when the tension was produced by means of a weight, and the string thus allowed to lengthen, there was still a notable fall of pitch, due, in all probability, to alteration of its elasticity.

Strings of catgut, being very hygrometric, are also materially affected by moisture, which swells the material laterally and tends to shorten it. In a hot damp concert-room violins vary rapidly and somewhat irregularly from this cause.

On Tuning forks.-The effect of heat on tuning-forks was noticed and roughly computed by Perronet Thompson. He states that they are made flatter by heat at the rate of a hundredth of a comma for every three degrees Centigrade (or 5.4 Fahrenheit). His experiment was conducted as follows. A steel tuning-fork sounding treble C was cooled to the freezingpoint in snow and the load noted which brought the harmonic double octave of his monochord into unison with it; this was 240 lbs. The fork was then heated in boiling water; upon which 2 lbs. had to be taken off the load to bring it again into unison. The increase of length may be estimated at 0147 of an inch to the foot, which is competent to flatten the pitch by less than the tenth of a comma. But the fork was found flatter by the third of a comma: more than twothirds of the effect must therefore have proceeded from some other cause. "For which nothing so readily presents itself as a relaxation of the elastic power of the metal at the shoulder or bend. In confirmation of which it was observable that the forks, when heated to boiling-point, lost much of their strength of tone, and did not entirely recover it on cooling." "" I

Perronet Thompson's observations give the coefficient of alteration as '000023. Scheibler gives as the results of his observations what is equivalent to 0000573 at 440 vibrations and 0000505 at 220 vibrations as the alteration of pitch per degree Fahrenheit. Professor MacLeod and the writer made the amount larger, namely 000125 for 1° Centigrade or 00007 for 1° Fahrenheit. Professor. Mayer says that an ncrease or diminution of 1° Fahrenheit lengthens or shortens the fork by part, thus making the coefficient of change 00004545. The latter determination was made by exposing a fork in a room with an open window during four days of cold weather, noting the temperature, and another in a room heated to 70° by steam pipes. These discrepancies are doubtless due to the kind of steel employed for making the forks.

On Just Intɔnation, p. 74.

Effect of Barometric Pressure.-When the barometer rises, the effect on strings, wires, and tuning-forks, is to increase the flattening attributable to the resistance of the air by for every inch of rise. From this cause the harmonic octaves are too sharp to the fundamental tone, an error which may be referred to the effects of the air's resistance on different lengths of string. To take the instance of the octave, the vibrations of this will be twice as many in a given time as those of the open string, but the extent of each vibration may be estimated as half; so that the whole space travelled over may be considered as unaltered. So far, therefore, the resistance of the air may be expected to produce the same effects on both. But there is another element, which is that the shortened lengths present proportionally less surface to the air. Hence the greater lengths will be more retarded than the smaller, and the increase of retardation of the longer strings, which is the same as over-sharpness of the shorter, may be expected to be as the differences in length. It is * experimentally found to be so.

On Free Reeds. In the harmonium reed the process of alteration by change of temperature is far more complicated. For whereas the reed itself is equivalent to a vibrating rod supported at one extremity, its vibrations slacken by its expansion, and by its diminution of elasticity; on the other hand, the air which it sets in motion becomes less dense, and transmits the sound-wave with increased velocity. The latter action predominates on the whole; hence free reeds do not flatten with heat as has been stated; but they sharpen so much less than flue-pipes in an organ as to produce the same effect. For this reason they are almost entirely disused in this combination. They are slightly inferior in this respect to tuning-forks, which, as above stated, alter, according to one of the higher estimates, 0125 per cent. for 1° Centigrade, whereas from experiments recently made with a brass reed, it seemed to vary about 0277 per cent. for 1° Centigrade of temperature. As standards of pitch, however, in spite of this trifling inferiority, they are far superior on account of their more incisive tone, the abundance of their upper partial tones, and the consequent loudness of the beats they produce.

