There is a bit of mathematical formalism about how to define numbers, from sound foundational principles. The usual method is set theory -- numbers are equivalent to sets, or sets of sets, a set being an object made of other objects grouped together in curly brackets: {x,y,z}.

The most basic set is the

If you make a set which contains the empty set, you get {{∅}}, or {{}}. The cardinality of this set is 1: it has one member, {∅}. If the empty set does well as be a definition of 0, this does well for 1 and so 1 = {0} = {{∅}}.

And then you can get 2 by having {0,1}, that is, { {∅}, {{∅}} }, cardinality 2.

3 is {0,1,2} = { {∅}, {{∅}}, {{∅},{{∅}}} } and so on. Keep doing this and you construct the

But there are other kinds of numbers: the two relevant ones for the purposes of this explanation are the

http://en.wikipedia.org/wiki/Integers#Construction

http://en.wikipedia.org/wiki/Rational_numbers#Formal_construction

Suffice it to say, that if you write these things out, you get things like:

`-5 = {{{{{}},{}},{{{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}},{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}}}}},{{{{{}}},{}},{{{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}},{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}},{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}},{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}}}}}}}`

and

`¾ = {{{{{{{{{{},{{}},{{},{{}}}}},{}},{{{}}}},{{{{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}},{}},{{{{}}}}}}},{}},{{{{{{{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}},{}},{{{}}}},{{{{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}},{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}}},{}},{{{{}}}}}}}}},{{{{{{{{}},{}},{{{{},{{}},{{},{{}}}}}}},{{{{{}}},{}},{{{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}}}}}},{}},{{{{{{{}},{}},{{{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}}}},{{{{{}}},{}},{{{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}},{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}}}}}}}}}}`

These have some interesting, if repetitive, structure to them and I wondered if they might sound nice. Suppose you go up an interval when you see a { and down one when you see a } (and, say, repeat the last note when you hit a comma)-- what would it sound like? The answer is produced by Peano, named after a mathematician who did work on this sort of thing, and which also sounds like the word 'piano'.

You can download it here. You will need numpy. Running it in linux, with the command line LAME encoder installed will also be a great help.

You feed it an integer or rational number (either between, or with numerators and denominators between -10 and 10, unless you modify it to take these safety locks off) and it produces an mp3 (all going well) of beeping noises. There are parameters you can twiddle with, including:

- How many intervals the octave is divided into (default: 12)
- What frequency it starts on (default: 110 Hz)
- How many semitones it moves by (default: 2 (1 is a little dull))
- Length of each tone (default: 0.2s)
- A series of bright organ beep tones (default: saworgan: odd harmonics of the sine, filtered exponentially to produce a sort of sawtooth effect)

Typical output sounds like this (mp3).

It is not unmusical.

I got the sine generation code from here. The bad additive synthesis is all my addition.