## Golden Ratio and Fibonacci numbers

Phi Page
Golden Section Ratio

Number Games

Construction by Golden ratio

Egypt: Cheops' pyramid

Alta Vista Golden Ratio

Lycos Fibonacci

Fibonacci and the Golden Section

Fibonacci and Golden Section in Nature

Lycos Golden Ratio

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".

See also Erno Lendvai *Bela Bartok : Analysis of His Music*

Dr. Sever Tipei, Professor of Music

Manager, Computer Music Project of the
University of Illinois Experimental Music Studios

Urbana, Illinois 61801, USA

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