THE EAR AND HEARING. 569 



rate are otherwise accounted for in sensation by loud ness 

 and pitch. 



It can be proved, by the employment of the higher mathe- 

 matics, that every periodic non-pendular movement can be 

 analyzed (as the dotted curve of Fig. 169 may be) into a 

 given number of pendular vibrations, that is,-every compound 

 vibration into a set of simple ones; and that every periodic 

 non-pendular vibration can be made by the combination of 

 pendular. Moreover, any given compound vibration can be 

 analyzed into but one set of simple ones; no other combina- 

 tion will produce it. Consequently a vibrational movement 

 of the air in the external auditory passage, producing a com- 

 pound musical tone sensation, can be exhibited in every case, 

 but only in one way, as the sum of a number of simple vibra- 

 tions, whose rates are multiples of that which determines the 

 pitch of the tone. 



Now when the trained ear listens to a tone with the ob- 

 ject of detecting upper partials if present, it hears them only 

 when the vibrations producing the tone are non-pendular, 

 i.e. when upper partials, theoretically, might be expected; 

 and those heard are exactly those demanded by theory; by 

 the help of instruments their detection is made easy even to 

 untrained ears. In ordinary circumstances we do not heed 

 secondary partial tones; we hear a note of the pitch of the 

 primary partial and of a certain timbre; and whenever the 

 upper partials present are different, or of different relative 

 intensities, the timVre of the note varies. Hence it becomes 

 probable that, jnst as the ear can at will follow any instru- 

 ment in an orchestra, analyzing the aerial movement so as to 

 select and follow the fraction of the whole due to that one, 

 so it can and does analyze compound tones when proceeding 

 from one instrument, and that the upper partials, not rising 

 into consciousness as definite tones, but present as subdued 

 sensations, give its character to the whole tone and determine 

 its timbre. It might be, however, that the composition of 

 non-pendular vibrations from pendular was a mere mathe- 

 matical possibility, having no real existence in nature. Be- 

 fore we nan accept the above explanation of timbre, we must 

 see if there is any evidence that, as a matter of fact, non- 

 pendnlar vibrations, not only mat/ be, but are, made up by 

 the combination of pendnlar. 



Sympathetic Resonance. Imagine slight taps to be 



