Prime neighborhood

All of us have no doubt noticed that the strangeness/poignancy/difficulty of
intervals increases as higher primes are introduced into ratios, up to and
including 13. After that, 17 and 19 are fairly tame/consonant sounding and
relatively easy to work into the rational scale fabric, then with 23 it
starts getting pretty bizarre-sounding again.

Here's a related observation I find interesting. Look at the neighboring
integers of prime numbers in terms of their octave-equivalent factor(s):

3, the most consonant ratio-maker lies above and below octaves of 1
5 lies above 1 and below 3
7 lies above 3 and below 1

OK. So far the neighbors are all octave-equivalents of 1 and 3. Perhaps
consequently, most people easily accept 7-limit harmony but have trouble with
higher numbers like:

11, which lies above 5 and below 3
13 lies above 3 but below 7

17, however, lies above 1 and below 3^2 - almost back to the simplicity of
base-10 single-digit primes
19 lies between 3^2 and 5. That might seem to be as complex as 11 but notice
that the order is swapped - above 3 and below 5 this time.

With 23 and beyond, we're back to more complex neighbors.

Does this mean anything? It seems a pretty wacky coincidence if that's all
it is.

David J. Finnamore
Nashville, TN



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