606 Proceedings of the Eoyal Society 
beat is equal to the error of frequency of one note multiplied by the 
harmonic number of the other. When in a harmony of three or 
four notes all are perfect except one, the heats due to the imperfec- 
tion of the false one are to he reckoned just as if the harmony were 
binary, according to the following rule : — 
For the two or more notes which are in perfect harmony imagine 
one whose period is the period of their harmony. Take this as if 
it were one tone of an approximate binary harmony, the false note 
of the given harmony being the other. Example : Let the 
frequencies of the three notes be 257, 320, and 384 : the common 
period of the two last-mentioned is of a second, and we have to 
calculate the beats on two notes whose frequencies are 64 and 257. 
The harmonic numbers of the harmonies to which these notes 
approximate are 1 and 4, and the error in frequency of the higher 
note is 1 per second ; hence the beats are at the rate of 1 per 
second. When there is error in two or more notes of a multiple 
harmony, two or more sets of beats in periods not commensurable 
with one another are heard ; but the general effect is apt to be too 
confused to allow any one of the sets to be distinctly counted. On 
a multiple harmony with only one note false the beats are in 
general exceedingly distinct, more so in general than in binary 
harmonies. 
Sometimes, as for distance in reckoning the beats in the imperfect 
harmonies of a tempered musical scale, it is convenient to regard 
the two notes of an imperfect harmony as in error from two notes 
of a perfect harmony differing but little from them ; then the rule 
for calculating the frequency of the beats is to take the difference of 
the products of the errors of the two notes, each multiplied into the 
harmonic number of the other. Thus, let n and n' be the harmonic 
numbers of the perfect harmony to which the given notes approxi- 
mate, and let e and e be the excesses of the vibrational frequencies 
of the two actual notes above two in perfect harmony nearly agree- 
ing with them. The frequency of the beat of the actual notes is 
ne - ne\ 
For example, take the following table of numbers of vibrations in 
a perfect diatonic scale, with 256 vibrations per second for C, and in 
the corresponding scale of equal temperament (founded on 12 equal 
semitones, in each of which the interval ratio is 2 tIj ) : — 