On Organ Pipes.-The general effect of heat on an organpipe is to sharpen the note it emits. The compensatory effect named in the case of the reed occurs in this instance also, but to a much less extent. For the pipe itself lengthens, especially if it be made of a very expansile metal, such as tin or pewter. But the pitch of the flue-pipe is far more dependent

on the contained column of air than on the envelope which surrounds it, and consequently the rarefaction of the medium tells upon the note with an influence quite predominant. In wooden pipes, the expansion of the tube is so very small as to be entirely inappreciable. Small pipes grow relatively sharper than large ones under the same increment of heat. Perronet Thompson states that the middle C pipe of the open diapason, two feet long, and two inches in diameter, sharpens a comma under a rise of 10° Centigrade, or 18° Fahrenheit, the barometer being stationary.

Different kinds of pipes, such as stopped and open diapasons, vary not only in the amount of sharpening, but also in the rate with which it takes place. Open pipes sharpen more quickly than stopped pipes, and metal pipes more rapidly than wooden ones. The smaller pipes are affected by changes of heat sooner than the large, as well as more.

In an organ of several stops, on suddenly raising the heat from 10° to 15° Centigrade (50° to 59° Fahrenheit), the smallest pipes of the metal open diapason grew sensibly too sharp for the others in the course of half an hour. This sharpening continued to increase for four hours, when they were too sharp for the largest pipes, by a quarter of a comma, the intermediate pipes in the meanwhile growing sharper, the smallest first. In the stopped diapason of wood, for four hours there was no perceptible alteration; but after that time the differences began to be sensible. During these processes, in consequence of smaller alterations in the wood, the open diapason sharpened upon the stopped; the greatest difference during the time of the experiment being at treble C, where the open diapason grew sharper than the stopped by three-eighths of a comma. In the enharmonic organ a rise in the thermometer of 10° Centigrade (18° Fahrenheit) raised the tuning C, termed the "master pipe," by a comma; a fall in the barometer of an inch, according to P. Thompson, did the same. He states generally that when the barometer and thermometer move the same way they act in opposition to each other, when they move different ways they act together. Stopped pipes are less affected by barometrical variations than open.

The writer has endeavoured to obtain some more accurate determinations on the subject of heat, as applied to reeds and organ-pipes in the following manner :-A wind-chest, capable of supplying a continuous stream of air at very equable pressure, is connected with two spirals of metal tubing, one of which is kept constantly at the freezing point of water by being immersed in melting ice, the other at boiling point by

means of a cistern of boiling water. The issuing air is used to feed two similar pipes, both of which are compared with a standard at the ordinary temperature. Thermometers are inserted in the respective currents, and the variations of pitch studied by means of the beats. In all cases the rise and fall is gradual, much more so with wooden and stopped metal pipes than with open metal ones. It seems to depend chiefly on the warming of the metal walls of the pipe itself, which rapidly radiate back to the contained air. In the same manner a reed fixed in a small wind-chest, is alternately blown by means of air at 32° and 212° Fahrenheit. The changes can be noted against the standard named above. In the reed the alterations of pitch are considerably less than those of pipes, for the reasons given above. The best result seems to follow from the use of German silver vibrators, which alter their molecular condition very slightly with moderate increments of heat, as is well known to students of dynamic electricity.

CHAPTER VII.

SCALES, CHORDS, TEMPERAMENT, AND TUNING.

HITHERTO Sounds have been treated as independent of one another, as bearing no mutual relations, and, except in the case of interference, as exercising no influence the one upon the other. This view represents only a limited, and what may be termed the physical side of acoustics. Beyond this lies the chief part, namely the aesthetic or musical conception of sound; which differs essentially from the former, in contemplating vibrations as intimately linked together, either in close series and succession, forming scales and melodies, or as simultaneously elicited, and furnishing the infinite varieties of chords and harmony. It is remarkable that whereas melody existed in ancient times, and has been cultivated by all nations, harmony in an extended sense, and possessing any pretension to exactness, is comparatively modern in its origin, and limited in its diffusion.

It was long known that the rapidity of vibration of a string under constant tension was inversely proportional to the length of the string, that is to say, that if we halve the length of the string, we double the number of its vibrations. To this we owe all power of playing on the violin, and also all knowledge of the relative pitch of the notes in the Greek and Arabic scales, for which the corresponding lengths of the string were given by Euclid the mathematician in the fourth century B.C. and by Abdul Kadir, the Persian theorist of the fourteenth century. Helmholtz points out that in the music of all nations so far as is known, alterations of pitch in melodies take place by intervals, and not by continuous transitions, and he further defines all melodies as motions within extremes of pitch.