Prime EDO: Difference between revisions

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

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

== Prime numbers in EDOs ==

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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:

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

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

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

* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone 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 are no [[Wikipedia: 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)

* There is no support for rank-2 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~~)

* 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 12-EDO)

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.

For these or similar reasons, some musicians do not like 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~~]]).

Primality may be desirable if you want, for example, a whole tone scale that is ''not'' absolutely uniform. In this case you might like [[19-EDO]] (with whole tone scale 3 3 3 3 3 4, [[MOS scale]] of type [[1L 5s]]) or [[17edo|17-EDO]] (with whole tone scale 3 3 3 3 3 2, MOS scale of type [[5L 1s]]). In general, making a chain of any interval of a prime ''n''-EDO, thus treating the interval as the generator of a MOS 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|12-EDO]] (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|3-EDO]]).

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.

A prime edo is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12-EDO. Since 12 is 2 × 2 × 3, it contains [[2edo|2-EDO]], 3-EDO, [[4edo|4-EDO]] and [[6edo|6-EDO]]. All EDOs with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12-EDO, 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|35-EDO]], will work just as well for this purpose.

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.

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|27-EDT]] is a non-prime system very similar to 17-EDO, while [[19edt|19-EDT (Stopper tuning)]] is a prime system very similar to the ubiquitous 12-EDO. (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.

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.

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

== The first 46 Prime EDOs ==

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-A '''prime edo''' is an [[EDO]] with a [[prime number]] of different pitches per [[octave]].
+A '''prime EDO''' is an [[EDO]] with a [[prime number]] of different pitches per [[octave]].
 == Prime numbers in EDOs ==
@@ Line 5: / Line 5: @@
 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:
-* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo|12EDO]])
+* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo|12-EDO]])
-* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the wholetone scale in 12EDO)
+* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone 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 are no [[Wikipedia: 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)
+* There is no support for rank-2 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)
+* 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 12-EDO)
-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.
+For these or similar reasons, some musicians do not like 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]]).
+Primality may be desirable if you want, for example, a whole tone scale that is ''not'' absolutely uniform. In this case you might like [[19-EDO]] (with whole tone scale 3 3 3 3 3 4, [[MOS scale]] of type [[1L 5s]]) or [[17edo|17-EDO]] (with whole tone scale 3 3 3 3 3 2, MOS scale of type [[5L 1s]]). In general, making a chain of any interval of a prime ''n''-EDO, thus treating the interval as the generator of a MOS 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|12-EDO]] (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|3-EDO]]).
-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.
+A prime edo is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12-EDO. Since 12 is 2 × 2 × 3, it contains [[2edo|2-EDO]], 3-EDO, [[4edo|4-EDO]] and [[6edo|6-EDO]]. All EDOs with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12-EDO, 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|35-EDO]], will work just as well for this purpose.
-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.
+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|27-EDT]] is a non-prime system very similar to 17-EDO, while [[19edt|19-EDT (Stopper tuning)]] is a prime system very similar to the ubiquitous 12-EDO. (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.
 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.
-[TODO: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here.]
 == The first 46 Prime EDOs ==