Prime EDO: Difference between revisions

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''~~todo: improve definition rearrange contents of pages~~ [[prime number]] ~~and~~ [[prime EDO]]''

A '''prime edo''' is an [[EDO]] with a [[prime number]] of different pitches per [[octave]].

== Prime numbers in EDOs ==

Whether or not a number ''n'' is prime has important consequences for the properties of the corresponding ''n''-[[EDO]], especially for lower values of ''n''. In these instances:

A '''prime ~~edo~~''' is an [[EDO]] with a [[prime ~~number~~]] of ~~different pitches per~~ [[~~octave~~]].

* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo|12EDO]])

* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the wholetone scale in 12EDO)

* There are no [http://en.wikipedia.org/wiki/Modes_of_limited_transposition modes of limited transpostion], such as used by the composer Olivier Messiaen

* There is no support for rank-two temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)

* Making a chain of any interval of the ''n''-EDO, one can reach every tone in ''n'' steps. (For composite EDOs, this works with intervals that are co-prime to ''n'', for example, 5 degrees of 12EDO)

For these or similar reasons, some musicians do not like the prime EDOs (e.g. the makers of [http://www.armodue.com/risorse.htm Armodue]) and others love them.

Primality may be desirable if you want, for example, a wholetone scale that is ''not'' absolutely uniform. In this case you might like [[19edo]] (with whole tone scale 3 3 3 3 3 4, MOS scale of type [[1L 5s|1L+5s]]) or [[17edo|17EDO]] (with whole tone scale 3 3 3 3 3 2, MOS Scale of type [[5L 1s|5L+1s]]). In general, making a chain of any interval of a prime ''n''-EDO, thus treating the interval as the generator of a [[MOSScales|Moment of Symmetry]] scale, one can reach every tone in ''n'' steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12edo|12EDO]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo|3EDO]]).

A prime edo is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12edo. Since 12 is 2*2*3, it contains [[2edo|2EDO]], [[3edo|3EDO]], [[4edo|4EDO]] and [[6edo|6EDO]]. All edos with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12edo, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo|35EDO]], will work just as well for this purpose.

~~== Examples ==~~

If you like a certain EDO for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[Edt|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt]] is a non-prime system very similar to [[17edo]], while [[19ED3|19edt (Stopper tuning)]] is a prime system very similar to the ubiquitous [[12edo]]. (See [[Edt#EDT-EDO correspondence|EDT-EDO correspondence]] for more of these.) Anyway, for every prime EDO system there is a non-prime [[Ed4|ED4]] system with identical step sizes.

~~macrotonal~~

The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.

* [[2edo]]

* [[3edo]]

* [[5edo]]

* [[7edo]]

* [[11edo]]

~~mesotonal~~

[TODO: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here.]

* [~~[13edo]]~~

* [[17edo]]

* [[19edo]]

* [[23edo]]

~~microtonal~~

== The first 46 Prime EDOs ==

Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below:

* [[29edo]]

[[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[11edo|11]], [[13edo|13]], [[17edo|17]],

* [[31edo]]

[[19edo|19]], [[23edo|23]], [[29edo|29]], [[31edo|31]], [[37edo|37]], [[41edo|41]], [[43edo|43]],

* [[41edo]]

[[47edo|47]], [[53edo|53]], [[59edo|59]], [[61edo|61]], [[67edo|67]], [[71edo|71]], [[73edo|73]],

* [[43edo]]

[[79edo|79]], [[83edo|83]], [[89edo|89]], [[97edo|97]], [[101edo|101]], [[103edo|103]], [[107edo|107]],

* [[53edo]]

[[109edo|109]], [[113edo|113]], [[127edo|127]], [[131edo|131]], [[137edo|137]], [[139edo|139]], [[149edo|149]],

[[151edo|151]], [[157edo|157]], [[163edo|163]], [[167edo|167]], [[173edo|173]], [[179edo|179]], [[181edo|181]],

[[191edo|191]], [[193edo|193]], [[197edo|197]], [[199edo|199]].

