Terry Pratchett, Ian Stewart & Jack Cohen wrote:
Let's come down to Earth and ask a simpler question. The wels of human creativity run deep, but if you take too much water from a well it runs dry. Once Beethoven had written the opening bars of his "SYmphony in C Minor" (5th, as I'm sure you're aware - Ed) that was one less tune for the rest of us. Given the amount of music that has been composed over the ages, maybe most of the best tunes have found already. Will the composers of the future be unable to match those of the past because the world is running out of tunes?
There is, of course, far more to a piece of music than a mere tune. There is melody, rhythm, texture, harmony, development ... But even Beethoven knew you can't beat a good tune to get your composition off the ground. By "tune" we mean a relatively short section of music - what the cogniscenti call a "motif" or "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 tunes that have been written, but all the tunes that could ever be written. How big is T-space (the phase space of tunes. For further info on phase spaces,
http://en.wikipedia.org/wiki/Phase_space - Ed)?
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. In contrast, a monkey pounding the keys of a piano might actually hit on a reasonable note every so often, so it looks as though the space of acceptably tuneful tunes is a reasonable-sized chunk of the space of
all tunes. And at that point, the mathematician's reflexes can kick in and we can do some combinatorics again.
(
http://en.wikipedia.org/wiki/Combinatorics )
To keep things simple, we'll consider only Eurpean-style music with 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'l 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 (Edit of irrelevant waffle for the purpose of this post - Ed). So the number of sequences of 30 notes is 25 x 25 x ... x 25, with 30 repetitions of that 25. Computerjob, that: it says that the answer is
867361737988403547205962240695953369140625
which has 42 digits. Adding in the 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. Assume that a million composers write music for a thousand years, each producing a thousand tunes per year. 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 sigificant inroads into T-space at all. Nearly all of it will be uncharted 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#, 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"
