Will we ever run out of new songs?

Be advised: this will be a lengthy and somewhat speculative (if not downright fanciful) post.

There’s a question related to my two biggest interests, math and music, that has bugged me for several years now. My curiosity was piqued further by the excellent short story Melancholy Elephants by Spider Robinson, which I heartily recommend to anybody interested in the future, the arts, and the future of the arts. The question is this: will there come a day when every effectively different song has been written & copyrighted? Going further, are there infinitely many truly different songs, or just finitely many? If finitely many, few enough that we could exhaust?

First, we must insist on restricting our attention to songs less than some fixed time T, because if the time duration is unbounded, of course there are infinitely many different songs. The majority of these are too long to ever listen through, so let’s take T to be five minutes - short enough for the average listener to tolerate, and long enough to generate interesting results.

But there’s a more pressing and basic question - what’s a “song”? Let’s temporarily adopt the philosophy of John Cage and take a “song” to be any sound recording whatsoever. Naturally, a lot of these recordings sound extremely weird and not musically conventional, but it’s that very weirdness we really desire, for the sake of variety.

Could there really be only finitely many sound recordings less than five minutes long?

Any sound recording can be broken down into its waveform, a function describing how the air molecules vibrate when the song is playing. This in turn can be described in terms of three fundamental ingredients: pitch, rhythm and timbre. Pitch is the collection of frequencies (typically given in units of Hertz) sounding at any given moment. Rhythm is the distribution of sounds and silences through the duration of the song. Timbre is the shape or tone quality/color of the waveform; it’s what differentiates a clarinet and an oboe both playing a 440 Hz concert A equally loudly. One other possibly independent parameter is volume, something like the average amplitude of the waveform in time, although, volume is related to timbre.

At this point some people (particularly physicists) may jump in and say “time and space are quantized by atomic radii and Planck’s constant, leading to finitely many five-minute songs”. That may be true, and quite profound, but I’m going to ignore this point and emphasize a more practical issue: human perception is rather limited. Nobody’s ears are fine enough to distinguish two sine tones playing 0.001 Hz apart, or the difference between a piano tone sustained for 3 seconds versus 3.001 seconds. And there’s undoubtedly a limit to how little the timbre or volume of a waveform can be adjusted before this change is perceived by human ears.

So we have all this perceptual quantization on scales much larger than the atomic level. However, timbre is a richly complex parameter having to do with the shape of the waveform, and conceivably a pitch like 440 Hz could sound infinitely many different ways. But consider: with the mathematical technology of Fourier analysis, any waveform can theoretically be broken down into an infinite sum of sine-waves of varying frequencies and wavelengths. Once again, our perception is limited and granularity enters the picture: many (seemingly all but finitely many) of these Fourier series will be too similar for human ears to distinguish.

Now it seems that the number of effectively different songs less than five minutes long, though vast, is surely finite. It’s not realistic to suppose our senses can detect infinitesimal changes, and music doesn’t really have as many physical dimensions as you might suppose just from hearing an orchestra.

All this said, there is quite a big difference between “finite” and “able to be exhausted in a time span accessible to humans”. There’s Bremermann’s limit, which is something like the maximum operating speed for any computer anybody could possibly build, given limitations from the laws of physics. Consider a composite number N so large that it would take a computer the size of the entire universe and running at Bremermann’s limit a whole trillion years to factorize N. (Such numbers exist, for numbers get as big as we need!). Now consider feeding that gargantuan number into the Ackermann function. The resulting number Ack(N) is just unreasonably large - far bigger than any computer could handle thinking about even given a billion billion billion billion years. And we could feed this latest beast back into the Ackermann function…etc.

I like to think the number of distinguishable songs less than five minutes long is finite, but incomprehensibly vast, akin to the monstrous number Ack(Ack(N)) just constructed (well, probably not nearly as big as Ack(Ack(N)), to be honest, but bigger than we can understand, anyway). There are at least 176 different pitches we can distinguish (all the keys of the piano, plus the quarter-tones between them), more than 50 traditional instruments used by orchestras, hundreds or thousands more unconventional & electronic instruments, and millions of ways to arrange things in time. Finding the number of distinct songs would involve exponentiation of these numbers, as well as taking factorials. We could potentially have every person on earth employed as a composer, all churning out songs without rest until the sun explodes, and still not cover one percent of all that’s possible.

On that note, here’s some radical variation in music from the last few hundred years.

Palestrina’s motet “Salve Regina”: