404 



Mr. A. J. Ellis on the Temperament 



[June 16, 



chords have been arranged in the forms 4, 5, 6 and 10, 12, 15, in accord- 

 ance with the usual practice of musicians. In the present paper the typical 

 1, 3, 5 and 3, 5, 15 have, for obvious reasons, been made the basis of the 

 arrangement. 



XXIV. ^^On the Temperament of Musical Instruments with Fixed 

 Tones.'' By Alexander J. Ellis, F.R.S., E.C.P.S.* Received 

 Junes, 1864. 



In the preceding paper on the Physical Constitution of Musical Chords 

 (Proceedings, vol. xiii. p. 392), of which the present is a continuation, I 

 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 con- 

 trivances necessary for this purpose have long been known under the name 

 of Temperament. I have shown that the musical scale which introduces 

 the comma consists of tones whose pitch is formed from the numbers 

 1, 3, 5, by multiplying continually by 2, 3, and 5. Hence to abolish the 

 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 representa- 

 tives of qualities of tone with a precisely defined root. It is also evident 

 that all the conjunct harmonics will be thus rendered pulsative, and that 

 therefore all the concords will be decidedly dissonant at all available 

 pitches. The result would be intolerable if the beats were rapid. Tem- 

 perament, therefore, only becomes possible because very slow beats are not 

 distressing to the ear. Hence temperament may be defined to consist in 

 slightly altering the perfect ratios of the pitch of the constituents of a 

 chord, for the purpose of increasing the number of relations between 

 chords, and facilitating musical performance and composition by the re- 

 duction of the number of tones required for harmonious combinations. 



The subject has been frequently treated f, but the laws of beats and 



* The Tables belonging to this Paper will be found after p. 422. 



t I have consulted the following works and memoirs. Huyghens, Cosmo- 

 theoreos, lib. i. ; Cyclus Harmonicus. Sauveur, Memoires de 1' Academie, 1701, 

 1702, 1707, 1717. Henjiing, Miscellanea Berolinensia, 1710, vol. i. pp. 265-294. 

 Smith, Harmonics, 2nd edit. 1769. Marpurg, Anfangsgruende der theoretischen 

 Musik, 1757. JEsteve, Mem. de Math, presentes a I'Acad. par divers Savans, 

 1755, vol. ii. pp. 113-136. Cavallo, Phil. Trans, vol. Ixxviii. Momieu, Mem. 

 de I'Acad., 1768. Lambert, Nouveaux Mem. de I'Acad. de Berlin, 1774, pp. 

 65-73. Br. T. Young, Phil. Trans. 1800, p. 143 ; Lectures, xxxiii. Rohison, 

 Mechanics, vol. iv. p. 412. Farey, Philosophical Magazine, 1810, vol. xxxvi. 

 pp. 39 and 374. Delezemie, Recueil des Travaux de la Societe des Sciences, &c. 

 de Lille, 1826-27. Woolhouse, Essay on Musical Intervals, 1835. De Morgan, 

 On the Beats of Imperfect Consonances, Cam. Phil. Trans, vol. x. p. 129. 

 Drohisch, Ueber musikalische Tonbestimmung und Temperatur, Abhandlmigen 



( 



