761edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|761}} == Theory == 761edo is consistent to the 9-odd-limit. It tempers out 32805/32768, 420175/419904 and {{monzo|3 13 -15 4}}..."
 
m Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|761}}
{{ED intro}}


== Theory ==
== Theory ==
761edo is [[consistent]] to the [[9-odd-limit]]. It [[tempers out]] [[32805/32768]], [[420175/419904]] and {{monzo|3 13 -15 4}} in the [[7-limit]]. The equal temperament is strong in the 2.3.5.13.29.31 [[subgroup]], tempering out 32805/32768, 21141/21125, 3627/3625, [[140625/140608]] and 45349632/45287125.
761edo is [[consistent]] to the [[9-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] [[32805/32768]] in the 5-limit; [[420175/419904]] and {{monzo| 3 13 -15 4 }} in the [[7-limit]]. The equal temperament is strong in the 2.3.5.13.29.31 [[subgroup]], tempering out 32805/32768, 21141/21125, 3627/3625, [[140625/140608]] and 45349632/45287125.


=== 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" | [[Subgroup]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Comma list]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | [[Mapping]]
! colspan="2" |Tuning Error
! rowspan="2" | Optimal<br />8ve stretch (¢)
|-
! colspan="2" | Tuning error
![[TE error|Absolute]] (¢)
|-
![[TE simple badness|Relative]] (%)
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo|-1206 761}}
| {{monzo|-1206 761}}
| {{mapping|761 1206}}
| {{mapping|761 1206}}
| 0.0778
| +0.0778
| 0.0778
| 0.0778
| 4.93
| 4.93
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| 32805/32768, {{monzo|69 81 -85}}
| 32805/32768, {{monzo|69 81 -85}}
| {{mapping|761 1206 1767}}
| {{mapping|761 1206 1767}}
| 0.0490
| +0.0490
| 0.0755
| 0.0755
| 4.79
| 4.79
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| 32805/32768, 420175/419904, {{monzo|3 13 -15 4}}
| 32805/32768, 420175/419904, {{monzo|3 13 -15 4}}
| {{mapping|761 1206 1767 2136}}
| {{mapping|761 1206 1767 2136}}
| 0.0925
| +0.0925
| 0.0998
| 0.0998
| 6.33
| 6.33
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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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| 498.292
| 498.292
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
== Music ==
; [[Francium]]
* "shortcrust" from ''wiloliquy'' (2025) – [https://open.spotify.com/track/7eX41UXoJhmlzFVe3hWP7d Spotify] | [https://francium223.bandcamp.com/track/shortcrust Bandcamp] | [https://www.youtube.com/watch?v=0vPnUo6X9IY YouTube]

Latest revision as of 02:30, 17 April 2025

← 760edo 761edo 762edo →
Prime factorization 761 (prime)
Step size 1.57687 ¢ 
Fifth 445\761 (701.708 ¢)
Semitones (A1:m2) 71:58 (112 ¢ : 91.46 ¢)
Consistency limit 9
Distinct consistency limit 9

761 equal divisions of the octave (abbreviated 761edo or 761ed2), also called 761-tone equal temperament (761tet) or 761 equal temperament (761et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 761 equal parts of about 1.58 ¢ each. Each step represents a frequency ratio of 21/761, or the 761st root of 2.

Theory

761edo is consistent to the 9-odd-limit. As an equal temperament, it tempers out 32805/32768 in the 5-limit; 420175/419904 and [3 13 -15 4⟩ in the 7-limit. The equal temperament is strong in the 2.3.5.13.29.31 subgroup, tempering out 32805/32768, 21141/21125, 3627/3625, 140625/140608 and 45349632/45287125.

Odd harmonics

Approximation of odd harmonics in 761edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.247 +0.020 -0.626 -0.493 +0.587 -0.055 -0.227 +0.695 +0.516 +0.704 -0.679
Relative (%) -15.6 +1.3 -39.7 -31.3 +37.3 -3.5 -14.4 +44.1 +32.7 +44.6 -43.1
Steps
(reduced) 		1206
(445) 		1767
(245) 		2136
(614) 		2412
(129) 		2633
(350) 		2816
(533) 		2973
(690) 		3111
(67) 		3233
(189) 		3343
(299) 		3442
(398)

Subsets and supersets

761edo is the 135th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-1206 761⟩ [⟨761 1206]] +0.0778 0.0778 4.93
2.3.5 32805/32768, [69 81 -85⟩ [⟨761 1206 1767]] +0.0490 0.0755 4.79
2.3.5.7 32805/32768, 420175/419904, [3 13 -15 4⟩ [⟨761 1206 1767 2136]] +0.0925 0.0998 6.33

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 316\761 498.292 4/3 Helmholtz

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

Music

Francium
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