1/8, and if we multiply 46 by 0.618..., the unit 28 is obtained, where the inverted motive begins. The analysis can be continued, and if we call positive the long portion and negative the short portion it can be said that a symmetric relationship exists among the positive and negative parts. This process goes accompanied with an increment in the pp dynamics to f or ff in the positive section and the negative goes accompanied by a decrease of the sound intensity. The whole work can be divided in slow-fast+slow-fast parts within the movements. The golden section should appear at the beginning of the second movement which happens if it is considered the total of the 6432 eighths that when multiplied by 0.618... gives the eighth 3975 that is exactly in the one where the second movement begins.

If we compare the Fibonacci sequence with the fugue (first movement) of the »Music for Strings, Percussion and Celesta« we can observe that the bars of the movement are divided in sections of and bars. These sections are subdivided respectively in sections of bars and bars. The climax in fff occurs in bar , and in the ends it begins and it concludes in pp. It is not by chance that the exposition concludes in bar , and that the last bars are divided in sections of bars.

The »Allegro Barbaro« (Bartok, 1918) is another composition for piano solo in which Bartók uses the Fibonacci numbers 2, 3, 5, 8, and 13 in several occasions, contrary to traditional music, which uses 8 bars in almost all themes and multiples of 2 in the motives and phrases. He also uses his circle of neighboring tonalities and the duration of the piece is of 3 minutes.

Bartók wrote that he followed nature in the act of composing and that he was guided indirectly by natural phenomena to discover these regularities. He constantly increased his collection of plants, insects and mineral specimens. The sunflower was his favorite plant, he was very happy when he found fir pineapples in his desk. He considered that folkloric music was also a phenomenon of nature and that its formations were developed as spontaneously as other living organisms: the flowers, the animals, etc. For this reason his music reminds some listeners of natural scenes. For example, the sunflower has 34 petals and its hairsprings have the values 21, 34, 55, 89,144.

Bartók’s use of chords is also based on the numbers of Fibonacci. For example, in semitones, 2 is a major second, 3 is a minor third, 5 is a fourth, 8 is a minor sixth and 13 is an augmented octave, etc. When Bartók uses chords in a chromatic movement, he places the minor third over the perfect fourth in such a way that the chord acquires the form 8:5:3 and considering a minor third, superimposing a fourth followed by another minor third, his major-minor characteristic chord is obtained. It would seem for Bartók that the golden section is not an external restriction but one of the most intrinsic laws in music.

5 Mazzola’s Mathematical Music Theory.

I want to mention very briefly, because it is thousands of pages long, one of the most interesting projects that is being developed. I refer to the Mathematical Music Theory of Guerino Mazzola. I have taken parts of his articles and books to give a rough idea of it.