221 }}
| −0.089
| 0.263
| 7.73
|-
| 2.5.7.11.13
| 3136/3125, 4394/4375, 6656/6655, 5767168/5764801
| {{mapping| 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
| {{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
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperament
|-
| 1
| 19\353
| 64.59
| 26/25
| [[Rectified hebrew]]
|-
| 1
| 34\353
| 115.58
| 77/72
| [[Subgroup temperaments#Apparatus|Apparatus]]
|-
| 1
| 152\353
| 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
* 18-Glacial[19] - same as above
* RectifiedHebrew[130] – 93L 37s
* RectifiedHebrew[130] - 93L 37s
* Austro-Hungarian Minor[9] – 57 38 38 38 38 38 38 38 30


== See also ==
== See also ==
Line 119: Line 233:
* [[Maximal evenness]]
* [[Maximal evenness]]


== Links ==
== Music ==
* [https://individual.utoronto.ca/kalendis/hebrew/rect.htm Rectified Hebrew Calendar]
; [[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]


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