Friday, July 3, 2009

The Fibonacci Sequence in Music - is music mathematical?

The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 . . .) occurs throughout the worlds of nature, art, music, and mathematics!

Portrait of FibonacciEach term in the series is produced by adding together the two previous terms, so that 1 + 1=2, 1 + 2=3, 2 + 3=5, and so on. The sequence takes its name from a famous thirteenth-century European mathematician, Leonard of Pisa (?1170-1250), also called Fibonacci. Fibonacci was one of the first Europeans to use Arabic numbers, whose use he explained in his 1202 Liber abaci.

The basic structures of certain instruments display the use of Fibonacci numbers and the Golden section. The most widely used instrument in music, the piano, displays the use of Fibonacci numbers. For instance, there are 13 notes that separate each octave of 8 notes in a scale. The foundation of a scale is based around the 3rd and the 5th tones. Both pitches are whole tones, which are 2 steps from the 1st note of the scale, also called the root.

Piano

The keys of a piano also portray the Fibonacci numbers. Within the scale consisting of 13 keys, 8 of them are white, 5 are black, which are split into groups of 3 and 2. Look familiar? Well, it should, it's Fibonacci!The keys of a piano also portray the Fibonacci numbers. Within the scale consisting of 13 keys, 8 of them are white, 5 are black, which are split into groups of 3 and 2. Look familiar? Well, it should, it's Fibonacci!

Treble clef

Not only the piano, but also the violin is constructed through the use to the golden section. Check out the link for the information in Russian:

ViolinThe proportions of the violin conform to the ratios of the golden section or the Fibonacci sequence.
The Fibonacci sequence can also display the preference of the human ear to music. The following is some Fibonacci music. It consists of the first eight Fibonacci numbers. For each new number that is performed, the note length is decreased rotationally by 1/2 or 1/3. After four steps of the sequence are completed the tune starts over at the root, one octave up, while the other one continues, so there is an overlapping effect.

Fibonacci numbers occur many times in the natural world. Plants tend to have a number of leaves that is a Fibonacci number, and flowers have a Fibonacci number of petals. Seeds in a flower head are often arranged in spiral patterns that are related to Fibonacci numbers (for example, the number of spirals that curve to the left and the number of spirals that curve to the right will be adjacent numbers in the Fibonacci sequence). Spiral shells also exhibit patterns related to the Fibonacci sequence.
Fibonacci numbers are also important in art and music. The ratio between successive Fibonacci numbers approximates an important constant called "the golden mean" or sometimes phi,which is approximately 1.61803. The higher you go in the Fibonacci sequence, the more closely the ratio between two successive numbers in the sequence approximates phi. (By the way phi2=phi + 1!)

A golden rectangle, with line dividing its longer side in proportion 1 : .61803 (golden section)A rectangle whose sides are in the proportion 1 : 1.61803 is supposed to be the most aesthetically perfect rectangle (the "golden rectangle"). The Parthenon in Athens has such a rectangle as its face, and phi is said to have figured in the construction of the Great Pyramids. The "golden section," in which a line is divided into segments of lengths in the ratio 1 : .61803 is supposed to be an aesthetically ideal way to divide a line.

Numerous artists have used the golden section in their works, as well as composers, including Bach, Mozart, Beethoven, Schubert, Debussy, Satie and the Hungarian Bela Bartok.

Fibonacci Fingers?

Look at your own hand:

handYou have ...
  • 2 hands each of which has ...
  • 5 fingers, each of which has ...
  • 3 parts separated by ...
  • 2 knuckles

Is this just a coincidence or not?????

However, if you measure the lengths of the bones in your finger (best seen by slightly bending the finger) does it look as if the ratio of the longest bone in a finger to the middle bone is Phi?
What about the ratio of the middle bone to the shortest bone (at the end of the finger) - Phi again?
Can you find any ratios in the lengths of the fingers that looks like Phi? ---or does it look as if it could be any other similar ratio also?

Why not measure your friends' hands and gather some statistics?

For some amazing pictures of examples of the Fibonacci series in nature please visit these sites:

The Golden Section Illustrated in Arts, Architecture and Music

Fibonacci Sequence Illustrated by Nature

Fibonacci series slides of Petals and Flowers

Fibonacci Numbers and Nature

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