Beauty and Tragedy In the Mathematics of Music

Some things are too good to be true. Others are too good not to be true. The elegant correspondence between musical consonance and simple mathematical ratios is too good not to be true. The infinitely creative system of musical harmony—based on a small set of canonical intervals and chordsarises out of this correspondence.

And yet the system has a flaw—a rip or tear in the musico-mathematical universe.

The most consonant musical interval is the octave, so much so that two tones an octave apart bear the same letter name, for example, A. The next most consonant interval is the fifth (for example, A to E). If you pluck the A string on a guitar and then pluck it again while placing your finger halfway up the fingerboard (allowing only half the string to vibrate and doubling the frequency), the pitch goes up by an octave to A. If instead you place your finger one-third the way up the fingerboard (allowing only the remaining two-thirds of the string to vibrate and increasing the frequency by a factor of 3/2) the pitch goes up by a fifth to E.

On a full size piano keyboard, if you play the lowest note, A, and then go up by twelve successive fifths, you reach A again—which happens to be seven octaves higher than the lowest A. Calculating by fifths, the highest A is (3/2)12times the frequency of the lowest A. Calculating by octaves, the highest A is 27 times the frequency of the lowest A.

It should therefore be the case that:

(3/2)12 = 27

Tragically, this is not the case: (3/2)12is 129.7463, while 27 is 128. Close but unequal. If you tune by octaves, the fifths are out-of-tune, and vice versa.

This discrepancy—a variant of the Pythagorian comma—shatters an edifice that rests on the beautiful principle that ratio simplicity underpins musical consonance. While each interval is most purely tuned using these simple ratios (the just tuning system: 2/1 = octave, 3/2 = perfect fifth, 4/3 = perfect fourth, 5/4 = major third and 6/5 = minor third), the mathematics falls apart when these intervals interact in music.

The Pythagorian comma has haunted string players for centuries, requiring annoying adjustments. On the violin, some notes must be played slightly sharper and others flatter to dodge dissonance. The comma held back the development of keyboard instruments, in which notes cannot be adjusted while playing, thus limiting a composition to the musical key for which the instrument was tuned.

The solution to this problem is to forego mathematical purity by fudging tuning. This discovery—which emerged in various forms over a period of centuries—unleashed new creative possibilities and inspired J.S. Bach to write The Well Tempered Clavier, which traverses all major and minor keys without discriminating on the basis of dissonance.

Today, several tuning systems achieve this end. In equal tempered tuning (used in some electronic synthesizers), the octave is divided into twelve equal semitones on a logarithmic scale. In this system, each successive semitone is 21/12 times the frequency of the previous one. Thus a fifth is 27/12 or 1.498 instead of 3/2 or 1.5; the perfect fourth is 25/12 or 1.3348 instead of 4/3 or 1.3333; the major third is 24/12 or 1.2599 instead of 5/4 or 1.25; and the minor third is 23/12 or 1.1892 instead of 6/5 or 1.2.

By anchoring the octaves and sprinkling the error elsewhere, consonance is compromised slightly but never disturbingly. The edifice of musical harmony is preserved by systematically fudging the very mathematical system upon which it is built. An elegant way to fix inelegance in an elegant musico-mathematical system.