XXIV. On the temperament of musical instruments with fixed tones
Alexander John Ellis · Proceedings of the Royal Society of London · 1864
Abstract In the preceding paper on the Physical Constitution of Musical Chord (Proceedings, vol. xiii. p. 392), of which the present is a continuation, drew attention to the importance of abolishing the distinction between tones which differ by the comma 81:80, on account of the number of fresh relations between chords that would be thus introduced. The contrivances necessary for this purpose have long been known under the nam of Temperament. I have shown that the musical scale which introduce the comma consists of tones whose pitch is formed from the number 1, 3, 5, by multiplying continually by 2, 3, and 5. Hence to abolish th comma it will be necessary to use other numbers in place of these. But this alteration will necessarily change the physical constitution of musical chords, which will now become approximate, instead of exact representatives of qualities of tone with a precisely defined root. It is also eviden that all the conjunct harmonics will be thus rendered pulsative, and tha therefore all the concords will be decidedly dissonant at all availabl pitches. The result would be intolerable if the beats were rapid. Temperament, therefore, only becomes possible because very slow beats are no distressing to the ear. Hence temperament may be defined to consist i slightly altering the perfect ratios of the pitch of the constituents of chord, for the purpose of increasing the number of relations between chords, and facilitating musical performance and composition by the reduction of the number of tones required for harmonious combinations. The subject has been frequently treated, but the laws of beats annposition of tones discovered by Prof. Helmholtz have enabled me to sent it in an entirely new form, and to determine with some degree of dainty what is the best possible form of temperament.