Nicholas ClayCommunicating Mathematics IIISupervisor: Mikhail Belolipetsky

FIBONACCI NUMBERS ANDMUSIC

The Fibonacci NumbersThe Fibonacci numbers are a famous series where each term isthe sum of the previous two:

0, 1, 1, 2, 3, 5, 8, 13, 21, ...

These are defined by the following recurrence relation, whereFn is the nth Fibonacci number

Fn = Fn−2 + Fn−1, n ≥ 2, F0 = 0, F1 = 1

Golden RatioIf we take a line and break it into two pieces in such a waythat the whole line has the same ratio to the larger segment asthe larger segment has to the smaller segment, then the twosegments are said to be in the golden ratio.

When you look at the sequence of ratios Fn+1Fn

of consecu-tive Fibonacci numbers, it converges to 1.618033... as n → ∞,which is also the golden ratio, ϕ.

MusicMusic has a foundation in the series of Fibonacci numbers:

• there are thirteen notes through an octave

• the most common scales in Western music consist ofeight notes

• the third and fifth notes of a scale form the basic foun-dation of chords

• major scales are based on whole tone steps from one noteto another, which is two steps from the previous note

all of which are Fibonacci numbers!

PianoThe layout of the piano keys has an obvious relation to the Fi-bonacci numbers. The white keys form the diatonic scale of Cmajor. The thirteen notes of an octave are split into eight whitekeys and five black keys, which are then split into a group ofthree and a group of two:

BeethovenIn his 5th Symphony, Beethoven appears to have used thegolden mean, although there is no evidence to suggest thatit was intentional. The famous five bar motto from this greatwork appears at the beginning and end of the piece, but is alsopresent at bar 377, out of a total of 610 bars.

610377≈ 1.618 = ϕ

That is, the motto divides the piece into the golden ratio.

Chromatic ScaleThe chromatic scale uses all of the semitones in an octave andforms the foundation of Western music. There are thirteennotes in a chromatic scale through the octave - a Fibonaccinumber!

All of the twelve different notes of the chromatic scale arefound by ascending in fifths and can be done from any startingnote. For example, if we start on C we obtain:

C-G-D-A-E-B-F#-C#-G#-D#-A#-F-C

which when rearranged becomes:

C-C#-D-D#-E-F-F#-G-G#-A-A#-B-C

a chromatic scale on C.

TriadsA major triad is composed of a Root, a Third, and a Fifth andcan be formed by using a recurrence relation similar to that ofthe Fibonacci numbers using note frequencies.

If we start with middle C (264 Hz) we get:

264 + 264 = 528 (C an octave higher)264 + 528 = 792 (G)528 + 729 = 1320 (E)

These three notes, C, E and G form the triad of C major. Al-though the frequencies of the notes obtained are from threedifferent octaves, it can be seen that the C, E and G familiesare related to each other by summations of frequencies.