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64 revolutions per second, the two notes corresponded to 51.2 and 115.2 vibrations per second, the difference-tone being 64.
The pitch was determined on different occasions by different methods. Firstly, by noting the beats between the higher note and a König's fork adjusted to 115.2 complete vibrations per second, and secondly, by watching the row of eight holes through a slit carried by a fork which gave 25.5 vibrations per second.
The effect was rather feebler than in the last experiment, but there was absolutely no doubt as to the objective reality of the difference-tone. The bands regularly disappeared when the required pitch was obtained, and reappeared when it was lost.
Again the 15 and 9 rows of holes were used. The difference-tone is thus proportional to 6, and König's beat-tone to 9×2-15-3. When the rate of revolution was 10-6 the two notes were 160 and 96 respectively. In this experiment the mirror-resonator which responds to 161 vibrations was employed to determine the rate of the siren. The bands and the spot of light were sometimes watched together: on another occasion one observer who could not see the bands raised his hand whenever the spot of light moved. The bands invariably disappeared at the instant that this signal was made.
The next experiment was directed to determine the objective reality of König's lower beat-tone when the interval was greater than an octave. The 8 and 18 rows of holes being kept open as before, the speed was increased until the lower note was that of 256 vibrations. The upper note was then 576, and König's lower beat-tone was of 576-2×256=64 vibrations.
We lay less stress on negative than on positive results; but we tried for a long time on two occasions to get evidence of the objective character of the note, but entirely failed. The pitch was determined by the beats with a 256 fork.
We next turn to observations on the summation-tone. The 8 and 10 rows of holes were opened, so that when the cover made 3.55 revolutions per second the summation-tone would be that of 18 x 3.5=64 vibrations,
The pitch of the notes given by the siren was again determined in different ways on different occasions. The summation-tone being produced in the lower box, the 15 row in the upper box was also opened, thus producing a note of 15 x 3.5=53.3 vibrations per second. The required speed was determined by making the beats vanish between this note and a König's fork tuned to give 53.3 vibrations. With this method it was difficult to keep the speed constant for a length of time sufficient to disturb the resonating fork appreciably. When the pitch was altered very slowly the bands disappeared just as the right note was reached, and did not disappear at any other time during the experiment.
On another occasion the 9 and 12 rows of holes were opened, so that the summation-tone of 64 vibrations would be given when the siren made 3·05 revolutions per second. The 18 row of holes was watched through a fork of 27.2 vibrations, so that 544 views would be obtained while a hole moved over 18 x 3.05-54.9 intervals. Hence the right pitch was obtained when the holes moved slowly forwards. The bands invariably disappeared when this state of things was attained.
On a third occasion the lower cover of the siren was covered with a thin piece of silvered glass as above described, carrying a concentric circle of black paper, the edge of which was divided into 18 equidistant cogs. An image of these was produced on a screen by a lens, and made intermittent by the 27-vibrations fork. The disturbance due to the summation-tone was again and again made evident when the images of the cogs appeared to be moving slowly. In the intervals the bands were beautifully steady.
The earlier of these experiments were performed before, and the later ones after, the apparatus had been taken down and set up again in another room. They left in the minds of those who saw them no shadow of doubt as to the objective reality of a note corresponding in frequency with the summation
. We now turn to experiments intended to throw light on the cause of the production of this note.
It has been suggested that the summation-tone may be the difference-tone of partials. König (Acoustique, p. 127) remarks that it may occasion some surprise that the particular harmonics whose difference-tone corresponds to the summation-tone should be especially prominent; but he points out
that in some cases the difference-tones of the lower harmonics correspond either to the fundamentals or to some of their upper partials. In the case of the fourth (3:4), however, König remarks that the 5th partials would give a difference-tone (5) which could be distinguished from the lower partials, and that the difference-tone of the 7th partials would give the summation-tone. Now we have already proved (Exp. IV.) that the summation-tone produced by two notes separated by the interval of a fourth (9: 12) is objective; and if this is due to the difference-tone of the 7th partials, there seems to be no reason why the difference-tone of the 5th partials should not be objective also, and probably more intense.
We therefore ir creased the velocity of revolution to 4.27 per second, the 9 and 12 rows of holes being opened as before. The frequencies of the two notes were thus 38.43 and 51.24. The pitch was determined by keeping the 12 holes nearly stationary when viewed 51 times a second by aid of the 25.5 fork. The first difference-tore was 12.81, and the differencetone of the 5th partials was 64:05. When the speed corresponding to this difference-tone was attained there were occasional flickers of the bands, so that it is possible that it has an objective existence. But, on the other hand, the effect was less than that produced by the summation-tone. The bands never disappeared for any considerable length of time, as they did when the fork responded to the summation-tone, and the experiment left no doubt in our minds that the greater effect was produced by the summation-tone.
