465edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|465}} == Theory == 465et is consistent to the 5-odd-limit. It can be considered for the 2.3.5.11.13.17 subgroup, tempering out 936/935,..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|465}}
{{ED intro}}


== Theory ==
== Theory ==
465et is consistent to the [[5-odd-limit]]. It can be considered for the 2.3.5.11.13.17 [[subgroup]], tempering out [[936/935]], 1377/1375, 71874/71825, 131648/131625 and 225000/224939. It [[support]]s [[birds]], [[counterschismic]] and [[belobog]].
465edo is only [[consistent]] to the [[5-odd-limit]], and the errors of [[harmonic]]s beyond [[3/1|3]] tend to be quite large. It can be considered for the 2.3.5.11.13.17 [[subgroup]], [[tempering out]] [[936/935]], 1377/1375, 71874/71825, 131648/131625 and 225000/224939. It [[support]]s [[counterschismic]] in the 5-limit, and [[birds]] and [[belobog]] in the 7-limit using the [[patent val]].


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
465 factors into 3 × 5 × 31, with subset edos {{EDOs|3, 5, 15, 31, 93, and 155}}. [[930edo]], which doubles it, gives a good correction to the harmonic 7.
Since 465 factors into 3 × 5 × 31, 465edo has subset edos {{EDOs| 3, 5, 15, 31, 93, and 155 }}. [[930edo]], which doubles it, gives a good correction to the harmonic 7.


== 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|-737 465}}
! rowspan="2" | [[Comma list]]
|{{mapping|465 737}}
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -737 465 }}
| {{mapping| 465 737 }}
| +0.0062
| +0.0062
| 0.0062
| 0.0062
| 0.24
| 0.24
|-
|-
|2.3.5
| 2.3.5
|{{monzo|25 15 -21}}, {{monzo|-22 30 -11}}
| {{monzo| 25 15 -21 }}, {{monzo| -22 30 -11 }}
|{{mapping|465 737 1080}}
| {{mapping| 465 737 1080 }}
| -0.1083
| −0.1083
| 0.1619
| 0.1619
| 6.27
| 6.27
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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
|193\465
| 193\465
|498.06
| 498.06
|4/3
| 4/3
|[[Counterschismic]]
| [[Counterschismic]]
|-
|-
|5
| 5
|322\465<br>(43\465)
| 322\465<br />(43\465)
|830.97<br>(110.97)
| 830.97<br />(110.97)
|80/49<br>(15/14)
| 80/49<br />(15/14)
|[[Qintosec]]
| [[Qintosec]] (465)
|}
|}
 
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if 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

Latest revision as of 13:33, 13 March 2026

← 464edo 465edo 466edo →
Prime factorization 3 × 5 × 31
Step size 2.58065 ¢ 
Fifth 272\465 (701.935 ¢)
Semitones (A1:m2) 44:35 (113.5 ¢ : 90.32 ¢)
Consistency limit 5
Distinct consistency limit 5

465 equal divisions of the octave (abbreviated 465edo or 465ed2), also called 465-tone equal temperament (465tet) or 465 equal temperament (465et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 465 equal parts of about 2.58 ¢ each. Each step represents a frequency ratio of 21/465, or the 465th root of 2.

Theory

465edo is only consistent to the 5-odd-limit, and the errors of harmonics beyond 3 tend to be quite large. It can be considered for the 2.3.5.11.13.17 subgroup, tempering out 936/935, 1377/1375, 71874/71825, 131648/131625 and 225000/224939. It supports counterschismic in the 5-limit, and birds and belobog in the 7-limit using the patent val.

Prime harmonics

Approximation of prime harmonics in 465edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.02 +0.78 -1.08 +0.94 +0.76 +0.85 -0.74 -1.18 +0.10 +0.77
Relative (%) +0.0 -0.8 +30.3 -42.0 +36.4 +29.6 +33.0 -28.6 -45.6 +3.9 +29.9
Steps
(reduced) 		465
(0) 		737
(272) 		1080
(150) 		1305
(375) 		1609
(214) 		1721
(326) 		1901
(41) 		1975
(115) 		2103
(243) 		2259
(399) 		2304
(444)

Subsets and supersets

Since 465 factors into 3 × 5 × 31, 465edo has subset edos 3, 5, 15, 31, 93, and 155. 930edo, which doubles it, gives a good correction to the harmonic 7.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-737 465⟩ [⟨465 737]] +0.0062 0.0062 0.24
2.3.5 [25 15 -21⟩, [-22 30 -11⟩ [⟨465 737 1080]] −0.1083 0.1619 6.27

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 193\465 498.06 4/3 Counterschismic
5 322\465
(43\465) 		830.97
(110.97) 		80/49
(15/14) 		Qintosec (465)

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

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