The science behind Europe's music scale

Abstract

Europe's first music scale was set out by Pythagoras some two and a half thousand years ago. It conformed to the algebraic theorem that two conditions can be satisfied with two variables, for its two sizes of tone were chosen to produce perfect octaves and perfect fifths. Pythagoras also discovered the simple definition of these concordant intervals as the ratios of the lengths of the vibrating strings which produce them, the ratios being 2/1, 3/2, and 4/3 for octave, fifth, and fourth respectively, the last two combining, as he showed, to make an octave. These discoveries quantified the physics of music much as Newton's laws did the physics of motion. Some two thousand years on, musicians began to exploit key changes and other intervals whose concordances had become appreciated, both kinds of exploitation requiring alteration of Pythagoras's tone sizes. Some partial success was obtained by employing the skill which had developed in the logarithmic handling of intervals to produce a tuning system called ‘meantone’, and musicians had to be content with it for about three centuries. It used two such markedly different sizes of semitone that the wrong size in a sequence destroyed a concordance. In consequence, only half the keys played satisfactorily, eventually provoking interest in equal semitones. However, the tuning of equal semitones could only be described in a qualitative way that was not accurate enough in practice. The seventeenth century revolution in science included an enormous advance in the physics of music initiated by Mersenne. He discovered the frequencies of vibration, that Pythagoras's simple ratios apply to frequencies, the physical laws of vibrating strings, that vibrating strings and pipes produce harmonics, and that ‘beats’ are heard when two frequencies are slightly out of unison. ‘Beats’ make possible the accurate mistuning the ear requires of equal semitones. Around AD1700 Werckmeister produced a satisfactory ‘equal temperament’ tuning which played all keys and chords, and soon J. S. Bach had appreciated it and famously demonstrated the new power it gave composers. Since Bach's role was much like that of later inventors who made successful applications of science in technological disciplines, it seems right to regard him as the first in a long line.