A sequence of numbers in which each is the sum of the previous two, thus: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 etc. It is found in nature (in numbers of petals in single flowers, for instance, or in the proportions of snail shells); it has also been used by composers to govern rhythms and forms (see Numbers and music).
The series was first described by Leonardo da Pisa (Liber abaci, 1202) as the successive population sizes of pairs of rabbits breeding each month from one parent pair. The ratio of successive numbers is an arithmetical expression of Euclid’s geometrical division into extreme and mean ratio (the ‘golden ratio’; see Golden number), but da Pisa was unaware of this. The first written connection between Fibonacci’s series and Euclid’s ratio appears in a handwritten comment in a copy of Pacioli’s 1509 edition of Euclid’s Elements. The mathematician Johannes Kepler also demonstrated a connection between the two in a letter of 12 May 1608 to Professor Joachim Tanckius.
It has been claimed that composers have used Fibonacci numbers in musical compositions as a deliberate attempt to reproduce the golden ratio. While this is undoubtedly the case in certain 20th-century compositions, it appears to be a historical impossibility for earlier composers. The first attested use of the term ‘golden number’ in a strictly mathematical context was by Martin Ohm in 1835, and it was not until 1843 that the explicit expression for fn in terms of G was published by J.P.M. Binet. Thus any composer using Fibonacci numbers before then would not have done so with the ‘golden number’ in mind. It could be argued that ‘naturally occurring’ Fibonacci sequences appear in compositions written before 1843, but in such cases the musicologist must maintain a clear distinction between an interpretation imposed on the composition and the composer’s conscious intention.
For bibliography see Numbers and music.
RUTH TATLOW