Prime numbers

日本穂

todo: improve definition rearrange contents of pages prime number and prime edo

Prime numbers in EDOs

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, especially for lower values of n. In these instances:

• There is no fully symmetric chord (such as the diminished seventh chord in 12EDO)
• Excepting the scale comprising all notes of the EDO, there is no absolutely uniform, octave-repeating scale (such as the wholetone scale in 12EDO)
• There are no 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 Armodue) and others love them.

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) 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-EDO, thus treating the interval as the generator of a 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 (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 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, 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 EDO-EDT 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.

[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:

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.

See also

• The Prime Harmonic Series
• Monzo - an alternative notation for interval ratios
• Prime limit or Harmonic Limit

Links

• Die Primzahlseite (German) by Arndt Brünner (helpful tools for prime factorization and ~test)
• Prime number the Wikipedia article