604 Proceedings of the Royal Society 
the harmony is, however, in every case the least common multiple 
of the period of its constituent tones. The number of times that 
the period of the harmony contains the period of any one of its con- 
stituent tones is called the harmonic number of that tone. This 
expression is only applicable to any particular tone when viewed as 
one constituent of a harmony. Following the usage of Lord Rayleigh 
and Professor Everett, I shall employ the word “ frequency ” to 
denote the number of periods per unit of time, — per second, let us 
say, generally in acoustical reckonings. Thus the “ frequency ” of 
a tone or of a harmony means the number of its periods per second. 
Similarly the frequency of any set of beats, according to the defini- 
tions and descriptions below, will mean the number of the beats 
per second, and in this application of the term it will designate some- 
times a proper fraction, and sometimes a small whole number with 
fraction. 
The quality of a harmony, when the periods of its several con- 
stituent tones are given, depends upon the amplitudes of the dif- 
ferent constituents, and on the relation of their phases. Thus, for 
example, consider a harmony of two tones. They may be so related 
in phase that at one of the instants of maximum pressure of one of 
the constituents there is also maximum pressure of the other con- 
stituent. The same phase-relation, if the harmonic numbers of the 
constituent tones be both odd, will give also coincident minimums. 
But when one of the harmonic numbers is even and the other odd 
the phase-relation of coincident maximums will also be such that 
there is a coincidence of minimum pressure due to one tone with 
maximum pressure due to the other ; and again there will be an 
opposite phase in which there will be coincidence of minimums, 
and in this opposite phase there will also be a coincidence of maxi- 
mum and minimum. (To avoid circumlocutions a harmony of two 
odd numbers will be called an odd binary harmony ; a harmony of 
even and odd numbers will be called an even binary harmony.) 
Thus we see that in an odd binary harmony there is a phase-rela- 
tion of coincident maximums and coincident minimums, and again 
an opposite phase-relation of coincident maximum minimum and 
minimum maximum. The former will be called the phase-relation 
of coincidences, the latter the phase-relation of oppositions. In an 
even binary harmony there is a phase-relation of coincident maxi- 
