1890.] on the Physical Foundation of Music. 207 



pitch the middle c' between the bass and treble staves is defined as 

 consisting of 256 complete vibrations. In concert pitch the usual 

 number given to the same note is 268 or 270. Further, the notes 

 of the major scale are always related to one another in the following 

 definite ratios : — 



9 5 4 3 5 15 



•8*4'3'2*3' 8 • 



In the execution of music the performer does not always adhere 

 rigidly to these ratios. For pure melodic purposes, the actual scale 

 is nearer to that called the Pythagorean ; but for the purpose of 

 harmony in any one key the above ratios must be precisely observed. 

 But, obviously, a scale is wanted which will serve both purposes, 

 melodic and harmonic ; hence the same scale is attempted to be used 

 for melody as for harmony. Further, the requirements of modern 

 music, ever since the time of Bach, have necessitated a further 

 modification of the scale to enable the performer to modulate into 

 harmonies, not in one key, but in all ; so that he may enjoy the 

 tonal relations of contrasted keys, modulating from key to key. In 

 order that this may be done, and yet that all the keys may be played 

 from one key-board, musicians have been content to spoil the exact 

 consonances in individual keys by departing from these exact ratios 

 by the device of tempering all the keys, so that twelve successive 

 fifths shall exactly equal seven successive octaves. 



I am not here, however, to fight over again the battle of the 

 temperaments, nor do I purpose to enter upon a discussion of the 

 origin of melody, which, indeed, I believe to be associative rather 

 than physical. I shall confine myself to two matters only — the 

 cause of harmony and the nature of timbres. 



Eeturning, then, to the ratios of the vibration-numbers 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 usually so re- 

 garded. It will also be noticed that these particular fractions are 

 more complex than those that represent the consonant intervals. 

 This naturally raises the question: Why is it that the consonant 

 intervals should he rejpresented hy ratios made up of the numbers 1 to 6, 

 and hy 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 number. The true answer, or rather the first approximation to 

 a true answer, was only given about forty years ago, when von Helm- 

 holtz, as the result of his ever-memorable researches on the sensa- 

 tions of tone, returned the reply : Because only by fulfilling numerical 

 relations which are at once exact and simple, can the " heats " be avoided, 

 which are the cause of dissonance. The phenomenon of beats is so 

 well known that I may assume the term to be familiar. An excellent 

 mode of making beats audible to a large audience is to place upon a 



