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Add draft for lead section of Perfect fourth
 
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== Edo ==
== Perfect fourth ==
An '''equal division of the octave''' ('''edo''' or '''EDO''') is a [[Musical tuning|tuning]] obtained by dividing the [[octave]] in a certain number of [[Equal-step tuning|equal steps]]. This means that each step corresponds to the same [[interval]].
{{Wikipedia|Perfect fourth}}
{{About|the [[interval category]]|the just perfect fourth|4/3}}
In Western music theory, a '''perfect fourth''' is an [[interval]] with a [[frequency ratio]] equal or approximately equal to [[4/3]] (≈498{{c}}). It is a strongly [[consonant]] interval, hence its [[interval quality]] "perfect". A perfect fourth always spans four degrees of a [[diatonic]] scale (e.g. C-F spanning C-D-E-F), hence its [[interval number]] "fourth". In particular, a perfect fourth spans two whole tones (e.g. C-D-E) and one diatonic semitone (e.g. E-F).


A tuning with <span><math>n</math></span> equal divisions of the octave is usually called "<span><math>n</math></span>edo" ("<span><math>n</math></span>-EDO"). For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO).
An ''imperfect fourth'' also spans four degrees, but with a different combination of steps. In particular, the augmented fourth spans three whole tones, hence its other name "[[tritone]]". Other fourths may occur in altered diatonic scales, such as the diminished fourth in the harmonic minor scale (e.g. B-E{{flat}}).


An edo is a specific case of [[EPD|equal pitch division]], which is a kind of [[equal-step tuning]]. Therefore, it is also a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
A ''wolf fourth'' (or ''imperfect fourth'', in a second sense) is an interval that attempts to close a [[circle of fourths]], but that fails to approximate a frequency ratio of 4/3. For example, in [[quarter-comma meantone]] temperament, an augmented third (e.g. A{{flat}}-C{{sharp}}) can be used to close the 12-tone circle of fourths, but it falls approximately 36{{c}} flat of a just perfect fourth.


=== History===
In microtonal music, an interval may be considered a perfect fourth even if it exhibits only one of the two defining features of a perfect fourth:
For a long time, tuning theorists used the term "equal temperament" for edos designed to approximate [[Low-complexity JI|low-complexity just intervals]]. The same term is still used today to designate more generally all rank-1 [[Regular temperament|temperaments]]. For example, [[15edo]] can be referred to as 15-tone equal temperament (15-TET, 15-tET, 15tet, etc.), or more simply 15 equal temperament (15-ET, 15et, etc.).
* As intervals in an [[interval region]], perfect fourths only need to have a size close to ≈498{{c}}, no matter the underlying scale or other harmonic context. According to [[Margo Schulter]]'s ''[http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt Regions of the Interval Spectrum]'', perfect fourths typically range from approximately 470{{c}} to 530{{c}}. Intervals that are too sharp or too flat to qualify as perfect fourths are often called [[superfourth]]s and [[subfourth]]s respectively.
* As intervals in a diatonic [[MOS scale]], perfect fourths only need to span two large steps and one small step. This approach allows for sizes ranging from [[7edo|4\7]] (≈686{{c}}) to [[5edo|3\5]] (720{{c}}). Intervals inflected by a [[quartertone]] alteration up or down from a perfect fourth are often called [[semi-augmented fourth]]s and [[semi-diminished fourth]]s respectively.


The acronym "EDO" (''EE-dee-oh'') was coined by [[Daniel Anthony Stearns]]<sup>[''year needed'']</sup>. More recently, the [https://en.wikipedia.org/w/index.php?title=Anacronym anacronym] "edo" (''EE-doh''), spelled in lowercase, has become increasingly widespread.
== Temperament databox ==
Using [[User:FloraC/Temperament data]].


With the development of [[Edonoi|equal divisions of non-octave intervals (edonoi)]], some musicians started using "ed2" ("ED2"), especially when naming a specific tuning. Furthermore, in order to distinguish equal pitch division from [[EFD|equal frequency division]] and [[ELD|equal length division]], "epd" ("EPD") is sometimes used in place of "ed" ("ED").
=== Septimal meantone ===
{{Main|Septimal meantone}}
{{Databox|Temperament data|defaultstate=expanded|
{{User:FloraC/Temperament data
| Subgroup = 2.3.5.7
| Comma list = 81/80, 126/125
| Mapping = {{mapping| 1 0 -4 -13 | 0 1 4 10 }}
| Mapping generators = ~2, ~3
| Wedgie = 1 4 10 4 13 12
| Optimal ET sequence = 12, 19, 31, 81, 112b, 143b
| Badness = 0.013707
}}
{{Databox|[[Optimal tuning]]s:|defaultstate=expanded|
text=&#10;
* [[CTE]]: ~3/2 = 696.952
* [[POTE]]: ~3/2 = 696.495
* [[TE]]: ~2 = 1201.242, ~3/2 = 698.458
}}}}
 