The same point was also investigated in another way. If the summation-tone of two notes of frequencies p and 9 corresponds to the difference-tone of the nth partial, we
(p+q) = n(p−q),
where n is an integer. If, however, the 9 and 16 rows of holes were opened,
so that the summation-tone could not be produced by partials of the same order. The 10th partial of the higher note beating with the 15th of the lower note (160-135=25) would indeed have the same frequency as the summation-tone, but it appears to us absurd to suppose that so improbable a combination should produce appreciable results. It is true that lower partials may give beat-tones near to the summation-tone,
Thus 5x 16-6x9=26. But if we are to assume that any pair of partials can thus produce objective tones, the number of combinations will be so great that the fork ought to have been disturbed frequently when the note of the siren was being raised to the required pitch. As a matter of fact, when once the C of 64 vibrations was passed, so that all the partials were higher than the pitch of the resonating fork, no such disturbances were ever observed except when the difference- or summation-tone of the primaries was produced. Putting, therefore, all such fantastic combinations aside, the experiment may be regarded as a test whether the summation-tone can be produced when it cannot be due to two partials of the same order.
When the velocity of revolution was 2-56 per second, the 16 and 9 holes gave notes of 40·96 and 23·04 vibrations. The sum of these is 64. The 12 holes were viewed through a slit alternately closed and opened by a fork of 15 vibrations per second, and when the holes appeared to move slowly the summation-tone caused the bands to disappear.
In this experiment, however, the third partial of the lower note corresponds to 69.12 vibrations, and we thought it desirable to make sure that the disturbance attributed to the summation-tone was not in reality due to this partial. This was the more important, because the difference in the speeds of the siren when the summation-tone and the partial in question corresponded to 64 vibrations was very small.
Thus, when the speed was 2.56 revolutions per second each of the 12 holes would advance through 30.72 intervals in a second, and since the fork gave 30 views per second the holes would appear to move slowly forwards.
When the speed was 2:37 revolutions per second the third partial of the lower note (9 row of holes) would be 3x9x2.3764, and each hole of the 12 rows would advance through 2844 intervals—that is, would appear to recede through 1.56 intervals per second. Thus the partial would be most efficient in promoting disturbance when the holes appeared to go backward with moderate speed.
The question to be answered was whether these two disturbances could be confused with each other.
When care was taken to keep the pressure in the windchest the same whether one or both sets of holes were opened, the effect of the partial produced by the 9 set of holes could hardly be detected. The bands were shaken a little when the row of 12 holes appeared to move backwards, but they did not disappear; whereas they were completely wiped out by the summation-tone when the two notes were sounded.
When the pressure on the wind-chest was increased, the rate of revolution being nevertheless maintained constant by pressing lightly on the axle of the siren with a straw, the effect of the partial was more marked, but it was always produced when the holes appeared to move backwards.
On the other hand, when both notes were sounded together and when the pitch was gradually reduced to the desired point, the disturbance always began when the holes moved slowly forwards. If the pitch fell very slowly it was possible to note a reduction of the disturbance, followed by an increase when the holes appeared to move backwards.
We thus convinced ourselves that the effects of the two sources of disturbance could be distinguished, and that the supposed summation-tone was not due to the partial of the lower note.
We have also succeeded in demonstrating the reality of the summation-tone with a mirror-resonator constructed by Professor Boys to respond to a vibration-frequency of 576.
The rows of 15 and 12 holes being opened, notes of 320 and 256 vibrations were produced. When they were sounded separately, the mirror moved slightly. When they were sounded together, the spot of light was driven off the scale when the upper note coincided with that of a 320-vibration fork, but immediately returned when this pitch was lost.
The experiment was varied by using the 16 and 12 rows, and also the 16 and 9 rows. The summation-tone corresponds to 576 vibrations when the upper note is of 329·15 and 360 vibrations in these two cases respectively. The 320-fork was used, and the disturbance occurred in the one case when the pitch of the note was nearly the same as before, and in the other when it was about a tone higher.
We attach great importance to this corroboration of our results by an instrument of a totally different construction from that first employed.
The attempt to obtain proof of the existence of a differencetone by means of the mirror-resonator of 161 vibrations has not been successful. The instrument is much less affected by the note to which it responds than is that which answers to 576 vibrations, even when that note is produced directly by the siren. It is, therefore, perhaps not wonderful that it gives no reliable evidence of the existence of a difference
We now sum up the results we have obtained in two tables.