of Edinburgh, Session 1877 - 78 . 
605 
mums and coincident maximum minimum ; and again an opposite 
phase-relation of coincident maximum minimum and coincident mini- 
mums. The former will be called the phase -relation Gf coincident 
maximums, the latter the phase-relation of coincident minimums. 
The annexed diagrams illustrate and prove these assertions. The 
horizontal line in each may be either regarded as representing space 
at any one instant in the direction of propagation of the sound, or 
it may represent times at which successive phases of the motion are 
perceived by an ear in a fixed position. The long vertical cross- 
bar in each case denotes maximum of air-pressure ; the short vertical 
cross-bar, minimum. 
For lecture illustrations it is convenient to use long slips of wood 
with paper pasted on one side, and the short and long cross-bars 
marked upon it, and to support these slips of wood on a board with 
nails to guide them so that they may be placed in groups of two, 
three, or four, one over the other, and any of them moved in the 
direction of its length to illustrate the different phase-relations of a 
harmony. 
Suppose now, one note of a perfect binary harmony to be very 
slightly sharpened or flattened : so slightly that during a large 
number of the periods of the perfect harmony, the phase-relation in 
the imperfect harmony experiences but little change. Let the two 
notes of the imperfect harmony be sustained long enough with 
perfect uniformity as to pitch and intensity : — the effect will be that 
of perfect harmony, modified by a slow change of its phase-relation 
through a cycle ; which in the case of an even binary harmony is 
from coincident maximums gradually to coincident minimums, and 
thence gradually round again to coincident minimums ; and in the 
case of an odd binary harmony is from oppositions to coincidences, 
and round to oppositions again ; and so on in cycles. In favourable 
circumstances, and with careful attention, a variation of the quality 
of the sound recurring periodically in these successive cycles is dis- 
tinctly heard, even by an unpractised ear, unless the duration of the 
cycle be too long or too short to suit its sensibility. It is this 
variation which is called the “beat” on the imperfect harmony. 
The period of the beat — that is to say, the duration of the cycle 
described above — is most easily found by taking the reciprocal of its 
frequency, calculated by the following rule : — The frequency of the 
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VOL. IX. 
