David Wright, Department of Mathematics, Washington University
Mathematics and Music
Abstract: Many people intuitively sense that there is a connection between mathematics and music. If nothing else, both involve counting. There is actually much more to the association. Mathematics has been observed to be the most abstract of the sciences, music the most abstract of the arts. Mathematics attempts to understand conceptual and logical truth and appreciates the intrinsic beauty of such. Music evokes mood and emotion by the audio medium of tones and rhythms without appealing to circumstantial means of eliciting such innate human reactions. It is therefore not surprising that the symbiosis of the two disciplines is an age old story. The Greek mathematician Pythagoras noted the integral relationships between frequencies of musical tones in a consonant interval; the 18th century musician J. S. Bach studied the mathematical problem of finding a practical way to tune keyboard instruments. In today's world it is not at all unusual to encounter individuals who have significant interest in both subjects. In this talk, several musical and mathematical notions will be brought together, such as scales/modular arithmetic, octave identification/equivalence relation, intervals/logarithms, equal temperament/exponents, overtones/integers, tone/trigonometry, timbre/harmonic analysis, tuning/rationality. Music concepts to be discussed include scales, intervals, rhythm, meter, melody, chords, progressions, note classes, overtones, timbre, formants, equal temperament, and just intonation. Mathematical concepts covered include integers, rational and real numbers, equivalence relations, geometric transformations, modular arithmetic, logarithms, exponentials, and periodic functions. Each of these notions enters the scene because it is involved in one way or another with a point where mathematics and music converge. A number of musical and sound examples will be played to demonstrate concepts.
