SECTION II: THE SCIENTIFIC BASIS OF MUSIC
Regularity, Pitch, Loudness, Quality.
Musical sounds reach our ears through the air. This is proved by showing that a sounding body placed in a space void of air, as under the receiver of an air-pump, is unable to emit sound. It will be sufficient for our present purpose to admit that musical sounds are transmitted through the air with substantially uncharged quality, even to great distances. The law according to which the apparent intensity of a musical sound diminished as it spreads itself over an increasing surface of air is as yet uncertain; we can only say with certainly that it does diminish.
The origin of musical sound consists in the regular periodic vibration of some surface in contact with air, whereby motion is imparted to the air, and thus transmitted to the ear. Experience tells us as follows: --
1. Regular repetition is characteristic of those motions which give rise to musical sounds.
2. The pitch of the note produced depends on the time in which the motion takes place.
3. The loudness or intensity of the note depends on the magnitude of the motion and on the pitch.
4. The quality of the note depends on the form or shape of the motion, that is to say, on the manner in which it is executed within the time in which it takes place.
1. Regular repetition characterizes musical notes; irregularity in the movements in successive periods characterizes unmusical noises. This is most usefully illustrated by cases in which false notes are obtained. Strings, for instance, sometimes cannot be tuned. In these cases the motion can frequently be seen to be irregular,
2. The time in which the motion takes place is defined conveniently by the number of times the whole motion is repeated in a second. The number which expresses this may be called the vibration number, or the frequency of the note. Pitch, then, is defined by frequency. Notes of different frequencies present sounds to the ear which are essentially different form one another. The physical analogy based on frequency would compare notes of different pitch to light of different colours; this analogy does not, however, extend to the nature of the perceptions. The notes of a uniform instrument present a closer analogy with a definite colour in virtue of their uniform quality, so far as perception goes. And from this point of view there is nothing in the perception of colour analogous to the perception of difference of pitch in music. In specking of the perception of combinations of notes we shall see that the ear possesses, in a more or less perfect form, the power of analyzing combinations and hearing the notes separately. If we admitted the physical analogy between pitch and colour this would correspond to a power of seeing compound colours analysed into their constituents. This of course does not exist. Though we may know by experience what a compound colour consists of no amount of experience will enable us to see the components separately in the same way in which we hear the notes of a combination separately. The modes of perception are therefore wholly different in the two cases.
When one note is produced by a motion whose frequency is twice that of another, a relation subsists between the sounds of the two notes which appears not to be capable of further explanations. They are said to form octaves with one another. It is easy to give some account of the formation of the octave regarded as a concord, but the inexplicable peculiarity of the relation consists in a short of quasi-identity between the sounds of the two notes. Many persons cannot distinguished with centainty two notes an octave apart, particularly if the quality of tone employed be one in the use of which the observer is not practised, and this is the case even with ears of considerable acuteness and cultivation; but it does not apply to those ears of the highest class which posses the power of the recollection of absolute pitch. Such ears can usually distinguish octaves with certainty. But in all cases the similarity of effect between notes differing in pitch by one or more octaves will be admitted. It is a purely mental phenomenon, and no explanation can be given of it. If for a moment we recur to the physical analogy of colour simply for the purpose of illustration, the whole range of visible colour corresponds to less than an octave. The musical phenomenon of the similarity of octaves is as if part of the invisible spectrum, say in the ultra-red, exited a sensation having some similarity to the sensation of its octave in the visible spectrum, the two sensations being such that there is a continuous change from the one to the other. Nothing of the kind actually exists in the case of light. The actual impression on the nervous system of the ear which is concerned with the perception of pitch is believed to be substantially the same in different individuals. But the mental processes attached to this perception differ so widely that for all practical purposes the results are different in individuals. The chief difference appears to consist in the different development of the memory of the actual sound of definite notes, which we may speak of as the perception of absolute pitch. This is developed in all degrees: from almost complete absence, in which case we have a want of musical ear arising from the failure to retain the pitch of a note heard even for the shortest time, to that highest degree of perfection in which the memory retains permanently the sound of every note that is once heard. It is not believed that the possession of this memory is capable of cultivation to any considerable extent. It appears at once in childhood where it is possessed at all. Roughly speaking, and in the absence of reliable statistics, we may say that the possession of the perception of absolute pitch is distributed as follows: -- say 1 per cent. possess it, 1 per cent. are entirely destitute of it, so as to be said t have no ear, and the remaining 98 per cent. or so possess it in a more or less modified form. Of this usual effect of a musical note can be retained in the memory for some minutes.
