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The '''323 equal divisions of the octave''' ('''323edo''') is the [[EDO|equal division of the octave]] into 323 parts of 3.7152 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
323et [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], 589824/588245 and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]]. In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, 5632/5625, [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as the 2.3.5.11 subgroup temperament tempering out 14641/14580, the semicanousma.  
323edo is a strong [[5-limit]] system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].  


323 = 17 × 19, and shares the excellent approximations of [[25/24]] in [[17edo]] and of the [[28/27]] and the [[6/5]] in [[19edo]].  
As an equal temperament, it [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the 5-limit; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], [[support]]ing 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].
 
In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. [[1375/1372]], [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], [[12005/11979]], and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament tempering out 14641/14580.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|323}}
{{Harmonics in equal|323|columns=11}}
{{Harmonics in equal|323|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 323edo (continued)}}
 
=== Subsets and supersets ===
Since 323 factors into primes as {{nowrap| 17 × 19 }}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[6/5]] and [[28/27]] in [[19edo]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 21: Line 29:
|-
|-
| 2.3
| 2.3
| {{monzo| 512 -323 }}
| {{Monzo| 512 -323 }}
| [{{val| 323 512 }}]
| {{Mapping| 323 512 }}
| -0.0669
| −0.0669
| 0.0669
| 0.0669
| 1.80
| 1.80
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 24 -21 4 }}, {{monzo| 38 -2 -15 }}
| {{Monzo| 24 -21 4 }}, {{monzo| 38 -2 -15 }}
| [{{val| 323 512 750 }}]
| {{Mapping| 323 512 750 }}
| -0.0538
| −0.0538
| 0.0577
| 0.0577
| 1.55
| 1.55
Line 36: Line 44:
| 2.3.5.7
| 2.3.5.7
| 4375/4374, 589824/588245, 703125/702464
| 4375/4374, 589824/588245, 703125/702464
| [{{val| 323 512 750 907 }}]
| {{Mapping| 323 512 750 907 }}
| -0.1146
| −0.1146
| 0.1165
| 0.1165
| 3.14
| 3.14
|-
|-
| 2.3.5.7.13
| 676/675, 4096/4095, 4375/4374, 16848/16807
| {{Mapping| 323 512 750 907 1195 }}
| −0.0431
| 0.1770
| 4.76
|-
| 2.3.5.7.13.17
| 442/441, 676/675, 2500/2499, 4096/4095, 4375/4374
| {{Mapping| 323 512 750 907 1195 1320 }}
| +0.0020
| 0.1905
| 5.13
|- style="border-top: double;"
| 2.3.5.7.11
| 1375/1372, 4375/4374, 5632/5625, 14641/14580
| {{Mapping| 323 512 750 907 1117 }} (323)
| −0.0066
| 0.2399
| 6.46
|-
| 2.3.5.7.11.13
| 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374
| {{Mapping| 323 512 750 907 1117 1195 }} (323)
| +0.0350
| 0.2380
| 6.40
|- style="border-top: double;"
| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 4375/4374, 12005/11979, 16384/16335
| 540/539, 4375/4374, 12005/11979, 16384/16335
| [{{val| 323 512 750 907 1118 }}] (323e)
| {{Mapping| 323 512 750 907 1118 }} (323e)
| -0.2213
| −0.2213
| 0.2375
| 0.2375
| 6.39
| 6.39
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11.13
| 1375/1372, 4375/4374, 5632/5625, 14641/14580
| 364/363, 540/539, 676/675, 4096/4095, 4375/4374
| [{{val| 323 512 750 907 1117 }}] (323)
| {{Mapping| 323 512 750 907 1118 1195 }} (323e)
| -0.0066
| −0.1440
| 0.2399
| 0.2773
| 6.46
| 7.47
|}
|}
* 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[289edo|289]] and followed by [[388edo|388]].


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 200/189
| 200/189
| [[Hemiluna]] (323)
| [[Hemiluna]] (323)
|-
| 1
| 27\323
| 100.31
| 675/637
| [[Heptacot]] (323)
|-
| 1
| 30\323
| 111.46
| 16/15
| [[Stockhausenic]] (323)
|-
| 1
| 31\323
| 115.17
| 77/72
| [[Semigamera]] (323)
|-
|-
| 1
| 1
Line 86: Line 142:
| 128\323
| 128\323
| 475.54
| 475.54
| 320/243
| 25/19
| [[Vulture]]
| [[Vulture]]
|-
|-
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| [[Enneadecal]]
| [[Enneadecal]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave]]
[[Category:Deuteromere]]
[[Category:Deuteromere]]
[[Category:Lambeth]]
[[Category:Lambeth]]
[[Category:Stockhausenic]]
Retrieved from "https://en.xen.wiki/w/323edo"