== HEJI2Text testing area ==
{| class="wikitable center-all"
|+ 8:16 scale on E{{HEJI|e}}
! HEJI (normal)
! Ratio
! HEJI (big)
|-
| {{HEJI|e}}E
| 1/1
| <big>{{HEJI|e}}E</big>
|-
| {{HEJI|n}}F
| 9/8
| <big>{{HEJI|n}}F</big>
|-
| {{HEJI|m}}G
| 5/4
| <big>{{HEJI|m}}G</big>
|-
| {{HEJI|4e}}A
| 11/8
| <big>{{HEJI|4e}}A</big>
|-
| {{HEJI|e}}B
| 3/2
| <big>{{HEJI|e}}B</big>
|-
| {{HEJI|0}}C
| 13/8
| <big>{{HEJI|0}}C</big>
|-
| {{HEJI|e}}D
| 7/4
| <big>{{HEJI|e}}D</big>
|-
| {{HEJI|m}}D
| 15/8
| <big>{{HEJI|m}}D</big>
|-
| {{HEJI|e}}E
| 2/1
| <big>{{HEJI|e}}E</big>
|}
 
== Lumatone mapping for 23edo ==
Derived from the [[Standard Lumatone mapping for Pythagorean]], assuming a perfect fifth of 13\23 (678{{cent}}) that generates an [[antidiatonic]] scale. 
{{Lumatone EDO mapping|n=23|start=4|xstep=3|ystep=-4}}
 
== Lua module testing area ==
* log(3) = {{#invoke:Utils|log|3}}
* round(123.4567) = {{#invoke:Utils|round|123.4567}}
* to_cents(1.5) = {{#invoke:Interval|to_cents|1.5}}
* parse_ET_size(17edo) = {{#invoke:Interval|parse_ET_size|17edo}}
* parse_ET_equave(17edo) = {{#invoke:Interval|parse_ET_equave|17edo}}
* to_cents(3/2) [6 s.f.] = {{#invoke:Interval|to_cents|3/2|6}}
* backslash_ratio(2\7) = {{#invoke:Interval|backslash_ratio|2\7}}
* backslash_ratio(5\9edf) = {{#invoke:Interval|backslash_ratio|5\9edf}}
* backslash_ratio(1\12-EDO) = {{#invoke:Interval|backslash_ratio|1\12-EDO}}
* backslash_ratio(8\13-edt) = {{#invoke:Interval|backslash_ratio|8\13-edt}}
* backslash_ratio(16\21ed12/7) = {{#invoke:Interval|backslash_ratio|16\21ed12/7}}
* backslash_ratio(\ed) [defaults to 1\12(edo)] = {{#invoke:Interval|backslash_ratio|\ed}}


Several alternate notations have been devised by some musicians more recently, including "edd" ("EDD"; equal divisions of the [[ditave]]), "DIV," and "EQ."
== Sidebar revision ==
=== Summary of proposed changes (2025-02-17 update) ===
; Remove ''Discussion'': The first part of that page would fit better in a "pedagogy" page, while the links should be directly added in the sidebar (see below).
; Merge ''General Theory'' and ''Mathematical Theory'' into ''Theory overview'': Since General Theory already has some high-level stuff, we might as well want a single overview of tuning theory, with the basic stuff on top. So far my suggestion for the title of that page is "Overview of tuning theory" (while "Theory overview" would be the shorthand for the sidebar).
; Add ''Links'': Many websites have a "links" page available through the site's menu, so it would make sense to do the same here, especially since we have the [[Links]] page. By the way, since we have many pages with similar lists of links but for specific aspects, we might want to make this a sort of curated list with a "see also" hatnote in each section.
; Rename ''Lists and Galleries'' to ''Lists and tables'': Most pages that are called galleries are not really galleries (of images); however we do have a decent amount of tables. I think ''Lists and tables'' sounds overall better and clearer than ''Lists'' alone. This would imply renaming that page, of course.


=== Formula===
=== Changes applied on 2023-08-04 ===
To find the step size for an <span><math>n</math></span>edo, take the <span><math>n</math></span>th root of 2. For example, the step of 12edo is <span><math>2^{\frac{1}{12}}</math></span> (<span><math>\approx 1.059</math></span>). So the formula for the <span><math>k</math></span>th step of an <span><math>n</math></span>edo is:
; Remove ''Introduction'': That page could be merged into the Main Page, since there's not a lot of information that's not already on the Main Page anyway.
; Add ''FAQ'' between ''Random page'' and ''Help'': I suggest renaming ''[[Ask Questions]]'' to ''Frequently asked questions''. That page should offer short answers in addition to links to dedicated articles, as needed. By the way, most of these dedicated articles that we already have are written in a personal/non-neutral style (and are often old, in case that matters for some of them), so we might want to warn readers accordingly.
; Remove ''Conventions'': I don't think that link is worth the space it takes, especially since the page is directly accessible from the top of the ''Help'' page.
; Rename ''Pedagogy'' to ''Guides'' and move it to the bottom of its section: "Pedagogy" is a bit vague and sounds like it's more oriented towards educators than learners. Most "pedagogical" pages are guides, since everything else is just a normal page from which readers can learn anyway, at least from my perspective.
; Rename ''Orientation'' to ''Approaches'': This seems less vague than "orientation", considering the goal of that page. See also [[Talk:MicrotonalTheory]].
; Remove ''Useful Tools'': See [[Talk:Useful Tools]]. We could have a duplicate link to ''Software'' for the sake of independent sections, but I don't think it's worth it.
; Misc. edits: Use sentence case, change some target page titles (already updated or suggested), move LANGAUGES to the bottom (where it's displayed anyway), etc.


