EDO: Difference between revisions

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An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') 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.~~

An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[tuning system]] obtained by dividing the [[2/1|octave]] into a whole number of [[equal-step tuning|equal steps]]. A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" (or "''n''-EDO"). In terms of frequency, the octave with frequency ratio 2/1 is logarithmically divided into ''n'' steps, each with frequency ratio 21/n. For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO), with consecutive steps having a frequency ratio of 21/12. This implies that the [[interval]] between any two consecutive pitches is identical. Equal divisions of the octave are the most common [[equal-step tuning]]s, with other [[nonoctave]] tunings existing as well.

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

~~An equal (pitch) division of the octave is~~ the ~~most common type of~~ [[~~EPD|equal (pitch) division~~]]~~, which~~ is ~~a kind~~ of [[equal-step tuning]]~~. Therefore~~, ~~it is also a kind of~~ [[~~arithmetic tuning|arithmetic~~]] ~~and [[harmonotonic tuning|harmonotonic]] tuning~~.

== History ==

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To find the step size of ''n''-edo in terms of [[cent]]s, divide 1200 by ''n''. The size ''s'' of ''k'' steps of ''n''-edo (''k''\''n'') is

~~<math>\displaystyle~~ s = 1200 \cdot k/n~~</math>~~

$$ s = 1200 \cdot k/n $$

To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 21/12 (≈ 1.059). So the ratio ''c'' of the ''k'' steps of ''n''-edo is

To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 21/12 (≈ 1.059). So the ratio ''r'' of the ''k'' steps of ''n''-edo is

~~<math>\displaystyle c~~ = 2^{k/n}~~</math>~~

$$ r = 2^{k/n} $$

In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''c'' is simply 2, because any number to the 1st power is itself.

In particular, when ''k'' is 0, ''r'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''r'' is simply 2, because any number to the 1st power is itself.

== Properties ==

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== Approaches to exploring EDOs ==

If you are interested in exploring the unique merits and challenges of each EDO, irrespective of any desire to approximate just intonation (or any other a priori musical goal), starting at the bottom and working your way up can be a most illuminating exercise.

If you are interested in exploring the unique merits and challenges of each EDO, irrespective of any desire to approximate just intonation (or any other a priori musical goal), starting at the bottom and working your way up can be a most illuminating exercise. [[Macrotonal EDO]]s have a step size larger than that of 12edo, and thus have fewer than 12 steps per octave, so they may be preferable to those who want simplicity.

If you're a classically trained musician and you'd like to start with some EDOs that have some relationship to common-practice tonal music, starting with reasonably-low EDOs that give a good approximation to the perfect fifth ([[3/2]]) can be rewarding. Classical music also relies on the fact that [[5/4]] is the major third of the diatonic scale, which occurs only when [[81/80]], the syntonic comma, is [[tempered out]]. EDOs {{EDOs| 12, 19, 24, 26, 31, 36, 38, 43, 45, 48, and 50}} temper out 81/80, and are thus [[meantone]] systems. EDOs {{EDOs| 17, 22, 27, 29, 34, 39, 41, 44, 46, 49, 51, and 53}} have an accurate perfect fifth, but do not temper out 81/80, and thus require new ways of thinking about harmony. Many EDOs, such as {{EDOs| 12, 17, 19, 22, 26, 27, 29, 31, 39, 41, 43, 45, 46, 49, and 53 }}, can be notated with some variant on the [[chain-of-fifths notation|A–G "circle of fifths" notation]], while other EDOs, including {{EDOs| 24, 34, 36, 38, 44, 48, or 51}}, involve multiple such circles.

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== Individual pages for EDOs ==

Note: Before creating an EDO page, please make sure that it satisfies the [[Xenharmonic Wiki:Notability guidelines|notability guidelines]]. Also, if the EDO is greater than or equal to 1000, please add it to the list below.

=== 0…999 ===

{| class="wikitable center-all mw-collapsible"

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=== 10000 and up ===

{{EDOs

| 10009~~, 10459~~, 10600, 10729, 11664, 12276, 12348, 12500~~, 13382~~, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16625, 16808, 17100, 17461, 18355, 20203, 20567, 28000, 28742, 30103, 30631, 31867, 31920, 32436, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 2547047, 2901533, 3159811, 6000000, 11358058, 402653184, 5407372813

| 10009, 10600, 10729, 11664, 12276, 12348, 12500, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16625, 16808, 17100, 17461, 18355, 20203, 20567, 28000, 28472, 28742, 30103, 30631, 31867, 31920, 32436, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 2547047, 2901533, 3159811, 6000000, 11358058, 402653184, 5407372813

