User:Fredg999/Sandbox: Difference between revisions

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== Edo ==

An '''equal division of the octave''' ('''edo''' or '''EDO''') is a [[Musical tuning|tuning ~~system~~]] ~~constructed~~ by dividing the [[octave]] in a certain number of [[Equal-step tuning|equal steps]].

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]].

A tuning with ''n'' equal divisions of the octave is usually called ''"n''edo" or "''n''-EDO." For instance, the predominant tuning system in the world today is [[12edo]] or 12-EDO.

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 ~~being an~~ [[equal-step tuning]], it is also an [[Arithmetic tunings|arithmetic]] and a [[Harmonotonic tunings|harmonotonic]] tuning.

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 ===

=== History===

For a long time, tuning theorists used the term "equal temperament" for edos designed to approximate 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.).

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.).

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

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 musicians started using "ed2" or "ED2" in place of "~~edo~~" or "~~EDO,~~" ~~especially when naming a specific tuning~~.

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").

~~A few more~~ alternate notations have been devised by some musicians more recently, including "edd" or "EDD" (equal divisions of the [[~~Ditave~~]]), "DIV," and "EQ."

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

=== Infoboxes ===

=== Formula===

To find the step size 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> (<math>\approx 1.059</math>). 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>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.

===Infoboxes===

{{Infobox ET

| Prime factorization = 2 × 3

@@ Line 1: / Line 1: @@
 == Edo ==
-An '''equal division of the octave''' ('''edo''' or '''EDO''') is a [[Musical tuning|tuning system]] constructed by dividing the [[octave]] in a certain number of [[Equal-step tuning|equal steps]].
+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]].
-A tuning with ''n'' equal divisions of the octave is usually called ''"n''edo" or "''n''-EDO." For instance, the predominant tuning system in the world today is [[12edo]] or 12-EDO.
+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 being an [[equal-step tuning]], it is also an [[Arithmetic tunings|arithmetic]] and a [[Harmonotonic tunings|harmonotonic]] tuning.
+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 ===
+=== History===
-For a long time, tuning theorists used the term "equal temperament" for edos designed to approximate 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.).
+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.).
-The acronym "EDO" (''EE-dee-oh'') was coined by [[Daniel Anthony Stearns]]<sup>[''year needed'']</sup>. Since then, the [https://en.wikipedia.org/w/index.php?title=Anacronym anacronym] "edo" (''EE-doh''), spelled in lowercase, has become increasingly widespread.
+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 musicians started using "ed2" or "ED2" in place of "edo" or "EDO," especially when naming a specific tuning.
+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").
-A few more alternate notations have been devised by some musicians more recently, including "edd" or "EDD" (equal divisions of the [[Ditave]]), "DIV," and "EQ."
+Several alternate notations have been devised by some musicians more recently, including "edd" ("EDD"; equal divisions of the [[ditave]]), "DIV," and "EQ."
-=== Infoboxes ===
+=== Formula===
+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:
+<math>
+c(k) = 2^{\frac{k}{n}}
+</math>
+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.
+===Infoboxes===
 {{Infobox ET
 | Prime factorization = 2 × 3</sup>