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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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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|467}}
{{ED intro}}


== 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. It [[support]]s [[mitonic]] and [[counterkleismic]], supplying the [[optimal patent val]] for the latter.
 
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]].
 
In the 17-limit, it supplies the optimal patent val for the rank-6 temperament tempering out [[375/374]].  


=== 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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|-740 467}}
! rowspan="2" | [[Comma list]]
|{{val|467 740}}
! rowspan="2" | [[Mapping]]
| 0.1439
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -740 467 }}
| {{mapping| 467 740 }}
| +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
| 6.05
| 6.05
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=== 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*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! 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 />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Music ==
; [[Francium]]
* "Cuckoo Mackerel" from ''Cursed Cuckoo Creations'' (2024) – [https://open.spotify.com/track/6d9Ip2EuZkqQdx0OUJkw86 Spotify] | [https://francium223.bandcamp.com/track/cuckoo-mackerel Bandcamp] | [https://www.youtube.com/watch?v=MjgQWmcKQB4 YouTube]
* "livemywarmlive" from ''albumwithoutspaces'' (2024) – [https://open.spotify.com/track/4jwETd5M7kbxkcR1TOflMk Spotify] | [https://francium223.bandcamp.com/track/livemywarmlive Bandcamp] | [https://www.youtube.com/watch?v=G2CH958nhOk YouTube] – counterkleismic[19] in 467edo tuning


<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:Counterkleismic]]
[[Category:Listen]]
[[Category:Ursulismic]]
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