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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|323}}
{{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 [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.  
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]].  


323edo is an excellent tuning when considered in the no-11 subgroup, with errors of 25% or less all the way into the 31-limit.
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 ===
{{Harmonics in equal|323|columns=11}}
{{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|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 24: 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 39: 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
| 2.3.5.7.13
| 676/675, 4375/4374, 4096/4095, 16848/16807
| 676/675, 4096/4095, 4375/4374, 16848/16807
| [{{val| 323 512 750 907 1195}}]
| {{Mapping| 323 512 750 907 1195 }}
| -0.0431
| −0.0431
| 0.1770
| 0.1770
| 4.76
| 4.76
|-
|-
| 2.3.5.7.13.17
| 2.3.5.7.13.17
| 442/441, 1275/1274, 2500/2499, 4096/4095, 4375/4374
| 442/441, 676/675, 2500/2499, 4096/4095, 4375/4374
| [{{val| 323 512 750 907 1195 1320}}]
| {{Mapping| 323 512 750 907 1195 1320 }}
| +0.0020
| +0.0020
| 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)
| −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;" | [{{val| 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)
| −0.2213
| 0.2375
| 6.39
|-
|-
| style="border-top: double;" | 2.3.5.7.11
| 2.3.5.7.11.13
| style="border-top: double;" | 1375/1372, 4375/4374, 5632/5625, 14641/14580
| 364/363, 540/539, 676/675, 4096/4095, 4375/4374
| style="border-top: double;" | [{{val| 323 512 750 907 1117 }}] (323)
| {{Mapping| 323 512 750 907 1118 1195 }} (323e)
| style="border-top: double;" | -0.0066
| −0.1440
| style="border-top: double;" | 0.2399
| 0.2773
| style="border-top: double;" | 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 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! 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
| 1
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| 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|###]] <!-- 3-digit number -->
[[Category:Deuteromere]]
[[Category:Deuteromere]]
[[Category:Lambeth]]
[[Category:Stockhausenic]]
[[Category:Stockhausenic]]
[[Category:Lambeth]]

Latest revision as of 12:11, 20 May 2026

← 322edo 323edo 324edo →
Prime factorization 17 × 19
Step size 3.71517 ¢ 
Fifth 189\323 (702.167 ¢)
Semitones (A1:m2) 31:24 (115.2 ¢ : 89.16 ¢)
Consistency limit 9
Distinct consistency limit 9

323 equal divisions of the octave (abbreviated 323edo or 323ed2), also called 323-tone equal temperament (323tet) or 323 equal temperament (323et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 323 equal parts of about 3.72 ¢ each. Each step represents a frequency ratio of 21/323, or the 323rd root of 2.

Theory

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.

As an equal temperament, it tempers out the vulture comma, [24 -21 4⟩ and the luna comma, [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 13-limit stockhausenic, and deuteromere, the 2.3.5.11-subgroup temperament tempering out 14641/14580.

Prime harmonics

Approximation of prime harmonics in 323edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.21 +0.06 +0.83 -1.47 -0.90 -0.93 -0.30 -0.41 -0.48 -0.76
Relative (%) +0.0 +5.7 +1.7 +22.4 -39.6 -24.2 -25.0 -8.1 -11.1 -12.8 -20.5
Steps
(reduced) 		323
(0) 		512
(189) 		750
(104) 		907
(261) 		1117
(148) 		1195
(226) 		1320
(28) 		1372
(80) 		1461
(169) 		1569
(277) 		1600
(308)
Approximation of prime harmonics in 323edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +1.29 -1.82 +1.18 -0.49 -0.44 -0.35 +1.38 -1.29 -1.37 -1.16 -0.45
Relative (%) +34.7 -48.9 +31.6 -13.2 -11.8 -9.4 +37.2 -34.7 -36.8 -31.3 -12.1
Steps
(reduced) 		1683
(68) 		1730
(115) 		1753
(138) 		1794
(179) 		1850
(235) 		1900
(285) 		1916
(301) 		1959
(21) 		1986
(48) 		1999
(61) 		2036
(98)

Subsets and supersets

Since 323 factors into primes as 17 × 19, 323edo shares the excellent approximations of 25/24 in 17edo and of 6/5 and 28/27 in 19edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [512 -323⟩ [⟨323 512]] −0.0669 0.0669 1.80
2.3.5 [24 -21 4⟩, [38 -2 -15⟩ [⟨323 512 750]] −0.0538 0.0577 1.55
2.3.5.7 4375/4374, 589824/588245, 703125/702464 [⟨323 512 750 907]] −0.1146 0.1165 3.14
2.3.5.7.13 676/675, 4096/4095, 4375/4374, 16848/16807 [⟨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 [⟨323 512 750 907 1195 1320]] +0.0020 0.1905 5.13
2.3.5.7.11 1375/1372, 4375/4374, 5632/5625, 14641/14580 [⟨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 [⟨323 512 750 907 1117 1195]] (323) +0.0350 0.2380 6.40
2.3.5.7.11 540/539, 4375/4374, 12005/11979, 16384/16335 [⟨323 512 750 907 1118]] (323e) −0.2213 0.2375 6.39
2.3.5.7.11.13 364/363, 540/539, 676/675, 4096/4095, 4375/4374 [⟨323 512 750 907 1118 1195]] (323e) −0.1440 0.2773 7.47
  • 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past 289 and followed by 388.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 26\323 96.59 200/189 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 52\323 193.19 352/315 Luna / lunatic (323e)
1 62\323 230.34 8/7 Gamera
1 128\323 475.54 25/19 Vulture
17 134\323
(9\323) 		248.92
(33.44) 		[-23 5 9 -2⟩
(100352/98415) 		Chlorine
19 134\323
(2\323) 		497.83
(7.43) 		4/3
(225/224) 		Enneadecal

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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