We have to consider the musical perception of the class possessing the power, and of the numerically much larger class who only possess it in a modified form. A simple tune or melody produces the same effect in all cases as far as we know, except that those who have the perception of absolute pitch know what notes are employed, while others are only conscious of the intervals, i.e., of the relative pitch. All the analytical perceptions are as a rule much better developed in the case of those who possess the perception of absolute pitch than in other cases. In those whose perceptions fall below a certain mark the analysis of combinations of notes by the ear fails more or less entirely. In this particular, however, cultivation is generally possible. And this agrees with the assumption that the mechanism of the ear is generally the same. For the perception of differences of pitch is conceived to be due to a mechanism universally present, which effects the analysis of combinations by the same means by which the pitch is distinguished. The phenomena which give rise to this belief may be states as follows. If we sound two notes together and cause one of them to change its pitch, we can examine the sensations produced by all sorts of binary combinations. If the two notes have the same pitch of starting, then when they separate beats are heard. Up to a certain point, which is not very far form the point where the two notes are half a semitone apart, the effect in the ear is as if one note was heard, having the alternations of intensity which constitute beats and a pitch generally intermediate between the constituent notes. At this distance of about half a semitone the ear begins to perceive the two separate notes beside the beats. As the distance between the constituent notes increases the two notes become more prominent, the beats grow fainter, and ultimately, when the constituent notes are a little more than a whole tone apart, the beats disappear and the two notes are recognized as separate and distinct sensations.
Now this observation is explained by Helmholtz’ hypothesis of the existence of a resonant mechanism in the ear. Just as a harp, or a piano with the dampers raised, will have the strings corresponding to any sound that reached it set in vibration, so we suppose that there exists a collection of resonant bodies in the ear which respond to vibrations of pitch nearly coincident with their own. The question then arises, Is the range of sympathy, the distance at which a certain response is excited in the vibrating mechanism of the ear, the same in all cases?
It is easily seen that, if the above phenomena are substantially the same in all cases, the character of the mechanism must also be the same. The interval at which the two notes of a binary combination begin to be separately distinguished can be observed with considerable accuracy, and is critical in the matter. So far as such observations have been hitherto attempted, this interval is very nearly the same in persons of the highest type of ear, who have the perception of absolute pitch in an advanced degree, and in persons occupying a position not high among those who only possess the power of relative perception. In all cases, moreover, where beats are observed, the beats are the same to all persons, so far as our present knowledge goes. And as some of those beats arise within the ear itself, they depend upon the properties of its mechanism. The result is that, so far as ordinary observation goes, the main features of the mechanism of the ear may be safely pronounced to be the same, -- in the great majority of cases, at all events. Some slight differences in this respect have been observed; but there can be little doubt that the great differences which exist in the endowment of the ears of different persons do not arise from differences in the receptive mechanism, but have their source in the nervous or mental actions lie behind the mechanism. In this we are not referring to the differences which exist between the range of hearing in different individuals. These undoubtedly arise from what may be spoken of as differences in the compass of the receptive instrument in different individuals. These differences exist both at the upper and lower extremities of the scale. Those in whom the upward range is defective fail to hear the noises of certain insects, and are also unable to hear organ-pipes of very high pitch. The absence of the lower of hearing is less well established.
The complete range of audible musical sounds comprises about nine octaves. It extends from the 32-ffot C, two octaves below the lowest note of a bass voice, to somewhere about three octaves above C in alt. The upper notes of this range are not audible to some persons. Organ-pipes are made having notes covering this whole range, except about the top half-octave. The position of notes is so frequently referred to the length of the corresponding organ-pipe that it is convenient here to give these lengths, with the usual notation for the notes to which they correspond.
The normal perception of pitch does not, however, cover all this range. Its extent varies much in different individuals. We cannot assume that pitch can be completely octave or the lowest octave of the foregoing range. The lower half of the uppermost octave is easily examined, whatever an organ is available, by drawing the fifteenth alone and sounding the notes of the upper half octave of the keyboard. These notes will generally be heard as sounds, but if a bit of a melody be sounded on them, for instance a simple scale passage, it will generally be found unrecognized. The same is the case with the lowest octave lying between the 16-foot and 32-foot notes. There has been considerable doubts as to the reality of the notes which profess to occupy this position. All instruments sound notes of a complex character, which include sounds of a higher pitch than the nominal note, in fact the harmonies, of which the nominal note is the fundamental. And doubts have been raised, which have in some cases proved to be well founded, whether the deep notes in question really contain any of the nominal note or fundamental at all, and do not rather consist entirely of harmonics, or of sounds whose real pitch is much higher than the nominal one.
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 betas per second. The notes of large-scale open 32-foot pipes when thus tested give one beat peer second. The notes of stopped pipes vary very much according 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 t he statement, long accepted on the authority of Helmholtz, that large-scale stopped pipes give nearly verified. 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 steps. 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 fundamentals 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 fundamental. The conclusion we may draw is that the enormous 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.
