PROFESSOR AT HEIDELBERG 195 



new chord of the final tone now gives a fifth, which is no longer 

 identical with the third of the original key; hence in keyed 

 instruments it is usual to substitute for these two slightly 

 different tones (one of which is the third of the original tonic), 

 one single tone, since an impure fifth is more readily perceived 

 than an impure third. On further progression by fifths the tonic 

 is not recovered ; in order to distribute the error evenly, all the 

 fifths must be slightly altered ; the deviation of the fifths in the 

 now general system of intonation will however be exceedingly 

 small, since the ratio of the pure to the tempered fifths will be 

 as 886 : 885. But since this produces errors in the thirds, and 

 modern music is harmonic throughout, the discord of the 

 false intervals of intonation makes itself unpleasantly heard 

 in the beats of their combination tones and harmonic over- 

 tones. In the instruments best adapted for artistic music, the 

 disadvantage of tempered music is least felt, because the 

 singing voice is independent of it, while the harshness can be 

 modified on bowed instruments, and the piano, the tones of 

 which soon die away, does not favour dissonances. The want 

 of true intonation is, on the other hand, apparent in all long- 

 sustained tones, particularly in the harmonium, where the beats 

 are too obvious when the instrument is played slowly, and the 

 difference between pure and tempered chords is so marked, 

 that the latter sound like dissonances in contrast with the former. 

 In order to obtain pure harmony, Helmholtz gives two distinct 

 values to each note of the scale, according as it is the third or 

 fifth, in relation to the tonic of a major chord, and gives the series 

 of major chords which satisfy the conditions. In practice, either 

 two keyboards must be introduced, or the instrument tuned 

 correctly for each key that occurs in the course of the 

 piece, by arranging the notes in eight groups, and providing 

 all the notes of each group with a separate supply of 

 wind from the bellows, when almost pure intervals can be 

 obtained. 



In this connexion we may refer to a later work of Helm- 

 holtz. Starting from the view that scales originated in the 

 desire to distinguish single tones clearly and sharply from one 

 another (whole tones only being recognized by the unpractised 

 ear of uncivilized peoples to this day), and that semitones 

 were only introduced into music as the ear became gradually 



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