Music and Mathematics: The Fibonacci sequence

fibonacci-numbers

 

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]

From http://www.goacom.com/entertainment/28277-music-and-mathematics-the-fibonacci-sequence

Steinway & Sons Marks A Historic Milestone

fibonaci

Steinway & Sons, manufacturer of the world’s finest pianos, has unveiled the company’s 600,000th piano, stunningly designed by master artisan, Frank Pollaro. Handcrafted using natural Macassar ebony, the end result is a feat of artistry, engineering and precision that celebrates this milestone for Steinway & Sons and demonstrates the company’s ongoing commitment to uncompromising craftsmanship.

Named “The Fibonacci,” the piano’s veneer features the iconic Fibonacci spiral made entirely from six individual logs of Macassar Ebony, creating a fluid design that represents the geometric harmony found in nature. The lines of The Fibonacci spiral on the top of the piano’s lid are projected down to its unique curved base. Synthetic ivory inlay adds a breathtaking effect to the design, which also features unique patinated bronze details. In the end, over 6,000 hours of work over a four-year period were devoted to the creation of The Fibonacci, from design to finish. The superior craftsmanship of the piano’s exterior is matched only by the unparalleled craftsmanship that is at the very core of all Steinway & Sons pianos.

According to Frank Pollaro, “Designing Steinway & Sons’ 600,000th piano was an honor 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.” Pollaro added, “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.”

“Steinway & Sons has a long tradition of world-class craftsmanship, so as we began planning the 600,000th piano, it quickly became apparent that Frank Pollaro was the artist we needed for creative design and perfect execution. His ability to create unforgettable images through woodwork demonstrates artistry at the highest level, something that spoke to the very core of what this company is all about,” stated Darren Marshall, Chief Marketing Officer of Steinway & Sons. Adding, “The Fibonacci spiral is a representation of perfect proportions and natural beauty. Without a doubt, Frank captured those qualities in this piano, creating a work of art for the eyes and the ears.”

The Fibonacci is destined to become an important piece of cultural history, taking its place of honor amongst previous milestone models that the company has created. The 100,000th was originally given to the United States White House and is now part of the permanent collection at the Smithsonian Institute in Washington D.C., while the 300,000 piano currently resides in the East Room of the White House.

In April The Fibonacci was previewed during a special VIP reception at Steinway & Sons’ new showroom in Beverly Hills, where the internationally celebrated Steinway Artist, Lang Lang, had the honor of being the first artist to perform on the piano. This June The Fibonacci will travel to the United Kingdom, where it will be showcased at the 2015 Masterpiece London art fair.

The Fibonacci is a Steinway & Sons Model D piano, nine foot concert grand, priced at $2.4 million. In addition to the original 600,000th version, Steinway & Sons will create up to six exclusive, limited edition Model B pianos (6’10.5″) inspired by the same design.

via Steinway & Sons Marks A Historic Milestone With The Unveiling Of The… — NEW YORK, June 15, 2015 /PRNewswire/ —.