467edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|467}}
{{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]].  


It [[support]]s [[mitonic]] and [[counterkleismic]].
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|Comma List]]
! rowspan="2" | [[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]] (%)
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| {{monzo| -740 467 }}
| {{monzo| -740 467 }}
| {{mapping| 467 740 }}
| {{mapping| 467 740 }}
| 0.1439
| +0.1439
| 0.1439
| 0.1439
| 5.38
| 5.38
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| {{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
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| 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
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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
|-
|-
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| [[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 />* [[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]]

Latest revision as of 06:14, 21 February 2025

← 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

467 equal divisions of the octave (abbreviated 467edo or 467ed2), also called 467-tone equal temperament (467tet) or 467 equal temperament (467et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 467 equal parts of about 2.57 ¢ each. Each step represents a frequency ratio of 21/467, or the 467th root of 2.

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. It supports 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

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 distinct

Music

Francium
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