208 Professor Silvanus P. Thompson [June 13, 



wind chest two organ-pipes tuned to uty = 128, and then flatten one 

 of them slightly by holding a finger in front of its mouth. Von 

 Helmholtz's theory of dissonance may be briefly summarised by say- 

 ing that any two notes are discordant if their vibration numbers are 

 such that they produce beats, maximum discordance occurring when 

 the beats occur at about 33 per second ; beats if either fewer than 

 these, or more numerous, being less disagreeable than beats at this 

 frequency. It is an immediate consequence that the degree of disso- 

 nance of any given interval will depend on its position on the scale. 

 For example, the interval of the major, whole tone, represented by 

 the ratio 9 : 8, produces four beats per second at the bottom of the 

 pianoforte keyboard, 32 beats per second at the middle of the key- 

 board, and 256 beats per second at the top. Such an interval ought 

 to be discordant, therefore, in the middle octaves of the scale only. 

 Von Helmholtz expresses elsewhere the opinion that beats only occur 

 between two tones when the intervals between these tones are within 

 a minor third of one another. 



To this view of von Helmholtz it was at first objected that if that 

 were the whole truth, all intervals should be equally harmonious 

 provided one got far enough away from being in a bad unison : fifths, 

 augmented fifths, and sixths minor and major ought all to be equally 

 harmonious. This no musician will allow. To account for this von 

 Helmholtz makes the further supposition that the beats occur, not 

 simply between the fundamental or prime tones, but also between the 

 upper partials which usually accompany prime tones. This leads 

 me to say a word about upper partial-tones and harmonics. I believe 

 many musicians use these two terms as synonymous ; but they ought 

 to be carefully distinguished. The term harmonics ought to be 

 rigidly reserved to denote higher tones which stand in definite har- 

 monic relations to the fundamental tone. The great mathematician 

 Fourier first showed that any truly periodic function, however comj)lex, 

 could be analysed out and expressed as the sum of a certain series of 

 periodic functions having frequencies related to that of the funda- 

 mental or first member of the series as the simple numbers 2, 3, 4, 

 5, &c. Thirty years later G. S. Ohm suggested that the human ear 

 actually performs such an analysis, by virtue of its mechanical 

 structures, upon every complex sound of a periodic character, resolving 

 it into a fundamental tone, the octave of that tone, the twelfth, the 

 double octave, &c. Von Helmholtz, arming himself with a series of 

 tuned resonators, sought to pick up and recognise as members of a 

 Fourier-series, the higher harmonics of the tones of various instruments. 

 In his researches he goes over the ground previously traversed by 

 Eameau, Smith, and Youug, who had all observed the co-existence in 

 the tones of musical instruments, of higher partial tones. These higher 

 tones correspond to higher modes of vibration, in which the vibratile 

 organ — string, reed, or air column, — subdivides into two, three, four, 

 or more parts. Such parts naturally possess greater frequency of 

 vibration, and their higher tones, when tliey co- exist along with the 



