Primes have very direct application in music theory -- tuning theory. But anything much over 23, you're probably kidding yourself. Each small prime has a characteristic sound.
Primes of all sizes also can be used to make special self-similar sequences, but again, only the smaller ones are certain to be useful in music theory.
And -- here I'm really getting out of my territory -- they can be used in experimental designs.
Costas sequences, generated from primes, are used to make optimal radar signals.
Sounds fascinating - I'd love to know more.
What units are we talking here - semitones, Hertz...?
http://en.wikipedia.org/wiki/Just_intonation
The primes occur in frequency ratios, which are usually written like fractions. So we're talking ratios of hertz (cps) which are typically converted back and forth into cents.
Because we're used to 12 pitches per octave, intervals which differ from those intervals have a novel sound. As you go up the series of primes -- 5,7,11,13 -- you get more and more out of tune with 12-tone. 12-tone (or 12EDO) tuning has "fifths" (G-C) at 700 cents, which are around 2 cents narrow if compared to the frequency ratio of 3:2, which is 701.955 cents.
http://www.sengpielaudio.com/calculator-centsratio.htm
So 12EDO is a very good tuning for ratios involving primes no higher than 3 (or "3-limit" tuning). (Note that 9:8 (or approximately D-C) is still within the 3-limit, because 9 is 3x3, and 8 is 2x2x2.)
The next set of EDO cycles that approximate a 3:2 are 17 and 19 tones per octave. These tunings have their own flavors, but, unfortunately, don't do everything that 12-tone can do, so there's a trade-off. For example, 17EDO doesn't have a "minor second" or something approximating a 16/15 that is in the vicinity of 100 or 100+ cents.
http://xenharmonic.wikispaces.com/17edo
There's
always a trade-off between practicality and novelty or richness.
Because of advances in technology, it's much easier to explore new tunings than ever before. There's a fairly large community of people making music with non-standard tunings. However, advocates of alternate tunings exaggerate when they sometimes claim that making music in non-standard tunings is as easy and practical as making music in good old 12EDO.
When I say that each prime has its own flavor, I mean that a frequency ratio like 7:2 (2168.826 cents) has a "harmonic" sound, but is around 31 cents narrow compared to 12EDO, 11:8 is around 49 cents narrow, and so on. In other words, each new prime-generated ratio deviates from 12EDO by an approximately characteristic amount.
