CHAPTER LXVI.

CONSONANCE, DISSONANCE, AND RESULTANT TONES.

929. Concord and Discord.-Every one not utterly destitute of musical ear is familiar with the fact that certain notes, when sounded together, produce a pleasing effect by their combination, while certain others produce an unpleasing effect. The combination of two or more notes, when agreeable, is called concord or consonance; when disagreeable, discord or dissonance. The distinction is found to depend almost entirely on difference of pitch, that is, on relative frequency of vibration; so that the epithets consonant and dissonant can with propriety be applied to intervals.

The following intervals are consonant: unison (1 : 1), octave (1 : 2), octave+fifth (1 : 3), double octave (1:4), fifth (2:3), fourth (3:4).

The major third (4:5) and major sixth (3:5), together with the minor third (5:6) and minor sixth (5:8), are less perfect in their

consonance.

The second and the seventh, whether major or minor, are dissonant intervals, whatever system of temperament be employed, as are also an indefinite number of other intervals not recognized in music.

Besides the difference as regards pleasing or unpleasing effect, it is to be remarked that consonant intervals can be identified by the ear with much greater accuracy than those which are dissonant. Musical instruments are generally tuned by octaves and fifths, because very slight errors of excess or defect in these intervals are easily detected by the ear. To tune a piano by the mere comparison of successive notes would be beyond the power of the most skilful musician. A sharply marked interval is always a consonant interval.

930. Jarring Effect of Dissonance.-According to the theory propounded by Helmholtz, the unpleasant effect of a dissonant interval consists essentially in the production of beats. These have a jarring effect upon the auditory apparatus, which becomes increasingly disagreeable as the beats increase in frequency up to a certain limit (about 33 per second for notes of medium pitch), and becomes gradually less disagreeable as the frequency is still further increased. The sensation produced by beats is comparable to that which the eye experiences from the bobbing of a gas flame in a room lighted by it; but the frequency which entails the maximum of annoyance is smaller for the eye than for the ear, on account of the greater persistence of visible impressions. The annoyance must evidently cease when the succession becomes so rapid as to produce the effect of a continuous impression.

We have already (§ 888) described a mode of producing beats with any degree of frequency at pleasure; and this experiment is one of the main foundations on which Helmholtz bases his view.

931. Beats of Harmonics.-The beats in the experiment above alluded to, are produced by the imperfect unison of two notes, and indicate the number of vibrations gained by one note upon the other. Their existence is easily and completely explained by the considerations adduced in § 888. But it is well known to musicians, and easily established by experiment, that beats are also produced between notes whose interval is approximately an octave, a fifth, or some other consonance; and that, in these cases also, the beats become more rapid as the interval becomes more faulty.

These beats are ascribed by Helmholtz to the common harmonic of the two fundamental notes. For example, in the case of the fifth (2:3), the third tone of the lower note would be identical with the second tone of the upper, if the interval were exact; and the beats which occur are due to the imperfect unison consequent on the deviation from exact truth. All beats are thus explained as due to imperfect unison.

This explanation is not merely conjectural, but is established by the following proofs:—

1. When an arrangement is employed by which the fifth is made false by a known amount, the number of beats is found to agree with the above explanation. Thus, if the interval is made to correspond to the ratio 200: 301, it is observed that there are 2 beats to every 200 vibrations of the lower note. Now the harmonics which

BEATS.

943 are in approximate unison are represented by 600 and 602, and the difference of these is 2.

2. When the resonator corresponding to this common harmonic is held to the ear, it responds to the beats, showing that this harmonic is undergoing variations of strength; but when a resonator corresponding to either of the fundamental notes is employed, it does not respond to the beats, but indicates steady continuance of its appropriate note.

3. By a careful exercise of attention, a person with a good ear can hear, without any artificial aids, that it is the common harmonic which undergoes variations of intensity, and that the fundamental notes continue steady.

