Golden Ratio

Golden Ratio

  • 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

    send me mail