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| A '''prime EDO''' is an [[EDO]] with a [[prime number]] of different pitches per [[octave]].
| | #REDIRECT [[Prime equal division]] |
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| == 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:
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| * There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo|12-EDO]])
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| * 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)
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| * There are no [[Wikipedia: Modes of limited transposition|modes of limited transpostion]], such as used by the composer Olivier Messiaen
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| * There is no support for rank-2 temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)
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| * 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)
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| 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.
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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 [[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]]).
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| 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.
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| 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.
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| 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.
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| == The first 46 Prime EDOs ==
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| 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:
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| [[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[11edo|11]], [[13edo|13]], [[17edo|17]], <br/>
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| [[19edo|19]], [[23edo|23]], [[29edo|29]], [[31edo|31]], [[37edo|37]], [[41edo|41]], [[43edo|43]], <br/>
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| [[47edo|47]], [[53edo|53]], [[59edo|59]], [[61edo|61]], [[67edo|67]], [[71edo|71]], [[73edo|73]], <br/>
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| [[79edo|79]], [[83edo|83]], [[89edo|89]], [[97edo|97]], [[101edo|101]], [[103edo|103]], [[107edo|107]], <br/>
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| [[109edo|109]], [[113edo|113]], [[127edo|127]], [[131edo|131]], [[137edo|137]], [[139edo|139]], [[149edo|149]], <br/>
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| [[151edo|151]], [[157edo|157]], [[163edo|163]], [[167edo|167]], [[173edo|173]], [[179edo|179]], [[181edo|181]], <br/>
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| [[191edo|191]], [[193edo|193]], [[197edo|197]], [[199edo|199]].
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| [[Category:EDO theory pages]]
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| [[Category:Prime EDO| ]] <!-- main article -->
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| [[Category:Prime]] | |