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

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

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

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

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

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

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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=== 10000 and up ===

{{EDOs

| 10009, 10600, 10729, 11664, 12276, 12348, 12500~~, 13382~~, 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

| 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
-<center><math>\displaystyle s = 1200 \cdot k/n</math></center>
+$$ s = 1200 \cdot k/n $$
-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
-<center><math>\displaystyle c = 2^{k/n}</math></center>
+$$ r = 2^{k/n} $$
-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 1,242: / Line 1,238: @@
 === 10000 and up ===
 {{EDOs
-| 10009, 10600, 10729, 11664, 12276, 12348, 12500, 13382, 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
+| 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
 }}