[[Category:EDO theory pages]]

[[Category:Prime EDO| ]]

[[Category:Prime]]

@@ Line 1: / Line 1: @@
-''todo: improve definition rearrange contents of pages [[prime number]] and [[prime EDO]]''
+A '''prime edo''' is an [[EDO]] with a [[prime number]] of different pitches per [[octave]].
+== Prime numbers in EDOs ==
+Whether or not a number ''n'' is prime has important consequences for the properties of the corresponding ''n''-[[EDO]], especially for lower values of ''n''. In these instances:
-A '''prime edo''' is an [[EDO]] with a [[prime number]] of different pitches per [[octave]].
+* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo|12EDO]])
+* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the wholetone scale in 12EDO)
+* There are no [http://en.wikipedia.org/wiki/Modes_of_limited_transposition modes of limited transpostion], such as used by the composer Olivier Messiaen
+* There is no support for rank-two temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)
+* Making a chain of any interval of the ''n''-EDO, one can reach every tone in ''n'' steps. (For composite EDOs, this works with intervals that are co-prime to ''n'', for example, 5 degrees of 12EDO)
+For these or similar reasons, some musicians do not like the prime EDOs (e.g. the makers of [http://www.armodue.com/risorse.htm Armodue]) and others love them.
+Primality may be desirable if you want, for example, a wholetone scale that is ''not'' absolutely uniform. In this case you might like [[19edo]] (with whole tone scale 3 3 3 3 3 4, MOS scale of type [[1L 5s|1L+5s]]) or [[17edo|17EDO]] (with whole tone scale 3 3 3 3 3 2, MOS Scale of type [[5L 1s|5L+1s]]). In general, making a chain of any interval of a prime ''n''-EDO, thus treating the interval as the generator of a [[MOSScales|Moment of Symmetry]] scale, one can reach every tone in ''n'' steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12edo|12EDO]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo|3EDO]]).
+A prime edo is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12edo. Since 12 is 2*2*3, it contains [[2edo|2EDO]], [[3edo|3EDO]], [[4edo|4EDO]] and [[6edo|6EDO]]. All edos with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12edo, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo|35EDO]], will work just as well for this purpose.
-== Examples ==
+If you like a certain EDO for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[Edt|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt]] is a non-prime system very similar to [[17edo]], while [[19ED3|19edt (Stopper tuning)]] is a prime system very similar to the ubiquitous [[12edo]]. (See [[Edt#EDT-EDO correspondence|EDT-EDO correspondence]] for more of these.) Anyway, for every prime EDO system there is a non-prime [[Ed4|ED4]] system with identical step sizes.
-macrotonal
+The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.
-* [[2edo]]
-* [[3edo]]
-* [[5edo]]
-* [[7edo]]
-* [[11edo]]
-mesotonal
+[TODO: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here.]
-* [[13edo]]
-* [[17edo]]
-* [[19edo]]
-* [[23edo]]
-microtonal
+== The first 46 Prime EDOs ==
+Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below:
-* [[29edo]]
+[[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[11edo|11]], [[13edo|13]], [[17edo|17]], <br/>
-* [[31edo]]
+[[19edo|19]], [[23edo|23]], [[29edo|29]], [[31edo|31]], [[37edo|37]], [[41edo|41]], [[43edo|43]], <br/>
-* [[41edo]]
+[[47edo|47]], [[53edo|53]], [[59edo|59]], [[61edo|61]], [[67edo|67]], [[71edo|71]], [[73edo|73]], <br/>
-* [[43edo]]
+[[79edo|79]], [[83edo|83]], [[89edo|89]], [[97edo|97]], [[101edo|101]], [[103edo|103]], [[107edo|107]], <br/>
-* [[53edo]]
+[[109edo|109]], [[113edo|113]], [[127edo|127]], [[131edo|131]], [[137edo|137]], [[139edo|139]], [[149edo|149]], <br/>
+[[151edo|151]], [[157edo|157]], [[163edo|163]], [[167edo|167]], [[173edo|173]], [[179edo|179]], [[181edo|181]], <br/>
+[[191edo|191]], [[193edo|193]], [[197edo|197]], [[199edo|199]].
+[[Category:EDO theory pages]]
 [[Category:Prime EDO| ]] <!-- main article -->
+[[Category:Prime]]