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 Supposition 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.
Thus 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. [Stainer, John; Barrett, W.A.; A Dictionary of Musical Terms; Novello, Ewer and Co., London, pre-1900]
Ramsay
which seems to show that not only has one part of a vibrating string sympathy with another part of it so as to go into harmonic partials, as we have just seen, but as if the very air itself had sympathy with harmoniously vibrating strings; for Tartini observed that two harmonious sounds being produced and sustained as they can be, for example, by a strong bow on the violin, a third sound will be heard. Tartini's name for it was simply "a third sound." This is not an overtone, as Helmholtz has called the harmonic partials of one sounding string, but an undertone, because it is a "grave harmonic," away below the sounds of the two strings which awaken it. The subject of these undertones has been carefully studied since Tartini's day, and more insight has been obtained since we are now able to count and register the vibration of any musical sound. Helmholtz has called these third sounds of Tartini's "difference sounds," because when awakened by two strings, for example, the vibration-number of the third tone is the difference of the vibrations-numbers of the two tones which awaken it. The note C with vibration-number 512, and another C whose vibration-number is 256, the octave, awakened no third sound, because there is no difference between the two numbers - the one is just the doubled or halved; but if we take C256 and G381, its fifth, the difference number is 128; this being a low octave of C256, it has the effect of strengthening the upper one. Helmholtz found this to be the law of the third sound as to its producing, and the effect of it when produced. This third sound, mysteriously arising in the air through the sympathy it has with all concordant things, is another among many more suggestions that the whole Creation is measured and numbered to be in sympathy one part with another. The Creation is a universe. [Scientific Basis and Build of Music, page 60]
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