BACH AND THE ART OF TEMPERAMENT

Abstract

EVEN Bach scholars would be surprised to learn that the great Thomas Cantor had written a book called The Art of Temperament. A theoretical work by an outstanding composer on one of the most controversial subjects of his age! What wouldn't the bibliophiles give for a treasure so rare that, although it is referred to by F. W. Marpurg in his Versuch iiber die musicalische Temperatur, 1776, it is not mentioned in Eitner or any other bibliographical dictionary. But let us not become too excited over the possible find: Marpurg informs us that The Art of Temperament is a collection of preludes and fugues in every major and minor key: in other words, The Well-Tempered Clavier. Evidently the correct title had escaped him and he had very happily christened the work by analogy with Bach's Art of Fugue. Explanations of the of the musical scale are usually so full of figures that the non-mathematician shies away from them in terror. This is unfortunate, for the subject is not only of interest to the musicologist and theorist, but of immediate and practical concern to the performer. (Possible exceptions are the pianist and organist, too often slaves to the vagaries of the professional tuner.) Now, if the figures cannot be avoided completely in a discussion of tuning, their number can be reduced to a small minimum, and that is what is attempted in this article. Those whose knowledge of arithmetic extends to logarithms will find additional material in the appendix. The words tuning and temperament may be used almost synonymously. Strictly, tuning should be used for systems in which the frequencies of the notes in the scale are proportional to integers. That is true of all the ancient Greek systems for the various genera, as presented by Ptolemy.' Among the tunings he gave for the diatonic genus, the most important for us are the ditonic and the syntonic. The former, better known as the Pythagorean tuning, links together the notes of the scale by a succession of pure fifths and oc-