THE PHYSICAL BASIS OF MUSICAL HARMONY. 337 



nature of timbres. Returning, then, to the ratios of tlie vibration num- 

 bers of the major scale, we may note that two of these, namely, the 

 ratios 9:8 and 15:8, which correspond to the intervals called the 

 major whole tone and the seventh, are dissonant — or, at least, are usu- 

 ally so regarded. It will also be noticed that tliese particular fractions 

 are more comi)lex than those that represent the consonant intervals. 

 This naturally raises the question : Why is it that theeonsonant intervals 

 should he represented by ratios made up of the numbers 1 to ij and by no 

 others ? 



To this problem the only answer for long was the entirely evasive 

 and metaphysical one that the mind instinctively delights in order and 

 nnmber. The true answer or rather the first approximation to a true 

 answer was only given about 40 years ago, when von Helmholtz, as the 

 result of his ever-memorable researches on the sensations of tone, 

 returned the reply: Because only by fulfilling numerical relations which 

 are at once exact and simplecan the " beats " be avoided which are the cause 

 of dissonance. The phenomenon of beats is so well known that I uuiy 

 assume the term to be familiar. An excellent mode of making beats 

 audible to a large audience is to place upon a wind-chest two organ-pipes 

 tuned to ut2=V2S, and then flatten one of them slightly by holding a 

 finger in front of its mouth. Yon Helmholtz's theory of dissonance may 

 be briefly summarized by saying 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 disagree- 

 able than beats at this frequency. It is an immediate consequence 

 that the degree of dissonance 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 i)ianoforte keyboard, 32 beats per second at the mid- 

 dle of the keyboard, and 250 beats per second at the top. Such an 

 interval ought to be discordant therefore in the middle octaves of the 

 scale only. 



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

 were all, all intervals should be equally harmonious provided one got 

 far enough away from being in a bad unison ; fifths, augmented fifths, 

 and sixths, minor aiul major, ought to be equally harmonious. This no 

 nnisician will allow. To account for this von Helmholtz makes the 

 further supposition that the beats occur, not simi)ly between the funda- 

 mental or piime tones, but also between the upper partials which 

 usually ac(;ompauy 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 terni harnu^nics ought to be rigidly reserved to denote higher 

 tones which stand in definite harmonic relations to the fundamental 

 tone. The great mathematican Fourier first showed that any truly 

 H. Mis. 129 22 



