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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]].  
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]].  


The equal temperament [[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]].  
It [[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 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.  


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]].  
Since 323 factors into {{factorisation|323}}, 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 29: Line 29:
| {{monzo| 512 -323 }}
| {{monzo| 512 -323 }}
| {{mapping| 323 512 }}
| {{mapping| 323 512 }}
| −0.0669
| −0.0669
| 0.0669
| 0.0669
| 1.80
| 1.80
Line 36: Line 36:
| {{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
| −0.0538
| 0.0577
| 0.0577
| 1.55
| 1.55
Line 43: Line 43:
| 4375/4374, 589824/588245, 703125/702464
| 4375/4374, 589824/588245, 703125/702464
| {{mapping| 323 512 750 907 }}
| {{mapping| 323 512 750 907 }}
| −0.1146
| −0.1146
| 0.1165
| 0.1165
| 3.14
| 3.14
Line 50: Line 50:
| 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
| −0.0431
| 0.1770
| 0.1770
| 4.76
| 4.76
Line 64: Line 64:
| 1375/1372, 4375/4374, 5632/5625, 14641/14580
| 1375/1372, 4375/4374, 5632/5625, 14641/14580
| {{mapping| 323 512 750 907 1117 }} (323)
| {{mapping| 323 512 750 907 1117 }} (323)
| −0.0066
| −0.0066
| 0.2399
| 0.2399
| 6.46
| 6.46
Line 78: Line 78:
| 540/539, 4375/4374, 12005/11979, 16384/16335
| 540/539, 4375/4374, 12005/11979, 16384/16335
| {{mapping| 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
Line 85: Line 85:
| 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
| −0.1440
| 0.2773
| 0.2773
| 7.47
| 7.47
Line 149: Line 149:
| [[Enneadecal]]
| [[Enneadecal]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Deuteromere]]
[[Category:Deuteromere]]
[[Category:Lambeth]]
[[Category:Lambeth]]
[[Category:Stockhausenic]]
[[Category:Stockhausenic]]

Revision as of 15:21, 16 January 2025

← 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

Template:EDO intro

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.

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.

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

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)

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 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\323 475.54 320/243 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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