932. Beating Notes must be Near Together.-In order that two simple tones may yield audible beats, it is necessary that the musical interval between them should be small; in other words, that the ratio of their frequencies of vibration should be nearly equal to unity. Two simple notes of 300 and 320 vibrations per second will yield 20 beats in a second, and will be eminently discordant, the interval between them being only a semitone (15:16), but simple notes of 40 and 60 vibrations per second will not give beats, the interval between them being a fifth (2:3). The wider the interval between two simple notes, the feebler will be their beats; and accordingly, for a given frequency of beats, the harshness of the effect increases with the nearness of the notes to each other on the musical scale.1 By taking joint account of the number of beats and the nearness of the beating tones, Helmholtz has endeavoured to express numerically the severity of the discords resulting from the combination of the note C of 256 vibrations per second with any possible note lying within an octave on the upper side of it, a particular constitution (approximately that of the violin) being assumed for both notes. He finds a complete absence of discord for the intervals of unison, the octave, and the fifth, and very small amounts of discord. for the fourth, the sixth, and the third. By far the worst discords are found for the intervals of the semitone and major seventh,

1 The explanation adopted by Helmholtz is, that a certain part of the ear-the membrana basilaris-is composed of tightly stretched elastic fibres, each of which is attuned to a particular simple tone, and is thrown into vibration when this tone, or one nearly in unison with it, is sounded. Two tones in approximate unison, when sounded together, affect several fibres in common, and cause them to beat. Tones not in approximate unison affect entirely distinct sets of fibres, and thus cannot produce interference.

and the next worst are for intervals a little greater or less than the fifth.

933. Imperfect Concord.-When there is a complete absence of discord between two notes, they are said to form a perfect concord. The intervals unison, fifth, octave, octave + fifth, and the interval from any note to any of its harmonics, are of this class. The third, fourth, and sixth are instances of imperfect concord. Suppose, for example, that the two notes sounded together are C of 256 and E of 320 vibrations per second, the interval between these notes being a true major third (4:5); and suppose each of these notes to consist of the first six simple tones.

In searching for elements of discord, we select (one from each line) two multiples differing by unity.

Those which satisfy this condition are

4 and 5;

16 and 15;

24 and 25.

But the first pair (4 and 5) may be neglected, because their ratio differs too much from unity. Discordance will result from each of the two remaining pairs; that is to say, the 4th element of the lower of our two given notes is in discordance with the 3d element of the upper; and the 6th element of the lower is in discordance with the 5th element of the higher. To find the frequencies of the beats, we must multiply all these numbers by 64, since 256 is 4 times 64, and 320 is 5 times 64. Instead of a difference of 1, we shall then find a difference of 64, that is to say, the number of beats per second is 64 in the case of each of the two discordant combinations which we have been considering.

934. Resultant Tones.-Under certain conditions it is found that two notes, when sounded together, produce by their combination other notes, which are not constituents of either. They are called resultant tones, and are of two kinds, difference-tones and summation-tones. A difference-tone has a frequency of vibration which is the difference of the frequencies of its components. A summation-tone has a frequency of vibration which is the sum of the

frequencies of its components. As the components may either be fundamental tones or overtones, two notes which are rich in harmonies may yield, by their combination, a large number of resultant tones.

The difference-tones were observed in the last century by Sorge and by Tartini, and were, until recently, attributed to beats. The frequency of beats is always the difference of the frequencies of vibration of the two elementary tones which produce them; and it was supposed that a rapid succession of beats produced a note of pitch corresponding to this frequency.

This explanation, if admitted, would furnish an exception to what otherwise appears to be the universal law, that every elementary tone arises from a corresponding simple vibration.1 Such an exception should not be admitted without necessity; and in the present instance it is not only unnecessary, but also insufficient, inasmuch as it fails to render any account of the summation-tones.

Helmholtz has shown, by a mathematical investigation, that when two systems of simple waves agitate the same mass of air, their mutual influence must, according to the recognized laws of dynamics, give rise to two derived systems, having frequencies which are respectively the sum and the difference of the frequencies of the two primary systems. Both classes of resultant tones are thus completely accounted for.

The resultant tones-especially the summation-tones, which are fainter than the others—are only audible when the primary tones are loud; for their existence depends upon small quantities of the second order, the amplitudes of the primaries being regarded (in comparison with the wave-lengths) as small quantities of the first order.

If any further proof be required that the difference tones are not due to the coalescence of beats, it is furnished by the fact that, under favourable conditions, the rattle of the beats and the booming of the difference-tones can both be heard together.

935. Beats due to Resultant Tones.-The existence of resultant tones serves to explain, in certain cases, the production of beats between notes which are wanting in harmonics. For example, if two simple sounds, of 100 and 201 vibrations per second respectively, are sounded together, one beat per second will be produced between

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1 The discovery of this law is due to Ohm.