Generalized Fibonacci Numbers and Music

Anthony G Shannon,Irina Klamka,Robert Van Gend

doi:10.24297/jam.v14i1.7323

Abstract

Mathematics and music have well documented historical connections. Just as the ordinary Fibonacci numbers have links with the golden ratio, this paper considers generalized Fibonacci numbers developed from generalizations of the golden ratio. It is well known that the Fibonacci sequence of numbers underlie certain musical intervals and compositions but to what extent are these connections accidental or structural, coincidental or natural and do generalized Fibonacci numbers share any of these connections?

Highlights

Mathematics seems to be a language through which scientists can understand some aspects of the nature of the universe, but the extent to which numbers can explain the ephemeral experience of music has yet to be fully explored
It is well known that the Fibonacci sequence of numbers and associated golden ratio underlie certain musical intervals and compositions [1]
Variations of results for the Golden Section include the idea that the Golden Ratio may be considered as the first member of a family which can generate a set of generalized Fibonacci sequences

Summary

INTRODUCTION

Mathematics seems to be a language through which scientists can understand some aspects of the nature of the universe, but the extent to which numbers can explain the ephemeral experience of music has yet to be fully explored. It is well known that the Fibonacci sequence of numbers and associated golden ratio underlie certain musical intervals and compositions [1] Are these connections accidental or structural, coincidental or natural? The mention of ‘fourths’ suggests Lucas numbers might be involved, and so the point of this paper is to ask whether generalized Fibonacci numbers have theoretical connections which are fundamentally related to music. It is beyond the scope of the paper to do more than tease out some of the issues

MATHEMATICS AND MUSIC

FIBONACCI NUMBERS AND THE OCTAVE SCALE

GENERALIZED FIBONACCI NUMBERS

FIBONACCI COMPOSITION FOR PIANO

CONCLUSION