There is a persistent belief that mathematicians have a particular affinity for music. However, such was not borne out in a quick poll taken among mathematicians at a recent conference, which suggested that mathematicians are no more musically inclined than, say, doctors or lawyers. However, what is true is that there is a close relationship between the two subjects, mathematics and music.

Over 2500 years ago, Pythagoras understood that two notes played simultaneously on stretched strings have a particularly pleasing sound when their lengths obey a simple mathematical relationship. For example, if one string is twice as long as the other (and they are otherwise identical as to material and tension), the shorter string sounds exactly one octave higher than the longer one (and as we now know, the frequency of the higher tone-the number of vibrations per second-is exactly twice that of the lower tone). In the case of a perfect fifth, the relationship is 2 to 3. The Pythagoreans built an entire musical scale on this principle, but it remains hidden to us what brings about the connection-to be found in all cultures-between simple mathematical relationships and the pleasures of music.

Unfortunately, the Pythagorean scale and related musical systems possess a decided drawback: when one attempts to modulate from one key to another, thereby setting a different note as the fundamental tone of the scale, the mathematical relationships in the new scale are no longer quite the same as in the old one.

Out of this difficulty was born the idea of dividing the octave democratically into twelve equal parts. From one semitone to the next, the frequency increases by the twelfth root of two, thus by a factor of 1.059463094…. It was over three hundred years ago that the equal-tempered scale was developed, and in his Well-Tempered Clavier, Johann Sebastian Bach (1685-1750) demonstrated-by presenting a collection of pieces (preludes and fugues) written in each of the twenty-four major and minor keys-that one could play in any key without having to retune the instrument.

This development by no means exhausted the possibilities of relating mathematics to music. In the twentieth century, many composers used a variety of mathematical relationships in their compositions, from the method of tuning to the large-scale compositional form. For example, the composer Iannis Xenakis (1922-2001) used probabilistic methods, game theory, and group theory as organizing principles in his compositions.

However, no matter how high a value is placed on mathematics, it will never be possible to understand our enjoyment of a Schubert sonata or our favorite pop song in terms of a mathematical formula.

Pythagorean versus Chromatic

Why did the twelfth root of two pop up in our discussion of equal temperament? Suppose that the octave is to be divided into n parts, where n is any positive integer. A guitar builder would then have to supply n frets up to the middle of the fingerboard, where the last fret would be exactly in the middle to sound the octave. See Figure . If all the musical intervals are to be of the same size, then the frequency relationship between the note on the first fret and that on the open string must be the same as that between the second fret and the first, and so on.

If this frequency relationship is denoted by x, then the calculations are straightforward. If two notes are played simultaneously (say on two identically tuned guitars) that are k semitones apart, then the frequency relationship is xk. In particular, the nth note, which should sound the octave, must have frequency twice that of the lower tone, and thus satisfy xn=2. In equal temperament, we have n=12, which leads to the equation x12=2, or x=12√{2}=1.0594….

 

  
Instruments with equal temperament.

 

Thus from C# to C the frequency ratio is 1.059…, and the same ratio holds for D to C#, and so on. One can also calculate the frequency ratio between any pair of notes. For example, the ratio from D to C is calculated as

DtoC= DtoC#timesC#toC= 1.0594…·1.0594… = 1.12246….

The following table shows the frequency relationships for the phythagorean and equal-tempered C-major scales:

 

C	1	1
D	1.12500	1.12246
E	1.26563	1.25992
F	1.33333	1.33484
G	1,50000	1.49831
A	1.68750	1.68179
B	1.89844	1.88775
C	2	2

The frequency ratios are almost identical, and untrained ears will hardly detect a difference. In popular music, equal temperament is almost universal, but when music is played on period instruments from earlier times, the performers frequently attempt to make the music sound as it did at the time of composition.

This is an article from the book "Five-minute mathematics" by Ehrhard Behrends which was published in 2008 by the American Mathematical Society (AMS). It is reproduced here with the kind permission of the AMS.