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 and 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 per cent) of the work. We have seen this to be true in the first movement of

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 per cent 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?[NT]


August 25 ~ Today in Music History



• 1879 ~ New York’s Madison Square Garden displayed a real floating ship in a gigantic water tank as Gilbert and Sullivan’s operetta, H.M.S. Pinafore, was performed.

• 1902 ~ Stefan Wolpe, German-born American composer

• 1909 ~ Ruby (Ethel Hilda) Keeler, Dancer, actress

• 1913 ~ Bob Crosby, Bandleader with The Bob Cats, brother of Bing Crosby

OCMS   1918 ~ Leonard Bernstein, American conductor, composer and pianist
Read quotes by and about Bernstein
Links to more information about Bernstein
Grammy winner

• 1941 ~ Skinnay Ennis and his orchestra recorded the tune Don’t Let Julia Fool Ya.

• 1942 ~ Walter Williams, Singer with The O’Jays

• 1955 ~ Elvis Costello (Declan McManus), Musician, songwriter

• 1961 ~ Billy Ray Cyrus, Singer

• 1964 ~ The Beatles received a gold record for their hit single A Hard Day’s Night. It was the third gold record for the Fab Four. They would collect 18 more through 1970.

• 1971 ~ Ted Lewis passed away.  He was an American entertainer, bandleader, singer, and musician.

• 1982 ~ The group, Fleetwood Mac, received a gold record for the album Mirage.

• 2001 ~ Aaliyah died at the age of 22. She was a R&B singer and budding actress who made her film debut in “Romeo Must Die” and was killed in a plane crash in the Bahamas.

• 2001 ~ Jazz musician John Nelson, the father of pop star Prince, died at the age of 85. Nelson was the model for a character in the 1984 Prince movie “Purple Rain.” He also co-wrote songs on several of his son’s hit albums.
In the 1950s, Nelson was a pianist in the jazz group Prince Rogers Trio featuring singer Mattie Shaw. Shaw and Nelson married, and they named their son Prince Roger Nelson.
Nelson left the household when Prince was about 10 and his sister Tyka was 8. The father and son reconciled after Prince began his climb to fame.
Nelson co-wrote Computer Blue on the Purple Rain album, The Ladder on Around the World in a Day; Christopher Tracy’s Parade and Under the Cherry Moon on Parade and Scandalous on the Batman soundtrack.