Geometry in music?? Part 2

 El número áureo y la sucesión de Fibonacci

The Fibonacci sequence and the golden ratio are present in musical compositions, in compositional relationships; for example, in the proportion of the different sections into which a piece is divided, as is the case with some Mozart sonatas. Contemporary composers have used this sequence and the golden ratio to compose their works, as in the case of Béla Bartók, who developed the 'Fibonacci scale' and used these mathematical resources to compose his work 'Music for stringed instruments, percussion and sky blue'; Schillinger, who devised a musical composition system where the notes followed in Fibonacci intervals, or even Joan Serra, who in the year 2000 composed an electronic work based on this sequence.

But to apply it in another way, it is much more interesting to see it in the proportions of some instruments than analyzing scores, a task that can be complex.

For example. the number phi 𝜑 was used by Stradivarius to calculate the exact location of the effects of the violins and the distances between different parts of the instrument, thus using geometry. These proportions can be seen in the following illustration:

In the case of the guitar, it is designed based on geometric and aesthetic concepts, it is the result of several ellipses and lines, and on an aesthetic level its design follows a relationship with the human body (Rodríguez, 2011) and therefore, traditionally linked with the divine proportion. However, as Rodríguez (2011) demonstrates in his thesis on the construction of the guitar, this proportion is not fulfilled in certain brands of guitar, but it does come close to another proportion, the Cordovan proportion (ratio between the radius of a circle and the side of the regular inscribed octagon, whose quotient is 1.30656...), used in Cordovan architecture and art.

Fibonacci appears in other instruments, for example, on the piano keyboard: an octave has 8 white and 5 black keys, in total 13 notes, in addition, the pentatonic scale has 5 notes, the diatonic 8 and the chromatic 13. They are all numbers of the Fibonacci sequence. We also find it in the frequency ratio of the intervals: for example, the note Do has a frequency of 264 Hz and La 440, they have a ratio of 3/5, the Mi-Do ratio is 330/528 Hz, which are 5/8, again the relationship is linked to Fibonacci numbers.