Musical Idea 28: Arithmetic Paronomasia
04.12.2014 § Leave a comment
The fundamental theorem of arithmetic states that every positive integer can be expressed as a unique product of prime numbers. In this sense, primes are the elemental building blocks of all natural numbers, and prime factorization is truly as definitive as place on the number line.
When two numbers with completely different prime factors are next-door neighbors on the number line, we can conflate them, and surprise audiences by quietly modulating between them while revealing the two factorizations. Such tricks can be fairly referred to as puns, because, like traditional linguistic puns, two interpretations are tapped at once in a surprising way. To avoid confusion with linguistic puns that happen to be about arithmetic, I propose officially referring to these tricks as “arithmetic paronomasia” (and really, how could we not jump at this opportunity to associate ourselves with such a big, cool, Greek word?!).
Here is an example. The fifth, sixth, and seventh primes are related to the eighth in an astonishing way: 11•13•17 = 2431, while 19•2^7= 2432. I used this pun to create a musical tuning system based on the 11th, 13th, 17th, and 19th partials of the harmonic series. By octave equivalence, powers of 2 are negligible — pitches related by ratios of 2 are generally given the same name and considered the same note — so tempering out this 2432/2431 comma, then, one move each by an 11th, 13th and 17th becomes the same thing as one move by a 19th. Nice!
Another example: 5*5*7 = 175; 2*2*2*2*2*11 = 176. I used this for a rhythm. The smaller the less interesting, but the larger, the less chance the listener can perceive them.
Probably you would want to have some constraints: which prime factors, how big or small their orders can be, how close they have to be (relatively) to consider them conflatable.
Another type I might call “Egyptian fraction puns” 1/15 + 1/14 + 1/13 + 1/12 + 1/11 + 1/9 ~= 1/2. You’d need to set the threshold here too, and also other constraints on the denominators, the number of them that sum, the range, how different they can be, etc. This example means that you can have a hexarhythm with 15 against 14 against 13 against 12 against 11 against 9 all in the same duration, offset them in a particular way, so that when each one gets two strikes in a row stressed, the durations between these each other are also one of the lengths within them, so that each interval gets expressed once itself and once as a shadow.