<math>
=== Proposition ===
c(k) = 2^{\frac{k}{n}}
* Navigation
</math>
** mainpage|mainpage-description
** recentchanges-url|recentchanges
** randompage-url|randompage
** Frequently asked questions|FAQ
** Help|Help
** https://en.xen.wiki/index.php?title=Special:Search&profile=advanced&search=&fulltext=1|Advanced Search
-----
* Practice
** Listen|Listen
** Microtonal instruments|Instruments
** Software|Software
** Scores|Scores
** People|People
** Projects|Projects
** Guides|Guides
-----
* Theory
** Approaches to musical tuning|Approaches
** Overview of tuning theory|Overview
-----
* Connect
** http://www.facebook.com/groups/xenharmonic2|XA Facebook group
** https://discord.com/invite/FSF5JFT|XA Discord server
** https://web.libera.chat/?channels=##xenharmonic|XA IRC chat room
-----
* Other
** Links|Links
** Lists and tables|Lists and tables
** Backups|Backing up this wiki
** Xenharmonic_Wiki_License|Licensing
* SEARCH
* TOOLBOX
* LANGUAGES


This way, when <span><math>k</math></span> is 0, <span><math>k</math></span> is simply 1, because any number to the 0th power is 1. And when <span><math>k</math></span> is <span><math>n</math></span>, <span><math>c(k)</math></span> is simply 2, because any number to the 1st power is itself.  
== Music ==
=== Plan for a music page template ===
See for example [[Desert Island Rain]] and [[Almighty Fractal]].


===Infoboxes===
=== Table ===
{{Infobox ET
 
| Prime factorization = 2 × 3</sup>
{| class="wikitable sortable"
| Step size = 200¢
!Title
| Fifth = 4\6 = 800¢
!Composer
| Major 2nd = 2\6 = 400¢
!Year
| Minor 2nd = -2\6 = -400¢
!Genre
| Augmented 1sn = 4\6 = 800¢
!Additional links
}}
|-
{{Infobox ET
|[https://soundcloud.com/overtoneshock/dose-of-familiarityode-to-microtonality-22-edo-studio-version Emancipate Pitch!]
| Prime factorization = 2<sup>2</sup>
|[[Stephen Weigel]]
| Step size = 300¢
|2016
| Fifth = 2\4 = 600¢
|Dancehall
| Major 2nd = 0\4 = 0¢
|
| Minor 2nd = 2\4 = 600¢
|-
| Augmented 1sn = -2\4 = -600¢
|[https://soundcloud.com/metaclown/couples-therapy Couples Therapy]
}}
|metaclown ([[Jacob Barton]])
{{Infobox ET
|2016
| Prime factorization = 3 (prime)
|Folk
| Step size = 400¢
|
| Fifth = 2\3 = 800¢
|-
| Major 2nd = 1\3 = 400¢
|[http://www.tallkite.com/words/Tibia.mp3 TIBIA]
| Minor 2nd = -1\3 = 400¢
|Paul Erlich
| Augmented 1sn = 1\3 = -400¢
|2016
}}
|Classical
{{Infobox ET
|[[:File:TIBIA.pdf|Sagittal score in F∥\]] (errors in m. 9, 19, 20)<br>[[:File:Tibia in g.pdf|Sagittal score in G]] (errors in m. 9, 19, 20)<br>Ups and Downs score in G: [[:File:Tibia in G CORRECTED-1.png|page 1]], [[:File:Tibia in G CORRECTED-2.png|page 2]] (no errors)
| Prime factorization = 2 (prime)
|}
| Step size = 600¢
 
| Fifth = 1\2 = 600¢
== Unused edo infoboxes ==
| Major 2nd = 0\2 = 0¢
| Minor 2nd = 1\2 = 600¢
| Augmented 1sn = -1\2 = -600¢
}}
{{Infobox ET
| Prime factorization = (empty product)
| Step size = 1200¢
| Fifth = 1\1 = 1200¢
| Major 2nd = 1\1 = 1200¢
| Minor 2nd = -2\1 = -2400¢
| Augmented 1sn = 3\1 = 3600¢
}}
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