The Science of Discworld II

mean a relatively short section of music – what the cognoscenti call a ‘motif’ or a ‘phrase’, between one and thirty notes in length, say. Tunes are important, because they are the building blocks for everything else, be it Beethoven or Boyzone. A composer in a world that has run out of tunes is like an architect in a world that has run out of bricks.
    Mathematically, a tune is a sequence of notes, and the set of all possible such sequences forms a phase space: a conceptual catalogue that contains not just all the tunes that have been written, but all the tunes that could ever be written. How big is T-space?
    Naturally, the answer depends on just what we are willing to accept as a tune. It has been said that a monkey typing at random would eventually produce Hamlet , and that’s true if you’re willing to wait a lot longer than the total age of the universe. It’s also true that along the way the monkey will have produced an incredible amount of airport novels. 2 In contrast, a monkey pounding the keys of a piano might actually hit on a reasonable tune every so often, so it looks as though the space of acceptably tuneful tunes is a reasonable-sized chunk ofthe space of all tunes. And at that point, the mathematician’s reflexes can kick in, and we can do some combinatorics again.
    To keep things simple, we’ll consider only European-style music based on the usual twelve-note scale. We’ll ignore the quality of the notes; whether played on a piano, violin, or tubular bells, all that matters is their sequence. We’ll ignore whether the note is played loudly or softly, and – more drastically – we’ll ignore all issues of timing. Finally, we’ll restrict the notes to two octaves, 25 notes altogether. Of course all these things are important in real music, but if we take them into account their effect is to increase the variety of possible tunes. Our answer will be an underestimate, and that’s all to the good since it will still turn out to be huge . Really, really huge, right? No – bigger than that .
    For our immediate purposes only, then, a tune is a sequence of 30 or fewer notes, each chosen from 25 possibilities. We can count how many tunes there are in the same way that we counted arrangements of cars and DNA bases. So the number of sequences of 30 notes is 25 × 25 × … × 25, with 30 repetitions of that 25. Computer job, that: it says that the answer is
867361737988403547205962240695953369140625
    which has 42 digits. Adding in the 29-note tunes, the 28-note ones, and so on we find that T-space contains roughly nine million billion billion billion billion tunes. Arthur C. Clarke once wrote a science fiction story about the ‘Nine billion names of God’. T-space contains a million billion billion billion tunes for every one of God’s names. Assume that a million composers write music for a thousand years, each producing a thousand tunes per year, more prolific even than The Beatles. Then the total number of tunes they will write is a mere trillion. This is such a tiny fraction of that 42-digit number that those composers will make no significant inroads into T-space at all. Nearly all of it will be unexplored territory.
    Agreed, not all of the uncharted landscape of tune-space consists of good tunes. Among its landmarks are things like 29 repetitions of middle C followed by F sharp, and
BABABABABABABABABABABABABABABA,
    which wouldn’t win any prizes for musical composition. Nevertheless, there must be an awful lot of good new tunes still waiting to be invented. T-space is so vast that even if good-tune-space is only a small proportion of it, good-tune-space must also be vast. If all of humanity had been writing tunes non-stop since the dawn of creation, and went on doing that until the universe ended, we still wouldn’t run out of tunes.
    It is said that Johannes Brahms was walking along a beach with a friend, who was complaining that all of the good music had already been written. ‘Oh, look,’ said Brahms, pointing out to sea. ‘Here comes the last wave.’
    Now we come to what may well be the chief function of art and music for us – but not for edge people or chimpanzees, and probably not for Neanderthals. This, if we are right, is what Rincewind has in mind.
    When we look at a scene we see only the middle five to ten degrees of arc. We invent the rest all around that bit, and we give