Fibonacci’s sequence and the Golden Ratio

The Fibonacci’s sequence is a numerical succession in which every number after the first two is the sum of the two preceding ones: 1, 1, 3, 5, 8, 13, 21, 34, etc. This mathematical pattern appears to be strictly related with the concept of section aurea or golden ratio (two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities). The golden ratio seems to be a mathematical pattern that rules the symmetry of the natural world: we can find it in leaves that grows on trees, the geometric formation of snowflakes, the spirals in pinecones and even in the dynamics of black holes and the shape of galaxies. Many artists throughout the history used the golden ratio to craft their pieces; one form of art in which the golden ratio is very present is music.

Many composers applied this pattern to their composition with a sublime result; one great example is the composition “Apparition” by György Ligeti, that is divided into sections that are proportional to the golden ratio. Even the harmonics of frequencies seem to be organised by the Fibonacci’s sequence. Another interesting musical fact is the arrangement of the keys on a piano: in a keyboard scale of C to C there are 13 notes, with 8 white keys and 5 black keys arranged into groups of 3 and 2; also in an octave there are 13 notes but the scale is composed just by 8 notes in which the 5th and the 3rd notes create the basic foundation of all chords.

The golden ratio is connected to the golden number (1,6180339885…), and endless and irrational figure that doesn’t represent a periodic repetition. In a musical composition that goes from A to B the element F (golden ratio) would represent a a change in the work (a bridge, or an arrangement with a different instrument,…). Also, many musicians used the Fibonacci’s sequence to enhance rhythmical patterns or to establish changes.

One of the most brilliant use of this mathematical patterns in music is given by the song Lateralus performed by the visionary American band, Tool; the syllables of the lyrics are clearly set into Fibonacci’s pattern growing from 1 to 8 and in the 2nd verse from 13 to 3; also the intro of the song finishes at the minute 1:37 that appears to be the approximation of the golden number converted into minute and seconds (1,618- 1 becomes 1 minute+ 0.618ths of a minute= approximately 1 minute and 37 seconds).

Here’s the lyrics of the song’s verses organised in Fibonacci’s sequence:

1 black

1 then

2 white are

3 all I see

5 in my in-fan-cy

8 red and ye-llow that came to be

5 rea-ching out to me

3 let me see

13 as be-low so a-bove and be-yond, I i-magine

8 drawn be-yond the lines of rea-son

5 push the enve-lope

3 watch it bend


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