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A Mathematical Look at Musical Plagiarism

May, 2004

A famous composer was once accused of plagiarizing from a far inferior one. His answer was (tongue-in-cheek, no doubt) something like, "After all, there are only 12 notes." Now this led me to wonder what the chances are of a melody being plagiarized by sheer chance rather than by design. So let us do some classy calculations.

We will assume that the melody in question lies within a single octave, which is to say it consists of only the 7 white and 5 black keys on a piano, 12 possible notes in all. The simplest scenario is that the melody consists of 12 notes, no two the same. If you remember your high school math, the answer is 12!—for which read, "twelve factorial."

You see, the first note can be any one of the 12, the second note any of the remaining 11, the third note any of the remaining 10, and so on. By the laws of what we call "counting" (or "permutations," if you will), that gives us 12x11x10x9x8x7x6x5x4x3x2x1 or 479,001,600 possible sequences of 12 different notes, each played only once. So the probability of repeating those 12 notes by chance is 1 out of 12! or approximately .0000002%.

If you are in the mood for doing the same calculation for the entire keyboard with its 88 keys, black and white, you will have to play with 88!, a number that most of your calculators can only estimate. (Mine just says ERROR.)

Now if we allow that the original melody could have used any of the 12 possible notes, repeating some, omitting others, this means that each of the 12 notes (not just the first one) can be played in 12 different ways. So instead of 12!, we have 12 to the power of 12, which comes to 8,916,100,448, 256 different ways. (You can play the same note 12 times, you can play the same note 11 times and another note once, and so on.) My hand-held calculator cannot give me the probability of stumbling upon any one of these ways as a percent, but believe me: it is very tiny.

Let’s get back to a melody of only 12 notes, all different. To make things more complicated, consider this. Each of those 12 notes can be played as a whole, half, quarter, eighth, sixteenth or thirty-second note. (Forget for a moment that it can also be played ffff, fff, ff, f, p, pp, ppp, pppp, and so on.) This means that our very first note, any one of the 12, can be played in 6 different durations (omitting the possibility of longer or shorter durations).

Alas, this means that if our first note of the 12 can be played in 6 ways, then we have 72 possible openings. The second note can be played in 11x6 or 66 ways, the third note in 10x6 or 60 ways, and so on. Well (choke), this comes to 72x66x60x…x6 ways or a grand total of 1,042,682,221,795,737,600 ways in which 12 notes can be played with 6 degrees of loudness.

To give some idea of the chances of plagiarizing a simple tune of 12 notes that can be played in any of 6 durations, think of it this way. If you could play each sequence of that last huge number of sequences, how long would it take you to stumble upon the original tune if you were unlucky enough to hit upon it LAST?

To make this simple, make believe you could play each new sequence in one second’s time. Well, that would take you 10,042,682, 222, 000,000,000 seconds or 17,378,037,030,000,000 minutes or 289,633,950,500,000 hours or 1,206,808,1270,000 days or 33,063,236,360 years—and you can figure that out in decades, centuries or millennia. But that figure is pretty nearly correct — if you never stop for any reason whatsoever.

So our composer (I will give a no-prize if you can name him) was absolutely wrong. By saying there are ONLY 12 notes as an explanation for unconscious plagiarism, the prosecution could have countered that it HAD to be plagiarism since the chances of his accidentally hitting upon that tune (which had, by the way, more than just 12 notes) was next to zero.

By the way, read about the Rum and Coca Cola case in Louis Nizer’s "My Life in Court" for an actual proof of plagiarism that uses none of my arguments.

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