Prime equal division: Difference between revisions

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A '''prime equal division''' is an [[equal tuning]] that divides a given [[equave]] into a [[prime number]] of pitches. The opposite of a prime equal division is a [[highly composite equal division]].

A '''prime edo''' therefore contains a prime number of pitches per [[octave]].

A '''prime edo''' therefore contains a prime number of pitches per [[octave]], such as {{EDOs|7edo, 13edo, and 41edo}}.

== Properties ==

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Primality may be desirable if you want, for example, a whole tone 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]]) or [[17edo]] (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''-equal division, thus treating the interval as the generator of a mos scale, one can reach every tone in ''n'' steps. For composite equal divisions, this will only work with intervals that are co-prime to the edo, for example 5 degrees of [[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]]).

A prime equal division is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12edo, a highly composite equal division. Since 12 is 2 × 2 × 3, it contains [[2edo]], 3edo, [[4edo]] and [[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]], will work just as well for this purpose.

A prime equal division is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12edo, a highly composite equal division. Since 12 is 2 × 2 × 3, it contains {{EDOs|2edo, 3edo, 4edo and 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]], 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 [[19edt|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]] system with identical step sizes.

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 A '''prime equal division''' is an [[equal tuning]] that divides a given [[equave]] into a [[prime number]] of pitches. The opposite of a prime equal division is a [[highly composite equal division]].
-A '''prime edo''' therefore contains a prime number of pitches per [[octave]].
+A '''prime edo''' therefore contains a prime number of pitches per [[octave]], such as {{EDOs|7edo, 13edo, and 41edo}}.
 == Properties ==
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 Primality may be desirable if you want, for example, a whole tone 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]]) or [[17edo]] (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''-equal division, thus treating the interval as the generator of a mos scale, one can reach every tone in ''n'' steps. For composite equal divisions, this will only work with intervals that are co-prime to the edo, for example 5 degrees of [[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]]).
-A prime equal division is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12edo, a highly composite equal division. Since 12 is 2 × 2 × 3, it contains [[2edo]], 3edo, [[4edo]] and [[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]], will work just as well for this purpose.
+A prime equal division is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12edo, a highly composite equal division. Since 12 is 2 × 2 × 3, it contains {{EDOs|2edo, 3edo, 4edo and 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]], 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 [[19edt|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]] system with identical step sizes.