| Emilio Lluis-Puebla |
| Departamento de Matemática |
| Facultad de Ciencias |
| Universidad Nacional Autónoma de México |
| Lluisp@servidor.unam.mx |
Text corresponding to the invited lecture at the Second International Seminar on Mathematical Music Theory, Sauen, Germany. (July 26, 2001)
This paper is intended for a general audience. We explain how mathematics and music are »related«, first the way common people think they are, and then, by a few concrete examples of how these disciplines developed through time, show some of what we know today. A parallel on how music and mathematics are created is explained, and finally a personal point of view of the author is given of why both disciplines should be, first of all, be called fine arts. In particular, a Musical Dice Game by Mozart K. 294 (Anh C) is presented and some of its characteristics are analyzed mathematically. George David Birkhoff’s Aesthetic Measure and its application to music are exposed. The Fibonacci sequence, the golden relation or divine proportion and its application in Bela Bartók’s music is presented. Guerino Mazzola’s Mathematical Music Theory is mentioned, and finally some reflections or thoughts about culture and about the relation between mathematics and music are exposed. |
1 Introduction
It is very common to hear that »there are mathematics in music because when one opens a score this is full with numbers« , that is, the numbers of measures, the fingerings, etc. Obviously, this observation is very simple. Also, it is said »there are mathematics in music« , that »music and mathematics are related« . But are there mathematics in music? Are they related? What relation is there between music and mathematics? I wish to present you certain reflections about these questions and illustrate them with some examples of what some artists or scientists have done about this through the history of mankind.
Leibniz describes music as »an unconscious exercise in arithmetic«. Maybe this assertion could be justified by saying that an interpreter counts the measures of music bars when he starts studying a piece, but that afterwards, he is not counting consciously but lets the magic of music flow. Almost all the »external elements« of music are defined numerically: 12 notes in the scale; bars of 3/4, 7/8,...; 5 lines in the staff; n decibels; half tone of 12th root of 2; frequency of 440 Hz (i.e. chamber
