Does the Golden Ratio appear in music?

Answer 1

Clearly, the Golden Ratio is used in music. Note the following quote:

Professor Sever Tipei sent this in response to a student who asked about the Golden Ratio in music:

The golden mean ratio can be found in many compositions mainly because it is a "natural" way of dealing with divisions of time. One can find it in a lot of works by Mozart, Beethoven, Chopin, etc., etc. It is a question if it was used in a deliberate way or just intuitively (probably intuitively). On the other hand, composers like Debussy and Bartok have made a conscious attempt to use this ratio and the Fibonacci series of numbers which produces a similar effect (adjacent members of the series give ratios getting closer and closer to the golden mean ratio). Bartok intentionally writes melodies which contain only intervals whose sizes can be expressed in Fibonacci numbers of semitones. He also divides the formal sections of some of his pieces in ratios corresponding to the golden mean. Without going into much detail, Debussy also does this in some of his music and so does Xenakis (a composer who writes exclusively by using stochastic distributions, set theory, game theory, random walks, etc.) in his first major work, "Metastasis". The idea is not new, already in the Renaissance composers used it and built melodic lines around the Fibonacci sequence -just like Bartok's "Music for strings, percussion and celesta".

THE GOLDEN RATIO AND SEMITONES

A semitone is the smallest change of pitch (frequency) used in conventional music and musical notation. Semitones correspond with the individual keys of a tuned piano. A semitone is defined as the "12th root of 2" (1.05946...).

The Golden Ratio (1.61803...) when cubed, almost exactly equals 25 semitones. Therefore the 25th root of a cubed golden ratio (1.05945...) equals 0.99998... semitones. This difference, when accumulated over the entire range of human hearing (about 120 semitones) is less than 3.5/100 of one semitone, and is undetectable by the human ear.