39) The overtones will be obvious.
But we have to remember that with strings, pipes and instruments generally the fundamental tone is accompanied by overtones, called also " upper partials," and beating within the dissonance range may occur between these overtones.
Let the two tones with their harmonic overtones be 256 512 768 1024 1280 1536 512 1024 1536.
The fundamental and overtones of the second all coincide with overtones of the first.
Take as a further example the fifth with harmonic overtones as under The fundamental and overtones of the second either coincide with or fall midway between overtones in the first, and there is no approach to a dissonant frequency of beats, and the concord is perfect.
The harshness of deep notes on instruments rich in overtones may be explained as arising from beats between successive overtones.
Thus, if a note of frequency 64 is sounded, and if all the successive overtones are present, the difference of frequency will be 64, and this is an unpleasant interval when we get to the middle of the scale, say to overtones 256 and 320 or to 512 and 576.
Koenig, Quelques experiences d'acoustique (1882) describes apparatus and experiments, intended to show, in opposition to Helmholtz, that beats coalesce into tones, and also that the quality of a note is affected by alteration of phase of one of its component overtones relative to the phase of the fundamental.
He showed, both by analysis and by synthesis, that quality depends on the order, number and intensity of the overtones or harmonics that may, and usually do, enter into the structure of a musical tone.
A " stopped pipe " in an organ is a pipe of this type, and both the fundamental and the overtones may occur simultaneously when it is blown.
The whole series of fundamental and overtones gives the complete set of harmonics of frequencies proportional to 1, 2, 3, 4, ..., and wave-lengths proportional to 1, 2, 3, 4 ..
It is evident that the overtones will follow the same rule as for a pipe opened at both ends.
But keeping r/X small we may as before form stationary waves, and it is evident that the series of fundamental and overtones will be just as with the air in pipes, and we shall have the same three types - fixed at one end, free at both ends, fixed at both ends - with fundamental frequencies respectively 41, p ' 21 V p, and I velocity in rod =velocity in air X distance between dust heaps.