Lewis Kaplan, co-founder and artistic director of the Bowdoin International Music Festival, has a deservedly famous talk on J.S. Bach’s use of numerology in his musical works. I was surprised to find out last week that Debussy (and maybe Beethoven and Mozart and Schubert)) also experimented with mathematical ratios and series in at least some of their music.

I was looking up Debussy on the Internet because of a remark that Maine composer Daniel Sonenberg made about his “Whistlesparks” for harp and flute, performed at the Portland Chamber Music Festival. In the program notes he acknowledged his debt to Debussy and hinted at quotations from one of that composer’s better-known works — not the flute and harp sonata. I still don’t know what piece he was thinking of.

After wading through too much information on the Net and listening to “Whistlesparks” again (“Afternoon of a Fawn”? a passage from Stravinski’s “Rite of Spring”?) I came across a reference to Debussy’s interest in the “Golden Section,” a natural ratio whose most famous example is the Chambered Nautilus — a root five rectangle imitated on the frieze of the Acropolis.

The Golden Section was widely used by the Impressionist painters to structure their works to look as if they were without structure, and Debussy was more influenced by his friends among the poets and painters than by musicians, whom he once said were interested primarily in their careers.

My art instructor in Pennsylvania, Myron Barnstone (who was born in Portland), was an expert on the Golden Section, which he claimed was discovered by one of the ancient Greek philosophers, who passed around a long staff and asked people to grasp it so that the two lengths separated by their hand formed the most pleasing proportion. He identified the most popular grip as a ratio in which the smaller part is to the larger as the larger is to the entire length, which works out to approximately 0.618 of the total length.

The ratio is an infinitely long number, like Pi, and is approximated by the so-called Fibonacci Series, in which each number is the sum of the preceding two.

Golden Section rectangles are easy to construct with a compass — simply find the diagonal of a square and add it to the base to make a root one rectangle (art paper); take the diagonal of that rectangle and add it to the new base to make a root three and so on to the root five Parthenon, also known as the rectangle of the whirling squares.

What has all this got to do with music? Probably not much, because the Golden Section can be seen but not heard, which comes under the German definition of augenmusic — music for the eye.

Still, I think Debussy used it, at least occasionally, as a substitute for traditional musical forms, simply to avoid what Robert Frost called free verse — “playing tennis without a net.” A composer needs limits of some kind, even if merely self-imposed.

The best-known example of the use of a Fibonacci Series, plus the Golden Section, is in the opening bars of “Dialog du vent et la mer” in “La Mer.” The measures of the 55-bar introduction are divided according to a Fibonacci Series, and the trombones enter (about) 0.618 of the total length in.

As a pianist, my favorite example is “La cathedral engloutie.” I could never understand why, when Debussy himself played bars 7-12 and 22-83 at double time, that instruction doesn’t appear in the final printed version. The printed version conforms to the Golden Section; the original manuscript does not.

The Golden Section doesn’t work very well for “The sunken cathedral” and Debussy seems to have abandoned it in his later work. It’s intriguing, however, to think that “La Mer” may have been influenced by the same constructive principle that guides the growth of the chambered nautilus.

Christopher Hyde is a writer and musician who lives in Pownal. He can be reached at: [email protected]