FIBONACCI

By Judith Eckelmeyer

(GRACE WOODS MUSIC SESSION APRIL 29, 2024)

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Statue of Fibonacci (1863) by Giovanni Paganucci in the Camposanto di Pisa

​The man known today as Fibonacci was born in Pisa about 1170 and died there about 1250. In his lifetime he was known as Lionardo (sic) Fibonacci (first used in 1506), Leonardo Bonacci, Leonardo Bigollo Pisano (Leonardo the wanderer of Pisa), and Leonardo Pisano (Leonardo the Pisan); the name Fibonacci was created in 1838 by an Italian historian from the term filius Bonacci, meaning the son of Bonacci. 

Fibonacci is generally not a household name in our time, but he was a major mathematician, one of the most important contributors to today’s Western culture. His dad, Guglielmo, was a merchant and customs official working in what we know as Algeria, and in his youth Leonardo travelled with him and received his education in Algeria. Of course, Arabian scholars were his teachers. From them he learned the Arabic numbering system, which the Arabs had learned from Indian Hindu scholars; this system later  replaced the Roman numeral system then in use, which was very limited in its functions. With the Hindu-Arabic system, calculation was immeasurably easier: merchants and banks had the capacity to calculate advanced banking and accounting throughout Europe. Fibonacci’s 1202 book, Liber Abaci (The Book of Calculation), the first publication of the revolutionary number system outside India, became widely known in Europe, popularizing the Hindu-Arabic number system; it eventually made calculus possible.

A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled with Latin numbers and Roman numerals and the value in Hindu-Arabic numerals

The so-called Fibonacci sequence of numbers refers to a pattern in which a number is the product of the two preceding numbers: 1 + 1 = 2; 2 + 1 = 3; 3 + 2 = 5; 5 + 3 = 8, etc. The sequence is very old, having been described by Indian mathematicians in 200bc, based on syllable patterns in Sanscrit poetry. (Wow! Who knew???) The pattern could also be construed visually, showing proportional relationships: the smaller is to the larger as the larger is to the whole. The point at which the two meet is called the Golden Section, or Golden Mean, or Golden Ratio, represented by the Greek letter Phi (F), easy to remember as it sounds like F, as in Fibonacci.

The Golden Section has been employed by artists and composers through the ages. I won’t delve into the world of art and architecture, but let’s look at music. Three composer who definitely used this proportion are Franz Schubert, Claude Debussy, and Béla Bartók (1881-1945). To figure out the GS (F) in Schubert’s and Debussy’s music one must count measures and multiply that number by .618, the numerical point of F. We will consider this as we hear two works: Schubert’s Lied (art song) Der Tod und das Mädchen (Death and the Maiden), D532 of 1817 

Franz Schubert, composer

Original manuscript of "Death and the Maiden"

Schubert's Der Tod und das Mädchen, D. 531 (Op. 7/3
Dietrich Fischer-Dieskau · Gerald Moore

and Debussy’s Nuages (Clouds, 1899), the first tone poem in Nocturnes.

Claude Debussy c. 1900 by Atelier Nadar

Debussy's La Mer and Nocturnes
​Orchestra of Radio Luxembourg - Louis De Froment

In both cases, at the measure equivalent of .618 there is a significant change in sound. We’ll listen for that.

How did Schubert and Debussy learn of the Golden Section?
Schubert’s father taught math at the school where he worked, and Debussy, interested in nature, symbolism, esotericism, and impressionism, hung out with visual artists who were working in those spheres and employed the technique.

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On another front, a different kind of orchestral music was being written by a group of amateurs in Russia; they were nicknamed the Russian Mighty Five, or the Russian Mighty Handful. They were all concerned with creating a national style of classical music, bowing out of the Western traditions of French and German composers of the time. Mily Balakirev was their leader; the others were Alexander Borodin, César Cui, Modest Mussorgsky and Nikolai Rimsky-Korsakov. (Their contemporary, Tchaikovsky, was not included in the Five as he was conservatory-trained, not an amateur.) Each had come to music “sideways” from other careers. Rimsky-Korsakov was trained in the Russian naval academy, but “found” classical music through his piano teacher who took him to concerts. ​

Portrait of Balakirev, c. 1900

Borodin, c. 1865

César Cui, 1910

Mussorgsky, c. 1870

Portrait of Rimsky-Korsakov in 1898 by Valentin Serov (detail)

As one might expect, the Five drew on themes and music authentic to Russia’s culture, but they also looked to Arabia and the Orient. In reflecting these various cultures, the composers employed their own versions of what they thought would capture the essence of the culture. Their musical treatments were usually imaginative rather that authentic, but the exoticism, beauty, and energy of their renditions enraptured the audiences. Rimsky-Korsakov, in particular, was a brilliant orchestrator, which enhanced the attractiveness and popularity of his works; but others, such as Mussorgsky, were less successful at orchestral colorism but achieved other unique features. Mussorgsky, for example, was very inventive in setting Russian text using the varying rhythms of the language realistically.

In the first 40 or so years of his life, Rimsky-Korsakov wrote settings and arrangements of Russian folk songs and Western-style instrumental works. Scheherezade was one of the last compositions of this phase of his career. Written in 1888, it is termed a “symphonic suite” in 4 movements—essentially four symphonic poems rendering tales from Arabian Nights (1001 Nights) which was long known in Europe. These colorful and varied Middle-Eastern tales—based on both actual people and folk tales--were taken from a wide-spread area from North Africa to South and Central Asia over many centuries. Rimsky-Korsakov “framed” the four movements in a story about Shahryar, a king who was betrayed by his unfaithful wife and avenged himself by taking a new wife each night and then executing her. Scheherezade, however, kept the king on tenterhooks each night with perhaps the first cliffhanger serial, and so endured 1001 nights, after which the king spared her. The musical “frame” describes Scheherezade’s story-telling with a haunting violin solo at beginning and end of the suite. The four stories are: 1)The Sea and Sinbad’s ship; 2)The story of the Kalandar prince; 3)The young prince and the young princess; and 4)The festival at Baghdad—the sea—and the shipwreck.

Orquesta Sinfónica de Galicia Leif Segerstam, director Slava Chestiglazov, concertino ​

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The following has lots of ways to understand the Fibonacci series of numbers; so much of it is technical—take what makes sense to you!

A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21
In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ....[1]

The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.[2][3][4] They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.[5]
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, and the arrangement of a pine cone's bracts, though they do not occur in all species.

Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences.

​The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling (see preceding image.)

Judith Eckelmeyer ©2024