102 ON THE -PHYSIOLOGICAL CAUSES OF 



Tn the same way two tones, which have their pitches 

 exactly in the ratios of 3 to 4, or 4 to 5, and consequently 

 form a perfect fourth or a perfect major third, sound 

 much better when sounded together, than two others of 

 which the pitches slightly deviate from this exact ratio. 

 In this manner, then, any given tone being assumed as 

 fundamental, there is a precisely determinate number of 

 other degrees of tone which can be sounded at the same 

 time with it, without producing any want of uniformity 

 or any roughness of tone, or which will at least produce 

 less roughness than any slightly greater or smaller inter- 

 vals of tone under the same circumstances. 



This is the reason why modern music, which is essen- 

 tially based on the harmonious consonance of tones, has 

 been compelled to limit its scale to certain determinate 

 degrees. But even in ancient music, which allowed only 

 one part to be sung at a time, and hence had no harmony 

 in the modern sense of the word, it can be shown that 

 the upper partial tones contained in all musical tones 

 sufficed to determine a preference in favour of pro- 

 gressions through certain determinate intervals. When 

 , an upper partial tone is common to two successive tones 

 in a melody, the ear recognises a certain relationship 

 between them, serving as an artistic bond of union. 

 Time is, however, too short for me to enlarge on this 

 topic, as we should be obliged to go far back into the 

 history of music. 



I will but mention that there exists another kind of 

 secondary tones, which are only heard when two or more 

 loudish tones of different pitch are sounded together, and 

 are hence termed combinational. 1 These secondary tones 



1 These are of two kinds, differential and summational, according as their 

 pitch is the difference or sum of the pitches of the two generating tones. 

 The former are the only combinational tones here spoken of. The dis- 

 covery of the latter was entirely due to the theoretical investigations of the 

 author. Tit. 



