EDO: Difference between revisions

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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]] in 1999, originally standing for "equidistant divisions of the octave"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_65#65 Yahoo! Tuning Group | ''Where F + f = O'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_117#117 Yahoo! Tuning Group | ''f + F and WFS/MOS'']</ref>. More recently, the ~~[[Wikipedia: Anacronym~~|anacronym]] "edo" (''EE-doh''), spelled in lowercase, has also become common.

The acronym "EDO" (''EE-dee-oh'') was coined by [[Daniel Anthony Stearns]] in 1999, originally standing for "equidistant divisions of the octave"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_65#65 Yahoo! Tuning Group | ''Where F + f = O'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_117#117 Yahoo! Tuning Group | ''f + F and WFS/MOS'']</ref>. More recently, the {{w|anacronym}} "edo" (''EE-doh''), spelled in lowercase, has also become common.

With the development of [[Edonoi|equal divisions of non-octave intervals (edonoi)]], some people started writing "ed2" ("ED2"), especially when naming a specific tuning.

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The largest physically possible EDO in a frequency range can be found through molecular physics such as the [[mean free path]] combined with the [[wikipedia: Speed of sound|speed of sound]] in a given substance.

== ~~The~~ Scale ~~Tree~~ ==

== Non-integer EDO ==

A non-integer edo can be defined as using a non-integer divisor to divide the octave. Typically, non-integer edos are understood as ''not'' containing the exact octave, so that they remain [[equal tuning]]s. All fractional EDOs are integer equal divisions of another integer interval. For example, (25/2)edo is equivalent to 25ed4. In general:

<math>\displaystyle (p/q) \text{edo} = p \text{-ed} 2^q</math>

for integers ''p'' and ''q''. Irrational EDOs cannot be converted to integer equal divisions of another integer interval, so they are things of their own.

Non-integer EDOs can be written in decimal form, such as 12.1edo. This is often meant to be approximate, used in the context of [[octave stretch]] of an integer EDO, rather than as a fractional EDO.

== Scale tree ==

The scale tree, or Stern-Brocot tree, provides a visual map of the world of EDOs, based on fifth size.

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== Pergens ==

{{See also| Pergen#~~Pergens_and_EDOs| label 1 =~~ Pergens and EDOs }}

[[Pergen]]s provide a JI-agnostic way to name the rank-2 scales of an EDO. This table lists every possible period/generator combination for EDOs 5-24, and for each coprime combination, the simplest pergen that it can represent. Non-coprime combinations such as P = 6\12, G = 4\12 are marked as "-".

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 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]] in 1999, originally standing for "equidistant divisions of the octave"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_65#65 Yahoo! Tuning Group | ''Where F + f = O'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_117#117 Yahoo! Tuning Group | ''f + F and WFS/MOS'']</ref>. More recently, the [[Wikipedia: Anacronym|anacronym]] "edo" (''EE-doh''), spelled in lowercase, has also become common.
+The acronym "EDO" (''EE-dee-oh'') was coined by [[Daniel Anthony Stearns]] in 1999, originally standing for "equidistant divisions of the octave"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_65#65 Yahoo! Tuning Group | ''Where F + f = O'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_117#117 Yahoo! Tuning Group | ''f + F and WFS/MOS'']</ref>. More recently, the {{w|anacronym}} "edo" (''EE-doh''), spelled in lowercase, has also become common.
 With the development of [[Edonoi|equal divisions of non-octave intervals (edonoi)]], some people started writing "ed2" ("ED2"), especially when naming a specific tuning.
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 The largest physically possible EDO in a frequency range can be found through molecular physics such as the [[mean free path]] combined with the [[wikipedia: Speed of sound|speed of sound]] in a given substance.
-== The Scale Tree ==
+== Non-integer EDO ==
+A non-integer edo can be defined as using a non-integer divisor to divide the octave. Typically, non-integer edos are understood as ''not'' containing the exact octave, so that they remain [[equal tuning]]s. All fractional EDOs are integer equal divisions of another integer interval. For example, (25/2)edo is equivalent to 25ed4. In general:
+<math>\displaystyle (p/q) \text{edo} = p \text{-ed} 2^q</math>
+for integers ''p'' and ''q''. Irrational EDOs cannot be converted to integer equal divisions of another integer interval, so they are things of their own.
+Non-integer EDOs can be written in decimal form, such as 12.1edo. This is often meant to be approximate, used in the context of [[octave stretch]] of an integer EDO, rather than as a fractional EDO.
+== Scale tree ==
 The scale tree, or Stern-Brocot tree, provides a visual map of the world of EDOs, based on fifth size.
@@ Line 474: / Line 483: @@
 == Pergens ==
-{{See also| Pergen#Pergens_and_EDOs| label 1 = Pergens and EDOs }}
+{{See also| Pergen #Pergens and EDOs }}
 [[Pergen]]s provide a JI-agnostic way to name the rank-2 scales of an EDO. This table lists every possible period/generator combination for EDOs 5-24, and for each coprime combination, the simplest pergen that it can represent. Non-coprime combinations such as P = 6\12, G = 4\12 are marked as "-".