}}

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 | ro = DEO
 }}
-An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') 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.
+An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[tuning system]] obtained by dividing the [[2/1|octave]] into a whole number of [[equal-step tuning|equal steps]]. A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" (or "''n''-EDO"). In terms of frequency, the octave with frequency ratio 2/1 is logarithmically divided into ''n'' steps, each with frequency ratio 2<sup>1/n</sup>. For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO), with consecutive steps having a frequency ratio of 2<sup>1/12</sup>. This implies that the [[interval]] between any two consecutive pitches is identical. Equal divisions of the octave are the most common [[equal-step tuning]]s, with other [[nonoctave]] tunings existing as well.
-A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" ("''n''-EDO"). For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO).
-An equal (pitch) division of the octave is the most common type of [[EPD|equal (pitch) division]], which is a kind of [[equal-step tuning]]. Therefore, it is also a kind of [[arithmetic tuning|arithmetic]] and [[harmonotonic tuning|harmonotonic]] tuning.
 == History ==
@@ Line 24: / Line 20: @@
 To find the step size of ''n''-edo in terms of [[cent]]s, divide 1200 by ''n''. The size ''s'' of ''k'' steps of ''n''-edo (''k''\''n'') is
-<math>\displaystyle s = 1200 \cdot k/n</math>
+$$ s = 1200 \cdot k/n $$
-To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 2<sup>1/12</sup> (≈ 1.059). So the ratio ''c'' of the ''k'' steps of ''n''-edo is
+To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 2<sup>1/12</sup> (≈ 1.059). So the ratio ''r'' of the ''k'' steps of ''n''-edo is
-<math>\displaystyle c = 2^{k/n}</math>
+$$ r = 2^{k/n} $$
-In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''c'' is simply 2, because any number to the 1st power is itself.
+In particular, when ''k'' is 0, ''r'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''r'' is simply 2, because any number to the 1st power is itself.
 == Properties ==
@@ Line 44: / Line 40: @@
 == Approaches to exploring EDOs ==
-If you are interested in exploring the unique merits and challenges of each EDO, irrespective of any desire to approximate just intonation (or any other a priori musical goal), starting at the bottom and working your way up can be a most illuminating exercise.
+If you are interested in exploring the unique merits and challenges of each EDO, irrespective of any desire to approximate just intonation (or any other a priori musical goal), starting at the bottom and working your way up can be a most illuminating exercise. [[Macrotonal EDO]]s have a step size larger than that of 12edo, and thus have fewer than 12 steps per octave, so they may be preferable to those who want simplicity.
 If you're a classically trained musician and you'd like to start with some EDOs that have some relationship to common-practice tonal music, starting with reasonably-low EDOs that give a good approximation to the perfect fifth ([[3/2]]) can be rewarding. Classical music also relies on the fact that [[5/4]] is the major third of the diatonic scale, which occurs only when [[81/80]], the syntonic comma, is [[tempered out]]. EDOs {{EDOs| 12, 19, 24, 26, 31, 36, 38, 43, 45, 48, and 50}} temper out 81/80, and are thus [[meantone]] systems. EDOs {{EDOs| 17, 22, 27, 29, 34, 39, 41, 44, 46, 49, 51, and 53}} have an accurate perfect fifth, but do not temper out 81/80, and thus require new ways of thinking about harmony. Many EDOs, such as {{EDOs| 12, 17, 19, 22, 26, 27, 29, 31, 39, 41, 43, 45, 46, 49, and 53 }}, can be notated with some variant on the [[chain-of-fifths notation|A–G "circle of fifths" notation]], while other EDOs, including {{EDOs| 24, 34, 36, 38, 44, 48, or 51}}, involve multiple such circles.
@@ Line 88: / Line 84: @@
 == Individual pages for EDOs ==
+Note: Before creating an EDO page, please make sure that it satisfies the [[Xenharmonic Wiki:Notability guidelines|notability guidelines]]. Also, if the EDO is greater than or equal to 1000, please add it to the list below.
 === 0…999 ===
 {| class="wikitable center-all mw-collapsible"
@@ Line 1,240: / Line 1,238: @@
 === 10000 and up ===
 {{EDOs
-| 10009, 10459, 10600, 10729, 11664, 12276, 12348, 12500, 13382, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16625, 16808, 17100, 17461, 18355, 20203, 20567, 28000, 28742, 30103, 30631, 31867, 31920, 32436, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 2547047, 2901533, 3159811, 6000000, 11358058, 402653184, 5407372813
+| 10009, 10600, 10729, 11664, 12276, 12348, 12500, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16625, 16808, 17100, 17461, 18355, 20203, 20567, 28000, 28472, 28742, 30103, 30631, 31867, 31920, 32436, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 2547047, 2901533, 3159811, 6000000, 11358058, 402653184, 5407372813
 }}