screws upwards, increasing the force of the springs. Once every semiperiod of the pendulum it is held back by either pallet, and the nut collar screws down as much as it rose during the preceding interval of freedom when the action is regular; and the central or main escapement-shaft turns in the same period as the tooth, being the period of the pendulum. If, through increase or diminution of the driving-power, or diminution or increase of the coefficient of friction between the governing masses and the ring on which they press, the shaft tends to turn faster or slower, the nut collar works its way down or up the screw, until the governor is again regulated, and gives the same speed in the altered circumstances. It is casy to arrange that a large amount of regulating power shall be implied in a single turn of the nut collar relatively to the central shaft, and yet that the periodic application and removal of about of this amount in the half period of the pendulum shall cause but a very small periodic variation in the speed. The latter important condition is sccured by the great moment of inertia of the governing masses themselves round the main shaft.

1

50

ON BEATS OF IMPERFECT

HARMONIES

[Being Paper read before the Royal Society of Edinburgh, April 1st, 1878.]

a

ACCORDING to a usage which has been adopted from the German of Helmholtz by the best English scientific writers on sound, a sound is called a "simple tone,” 1 or without qualification "tone," when the variation of pressure of the air in the neighbourhood of the car, which is the immediate excitant of the sense, is according to a simple harmonic function of the time; that is to say, when the deviation from the mean pressure of the air varies in simple proportion to the distance, from a fixed plane, of a

1 The old musical usage, according to which the word tone denotes an interval (the major tone or minor tone, or the mean tone of the tempered scale), though it unfortunately clashes with this recent scientific use of the word tone, can scarcely be abandoned.

point moving uniformly in a circle. Considering the actual sensibility of the human car to musical sounds, we must introduce farther as a practical restriction that the period of the variation of the

pressure must be less than of a second, and

30

greater than 10000 or 2000 of a second. The vibrations of the air produced by a simple harmonic vibrator are either simple harmonic, or are in circular or elliptic orbits, resulting from the composition of two simple harmonic motions; and the consequent change of air-pressure in the neighbourhood of the ear follows the simple harmonic law, provided the maximum velocity of the vibrator and of the air in its neighbourhood be infinitely small in comparison with the velocity of sound. Hence the more nearly this condition is fulfilled the more exactly a simple tone is the sound heard; but it is far from being fulfilled when the vibrator, though itself performing simple harmonic motion, has sharp edges round which the air is forced to rush with great velocity, or when, as in the case of free-reed organ pipes or the reeds of a harmonium, the vibrator is

an elastic solid moving to and fro in a very narrow aperture. (In the case of a slapping reed, as of trumpet stops in an organ, the motion of the vibrator itself is not simple harmonic, and the sound is excessively rich in overtones, giving it its peculiarly rich or harsh character.)

A harmony is any sound of which the excitant change of air-pressure is strictly periodic, and is Fourier's not a simple tone. According to

beautiful analysis1 of periodic variations, to which the name of the harmonic analysis has been given, any periodically varying quantity may be regarded as the sum of quantities varying separately according to the simple harmonic law, in periods respectively equal to the main period, half the main period, a third of the main period, and so on. According to this analysis we see that the variation of air-pressure constituting a harmony may be regarded as the sum of variations constituting simple tones, one having its period equal to the

1 Compare Trans. R.S.E., April 30th, 1860; re-published in "Reduction of Vol. III. of my Mathematical and Physical Papers, Observations of Underground Temperature," where a short descrip

tion of Fourier's analysis is to be found.

period of the harmony; a second, half that of the harmony; a third, one-third that of the harmony, and so on; in other words, we may regard the harmony as compounded of these simple tones.

Practically, in musical language the term harmony is not applied when the tone of the main period predominates in the sensory impression, and in this case the sound is simply called a note; its pitch is reckoned according to the main period ; and the effect of the other tones, now called overtones, which enter into its composition, are merely felt as giving it its character or quality of sound. Thus the name harmony is in musical practice restricted to cases in which there is either no tone of the main or fundamental period, or not enough to produce a predominating impression; and a sound compounded of two, three, four, or more simple tones, having commensurable periods, is heard. In ordinary musical language a harmony is not regarded as having any one pitch, but is thought of as compounded of its known constituents. The true period of the harmony is, however, in every case the least common multiple of the