# Fibonacci hierarchy

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## Fibonacci hierarchy

Most people are very familiar with the series 0, 1, 1, 2, 3, 5, 8, 13, 21… It's known as the Fibonacci series and examples of it can be found all over nature. Less commonly known are its links to music, despite one of the earliest examples of its use being a rhythm. Have a look at this video:

These are some of the more commonly known visual Fibonacci examples:

- Trees, in the pattern of branches spiraling out of the trunk
- Number of petals on a flower
- Arrangements of sunflower seeds
- Shells (nautilus)
- Hurricanes

The Fibonacci series is named after Leonardo of Pisa (c1170—c1240-50), who was known as Fibonacci. He wrote about them in his book Liber Abaci (published in 1202), but he was not the first person to discover them. Acharya Hamachandra (c1088—c1173) investigated the rhythms that could be created from the syllables used in Sanskrit poetry. Two types of syllables exist – a long syllable and a short one. These could be considered equivalent to the quaver (♪) and crotchet (♩) used in Western musical notation. He found out that the number of possible combinations of short and long notes for an equivalent number of short notes followed the same series (see the table below):

This is not the only link between the Fibonacci series and music. In Western music, there is an important link in the frequency (pitch) of notes that sound the same and therefore have the same name. In the piano keyboard below, the C shown on the right would have twice the frequency of the C shown on the left. There are 13 notes in total, including the two Cs. The distance between each adjacent note (e.g. C to C#, D# to E, or E to F) is called a semitone.

Going back to the keyboard, the 13 notes are divided into eight white notes (C, D, E, F, G, A, B) and five black notes (C#, D#, F#, G# and A#). The black notes are themselves divided into a group of two and a group of three. All these groupings are numbers that can be found in the Fibonacci series.

The notes are combined in different ways as scales. Each scale has the same pattern of semitones and tones (two semitones) between adjacent notes, but starts on a different note on the keyboard. The easiest scale to visualize is C major, which consists of the notes C, D, E, F, G, A, B, C.

There are chords (combinations of notes) associated with each of the scales. The most important of these is the tonic chord (see below). Pieces of music often start and end with this chord, and it often features more than other chords during the piece. The intervals in the tonic chord are 5 semitones, 8 semitones and 13 semitones, numbers that should now be very familiar!

These are all examples of Fibonacci numbers, but there are also examples of the Fibonacci series being used in music. One of the most famous is the xylophone solo in movement 3 of Bartok’s *Music for Strings, Percussion and Celesta*, where the series can be found in the rhythms. The xylophone plays an 11 beat long phrase, all at the same pitch (an F), where the beats are divided into 1,1,2,3,5,8,5,3,2,1 and 1 notes.

There are many other claims made about Bartok’s use of the Fibonacci sequence in his music, including his use of it in the structure of pieces and for the places at which instruments enter. However, Bartok himself never gave explanations for his pieces, and many of the claims about examples of the Fibonacci series have been disputed.

For those of you with a (slightly) more modern taste in music, further examples can be found in one of rock band Tool’s songs. Pauses between words in the first verse of *Lateralus* are deliberately arranged to group the syllables in the counts: *1, 1, 2, 3, 5, 8, 5, 13, 13, 8, 5, 3*. An even more obvious example is in the hip hop duo Black Star’s song *Astronomy*, where the numbers from the Fibonacci sequence feature in the chorus.

Do you have an interesting example of use of the Fibonacci structure in music? You can tweet us @CambridgeMaths, or comment below.