Xenharmonic Wiki

353edo: Difference between revisions

← Older edit
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,351 edits
mNo edit summary
 
(43 intermediate revisions by 6 users not shown)
Line 1: Line 1:
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.  


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. In the 13-limit, the it tempers out [[3136/3125]], [[4394/4375]], [[10985/10976]], and [[1968512/1953125]]. This gives it a few more unique intervals.
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, {{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**
|-
| 0
| C
| 1/1
|-
| 1
| C-C#
|
|-
| 2
| C-Db
|
|-
| 3
| C-D
| [[196/195]]
|-
| 4
| C-D#
|
|-
|-
|19
| 19
|C#
| C#
|
| [[26/25]]
|-
|-
|38
| 38
|Db
| Db
|[[14/13]]
| [[14/13]]
|-
|-
|57
| 41
|D
| Db-D
|
| [[13/12]]
|-
|-
|76
| 46
|D#
| Db-F
|
| [[35/32]]
|-
|-
|95
| 57
|Eb
| D
|
|  
|-
|-
|114
| 76
|E
| D#
|[[6/5]]
|  
|-
|-
|133
| 95
|E#/Fb
| Eb
|[[13/10]] minor (best approximation is 134)
|  
|-
|-
|152
| 114
|F
| E
|
| [[5/4]]
|-
|-
|171
| 133
|F#
| E#
|[[7/5]]
| [[13/10]] I (patent val approximation)
|-
|-
|190
| 134
|Gb
| E#-C#
|
| 13/10 II (direct approximation)
|-
|-
|209
| 152
|G
| F
|98/65
|  
|-
|-
|228
| 171
|G#
| F#
|
| [[7/5]]
|-
|-
|247
| 190
|Ab
| Gb
|[[13/8]]
|  
|-
|-
|266
| 206
|A
| Gb-Bb
|
| 3/2
|-
|-
|285
| 209
|A#
| G
|[[7/4]]
| [[98/65]]
|-
|-
|304
| 228
|Bb
| G#
|
|  
|-
|-
|323
| 247
|B
| Ab
|
| [[13/8]]
|-
|-
|342
| 266
|B#/Cb
| A
|
|  
|-
|-
|353
| 285
|C
| A#
|2/1
| [[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 114: 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"