394 



Mr. A. J. Ellis on Musical Chords. 



[June 16, 



tone. Its intensity is generally very small, but it becomes distinctly 

 audible in beats. The differential tone is frequently acuter than the lower 

 generator, and hence the ordinary name "grave harmonic" is inapplicable. 

 As its pitch is the beat number of the combination, Dr. T. Young attri- 

 buted its generation to the beats having become too rapid to be distin- 

 guished. This theory is disproved, first, by the existence of differential 

 tones for intervals which do not beat, and secondly, by the simultaneous 

 presence of distinct beats and differential tones, as I have frequently 

 heard on sounding/^, or even /^/^J together on the concertina, when 

 the beats form a distinct rattle, and the differential tone is a peculiar pene- 

 trating but very deep hum. 



The object of this paper is to apply these laws, partly physical and 

 partly physiological, to explain the constitution and relations of musical 

 chords. It is a continuation of my former paper on a Perfect Musical 

 Scale*, and the Tables are numbered accordingly. 



Two simple tones which make a greater interval than 6 : 5, and there- 

 fore never beat, will be termed disjunct. Simple tones making a 

 smaller interval, and therefore generally beating, will be termed pulsative. 

 The unreduced ratio of the pitch of the lower pulsative tone for which the 

 beat number is 70 to that for which it is only 10, will be termed the range 

 of the beat. The fraction by which the pitch of the lower pulsative tone 

 must be multiplied to produce the beat number, will be termed the beat 

 factor. The ratio of the pitches of the pulsative tones, on which the 

 sharpness of the dissonance depends, will be termed the beat interval. 



A compound tone will be represented by the absolute pitch of its primary 

 and the relative pitches of its partial tones, as C (1, 2, 3, 4, . . . .). As 

 generally only the relative pitch of two compound tones has to be con- 

 sidered, the pitches will be all reduced accordingly. Thus, if the two 

 primaries are as 2 : 3, the two compound tones will be represented by 2, 4, 

 6, 8, 10, ... ., and 3, 6, 9, 12, 15 ... . The intensity of the various 

 partial tones differs so much in different cases, that any assumption which 

 can be made respecting them is only approximative. In a well-bowed 

 violin we may assume the extent of the harmonics to vary inversely as the 

 number of their order. Hence, putting the extent arid intensity of the 

 primary each equal to 100, we shall have, with sufficient accuracy — 



Harmonics... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 



Extent... 100, 50, 33, 25, 20, 17, 14, 12, 11, 10. 



Intensity... 100, 25, 11, 6, 4, 3, 2, 1, 1, 1. 



It will be assumed that this law holds for all combining compound 



* Proceedings of the Royal Society, vol. xiii. p. 98. The following mis- 

 prints require coiTection : — P. 97, line 7 from bottom, for read b. Table L, 

 p. 105, diminished 5th, example, i^ead f : B j minor 6th, logarithm, read -20412 ; 

 Pythagorean Major 6th, read 27:16, 3^:2*^ Table V., col. VI., last line, reai^' 