The determination of the frequency or vibration numbers of particular notes was first effected by calculations depending on the mechanical theory of strings, Subsequently the method of beats was employed, and the first reliable determinations appears 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 frequently 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 "siren." 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 determination of frequency could be effected by bringing the note of the instrument into coincidence with that to be determined, and counting the impulses delivered 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 principle pitches may be summarized as follows: --
___________ = 512 – old theoretical pitch
518 – equal temperament equivalent of French diapason 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 performance. 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 controlled by a clock so as to revolve exactly once per second. A disk is mounted on the machine, having say 135 radial slits spokewise. 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 vibration number of a vibrating body is referred directly to the clock, and the adjustment t the standard note is one easily made, and not requiring great delicacy of observation.
3. The loudness or intensity of notes undoubtedly increases with the magnitude of the displacements of which the vibrations consist, or rather perhaps with the magnitude of the changes of pressure which occur in the neighbourhood of the ear. It has been customary to speak of the energy of the vibration as affording a measure of intensity, 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 takes 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 intensity. The experiments on which this view is based consists of dropping weights from different heights. A weights m from a height h gives a certain loudness. Now let the weight be doubted, the question is whether the loudness remains the same when the height is halved, or when it is divided by square root of 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, 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 consists in doubling it; then every time that it is doubted 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 then different magnitudes under which sounds may be classified; these represent fairly equal differences of sensation: --
== TABLE ==
Magnitude 1 serves for sounds louder than those used in music, 10 for sounds softer than those used in music, -- microscopic sounds, so to speak. It seems reasonable, without going into detail, to assume that the mechanical ratio of any two consecutive magnitudes is the same. The general expression of this law is, The measure of sensation is the logarithm of the mechanical excitation. It appears probable that the ratio of energy corresponding to one of the above differences of magnitude is somewhere about 2 or 3. The corresponding ratio depending on amplitude or compression would be from 1·4 to 1·7, but these quantities are not known with any accuracy.
4. The pitch depending on the period or frequency and the loudness on the amplitude or magnitude of the changes, there remains on the one hand the quality of tone, and on other the form of the vibration or the manner in which the motion takes place between the prescribed limits. These may be expected to correspond with ach other, and in fact they do so. The peculiarities of form of vibration are most easily discussed in the case of a musical string whose vibrations are started or maintained in a given manner. The smoothest and purest quality of tone that can be produced is known as a simple tone. When a string produces a simple tone its motion is such that its shape at any moment is that of a curve of sines, and that every point of the string executes oscillations according to the pendulum law.
Simple tones are also produced by any vibrating surface which moves according to the pendulum law. The method for producing simple tones given by Helmholtz, and commonly employed, is to use tuning-forks as the sources of sound, and present their extreme faces to the opening of a resonator or air-chamber arranged so as to vibrate to the same note as to fork. Resonators may be conveniently made from wide-mouthed bottles with flat corks having holes bored in them. The dimensions are usually found by trial, through data exist for their calculation. Simple tones have also been produced by fitting a sort of organ-pipe mouthpiece into the corks of such bottles. The mouths require to be cut up much higher than usual; the notes produced are of an exceedingly full and pure character. Such bottle-notes can be blown from an organ-bellows, and being easily manipulated are very suitable for experiments on the properties of simple tones.
The law of Ohm states that the simple tone or pendulum vibration is that to which the sensation of pitch is attached in its simplest form. If the motion which constitutes the vibration of a note be of any other type, it is capable of being analysed by the ear into a series of simple tones according to what is called Fourier’s Theorem. This is most simply described in connexion with stretched strings, assuming that the notes which are exhibited in the form of the string pass over into the air through the sound-board without essential alteration of quality, which appears to be true in a general way. Fourier’s Theorem, as applied to a string, states that the motion of the string is equivalent to the sum of the motions which would result if there were a curve of sines of the whole length, two curves of sines each of half the length, three each of one-third the length, and so on, -- the amplitudes being determined when the total motions to be represented is given. This equivalence is true mechanically; the law of Ohm says that it is also true for the ear. Hence a great presumption that the ear acts by a receptive mechanism obeying the laws of mechanics.
The notes formed by the division of a string into two, three, or more parts are commonly called harmonics, They are also called overtones; but this word includes such cased as those of bars, &c., where the notes produced by these divisions are not harmonious with the fundamental. Harmonics play an important part in the theory of consonant combinations, but the theory of consonance cannot be rested entirely upon the properties of harmonics.
Footnotes
104-1 Nürr, Zeitschrift für Biologie, 1879, p. 297.
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