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 a violinist's notes are not fixed, but vary with the key in which he or she is playing at the moment.

Roger Penrose has come up with a visual, remarkably accessible 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 just 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 ✕ log (ratio)/log 2. 

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

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

Suppose you change 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 according 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 a peek into the mysteries of intonation and temperament.

Originally posted November 24, 1997