Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/116

Rh MUSIC [SCIENTIFIC BASIS. These doubts have been settled in the case of the low notes of the -organ by the process of analysis by beats. Where two notes differing slightly in pitch form beats, the number of beats in a second is equal to the difference of the frequencies. If, then, the frequencies of the notes sounded differ by one vibration per second nearly, there will be one beat per second ; if they differ by two or three per second there will be two or three beats. The frequencies of the two lowest notes of the 32-foot range are sufficiently nearly C =17 C$=18. If, therefore, C and C$ are true notes they should give about one beat per second. The octaves of these notes give about two beats per second, and the twelfths give about three beats per second. The notes of large-scale open 32 -foot pipes when thus tested give one beat per second. The notes of stopped pipes vary very much ac cording to the scale and voicing, but in these low notes no fundamental has as yet been detected. They always present the three beats characteristic of the twelfth. This is contrary to the statement, long accepted on the authority of Helmholtz, that large-scale stopped pipes give nearly pure tones. It does not appear that that statement was ever verified, and it appears not to be correct. 32-foot stops are not very common, but these principles can be illustrated with 16 -foot stops. Here the fundamental beat is about two per second, the octave four, and the twelfth six for adjoining semitones at the bottom of the range. The same results are easily found. As a rule the large open 16 -foot stops of the pedal give their funda mentals quite pure, while stopped pipes professing to speak the same notes almost invariably present the six beats of the twelfth, sometimes with and sometimes without funda mental. The conclusion we may draw is that the enor mous power laid on to the lower notes of the organ enables the test of audibility to be made under most favourable circumstances, and that under these circumstances the limit of audible sounds can be carried down to a point close to the 32-foot C, or to a frequency of about 17. Deter- The determination of the frequency or vibration number of par- mmation ticular notes was first effected by calculations depending on the of fre- mechanical theory of strings. Subsequently the method of beats quency. was employed, and the first reliable determinations appear to have been made by this method, which was developed by Scheibler. The usual process consists of providing a series of notes each of which makes four beats with its next neighbour, whence every such pair has vibration numbers differing by four. The series extends over an octave, whence the total difference of frequency between the extreme notes which form the octave is known. And this number is equal to the frequency of the lower note of the octave. This method, however, is difficult of execution, and depends on a number of observations, each of which is liable to error. The method described in most of the books depends on the employment of the &quot;siren.&quot; This consists essentially of a circular plate, revolving on an axis through its centre at right angles to its plane. Series of holes are arranged in circles, and puffs of air are sent through the holes as they move over fixed holes. In this way a known number of impulses is produced at each revolution. The revolutions are counted by a wheel work. With the more perfect forms of this instrument fair determinations of frequency could be effected by bringing the note of the instrument into coin cidence with that to be determined, and counting the impulses de livered during a certain time. But until quite recently there was great uncertainty as to the actual frequency of the notes in use. In particular, the forks sold some time ago as 512 for treble C were for the most part several vibrations higher. And the various forks sold as philharmonic have at different times represented a great variety of pitches. The pitch of treble C has in recent times varied between the limits 512 and 540, being almost exactly a semitone. A few of the principal pitches may be summarized as follows : = 512 old theoretical pitch. 518 equal temperament equivalent of French dia pason normale. A = 435. 528 Society of Arts. Helmholtz s theoretical pitch. 540 Modern concert pitch. There is a tendency in practice to keep the pitch rising. This appears to arise from the habit among musicians of considering flatness in the orchestra or in singing a more heinous offence than sharpness. Everybody tries, at all events, not to be flat. Wind instruments made to concert pitch force the pitch up at all public performances. A rise is easily made, but a fall only with great difficulty. It will be seen that the pitch of the French diapason normale is the best part of a semitone below modern concert pitch, and the difficulty of getting it adopted is well known. Forks stamped with the numbers of vibrations are now issued privately by some of the principal musical firms, and they appear to be fairly accurate. Probably they are copied from certain series of forks beating four per second which have been constructed according to Scheibler s process, so as to furnish the vibration numbers. The most easy and convenient way of settling the frequency of tuning-forks, or rather of adjusting any vibrating body to a standard note, appears to be by means of a uniform rotation machine con trolled by a clock so as to revolve exactly once per second. A disk is mounted on the machine, having say 135 radial slits spoke- wise. A light behind the disk then throws 135 flashes per second. If a tuning-fork or other vibrating body be placed in front of the disk and looked at against the illuminated background, it presents a pattern which will be stationary if the fork be 135 or 270 or 405 or 540, or any other multiple of 135. If the fork is sharp the pattern moves one way, if flat the other. In this way the vibra tion number of a vibrating body is referred directly to the clock, and the adjustment to the standard note is one easily made, and not requiring great delicacy of observation. 3. The loudness or intensity of notes undoubtedly in- Intensil creases with the magnitude of the displacements of which the vibrations consist, or rather perhaps with the magni tude of the changes of pressure which occur in the neigh bourhood of the ear. It has been customary to speak of the energy of the vibration as affording a measure of in tensity, and this is true from a mechanical point of view. But the subjective intensity or loudness is certainly not correctly measured by any of these quantities. Further, the same changes of pressure or the same mechanical intensity cause sounds which vary in loudness according to the pitch. Taking this last point first, it is easy to show that a given mechanical intensity produces a very much louder sound in the higher parts of the scale than in the lower. The simplest way of looking at this is to consider the work employed in exciting the pipes of an organ-stop. The upper pipes take only a small fraction of the wind, and consequently of the power, used by the lower ones, and yet the upper pipes appear quite as loud. It has been shown that with a particular stop the work consumed was proportional to the length of the pipe, and so inversely as the vibration number. It has been maintained lately that the loudness of sound is measured by the amplitude of the motion, or by the changes of pressure, rather than by the mechanical inten sity. The experiments on which this view is based consist of dropping weights from different heights. A weight m from a height h gives a certain loudness. Now let the weight be doubled, the question is whether the loudness remains the same when the height is halved, or when it is divided by /2. The experiments appear to prove that the latter is the case. The estimation of the loudness is difficult on account of the apparent change of timbre, but the experiments are carefully arranged and discussed, and appear to establish a prima facie case. 1 The experiments are based upon Fechner s law, and appear to afford proof of its applicability. Fechner s law may be stated thus : equal differences of sensation are produced by changes which are equal fractions of the whole excitation. Thus we may take the change in the mechanical excitation to consist in doubling it ; then every time that it is doubled a change will be made in the sensation which is in all cases equally recognizable. The general probability of the truth of this will be seen by enumerating ten different magnitudes under which sounds may be classified ; these represent fairly equal differences of sensation : Niirr. Zeitschrift fur Biologic, 1879, p. 297.