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##### November 22, 1997

Circles of Dissonance

Much of today’s music rests upon ancient tradition going back thousands of years to the time of the Greek mathematician and mystic Pythagoras.

The Pythagoreans observed that tones an octave apart are pleasing to the ear. In modern terms, one can say that consonance arises when the frequency of one tone is precisely twice the frequency of the other. Hence, a ratio of 2:1 produces harmonious tones. Similarly, musical intervals involving tones in the ratios of 3:2 (a fifth) and 4:3 (a fourth) are also pleasing.

Out of these ratios, the Pythagoreans and others put together sequences of tones to create various musical scales spanning an octave. The so-called major scale consists of seven notes, which nowadays are designated by the letters C, D, E, F, G, A, and B, or sounded out as do, re, mi, fa, sol, la, and ti. The chromatic scale includes five additional notes -- the sharps and flats (black keys of a piano).

The chromatic scale consists of a progression of half steps from one note to the next, while the major scale is an irregular combination of whole steps and half steps (whole, whole, half, whole, whole, whole, half).

Those steps can be associated with particular numerical ratios, which in turn correspond to ratios of frequencies. Thus, in a system known as just intonation, if C = 1, then D = 9/8, E = 5/4, F = 4/3, G = 3/2, A = 5/3, B = 15/8, and C (one octave higher) = 2. Notice that the scale includes the particularly harmonious Pythagorean ratios.

A violinist playing in the key of C would produce notes with frequencies corresponding to these ratios relative to the frequency of C. However, if the violinist modulates to, say, the key of G (in which the scale starts with G instead of C), then A is no longer 5/3 times the frequency of C, but 9/8 times the frequency of G and, as a result, 27/16 times the frequency of C. Thus, in this scheme, the pitches of the violinist’s notes are not fixed, but vary with the key in which he or she is playing.

Roger Penrose of the Mathematical Institute at the University of Oxford in England has come up with a remarkably accessible, visual way of illuminating this curious aspect of musical scales. Penrose is best known for his theoretical work on black holes, quantum gravity, and aperiodic tilings. He is also a strong advocate of the notion that mathematics is fun and a firm believer in the value of recreational mathematics.

Penrose’s idea is to convert the frequency ratios of the major scale into visual form as points along the circumference of a circle. Because going through one octave is equivalent to going once around the circle, the pattern repeats itself with each successive octave. The result is a kind of pie chart for musical tones.

The position of C is established first, then all other notes are placed according to the following formula, where the angle determines the size of the segment in the pie chart relative to the position of C:

angle = 360 x log (ratio)/log 2.

For example, for D, the ratio is 9/8, and the angle is 360 x log (9/8)/log 2 = 61.18. So the point for D is 61.18° away from the position of C along a circle’s circumference.

 Pie chart representation of the major scale for the key of C.

Suppose one changes to the key of G. The consequences of such a shift can be seen by making a copy of the original circle representation on a transparent plastic sheet, then placing it atop the original, with C on the transparent sheet lined up with G on the original. Every other note on the transparent copy coincides with a note on the original -- except A. The discrepancy is clearly visible.

 Discrepancy (shaded area) between A in the key of C and A in the key of G.

Thus, A tuned to the C major scale would not be in exactly the right place to give true harmony in the key of G. Similar anomalies can be found in other keys simply by looking at various placements of the transparent sheet.

Penrose’s device offers a way for anyone to see the harmony and dissonance that musicians can readily hear. It serves as a marvelous introduction to the mathematical relationships underlying musical scales, keys, intervals, chords, and harmonies and provides a peek into the mysteries of intonation and temperament.

### Meet the Author

References

James, J. 1995. The Music of the Spheres: Music, Science, and the Natural Order of the Universe. New York: Copernicus.

Jeans, J. 1968. Science & Music. New York: Dover.

Levenson, T. 1995. Measure for Measure: A Musical History of Science. New York: Simon & Schuster.

Penrose, R. 1997. The Large, the Small and the Human Mind. Cambridge, England: Cambridge University Press.

Peterson, I. 1997. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.

______. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman.

Rothstein, E. 1995.Emblems of Mind: The Inner Life of Music and Mathematics. New York: Times Books.

Ivars Peterson is the mathematics and physics writer and on-line editor at Science News. He is the author of The Mathematical Tourist, Islands of Truth, Newton’s Clock, Fatal Defect, and *The Jungles of Randomness: A Mathematical Safari. His current works in progress are an updated, 10th anniversary edition of The Mathematical Tourist (to be published in 1998 by W.H. Freeman) and The House at Infinity: Imagination, Mathematics, and Art (to be published in 1999 by Wiley).

*NOW AVAILABLE: The Jungles of Randomness: A Mathematical Safari by Ivars Peterson. New York: Wiley, 1997. ISBN 0-471-16449-6. \$24.95 US.