. . . The . . . Fourth . . . its apparent value . . . consists of two and a half tones. [Aristoxenus, The Harmonics of Aristoxenus, translated by Henry S. Macran, M.A., Oxford At The Clarendon Press, Henry Frowde, M.A., Publisher to the University of Oxford, New York, 1902, p182]

. . . there is a problem about what exactly is meant by a 'tone'. The Greek writers define it as the interval by which a fifth is greater than a fourth. Strictly speaking, that is the interval given by the ratio 9 : 8, or 204 cents. But Aristoxenus regards it as being at the same time a unit of which a fourth (properly 498 cents) contains exactly two and a half. In effect he is operating with a tempered tone of 200 cents and a tempered fourth of 500 cents. . . . [West, M.L., Ancient Greek Music, Clarendon Press, Oxford, 1994, p167]

72T-ET Keyboard #0 Showing Harmonic Limit <32/19 With Errors <4.9¢


18/17 17/16



19/16 25/21



17/12 24/17



19/12 27/17



16/9 25/14





9/8 28/25


24/19 29/23










Willi Appel, Harvard Dictionary of Music, Harvard University Press, Cambridge, MA 1964

III. Equal Temperament. The principle of equal temperament is to divide the octave into twelve equal semitones. [p735]

Temperament [G. Temperatur ]. The term denotes those systems of tuning in which the intervals deviate from the "pure," i.e., acoustically correct intervals as used in the Pythagorean system and in Just intonation. . . . It follows that compromise methods are necessary which, instead of being perfect in the simple keys and intolerably wrong in the others, spread the inevitable inaccuracy over all the tones and keys. The most consistent realization of this principle is the equal temperament which is universally used today. . . . [p734]

In equal temperament no interval other than the octave is acoustically correct or pure. The deviation of the fifth (2 cents) is too small to be noticed at all. With the thirds, the difference is considerably greater, the well-tempered third (400 cents) being 14 cents (one-eighth of a semitone) larger than the pure third (386 cents). However, our ear has become completely accustomed to this "error," and the advantages of the system far outweigh its flaws. [p735]

The history of equal temperament can be traced back to 1518, when H. Grammateus recommended dividing the octave into 10 equal semitones and two of somewhat smaller size. V. Galilei, in his Dialogo (1581), proposed to use a semitone of the frequency 18/17 (99.3 cents) which is a very good approximation of the well-tempered semitone. The principle of equal temperament was clearly expounded by the Chinese prince Tsai-yu in 1596, and by Mersenne in 1635. . . . At any rate, the system was not universally adopted in Germany until c. 1800, in France and England until c. 1850., p736]

Simplified Sexagesimal Approximation to 12edo by Joe Monzo

Henry Cowell


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December 4, 2003