v THE SENSE OF HEAEING 249 



taneously are known as concords or discords, according to 

 whether they produce agreeable or disagreeable sensations which 

 varies with different races, and also in different epochs and 

 individuals. The musical theory of the Greeks was acquainted 

 through the Pythagorean School, perhaps even through the 

 ancient Egyptians, with the distinction of intervals into symphonic 

 and diaphonic, which corresponded to concords and discords. They 

 held the octave and the fifth to be symphonic; all the rest 

 including the third to be diaphonic. In the Middle Ages the 

 major and minor third and the sixth were added to the symphonic 

 intervals or concords, and in the year 900 the fourth as well, 

 though later on it was once more relegated to the discords. 

 Modern musicians arrange the series of musical intervals commonly 

 employed, according to the diminishing degree of consonance, as 

 follows : octave, fifth, fourth, major third, major sixth, minor 

 third, minor sixth. 



Helmholtz was the first who attempted to give a strictly 

 scientific explanation of the consonance and dissonance of 

 intervals. For him a consonant interval is one that produces a 

 uniform sensation of sound, a dissonant interval one that produces 

 an intermittent sensation. Consonance relates to the affinity of 

 tones, dissonance to the frequency of beats. Two fundamental 

 tones are in greater affinity according to the greater number of 

 partial tones they have in common. Consonance is greatest 

 when the fundamentals of both tones are in the ratio of an 

 octave, because in this case their partial tones are fused and the 

 beats disappear. In examining the series of diminishing con- 

 sonances, fifth, fourth, third, etc., the number of coincident 

 harmonics is seen to decrease, while the possibility of beats 

 increases. The diminution of consonance therefore goes parallel 

 with the diminution of affinity. 



This is plain from the following table, in which the relative number of 

 the vibrations of the different tones of the scale and their corresponding 

 harmonic partial tones is shown in series. The first row of figures for each 

 interval represents the partial tones contained in the fundamental ; the second, 

 the partial tones of the second tone associated with the first. The partials 

 of the two tones which coincide are given in black type. 



(I 2.3.45 6 7 8 . 9 . 10 

 Octave 1 : 2 [ 2 4 6 8 10 



(2 . 4 . 6 . 8 . 10 . 12 . 14 . 16.18. 20 

 [3 6 9 12 15 18 



, f 3 6 9 . 12 . 15 . 18 . 21 . 24 . 27 . 30 

 r ourtn 3 



'M 



Major Third 4 : 5 j 



4 . 8 .12 .16 . 20 . 24 .28 . 32 . 36 . 40 

 5 10 15 20 25 30 35 40 



