to B, or to C'# or thereabouts, beats will be heard when they are sounded together, although they are far beyond the Beating Distance.
This fact, though at first sight inconsistent with the foregoing, is, in reality, not so; for the beats in question are not produced by the two simple tones, but by one of them and a differential tone generated by them. The following figures show tins—
C and B generate a Differential, the vibration number of which is 224, and this with the tone 0 will produce 256 — 224 = 32 beats per second; similarly C and generate the Differential 284,
which with C gives 284 — 256 = 28 beats per second; and on reference to the table on page 159, we see that both 32 and 28 beats per second, are well within beating distance at this part of the musical scale.
Again, if we sound together two forks, one tuned to 0 and the other tuned only approximately to G, beats may be heard, but only when the forks are vigorously excited. Thus taking C = 256, G should be 384 : let the G fork, however, be mistuned to 380, then
and these two differential tones will produce 132 — 124 = 8 beats per second. These beats, however, will be faint, inasmuch as the differential tone of the second order is itself very weak.
With other intervals beyond the beating distance, no dissonance will be heard between simple tones. Two forks, forming any interval between a minor and a major third for example, in the middle or upper part of the musical scale, produce no roughness when sounded together; the interval may sound strange to musical ears, but there is no trace of dissonance.
To sum up, therefore: if the interval between two simple tones be gradually increased beyond the beating distance, no roughness or dissonance will be heard, till we are approaching the Fifth; and only then, if the tones are sufficiently loud to produce a Differential of the second order: on still further widening the interval, beats may be heard in the neighbourhood of the octave, due to a Differential of the first order.