288 DYNAMICAL THEOBY OF SOUND 



case when the interval between the two tones is sufficiently 

 great ; but when the interval is small we have " interference," 

 as we should expect from the analysis of 10, and the sensation 

 is in whole or in part intermittent. The phenomenon of" beats" 

 hardly needs description ; it is often met with in mistuned pairs 

 of piano-wires, in the vibrations of finger-bowls, and so on. For 

 methodical study two pure tones are required of equal intensity, 

 as e.g. from two tuning forks (with resonators), or two stopped 

 organ pipes, which can be made to differ in pitch by a variable 

 amount. As unison is departed from, the beats (whose 

 frequency is always equal to the difference of the frequencies 

 of the primary tones) are at first slow and easily counted. As 

 the interval widens they become more rapid, and a sensation of 

 roughness or discord is experienced ; moreover, the primary 

 tones are now heard along with the beats. Finally, as the 

 interval is continually increased, the beats and the consequent 

 roughness gradually cease to be perceptible. 



The intervals at which roughness begins and ceases, vary in 

 different parts of the scale. For the same interval the rough- 

 ness is less, the higher the pitch ; on the other hand for a given 

 number of beats per second the roughness is greater in the 

 higher octaves. 



In the case of two (or more) compound musical notes, we 

 may have beats and eventual roughness between any constituent 

 tones which are sufficiently near in the scale. We may even 

 have interference between the higher overtones of the same 

 note ; and it is for this reason that harmonics of higher order 

 than the sixth are prejudicial to good musical quality. 



It is through the interference of pairs of overtones that 

 deviations from the consonant intervals ( 3) usually make 

 themselves felt. Thus in the case of the Octave cc' we have 

 tones with the frequencies 



c = 132, 264, 396, 528, 660, 792, ..., 

 c' = 264, 528, 792, ..., 



and if this be mistuned all the harmonics of c' are interfered 

 with by the even harmonics of c. 



