Melancholy Mathematics

In a thread over on the Free Keene Forum Bill St Clair comments on the Spider Robinson short story entitled Melancholy Elephants

Interesting Spider Robinson story. But let's do the math. Most popular melodies are only about 8 bars long. Assuming all 8th notes in 4/4, we have 64 notes. It's a rare melody that takes more than two octaves. That's 24 possible pitches in the standard well-tempered scale for each note. But in reality, songs are usually in a major or minor key, so that's 16 possible pitches for each of major and minor. Total 16**24 + 16**24 = 1.6E29. That's a bit less than 3E19 melodies for each of the earth's 6 billion (6E9) souls. That's 30 billion billion songs per soul. Yes, many of those will not be pleasant tunes, but I still think there are enough to last us for at least the expected lifetime of our sun.

(He corrects the obvious error in the above here)

The biggest problem with the above anaysis is I don't have to copy an entire work to commit plagarism but only reconizable parts of it.

Remember the plagarism lawsuit over My Sweet Lord and He's So Fine? (If not here is a quick overview) The suit was based on two musical motifs used together in almost identical ways in both compositions.

A musical motif is a short bit of the entire composition. It may be of any size, and is the shortest subdivision of a theme or phrase that still maintains a recognizable identity.

Think of the first four notes from Beethoven's Fifth Symphony.

A motif must be at least two notes but there is no maximum length. The longest I can think of offhand is the descending eight-note sequence in Dies irae (Day of wrath), a chant later incorporated into the Catholic requiem mass. There may be longer ones.

If I double eight and use sixteen as a working limit then the total number of possible phrases spanning two octaves on a 12 note scale becomes:

24^2 + 24^3 + ... + 24^16 = 1.968e+19

That is not a small number but no means is it an impossible one. If each note is represented by a single octet on a hard drive, the entire possible phrase space represents about four exabytes -- much less with some basic compression.

If plagarism requires at least two phrases be copied then the total is the above number taken two at a time without duplication but without regard to order (combination) which is calculated by:

1.968e+19*(1.968e+19-1)/2 = 1.937e+38

Still several order of magnitude less than Mr. St. Claire's estimate.