HERMANN VON HELMHOLTZ 



the interval be small, say a semitone, the beating, in 

 the lower part of the scale, may be so slow as not to 

 be disagreeable, whereas in the higher part it may 

 cause harsh and unpleasant dissonance. The sensa- 

 tional effect of beats then depends rather on the 

 difference of the vibration numbers than on the interval. 

 As a rule, the partials up to the seventh are beyond 

 beating distance, but above this they soon come close 

 together. In the neighbourhood of the tenth, the 

 interval may be about a tone, of the sixteenth, a 

 semitone, and still higher they come together so as 

 to cause dissonance. This fact explains why intervals 

 sound so harsh when produced by reeds, the sounds 

 of which are rich in upper partials, and also the 

 harsh but brilliant quality of intervals sounded on 

 two trumpets. Intervals even when produced on 

 instruments giving compound tones with few har- 

 monic constituents, such as flutes, have still their 

 own character. Helmholtz applied his theory with 

 consummate skill, not only as an explanation of 

 the quality of musical tones in many instruments, 

 the human larynx included, but also of the satisfying 

 character of certain musical intervals, as contrasted 

 with the discordant character of others. Thus 

 unison y, minor third , major third , fourth ^, fifth 

 -f , minor sixth -f , major sixth f , and octave f , are all 

 concords ; while a second , minor seventh y, and 

 major seventh y, are discords. The smoothest in- 

 terval is the octave, next the fifth, then the fourth, 

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