Math Music

By Ivars Peterson

Muse, May/June 2005, p. 16-17.

In the right hands, mathematics can be a musical instrument, a spooky player piano that turns numbers, number sequences, or mathematical functions into mysterious and haunting melodies.

Composers create music out of math by inventing special recipes, or formulas, that match numbers or mathematical patterns with musical notes. One example is Daniel Cummerow's recipe for a pi tune. (Pi is the number you get when you divide a circle's circumference by its diameter.)

Cummerow assigned each digit from 1 to 8 to a note of a particular, eight-note musical scale (O.K., for you music nuts, it was A harmonic minor). The digit 0 signaled a pause, and 9 meant a pause or a repeat of the previous note. Identical tones in a row were tied together into a longer note.

Suppose that 1 = do, 2 = re, 3 = mi, 4 = fa, 5 = sol, 6 = la, 7 = ti, 8 = do* (one octave higher). The number pi would become the following tune:

mi, do, fa, do, sol, sol, re, la, sol, mi, sol, do*, do*, ti, ti, mi, re, mi, do*, fa, la, re, la, fa, mi, mi, do*, mi, re, ti, ti, sol, rest, re do*, do*, fa, do, do, ti, do, la, la.

Try singing or playing it. What do you think? Maybe it would sound better with rhythm or dynamics.

Of course, you can change the recipe and make an entirely different kind of music from the same digits. Cummerow, for example, assigned each letter of the alphabet from A to Z to a note with its own pitch, octave, and duration, so he had 26 different notes. He then went through the first 255 digits of pi in pairs, assigning one of these 26 notes to each pair.

And pi is just one of many possible sources of musical inspiration. All sorts of special numbers and mathematical objects can be converted into music. You can use the numbers of an arrangement known as Pascal's triangle or even a geometric pattern, such as the Sierpinski triangle or Lorenz's butterfly, to create music.

A Sierpinski triangle is a large upright triangle, which consists of three smaller upright triangles, each of which consists of three smaller upright triangles, and so on . . . .

Maybe it's not all that surprising that math makes hypnotic music. After all, both music and math are about interesting patterns. Still, hearing pi in A harmonic minor sends shivers up your spine.

If you want to try making your own pi tune, go to Felix Jung's site, www.avoision.com/experiments/pi10k/pi10k.html. You might also want to check out some of the other experimental art pieces at the bottom of the page.

Advanced Math Music: If you have Web browser software that can play MIDI files, you can sample Cummerow's compositions for various mathematical constructs at www.geocities.com/Vienna/9349.

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