''todo: improve definition rearrange contents of pages [[prime_number|prime number]] and [[prime_edo|prime edo]]''
A '''prime number''' is an integer (whole number) greater than one that is divisible only by itself and one. There are an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13, ….
== Prime numbers in [[EDO|EDO]]s ==
== Prime factorization ==
{{Wikipedia|Integer factorization}}
A ''prime number'' is an integer (whole number) greater than one that is divisible only by itself and one. There are an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13 ... . Whether or not a number ''n'' is prime has important consequences for the properties of the corresponding ''n''-[[EDO|EDO]], especially for lower values of ''n''. In these instances:
By the {{w|fundamental theorem of arithmetic}}, any [[ratio]] can be uniquely represented by a product of prime numbers through prime factorization. It enables the notation of ratios as [[monzo]]s.
* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo|12EDO]])
== Prime equal division ==
* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the wholetone scale in 12EDO)
{{Main| Prime equal division }}
* There are no [http://en.wikipedia.org/wiki/Modes_of_limited_transposition modes of limited transpostion], such as 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.
A prime equal division is an [[equal-step tuning]] that divides a given [[interval]] into a prime number of pitches. They are notable because of many interesting properties.
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]]).
== Coprime numbers ==
{{Wikipedia|Coprime integers}}
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.
Two integers are '''coprime''' if they have no divisor in common except 1.
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 [[19edt|19edt (Stopper tuning)]] is a prime system very similar to the ubiquitous [[12edo]]. (See [[edt#EDO-EDT correspondence|EDO-EDT 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 ==
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:
* [[Monzo]] - an alternative notation for interval ratios
* [[Harmonic limit]]
* [[Prime limit]] or [[Harmonic Limit]]
* [[List of integer factorizations]]
==Links==
== External links ==
* [http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm Die Primzahlseite] (German) by Arndt Brünner (helpful tools for prime factorization and ~test)
* [http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm Die Primzahlseite] (German) by Arndt Brünner (helpful tools for prime factorization and ~test)
* [http://en.wikipedia.org/wiki/Prime_number Prime number] the Wikipedia article
A prime number is an integer (whole number) greater than one that is divisible only by itself and one. There are an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13, ….
By the fundamental theorem of arithmetic, any ratio can be uniquely represented by a product of prime numbers through prime factorization. It enables the notation of ratios as monzos.
A prime equal division is an equal-step tuning that divides a given interval into a prime number of pitches. They are notable because of many interesting properties.
