So. The really big flaw in my analysis of an “Infinite Jukebox” tune — one in which the song is free to jump between two points, with a probability of of jumping from the one-minute mark to the two-minute mark, and an equal likelihood of jumping from the two-minute mark to the one-minute mark — and my conclusion that, on average, the song would lose a minute just as often as it gained one and so we could expect the song to be just as long as the original, is that I made allowance for only the one jump. The three-minute song with two points at which it could jump, which I used for the model, can play straight through with no cuts or jumps (three minutes long), or it can play jumping from the one-minute to the two-minute mark (a two minute version), or it can play from the start to the second minute, jump back to the first, and continue to the end (a four minute version). But if you play any song on the Infinite Jukebox you see that more can happen.
Working through my circle of friends have been links to The Infinite Jukebox, an amusing web site which takes a song, analyzes points at which clean edits can be made, and then randomly jumps through them so that the song just never ends. The idea is neat, and its visual representation of the song and the places where it can — but doesn’t have to — jump forward or back can be captivating. My Dearly Beloved has been particularly delighted with the results on “I Am A Camera”, by the Buggles, as it has many good edit points and can sound quite natural after the jumps if you aren’t paying close attention to the lyrics. I recommend playing that at least a bit so you get some sense of how it works, although listening to an infinitely long rendition of the Buggles, or any other band, is asking for a lot.
One question that comes naturally to mind, at least to my mind, is: given there are these various points where the song can skip ahead or skip back, how long should we expect such an “infinite” rendition of a song to take? What’s the average, that is the expected value, of the song’s playing? I wouldn’t dare jump into analyzing “I Am A Camera”, not without working on some easier problems to figure out how it should be done, but let’s look.