Prime equal division
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, such as 7edo, 13edo, and 41edo.
Properties
Prime equal divisions have many musical properties, which are especially significant for small primes. For larger primes, these properties are not as significant, since the difference between an absolutely uniform scale and an approximated, nearly uniform scale eventually become inaudible.
- There is no fully symmetric chord (such as the diminished seventh chord in 12edo).
- Excepting the scale comprising all notes of the tuning, there is no absolutely uniform scale that repeats at the equave (such as the whole tone scale in 12edo, which only has whole steps and repeats at the octave).
- There are no modes of limited transposition, such as used by the composer Olivier Messiaen.
- There is no support for rank-2 temperaments whose period is a fraction of the equave (all octave-periodic temperaments are linear temperaments).
- Making a chain of any interval of the n-equal division, 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 prime equal divisions (e.g. the makers of Armodue) and others love them.
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.
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 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 (Stopper tuning) is a prime system very similar to the ubiquitous 12edo. (See edt-edo correspondence for more of these.) Anyway, for every prime edo system there is a non-prime ed4 system with identical step sizes.
Prime edo
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:
2, 3, 5, 7, 11, 13, 17, 19,
23, 29, 31, 37, 41, 43, 47, 53,
59, 61, 67, 71, 73, 79, 83, 89,
97, 101, 103, 107, 109, 113, 127, 131,
137, 139, 149, 151, 157, 163, 167, 173,
179, 181, 191, 193, 197, 199.