323edo: Difference between revisions

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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
Line 146: Line 132:
| 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]]

Revision as of 05:15, 16 November 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

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 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 = 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

Template:Comma basis begin |- | 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 |- style="border-top: double;" | 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 |- style="border-top: double;" | 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 Template:Comma basis end

  • 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past 289 and followed by 388.

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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