323edo: Difference between revisions

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


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.   
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]].   


=== Prime harmonics ===
=== Prime harmonics ===
Line 25: Line 25:
| 2.3
| 2.3
| {{monzo| 512 -323 }}
| {{monzo| 512 -323 }}
| [{{val| 323 512 }}]
| {{mapping| 323 512 }}
| -0.0669
| -0.0669
| 0.0669
| 0.0669
Line 32: Line 32:
| 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
Line 39: Line 39:
| 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
Line 46: Line 46:
| 2.3.5.7.13
| 2.3.5.7.13
| 676/675, 4096/4095, 4375/4374, 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
Line 53: Line 53:
| 2.3.5.7.13.17
| 2.3.5.7.13.17
| 442/441, 676/675, 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
Line 60: Line 60:
| style="border-top: double;" | 2.3.5.7.11
| style="border-top: double;" | 2.3.5.7.11
| style="border-top: double;" | 540/539, 4375/4374, 12005/11979, 16384/16335
| style="border-top: double;" | 540/539, 4375/4374, 12005/11979, 16384/16335
| style="border-top: double;" | [{{val| 323 512 750 907 1118 }}] (323e)
| style="border-top: double;" | {{mapping| 323 512 750 907 1118 }} (323e)
| style="border-top: double;" | -0.2213
| style="border-top: double;" | -0.2213
| style="border-top: double;" | 0.2375
| style="border-top: double;" | 0.2375
Line 67: Line 67:
| 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
| [{{val| 323 512 750 907 1118 1195 }}] (323e)
| {{mapping| 323 512 750 907 1118 1195 }} (323e)
| -0.1440
| -0.1440
| 0.2773
| 0.2773
Line 74: Line 74:
| style="border-top: double;" | 2.3.5.7.11
| 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;" | 1375/1372, 4375/4374, 5632/5625, 14641/14580
| style="border-top: double;" | [{{val| 323 512 750 907 1117 }}] (323)
| style="border-top: double;" | {{mapping| 323 512 750 907 1117 }} (323)
| style="border-top: double;" | -0.0066
| style="border-top: double;" | -0.0066
| style="border-top: double;" | 0.2399
| style="border-top: double;" | 0.2399
Line 81: Line 81:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374
| 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374
| [{{val| 323 512 750 907 1117 1195 }}] (323)
| {{mapping| 323 512 750 907 1117 1195 }} (323)
| +0.0350
| +0.0350
| 0.2380
| 0.2380
Line 91: Line 91:
|+Table of rank-2 temperaments by generator
|+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
|-
|-
Line 144: Line 144:
| [[Enneadecal]]
| [[Enneadecal]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Deuteromere]]
[[Category:Deuteromere]]
[[Category:Stockhausenic]]
[[Category:Stockhausenic]]
[[Category:Lambeth]]
[[Category:Lambeth]]

Revision as of 10:02, 21 February 2024

← 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

The equal temperament 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.

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

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.

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
2.3.5.7.11 540/539, 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
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

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 it is distinct

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