1864.] 



Mr. A. J. Ellis on Musical Chords. 



399 



joint harmonics. It must be remembered that when there are several dis- 

 junct harmonics, their unbroken sound tends to obliterate the action of the 

 beats. There is no sensible silence between the beats unless the tones are 

 simple and the intensities nearly equal. The intensities of the beats be- 

 tween joint harmonics and differential tones cannot be reduced to figures. 

 It is not large. The history of a beat is therefore given by four fractions, 

 which in this case are the interval 9 : 8, the factor 1 : 3, the range 210 : 30, 

 and the intensity 69 : 3. 



These calculations have been made for concordant dyads in Table VIII., 

 and for concordant or major triads in Table IX. An attempt has been 

 made to arrange the 13 forms of the first, and the 20 forms of the second 

 in order of sonorousness, by considering the distribution of the intensities 

 among the several joint harmonics, the development of pulsative differential 

 tones, and the nature of the beats, omitting those due to the seventh har- 

 monic of an isolated constituent. It has not been thought necessary to 

 give the history of every beat. The intervals of all the beats are seen at 

 a glance by the list of intensities of the joint harmonics. 



By Table VIII. we see that the only unisonant dyad is the octave 1,2*, 

 which will be as unisonant as the constituents themselves. All other dyads 

 are occasionally dissonant. Thus the fifth itself is decidedly dissonant 

 when the pitch of the lower constituent lies between 20 and 140. On a 

 bass concertina tuned justly, I find the fifth, C^(?^, quite intolerable, the fifth, 

 C G, rough, but D fA nearly smooth, and at higher pitches there is no per- 

 ceptible dissonance. The beat interval of the major third is 16 : 15, and 

 the range of dissonance is much greater. The roughness can be distinctly 

 heard as high asc e ; in the lower octaves CF is quite discordant, and C^E* 

 intolerable. This Table VIII., therefore, establishes the fact that con- 

 cordance does not depend on simplicity of ratio alone ; but when the de- 

 nominator of the beat factor is small the range is lower, and therefore the 

 dissonance less felt. Dissonance also arises from the pulsative differential 

 tones 7 and 11, so that if the relative pitches are expressed in terms high 

 enough to differ by 7 and 11, the combination will be dissonant. The ear 

 is also not satisfied with forms in which great intensities of joint harmonics 

 are widely separated by many small intensities. The four last forms in 

 Table VIII., namely, the minor tenth 5, 12, the eleventh 3, 8, and the 

 two thirteenths 3, 10 and 5, 16, should therefore be treated as discords. 

 The Table also suggests how defects may be remedied by introducing new 

 constituents to fill up gaps, or by duplications. 



Similar observations apply to the triads in Table IX. None of them can 

 be unisonant at all pitches. Some of them, as the last seven, are really 

 discordant. The gaps may be generally filled up by duplication. Thus 



* That is, within the limits of the Table. Dyads such as 1, 2 ; 1, 3 ; 1, 4 ; 

 1, 5 ; 1, 6 are all unisonant ; but when the interval is very large, the want of con- 

 nexion between the tones renders them unpleasant. The dyad 1, 8 which 

 developes the differential tone 7 is dissonant. 



