]864.] 



Mr. A. J. Ellis on Musical Chords, 



393 



tones or harmonics, and the result of their combination a compound tone. 

 By the pitch of a compound tone will be meant the pitch of the lowest 

 partial tone ox primary. 



When two simple tones which are not of the same pitch are sounded 

 together, they will alternately reinforce and enfeeble each other's effect, 

 producing a libration of sound, termed a beat. The number of these beats 

 in one second will necessarily be the difference of the pitches of the two 

 simple tones, which may be termed the beat number. As for some time 

 the two sets of vibrations concur, and for some time they are nearly oppo- 

 site, the compound extent will be for some time nearly the sum, and for 

 some time nearly the difference of the two simple extents, and the intensity 

 of the beat may be measured by the ratio of the greater intensity to the less. 



But the beat will not be audible unless the ratio of the greater to the 

 smaller pitch is less than 6 : 5, according to Professor Helmholtz. This 

 is a convenient limit to fix, but it is probably not quite exact. To try the ex- 

 periment, I have had two sliding pipes, each stopped at the end, and having 

 each a continuous range of an octave, connected to one mouthpiece. The 

 tones are nearly simple ; and when the ratio approaches to 6:5, or the 

 interval of a minor third, the beats become faint, finally vanish, and do 

 not reappear. But the exact moment of their disappearance is difiicult to 

 fix, and indeed seems to vary, probably with the condition of the ear. The 

 ear appears to be most sensitive to the beats when the ratio is about 

 16 : 15. After this the beats again diminish in sharpness ; and when the 

 ratio is very near to unity, the ear is apt to overlook them altogether. The 



effect is almost that of a broken line of sound, as , 



the spaces representing the silences. 



Slow beats are not disagreeable ; for example, when they do not exceed 

 3 or 4 in a second. At 8 or 10 they become harsh ; from 15 to 40 they 

 thoroughly destroy the continuity of tone, and are discordant. After 40 

 they become less annoying. Professor Helmholtz thinks 33 the beat 

 number of maximum disagreeableness. As the beats become very rapid, 

 from 60 to 80 or 100 in a second, they become almost insensible. Pro- 

 fessor Helmholtz considers 132 as the limiting number of beats which can 

 be heard. They are certainly still to be distinguished even at that rate, 

 but become more and more like a scream. Though /^Jf and should give 

 198 beats in a second if c=264, and the interval is that for which the ear 

 is most sensitive, I can detect no beats when these tones are played on two 

 flageolet-fifes. Hence beats from 10 to 70 may be considered as discord- 

 ant, and as the source of all discord in music. Beyond these limits they 

 produce a certain amount of harshness, but are not properly discordant. 



When the extent of the tones is not infinitesimal. Professor Helmholtz 

 has proved that on two simple tones being sounded together, many other 

 tones will be generated. The pitch of the principal and only one of these 

 co?nbinational tones necessary to be considered, is the difference of the 

 pitch of its generating tones. It will therefore be termed the differential 



