Music and Mathematics - The Golden Ratio and Fibonacci Numbers

The Golden Ratio and Fibonacci Numbers

James Tenney reconceived his piece "For Ann (Rising)", which consists of up to twelve computer-generated tones that glissando upwards (see Shepard tone), as having each tone start so each is the golden ratio (in between an equal-tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.

Ernő Lendvaï analyzes Béla Bartók's works as being based on two opposing systems: those of the golden ratio and the acoustic scale. In Bartok's Music for Strings, Percussion, and Celesta, the xylophone progression at the beginning of the 3rd movement occurs at the intervals 1:2:3:5:8:5:3:2:1. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose Croix.

The golden ratio is also apparent in the organization of the sections in the music of Debussy's Image, "Reflections in Water", in which the sequence of keys is marked out by the intervals 34, 21, 13, and 8 (a descending Fibonacci sequence), and the main climax sits at the φ position. "Prelude to the Afternoon of a Faun" also reaches a climax point, marked by the entrance of the antique cymbal, at the φ position.

Many of the important musical events in Krzysztof Penderecki's "Threnody for the Victims of Hiroshima" occur at φ positions.

Boards of Canada have discussed using the golden ratio and Fibonacci numbers in their work. Song titles such as "Music is Math", and numerous songs featuring samples of people counting or discussing numbers, also illustrate the influence of mathematics on their music.