Golden Proportions

There are a number of synonyms for this mathematical ratio: Golden Mean, Golden Ratio, Golden Proportion, and Golden Section. It is denoted by the Greek symbol 𝛗 (phi, pronounced fee).

The golden ratio can be found in the structure of the Egyptian pyramids, the balance of the human form in De Vinci’s paintings of the human body, and in nature. The ratio is an irrational number, approximately 0.618, somewhat less than 5/8.

It has a unique mathematically recursive property stemming from the fact that:

1/𝛗 = 1+𝛗.
1-𝛗 = 𝛗²

The Fibonacci number series converges to 𝛗 as it approaches infinity along the progression when taking the penultimate value divided by the last value of the series. This progression was developed in Liber Abaci, which Leonardo Fibonacci of Pisa wrote in 1202. The series is generated by taking the sum of the two final numbers in the series as the next value in the series. The relevant Fibonacci series values I use musically are:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610.

The golden ratio’s recursive feature provides the potential to nest golden sections. That is, one can take the golden section of the golden section, which is the overall length minus the original golden section. Depending on the length of the overall work, one can nest golden sections down to the pulse level of composition.

Use of the golden ratio in music composition has a history going back at least to the 16th century, so my use of it is not new. It is a formal structure, and like any other formal structure, it lends itself as a compositional guide. It can be likened to sonata form, providing a structural framework within which musical material can be composed. It does not provide or suggest content, only structure. It is also possible to superimpose traditional forms (like rondo or sonata allegro) onto the golden ratio. This is explored in my Five Meditations.

I have written over 20 compositions using the Golden Ratio and the Fibonacci series.