467edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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. | 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 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" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 29: | Line 30: | ||
| {{monzo| -740 467 }} | | {{monzo| -740 467 }} | ||
| {{mapping| 467 740 }} | | {{mapping| 467 740 }} | ||
| 0.1439 | | +0.1439 | ||
| 0.1439 | | 0.1439 | ||
| 5.38 | | 5.38 | ||
| Line 36: | Line 37: | ||
| {{monzo| -36 11 8 }}, {{monzo| -16 35 -17 }} | | {{monzo| -36 11 8 }}, {{monzo| -16 35 -17 }} | ||
| {{mapping| 467 740 1084 }} | | {{mapping| 467 740 1084 }} | ||
| 0.2215 | | +0.2215 | ||
| 0.1608 | | 0.1608 | ||
| 6.02 | | 6.02 | ||
| Line 43: | Line 44: | ||
| 4375/4374, 2100875/2097152, {{monzo| -16 4 9 -4 }} | | 4375/4374, 2100875/2097152, {{monzo| -16 4 9 -4 }} | ||
| {{mapping| 467 740 1084 1311 }} | | {{mapping| 467 740 1084 1311 }} | ||
| 0.1741 | | +0.1741 | ||
| 0.1617 | | 0.1617 | ||
| 6.05 | | 6.05 | ||
| Line 50: | Line 51: | ||
=== 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> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 69: | Line 71: | ||
| [[Counterhanson]] | | [[Counterhanson]] | ||
|} | |} | ||
<nowiki>* | <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 | |||
[[Category:Counterkleismic]] | |||
[[Category:Listen]] | |||
[[Category:Ursulismic]] | |||