Music and Mathematics: The Fibonacci Sequence
Life very often throws some curious coincidences my way. Just as I was preparing a presentation for architecture students at the Goa College of Architecture on ‘Architecture and Music’ and looking at the relationship of the Fibonacci sequence to music, what should appear in my newsfeed but the announcement of the famed piano firm Steinway and Sons unveiling its 600,000th piano, incorporating the iconic Fibonacci spiral in its design.
The veneer of the “Fibonacci” piano features the eponymous spiral made from six individual logs of Macassar Ebony, “creating a fluid design that represents the geometric harmony found in nature.”
In the words of designer Frank Pollaro, who spent over 6000 work-hours over four years in its creation: “Designing Steinway & Sons’ 600,000th piano was an honour and a challenge. To me, knowing that this piano would become part of history meant that it had to be more than just a beautiful design, but also needed to visually convey a deeper message….As I considered the number 600,000, the Fibonacci spiral came to mind. The way in which it continues to grow but stay true to its form is very much like Steinway & Sons over these many years. Combining the universal languages of music and mathematics suddenly made perfect sense.”
Mind you, 600,000 is not a number in the Fibonacci sequence; I checked. 600,000 is between the 29th and 30th numbers in the Fibonacci series, which are 514,229 and 832,040 respectively. But Pollaro was nevertheless highlighting an interesting relationship between music and mathematics.
Named after the Italian mathematician Leonardo Fibonacci (c. 1170- c. 1250) who brought the Indian-Arabic numeral system to Europe, the Fibonacci series appear in nature and in music, and finds application in architecture and in instrument design, much before the Fibonacci Steinway.
The basic ideas of the Fibonacci progression are contained in the writings of Indian scholar Pingala (300-200 BC) in his treatise on Sanskrit prosody.
The Fibonacci numbers have the following integer sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 and onward. Each added number is the sum of two previous numbers before it.
In nature, the Fibonacci sequence underpins phyllotaxis (arrangement of leaves on a stem), branching in trees, fruit sprouts of a pineapple among many other examples, and even the shape of the human external ear, and the cochlear apparatus of the inner ear.
It can be applied to the western musical scale as well, with the caveat that the starting note one makes the measurement from (or the ‘root’ note) is designated as 1 and not 0. By this token, there are 13 notes in a scale through its octave. There are 8 notes in a diatonic scale (hence the top note is called an ‘oct’ave). The 5th and 3rd notes create the basic foundation of musical chords. All these are Fibonacci numbers.
The very notes in the scale are based on natural harmonics created by ratios of frequencies. Ratios found in the first seven numbers of the Fibonacci series (0, 1, 1, 2, 3, 5, 8) are related to key frequencies of musical notes. Thus if we take an arbitrary frequency of 440 Hz, the root note has a ratio of 1/1, but the octave above it has a frequency of 880 Hz (2/1 of 440); a fifth above has a frequency of 660 Hz (3/2 of 440), and so on for other notes in the scale.
In last Sunday’s article, I had mentioned the “golden proportion” or phi, which underpins the proportions of the Parthenon temple in the Acropolis in Athens, Greece. This “golden ratio” (also called the “golden section”, “golden mean” or the “divine proportion”) of 1:1618 or 0.618 has influenced composition in painting and photography, prompting the notion of dividing a canvas into thirds vertically and horizontally, and to position a subject of interest “about one-third” of the way across instead of in the centre.
This “golden ratio” can be obtained by dividing a Fibonacci number (in the higher reaches, not the first few) by its immediate predecessor. The quotient approximates phi (φ). Thus 987/610= 1.61803, and its inverse is 0.618.
The climax or high point of many songs and other compositions is often found at the ‘phi’ (φ) point (61.8%) of the work. We have seen this to be true in the first movement of J. S. Bach’s G minor sonata for solo violin.
In many compositions in sonata form, the addition of a coda causes the recapitulation (the return of the original idea that started the work) to begin at the 61.8% point.
The legendary violin maker Antonio Stradivari seemed to be aware of the “golden section” and used it in the placement of the f-holes on his violins. The proportions of the violin conform to the ratios of ‘phi’ (φ). The spiral of a violin scroll also obeys the Fibonacci progression.
Isn’t it amazing, how the visual and aural world, indeed Nature itself can all be unified by the same mathematical sequence?
