THE CAUSE AND CHARACTER OF THE SOUND WAVES 761 



quently interfere with one another so as to give rise either to a reinforce- 

 ment or a neutraUzation of the individual undulations. In quite the 

 same manner the simultaneous transfer of two sounds through the same 

 medium may give rise to waves which may be either the' sum total 

 or the difference of the two systems of undulations. The complete 

 neutralization of the two sets necessitates, of course, the coming to- 

 gether of the condensed molecules of one system with the rarefied 

 molecules of the other system. 



If two tuning forks, the vibrations of which differ slightly per unit of 

 time, are being sounded simultaneously, the two systems of undulations 

 must interfere with one another. Consequently, the sound emitted by 

 them must vary from moment to moment, becoming louder when they 

 reinforce and softer when they neutralize one another. This consti- 

 tutes the phenomenon of ^' beats." If the difference in the number of 

 vibrations per unit of time is increased, the effect produced on the 

 ear becomes increasingly disagreeable. The sound then assumes 

 a harsh grating character and is said to be discordant or dissonant. 

 Helmholtz states that the dissonance assumes an intolerable character, 

 when the ''beats," or the difference in the vibration frequency of two 

 sounds, reaches 33 to the second. 



In the absence of "beats" the general sound becomes consonant or 

 harmonic. This implies that the two sets of undulations correspond in 

 rhythm and amplitude, enabling them to be combined into an evenly 

 balanced compound wave. It must be evident, therefore, that a 

 perfect consonance can only be gotten if the two sets of waves are 

 identical in character. An almost complete consonance is also obtained 

 if a sound is elicited in conjunction with its octave. It is a well-known 

 fact that two sounds possessing a numerical relationship of 2:1, 4:1, 

 etc., must be closely allied. Thus, if the first is designated as C, the 

 second is called CI, and the interval between them an octave. If we 

 now strike the octave note of the second and then the octave of this 

 one, it will be found that their entire series of octaves or eighth notes 

 become fused into a sound which gives an agreeable sensation. Other 

 intervals giving consonance are the following: 



1:2 octave 

 ^ 2:3 fifth 



3:4 fourth 

 4:5 major 

 5 : 6 minor third 

 5 : 8 minor sixth 

 3 : 5 major sixth 



Sympathetic Vibration or Resonance. — If the end of the handle 

 of a vibrating tuning fork is placed upon a table or other elastic body, its 

 vibrations are immediately communicated to a large area of this 

 vibrator. Moreover, since the latter generally vibrates synchro- 

 nously with the tuning fork, its sound will be intensified. In a similar 

 way, it is possible to produce vibrations in a certain string of a piano 



