EDO: Difference between revisions

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* '''pentatonic''' EDOs ({{EDOs| 5, 10, 15, 20, 25 & 30 }}) have a fifth of three-fifths of an octave = 3\5 = 720¢

* '''supersharp''' EDOs ({{EDOs| 8, 13 & 18 }}) have a fifth wider than 3\5 = 720¢

* '''trivial''' EDOs ({{EDOs| 1, 2, 3, 4 and 6 }}) have a fifth about 100¢ from just, and are contained in ~~12 EDO~~

* '''trivial''' EDOs ({{EDOs| 1, 2, 3, 4 and 6 }}) have a fifth about 100¢ from just, and are contained in 12edo

=== Non-tuning properties ===

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You will quickly find that the ''factorization'' of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, 6 = 2 x 3, so 6edo contains all of the intervals in both 2edo and 3edo. On the other hand, 7 is a prime number, so no 7edo intervals are redundant with those of smaller EDOs. See [[Prime EDO]] for more details.

The [[Moments of ~~Symmetry~~]] paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.

The [[Moments of symmetry]] paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.

=== Adding EDOs ===

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When an edo divides the octave into fewer than 12 divisions (so that each step exceeds 100 cents), you might call it a [[macrotonal EDO]]. Of these, 1, 2, 3, 4 and 6 divide 12 and so are already available to anyone wishing to explore them. [[5edo|5]], [[7edo|7]] and [[9edo|9]] have arguably been used in various kinds of musical traditions in different parts of the world. [https://soundcloud.com/scottthompson-3/the-13-edos-of-xmas ''The 13 EDOs of Xmas'' by Scott Thompson] is a humorous demonstration of EDOs 1–13.

On the other hand, if you use the edo to tune a scale or [[regular temperament]], the size of the edo does not matter so much (at least conceptually), as you don't need to use all of it. Some of the EDOs which can be used to tune various temperaments are listed on the [[optimal patent val]] page. Tuning a scale in just intonation by one of these EDOs can be regarded as automatically tempering it to the corresponding regular temperament.

To practically tune large edos through software tuning, one may take advantage of MIDI channels. See [[Tuning per channel]].

All of these tools are also applicable to equal divisions of other ([[nonoctave]]) intervals as well.

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* [[Minimal consistent EDOs]]

* [[Consistency levels of small EDOs]]

* [[Monotonicity levels of small EDOs]]

* [[Relative errors of small EDOs]]

* [[Distinct EDO Scales]]

@@ Line 60: / Line 60: @@
 * '''pentatonic''' EDOs ({{EDOs| 5, 10, 15, 20, 25 &amp; 30 }}) have a fifth of three-fifths of an octave = 3\5 = 720¢
 * '''supersharp''' EDOs ({{EDOs| 8, 13 &amp; 18 }}) have a fifth wider than 3\5 = 720¢
-* '''trivial''' EDOs ({{EDOs| 1, 2, 3, 4 and 6 }}) have a fifth about 100¢ from just, and are contained in 12 EDO
+* '''trivial''' EDOs ({{EDOs| 1, 2, 3, 4 and 6 }}) have a fifth about 100¢ from just, and are contained in 12edo
 === Non-tuning properties ===
@@ Line 66: / Line 66: @@
 You will quickly find that the ''factorization'' of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, 6 = 2 x 3, so 6edo contains all of the intervals in both 2edo and 3edo. On the other hand, 7 is a prime number, so no 7edo intervals are redundant with those of smaller EDOs. See [[Prime EDO]] for more details.
-The [[Moments of Symmetry]] paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.
+The [[Moments of symmetry]] paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.
 === Adding EDOs ===
@@ Line 80: / Line 80: @@
 When an edo divides the octave into fewer than 12 divisions (so that each step exceeds 100 cents), you might call it a [[macrotonal EDO]]. Of these, 1, 2, 3, 4 and 6 divide 12 and so are already available to anyone wishing to explore them. [[5edo|5]], [[7edo|7]] and [[9edo|9]] have arguably been used in various kinds of musical traditions in different parts of the world. [https://soundcloud.com/scottthompson-3/the-13-edos-of-xmas ''The 13 EDOs of Xmas'' by Scott Thompson] is a humorous demonstration of EDOs 1–13.
 On the other hand, if you use the edo to tune a scale or [[regular temperament]], the size of the edo does not matter so much (at least conceptually), as you don't need to use all of it. Some of the EDOs which can be used to tune various temperaments are listed on the [[optimal patent val]] page. Tuning a scale in just intonation by one of these EDOs can be regarded as automatically tempering it to the corresponding regular temperament.
+To practically tune large edos through software tuning, one may take advantage of MIDI channels. See [[Tuning per channel]].
 All of these tools are also applicable to equal divisions of other ([[nonoctave]]) intervals as well.
@@ Line 1,166: / Line 1,168: @@
 * [[Minimal consistent EDOs]]
 * [[Consistency levels of small EDOs]]
+* [[Monotonicity levels of small EDOs]]
 * [[Relative errors of small EDOs]]
 * [[Distinct EDO Scales]]