produced between overtones, never exists, except where it is already present by virtue of direct action among the overtones themselves." Thus we may pass on to the last cause of beats between compound tones, viz:—
(5) Beats due to Summation Tones. Summation Tones, though certainly not very loud, are much louder than they are commonly supposed to be. On the Harmonium and American Organ, Summation Tones, generated by any pair of tones on the lower half of the key-board, may be readily heard without the use of resonators, and therefore cannot but have some effect on the resulting sound.
From the table at the end of Chap. XII, page 134, we see that the fundamentalsgenerate the Summation Tone (s) which coincides with the 3rd partial of the lower tone, so that in the case of the Octave, no new element is introduced.
In the Fifththe Summation Tone (n) is a new tone introduced
between the 2nd and 3rd partials of the (d|), but it forms no dissonance with them.
In the Fourththe Summation Tone (approximately ta) comes very near the Beating Distance with the 2nd partial of d. For, taking d = 400, then S| = 300, the Summation lone = 300 + 400 = 700, and 2nd partial of d1 = 800; therefore the Summation Tone and (d1) will produce 800 — 700 = 100 beats per second, which, as may be seen from the table on page 159, is just beyond beating distance at this pitch.
In the Major Third, the Summation Tone (r) will dissonate
with the 2nd partials (d) and (n) of both tones; the same is true in the case of the Minor Third, but the dissonance is harsher; for take s, = 300; then pi| == 250, the Summation Tone is 300 + 250 = 550, and the 2nd partials (s) and (n) are 600 and 500 respectively. Thus the number of beats per second is 600 — 550 = 50, and 550 — 500 = 50, which, at this pitch, is less than the number due to the whole tone.
In the Major Sixth,the Summation Tone (f) dissonates at the
interval of a tone, with the 3rd partial of the (d|). The Minor Sixth, is better in this respect, for take
250, Summation Tone = 650, which with the 3rd partial t (= 750) will produce 750 — 650 = 100 beats per second, which at this pitch is only just on the borders of the Beating Distance.