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| == Edo ==
| | == Unused edo infoboxes == |
| 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]].
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| 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).
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| 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.
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| === History===
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| 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.).
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| 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.
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| 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").
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| Several alternate notations have been devised by some musicians more recently, including "edd" ("EDD"; equal divisions of the [[ditave]]), "DIV," and "EQ."
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| === Formula===
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| 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:
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| <math>
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| c(k) = 2^{\frac{k}{n}}
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| </math>
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| 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.
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| ==Unused edo infoboxes== | |
| {{Infobox ET | | {{Infobox ET |
| | Prime factorization = 2 × 3</sup> | | | Prime factorization = 2 × 3</sup> |