A crucial link between mathematics and music is nicely illustrated by the Tonnetz, a geometric diagram representing the harmonic relationships between the notes of the musical scale.
An early version of the Tonnetz appeared in Leonhard Euler’s book Tentamen Novae Theoriae Musicae (a new theory of music), published in 1739. A modern development of the diagram appears in the accompanying illustration. Closely related notes are grouped together: the C major triad, C–E–G, appears as a dark red triangle. The relative minor chord, A minor, is in dark blue beside it.
The Pythagoreans observed that two consonant notes are related by small whole numbers. Inspired by the sonorous ringing of anvils struck by hammers, Pythagoras went on to show there are simple numerical relationships between harmonically related notes. By halving the length of a vibrating string, he could double its pitch. Reducing the length by one third produces a perfect fifth.
The problem in tuning is how to divide the octave into smaller, musically pleasing steps. Pythagoras produced the seven notes of a diatonic scale (the familiar do-re-mi) using only the ratios 2:1 and 3:2. However, as intervals are combined, the numbers in the ratios soon grow large.
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A simplification of this scheme leads to a tuning method – “just intonation” – that involves smaller ratios and is more harmonious. In just intonation, the three notes of the C major triad have pitches in the ratios 4:5:6.
A serious shortcoming of just intonation is that, when the notes are tuned for one key, they sound discordant in another. As music developed, with multiple changes of key, the defects of this tuning method became evident and the more flexible “well-tempered” tuning scheme was devised. This ensures the ratio of pitch between every two adjacent notes is precisely the same. With 12 tones in the chromatic scale, the common ratio is the 12th root of two.
Johann Sebastian Bach wrote a set of 24 preludes and fugues, one for each of the major and minor keys, to demonstrate the power and versatility of the well-tempered tuning system.
However, equal-tempered tuning is a compromise: although the notes are in perfect geometric progression, intervals like fifths are no longer perfect. The deviations are so small that the method has become almost universal, although a purist has remarked “it makes all intervals equally imperfect”. In truth, the equal-tempered scale is a substantial gift of mathematics to music.
Efforts have been made to compose music using strict mathematical principles and to break free from the constraints of the tonal system. Perhaps the most extreme case of this was the serial music of Arnold Schoenberg, who insisted that each of the 12 notes of the chromatic scale should have equal status and prominence.
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Schoenberg dismantled the tonal foundation of classical music. Mark Twain was not around to hear 12-tone music, but his description of Wagner might well be applied to Schoenberg: “I am told his music is much better than it sounds!” Einstein went further, calling it crazy.
Dodecaphonic music could be called a heroic failure: today, 12-tone music is more written about than listened to. Music stirs our emotions, and mathematical principles alone do not suffice. Leonard Bernstein said that while music is born of science, it is a mysterious and metaphorical art, and “any explication of music must combine mathematics with aesthetics”.
Peter Lynch is emeritus professor at the School of Mathematics & Statistics, University College Dublin. He blogs at thatsmaths.com
