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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 temperament tempering out 14641/14580.  
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 temperament tempering out 14641/14580.  


323 = 17 × 19, and shares the excellent approximations of [[25/24]] in [[17edo]] and of the [[28/27]] and the [[6/5]] in [[19edo]].  
Since {{nowrap|323 {{=}} 17 × 19}}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[28/27]] and the [[6/5]] in [[19edo]].  


=== Prime harmonics ===
=== Prime harmonics ===
Line 15: Line 15:


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 512 -323 }}
| {{monzo| 512 -323 }}
| {{mapping| 323 512 }}
| {{mapping| 323 512 }}
| -0.0669
| &minus;0.0669
| 0.0669
| 0.0669
| 1.80
| 1.80
Line 35: Line 27:
| {{monzo| 24 -21 4 }}, {{monzo| 38 -2 -15 }}
| {{monzo| 24 -21 4 }}, {{monzo| 38 -2 -15 }}
| {{mapping| 323 512 750 }}
| {{mapping| 323 512 750 }}
| -0.0538
| &minus;0.0538
| 0.0577
| 0.0577
| 1.55
| 1.55
Line 42: Line 34:
| 4375/4374, 589824/588245, 703125/702464
| 4375/4374, 589824/588245, 703125/702464
| {{mapping| 323 512 750 907 }}
| {{mapping| 323 512 750 907 }}
| -0.1146
| &minus;0.1146
| 0.1165
| 0.1165
| 3.14
| 3.14
Line 49: Line 41:
| 676/675, 4096/4095, 4375/4374, 16848/16807
| 676/675, 4096/4095, 4375/4374, 16848/16807
| {{mapping| 323 512 750 907 1195 }}
| {{mapping| 323 512 750 907 1195 }}
| -0.0431
| &minus;0.0431
| 0.1770
| 0.1770
| 4.76
| 4.76
Line 59: Line 51:
| 0.1905
| 0.1905
| 5.13
| 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)
| &minus;0.0066
| 0.2399
| 6.46
|-
|-
| style="border-top: double;" | 2.3.5.7.11
| 2.3.5.7.11.13
| style="border-top: double;" | 540/539, 4375/4374, 12005/11979, 16384/16335
| 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374
| style="border-top: double;" | {{mapping| 323 512 750 907 1118 }} (323e)
| {{mapping| 323 512 750 907 1117 1195 }} (323)
| style="border-top: double;" | -0.2213
| +0.0350
| style="border-top: double;" | 0.2375
| 0.2380
| style="border-top: double;" | 6.39
| 6.40
|- style="border-top: double;"
| 2.3.5.7.11
| 540/539, 4375/4374, 12005/11979, 16384/16335
| {{mapping| 323 512 750 907 1118 }} (323e)
| &minus;0.2213
| 0.2375
| 6.39
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 364/363, 540/539, 676/675, 4096/4095, 4375/4374
| 364/363, 540/539, 676/675, 4096/4095, 4375/4374
| {{mapping| 323 512 750 907 1118 1195 }} (323e)
| {{mapping| 323 512 750 907 1118 1195 }} (323e)
| -0.1440
| &minus;0.1440
| 0.2773
| 0.2773
| 7.47
| 7.47
|-
{{comma basis end}}
| style="border-top: double;" | 2.3.5.7.11
| style="border-top: double;" | 1375/1372, 4375/4374, 5632/5625, 14641/14580
| style="border-top: double;" | {{mapping| 323 512 750 907 1117 }} (323)
| style="border-top: double;" | -0.0066
| style="border-top: double;" | 0.2399
| style="border-top: double;" | 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
|}
* 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[289edo|289]] and followed by [[388edo|388]].
* 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"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
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| 4/3<br>(225/224)
| 4/3<br>(225/224)
| [[Enneadecal]]
| [[Enneadecal]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


[[Category:Deuteromere]]
[[Category:Deuteromere]]
[[Category:Lambeth]]
[[Category:Stockhausenic]]
[[Category:Stockhausenic]]
[[Category:Lambeth]]
Retrieved from "https://en.xen.wiki/w/323edo"