# Prime EDO

A **prime EDO** is an EDO with a prime number of different pitches per octave.

## Prime numbers in EDOs

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 12-EDO) - Excepting the scale comprising all notes of the EDO, there is
*no absolutely uniform, octave-repeating scale*(such as the whole tone scale in 12-EDO) - There are no modes of limited transpostion, such as used by the composer Olivier Messiaen
- 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 12-EDO)

For these or similar reasons, some musicians do not like prime EDOs (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 19-EDO (with whole tone scale 3 3 3 3 3 4, MOS scale of type 1L 5s) or 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 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 3-EDO).

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 2-EDO, 3-EDO, 4-EDO and 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 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 tritave (3/1) instead of the octave, can be an option. For example, 27-EDT is a non-prime system very similar to 17-EDO, while 19-EDT (Stopper tuning) is a prime system very similar to the ubiquitous 12-EDO. (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.

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.

## 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.