Books › A dictionary of musical terms, 1889 › scan 22
Scan 22
A dictionary of musical terms, 1889
14 lines. Read by computer on 9 September 2026. No printed page number was found. See the scan on archive.org.
1the beats are most distinctly heard succeeding each other at perceptible intervals—whereas according to what was said above they should occur so rapidly as not to be heard at all. Two explanations of this phenomenon have been given, of which by far the most simple is due to Helmholtz—and which here follows. It appears that when the tones are simple and at a sufficiently large interval the beats should occur too rapidly to be heard, whereas when the interval is played on a violin they are easily distinguishable. The reason of this fact is that in the latter case the tones are no longer simple but compound—and the beats which are heard are not due to the fundamental tones themselves but arise from two of their harmonic components which are nearly in unison. Suppose the ratio of the interval between the fundamental tones to be \( \frac{m}{n} \), that is, let \( \frac{m}{n} \) be the fraction, reduced to its lowest terms, which is formed by putting in the numerator the number of vibrations per second of the upper tone, and in the denominator those of the lower. Then it is plain that the \( n \)th harmonic component of the tone \( m \), will be of the same pitch as the \( m \)th harmonic component of the tone \( n \); for they will each have exactly \( mn \) vibrations per second. Now let \( \frac{M}{N} \) be the ratio, expressed in the same way, of another interval, nearly, but not quite, equal to \( \frac{m}{n} \); then the \( n \)th harmonic component of \( M \) will have \( Mn \) vibrations per second, while the \( m \)th component of \( N \) will have \( Nm \). Now since \( \frac{M}{N} \) is nearly equal to \( \frac{m}{n} \), the difference between \( Mn \) and \( Nm \) will be a small number; and when the two notes are sounded together the number of beats per second will be equal to that difference.
3For example, let \( \frac{m}{n} \) be the ratio of a fifth, that is the fraction \( \frac{3}{2} \), and let \( \frac{M}{N} \) represent very nearly the same interval, say \( \frac{397}{284} \); then the difference between \( Mn \) and \( Nm \), or 794 and 792, is 2; hence if two stringstuned apart at an interval represented by \( \frac{397}{284} \) are sounded simultaneously there will be two beats heard per second.
519. When the vibrations of the air due to a number of different sounds which co-exist at the same time are infinitely small, they are merely superposed one on another, so that each separate sound passes through the air as if it alone were present; and this law of superposition holds, though only approximately, until the vibrations have increased up to a certain limit, beyond which it is no longer true. Vibrations which give rise to a large amount of disturbance produce secondary waves; and it is to these that the phenomena of resultant tones are due.
7Thus if two notes a fifth apart, for instance, are forcibly sounded together, a third tone is heard an octave below the lower of the two, and this ceases to be perceptible when the loudness of the concord diminishes. In general the resultant tone of any combination of two notes is produced by a number of vibrations per second equal to the difference of the numbers per second of the notes. This fact formerly led to the supposition that the resultant tone was produced by the beats due to the consonance, which, when they occurred with sufficient rapidity, linked themselves together so as to form a continuous musical note. If this were so it is clear that the resultant ought to be heard when the original notes are sounded gently as well as forcibly; and it was the failure of this condition that led Helmholtz to the re-investigation of their origin. These resultant tones have been named by him difference tones; he has also discovered the existence of resultant tones formed by the sum of the numbers of vibrations of the primaries. These summation tones as they are called cannot be explained on the old theory.
920. The theory of beats explains the law that the smaller the two numbers are, which express the ratio of their vibrations, the smoother is the combination of any two tones. When two simple tones are sounded together whose rates of vibration per second differ by more than 132, the beats, according to Helmholtz, totally disappear. As the difference grows less the beats become more and more audible, the interval meanwhile growing proportionately dissonant, till they number 33 per second, at which point the dissonance of the interval is at its maximum.
11This, however, depends upon the position of the interval as regards its pitch. For it should be remembered that though the ratio of any given interval remains the same whatever the absolute pitch of its tones may be, yet the difference of the actual numbers of their vibrations, and therefore the number of beats due to their consonance, alters with it. And vice versa, if the difference of the number of vibrations remains constant, the interval must diminish as its pitch rises. For instance, either of the following combinations
13would give rise to 33 beats per second, since the numbers of vibrations of their tones per second, are 99-66, and 528-495, respectively. Now it is obvious that in the latter case the dissonance would be far greater than in the former.
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