Xenharmonic Wiki

353edo: Difference between revisions

← Older edit
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,351 edits
mNo edit summary
 
(33 intermediate revisions by 6 users not shown)
Line 1: Line 1:
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 353 (is prime)
{{ED intro}}  
| Step size = 3.3994
| Fifth = 206\353 (700.28¢)
}}
The '''353 equal divisions of the octave''' ('''353edo''') divides the [[octave]] into parts of 3.3994 [[cent]]s each.


== 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.  
=== 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.  
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.


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]].
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.  


While the just 3/2 is on 206 in 353edo, the more rational way would be to use the perfect fifth that is provided by the 19-tone scale resulting from the generator. This produces the 353bbbbb val: [353 '''562''' 820 991 1306⟩, where the fifth is on 11*19 = 209 steps and measures about 710 cents. Such an usage of the rectified Hebrew temperament tempers out [[91/90]], [[169/168]], [[196/195]], [[625/624]], [[686/675]], [[875/864]], [[2197/2160]], and [45,-27,6,-7,1⟩. 
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}}.
 
=== Specific chords and intervals ===
353bbbbb val offers the following resolution sequence: 7/4 D7 - 13/8 D4/3 - D53 - T53. This has a very pleasant sound, with 13/8 acting as a "doubled resolvant" or "resolution into resolution". In the patent val, 169/168 amounts to 3 steps, which is the L step of the full 93L 37s rectified Hebrew scale.
 
18L 1s of Rectified Hebrew 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. 7\19 corresponds to [[13/10]] when measured using the patent val (1306 - 820 - 353 = 133), however the direct approximation using the number is 134 steps. Since patent val is used to define if a comma is "tempered out", repeatedly stacking 7\19 3 times and reducing arrives at 46\353, an approximation for [[35/32]]. The approach using 134 is inconistent by itself already, so therefore it can't be used.
 
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]]
|-
|-
|1
| 46
|C-C#
| Db-F
|
| [[35/32]]
|-
|-
|2
| 57
|C-Db
| D
|
|  
|-
|-
|3
| 76
|C-D
| D#
|[[169/168]]
|  
|-
|-
|4
| 95
|C-D#
| Eb
|
|  
|-
|-
|19
| 114
|C#
| E
|
| [[5/4]]
|-
|-
|38
| 133
|Db
| E#
|[[14/13]]
| [[13/10]] I (patent val approximation)
|-
|-
|41
| 134
|
| E#-C#
|[[13/12]]
| 13/10 II (direct approximation)
|-
|-
|46
| 152
|Db-F
| F
|[[35/32]]
|  
|-
|-
|57
| 171
|D
| F#
|
| [[7/5]]
|-
|-
|76
| 190
|D#
| Gb
|
|  
|-
|-
|95
| 206
|Eb
| Gb-Bb
|
| 3/2
|-
|-
|114
| 209
|E
| G
|[[5/4]]
| [[98/65]]
|-
|-
|133
| 228
|E#
| G#
|[[13/10]] I (patent val approximation)
|  
|-
|-
|134
| 247
|E#-C#
| Ab
|13/10 II (direct approximation)
| [[13/8]]
|-
|-
|152
| 266
|F
| A
|
|  
|-
|-
|171
| 285
|F#
| A#
|[[7/5]]
| [[7/4]]
|-
|-
|190
| 304
|Gb
| Bb
|
|  
|-
|-
|206
| 323
|Gb-Bb
| B
|
|  
|-
|-
|209
| 342
|G
| B#/Cb
|98/65,
|  
|-
|-
|228
| 353
|G#
| 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"
|-
|-
|247
! rowspan="2" | [[Subgroup]]
|Ab
! rowspan="2" | [[Comma list]]
|[[13/8]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|266
! [[TE error|Absolute]] (¢)
|A
! [[TE simple badness|Relative]] (%)
|
|-
|-
|285
| 2.5
|A#
| {{monzo| 820 -353 }}
|[[7/4]]
| {{mapping| 353 820 }}
| −0.263
| 0.263
| 7.74
|-
|-
|304
| 2.5.7
|Bb
| 3136/3125, {{monzo| 209 -9 -67 }}
|
| {{mapping| 353 820 991 }}
| −0.177
| 0.247
| 7.26
|-
|-
|323
| 2.5.7.11
|B
| 3136/3125, 5767168/5764801, {{monzo| -20 -6  1 9 }}
|
| {{mapping| 353 820 991 1221 }}
| −0.089
| 0.263
| 7.73
|-
|-
|342
| 2.5.7.11.13
|B#/Cb
| 3136/3125, 4394/4375, 6656/6655, 5767168/5764801
|
| {{mapping| 353 820 991 1221 1306 }}
| −0.024
| 0.268
| 7.89
|-
|-
|353
| 2.5.7.11.13.17
|C
| 3136/3125, 4394/4375, 7744/7735, 60112/60025, 64141/64000
|2/1
| {{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 162: 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 }}
Retrieved from "https://en.xen.wiki/w/353edo"