EDO: Difference between revisions

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| ja = オクターブ平均律

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~~The abbreviation~~ '''~~EDO~~''' ~~stands for~~ '''equal divisions of the octave~~'''~~ (~~not~~ to ~~be confused with the~~ [[~~Wikipedia:Edo period~~|~~Edo period~~]] ~~in Japanese history~~). The acronym was coined by [[Daniel Anthony Stearns]]~~. Other abbreviations in use include~~ ''~~'ET'~~'', ''~~'TET'~~'', ~~'''~~ED2~~'''~~, [[~~Ditave~~]] (the ~~Spanish name~~), DIV, and EQ. ~~Among~~ the ~~various systems based on~~ [[~~equal~~]]~~-distance pitches~~, ~~EDOs have perfect [[octave]] mappings in common~~.

An '''equal division of the octave''' ('''edo''' or '''EDO''') is a [[Tuning system|tuning]] obtained by dividing the [[octave]] in a certain number of [[Equal-step tuning|equal steps]]. This means that the [[interval]] between any two consecutive pitches is identical.

A tuning with <math>n</math> equal divisions of the octave is usually called "<math>n</math>edo" ("<math>n</math>-EDO"). For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO).

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.

== History ==

Tuning theorists first used the term "equal temperament" for edos designed to approximate [[Low-complexity JI|low-complexity just intervals]]. The same term is still used today for 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.).

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

With the development of [[Edonoi|equal divisions of non-octave intervals (edonoi)]], some people started writing "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").

Several alternate notations have been devised, including "edd" ("EDD"; equal divisions of the [[ditave]]), "DIV," and "EQ."

== Formula ==

To find the step size ([[Interval_size_measure#Ratio|interval ratio]]) for an <math>n</math>edo, take the <math>n</math>th root of 2. For example, the step of 12edo is <math>2^{\frac{1}{12}}</math> (≈1.059). So the formula for the <math>k</math>th step of an <math>n</math>edo is:

<math>

c(k) = 2^{\frac{k}{n}}

</math>

This way, when <math>k</math> is 0, <math>c(k)</math> is simply 1, because any number to the 0th power is 1. And when <math>k</math> is <math>n</math>, <math>c(k)</math> is simply 2, because any number to the 1st power is itself.

== EDO FAQ ==

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-The abbreviation '''EDO''' stands for '''equal divisions of the octave''' (not to be confused with the [[Wikipedia:Edo period|Edo period]] in Japanese history). The acronym was coined by [[Daniel Anthony Stearns]]. Other abbreviations in use include '''ET''', '''TET''', '''ED2''', [[Ditave]]  (the Spanish name), DIV, and EQ. Among the various systems based on [[equal]]-distance pitches, EDOs have perfect [[octave]] mappings in common.
+An '''equal division of the octave''' ('''edo''' or '''EDO''') is a [[Tuning system|tuning]] obtained by dividing the [[octave]] in a certain number of [[Equal-step tuning|equal steps]]. This means that the [[interval]] between any two consecutive pitches is identical.
+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 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.
+== History ==
+Tuning theorists first used the term "equal temperament" for edos designed to approximate [[Low-complexity JI|low-complexity just intervals]]. The same term is still used today for 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.).
+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.
+With the development of [[Edonoi|equal divisions of non-octave intervals (edonoi)]], some people started writing "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").
+Several alternate notations have been devised, including "edd" ("EDD"; equal divisions of the [[ditave]]), "DIV," and "EQ."
+== Formula ==
+To find the step size ([[Interval_size_measure#Ratio|interval ratio]]) 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> (≈1.059). So the formula for the <span><math>k</math></span>th step of an <span><math>n</math></span>edo is:
+<math>
+c(k) = 2^{\frac{k}{n}}
+</math>
+This way, when <span><math>k</math></span> is 0, <span><math>c(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.
 == EDO FAQ ==