467edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|467}} == Theory == 467et is consistent to thr 9-odd-limit. Using the patent val, it tempers out 4375/4374, 1640558367/1638400000, 52509..."
 
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== Theory ==
== Theory ==
467et is consistent to thr [[9-odd-limit]]. Using the patent val, it tempers out [[4375/4374]], [[1640558367/1638400000]], 5250987/5242880 and [[2100875/2097152]] in the 7-limit; 25165824/25109315, 1019215872/1019046875, 2097152/2096325, 26214400/26198073, 104162436/103984375, 166698/166375, 12005/11979, 151263/151250, 117649/117612, 514714375/514434888, 226492416/226474325, 104857600/104825259, 472392/471625, [[540/539]], [[6250/6237]], [[1953125/1948617]], 825000/823543, 85937500/85766121, 47265625/47258883 and 9453125/9437184 in the 11-limit. It [[support]]s [[counterkleismic]] and [[minos]].
467edo is [[consistent]] to the [[9-odd-limit]] with [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] all tuned flat. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[4375/4374]], [[2100875/2097152]], 5250987/5242880, and {{monzo| -16 4 9 -4 }} in the 7-limit.
 
In the 11-limit, the 467e [[val]] scores much better than the [[patent val]]. The 467e val tempers out 1375/1372, 24057/24010, 35937/35840, and 41503/41472, and in the 13-limit, [[625/624]], [[729/728]], [[1716/1715]], and [[2200/2197]]. The patent val tempers out [[540/539]], [[6250/6237]], 12005/11979, and 14700/14641, and in the 13-limit, 625/624, 729/728, and [[2080/2079]].  
 
It [[support]]s [[mitonic]] and [[counterkleismic]].


=== Odd harmonics ===
=== Odd harmonics ===
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== 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|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|-740 467}}
| {{monzo| -740 467 }}
|{{val|467 740}}
| {{mapping| 467 740 }}
| 0.1439
| 0.1439
| 0.1439
| 0.1439
| 5.38
| 5.38
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-36 11 8}}, {{monzo|-16 35 -17}}
| {{monzo| -36 11 8 }}, {{monzo| -16 35 -17 }}
|{{val|467 740 1084}}
| {{mapping| 467 740 1084 }}
| 0.2215
| 0.2215
| 0.1608
| 0.1608
| 6.02
| 6.02
|-
|-
|2.3.5.7
| 2.3.5.7
|4375/4374, 2100875/2097152, 5250987/5242880
| 4375/4374, 2100875/2097152, {{monzo| -16 4 9 -4 }}
|{{val|467 740 1084 1311}}
| {{mapping| 467 740 1084 1311 }}
| 0.1741
| 0.1741
| 0.1617
| 0.1617
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! Temperaments
! Temperaments
|-
|-
|1
| 1
|71\467
| 71\467
|182.441
| 182.441
|10/9
| 10/9
|[[Minortone]] / [[Mitonic]]
| [[Mitonic]]
|-
|-
|1
| 1
|123\467
| 123\467
|316.060
| 316.060
|6/5
| 6/5
|[[Counterhanson]]
| [[Counterhanson]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Revision as of 12:51, 17 February 2024

← 466edo 467edo 468edo →
Prime factorization 467 (prime)
Step size 2.56959 ¢ 
Fifth 273\467 (701.499 ¢)
Semitones (A1:m2) 43:36 (110.5 ¢ : 92.51 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

Theory

467edo is consistent to the 9-odd-limit with harmonics 3, 5, and 7 all tuned flat. Using the patent val, the equal temperament tempers out 4375/4374, 2100875/2097152, 5250987/5242880, and [-16 4 9 -4⟩ in the 7-limit.

In the 11-limit, the 467e val scores much better than the patent val. The 467e val tempers out 1375/1372, 24057/24010, 35937/35840, and 41503/41472, and in the 13-limit, 625/624, 729/728, 1716/1715, and 2200/2197. The patent val tempers out 540/539, 6250/6237, 12005/11979, and 14700/14641, and in the 13-limit, 625/624, 729/728, and 2080/2079.

It supports mitonic and counterkleismic.

Odd harmonics

Approximation of odd harmonics in 467edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.46 -0.87 -0.09 -0.91 +1.14 -0.27 +1.24 +0.40 +0.56 -0.55 +1.28
Relative (%) -17.7 -34.0 -3.5 -35.5 +44.5 -10.5 +48.2 +15.5 +21.8 -21.2 +49.7
Steps
(reduced) 		740
(273) 		1084
(150) 		1311
(377) 		1480
(79) 		1616
(215) 		1728
(327) 		1825
(424) 		1909
(41) 		1984
(116) 		2051
(183) 		2113
(245)

Subsets and supersets

467edo is the 91st prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-740 467⟩ [⟨467 740]] 0.1439 0.1439 5.38
2.3.5 [-36 11 8⟩, [-16 35 -17⟩ [⟨467 740 1084]] 0.2215 0.1608 6.02
2.3.5.7 4375/4374, 2100875/2097152, [-16 4 9 -4⟩ [⟨467 740 1084 1311]] 0.1741 0.1617 6.05

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 71\467 182.441 10/9 Mitonic
1 123\467 316.060 6/5 Counterhanson

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

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