May 29, 1908] 



SCIENCE 



843 



in opposite phase. The result is the 

 familiar phenomenon of beats. Such beats 

 when slow are not disagreeable and not 

 without musical value. If the difference 

 between the two notes is increased the beats 

 become more rapid and more disagreeable. 

 To this violent disturbance, to the starting 

 and stopping of the vibration of the fibers 

 of Corti, Helmholtz ascribed the sense of 

 roughness which we call discord. As the 

 notes are more widely separated in pitch 

 the overlapping of the affected areas 

 diminishes. Between pure notes the sense 

 of discord disappears with sufficient sepa- 

 ration in pitch. When the two vibrating 

 areas exactly match because the two notes 

 are of exactly the same pitch, and when 

 the two areas do not in the least overlap 

 because of a sufficiently wide separation in 

 pitch, the result, according to Helmholtz, 

 is harmony. Partial overlapping of the 

 affected areas produces beats, and the 

 roughness of beats is discord. Such, re- 

 duced to its fewest elements, is Helmholtz 's 

 explanation of the harmony and discord of 

 tones which are pure notes. 



But no musical tone is a pure note. A 

 musical tone always consists of a combina- 

 tion of so-called partial tones which bear 

 to each other a more or less simple relation- 

 ship. Of these partial tones, one is called 

 the fundamental — so called because it is the 

 loudest or lowest or, better still, because 

 it is that to which the other partial tones 

 bear the simplest relation. A mitsical tone, 

 therefore, affects not one, but, through its 

 fundamental and upper partial tones, 

 several areas of the diaphragm in the 

 cochlea. Two musical tones, each with its 

 fundamental and upper partials, therefore 

 affect areas of the diaphragm which over- 

 lap each other in a more or less com- 

 plicated manner, depending on the relative 

 frequencies of the fundamental tones and 

 the relationships of their upper partials. 

 The exact matching of the areas affected 



by these two systems of partial tones, or 

 the entire separation of the affected areas, 

 gives harmony. The overlapping of these 

 affected areas, if great, produces discord, 

 or if slight in amount, modifications and 

 color of harmony. 



In the great majority of musical tones 

 the upper partials bear simpler relation- 

 ships to the fundamentals, being integral 

 multiples in vibration frequency. Helm- 

 holtz showed that if of two such tones one 

 continued to sound unchanged in pitch, and 

 the other, starting in iinison, was gradually 

 raised in pitch, the resulting discord would 

 pass through maxima and minima, and 

 that the minima would locate the notes of 

 the pentatonic scale. The intermediate 

 notes of the complete modern musical scale 

 are determined by a repetition of this- 

 process, starting from the notes thus de- 

 termined. 



If to this is added a similar considera- 

 tion of the mutual interference of the com- 

 binational tones which are themselves due 

 to the interaction of the partial tones, we 

 have the whole, though, of course, in the 

 briefest outline, of Helmholtz 's theory of 

 the harmony and discord of simultaneously 

 sounding musical tones. 



Having thus in parts I. and II. de- 

 veloped a theory for the harmony and 

 discord of simultaneous sounds, and having 

 developed a theory which explains the 

 modern use of the musical scale in chords 

 and harmonic music, Helmholtz pointed 

 out in part III. that the musical scale in 

 its present form existed before the inven- 

 tion of harmonic music and before the use 

 of chords. Music may be divided into 

 three principal periods : 



1. "Homophonie or unison miisic of the 

 ancients," including the music of the 

 Christian era up to the eleventh century, 

 "to which also belongs the existing music 

 of Oriental and Asiatic nations." 



2. "Polyphonic music of the middle 



