i ? 4 HERMANN VON HELMHOLTZ 



The discovery made by Pythagoras, that vibrations which 

 bear to each other simple ratios of number, e. g. the octave, 

 fifth, twelfth, major third, produce a pleasing impression, 

 while tones with more complex ratios of vibration-frequency 

 are dissonant, had not been adequately explained by the 

 assumption that the contemplation of simple ratios of vibration 

 affords a pleasurable sensation to the mind. 



' Mathematics and music, the sharpest antithesis of intellectual 

 activity that can be found, are yet interrelated, mutually helpful, 

 as if to show the secret consistency that is implicated in all 

 the activities of our mind, and which leads us to surmise 

 unconscious expressions of a mysterious law of reason in the 

 revelations of artistic genius/ 



It had been proved in the earlier acoustic work of Helmholtz 

 and others, that when two tones have only an approximately 

 equal vibration-period, and their crests coincide at the outset, so 

 as to reinforce each other, the undulations of the one will 

 gradually outrun those of the other, and produce an alternating 

 ebb and flow in the tone, which are known as beats, and 

 which, if they become more rapid, are converted into a con- 

 tinuous tone-sensation. When the ratio of the prime tones 

 is 2 to 3, the two over-tones of six vibrations (whose existence 

 was previously ascertained) are exactly equal, and do not 

 disturb the harmony of the fundamentals, but when the ratio 

 is only approximately that of 2 to 3, the two partials are not 

 exactly equal, but produce beats with one another, and the 

 tone becomes harsh. Consonance and dissonance are there- 

 fore distinguished by the even flow of tones in the former, as 

 undisturbed as if each tone were sounding alone, while in 

 dissonance there is an incompatibility, and the tones are broken 

 up by their mutual action into separate impacts, which disturb the 

 listener and make him wish for harmony. In conclusion, Helm- 

 holtz drew a contrast between eye and ear, pointing out that 

 the eye cannot analyse a compound system of light waves, i. e. 

 composite colours, and is indifferent, in a mixed colour, whether 

 the component colours are or are not in simple ratios of 

 vibration-frequency. The eye has no harmony in the same 

 sense as the ear ; it has no music. ' The phenomena of purely 

 sensual harmony are indeed only the lowest grade of musical 

 beauty. Consonance and dissonance are but the means, albeit 



