ear. I now replaced the Ut, resonator and fork by an Ut, resonator and its corresponding fork, and again rotated the disk with the same velocity that it had during the above-described experiment. In these circumstances I no longer experienced a continuous sensation, but one which reminded me somewhat of the clatter of frogs in a marsh. This fact at once showed that a greater number of beats per second were required to blend the separated pulses of a sound of higher pitch; and this blending I actually obtained on sending into my ear about one hundred beats per second of Ut4.

I now prepared a series of disks adapted to four octaves of resonators and forks, and made many experiments to determine the durations of the residual sonorous sensations of several simple sounds-Ut, of 64 vibrations per second being the lowest note of the series, and the highest being Ut, of 1024 vibrations. I was not able to use an Ut, fork and resonator; so I substituted for the former an Ut, closed wooden organ-pipe, gently blown, and for the latter a small funnel of gutta percha, whose mouth was placed close to the perforated disk, while a rubber tube connected the funnel with the ear. I will here remark that in some series of experiments the resonators were replaced by this funnel of gutta percha, and the determinations thus made were the same as those reached by the use of the resonators.

The above-mentioned determinations I published in the American Journal of Science for October 1874, and embraced them

53248

in a law which has for its expression D=(532 +24) .0001, in

N+23

+24)·0001,

which D equals the duration of the residual sonorous sensation corresponding to N number of vibrations per second.

The precise determinations of the durations of the residual sonorous sensations are difficult, by reason of the complex character of the sound perceived when vibrations of a tuning-fork are sent intermittently into a resonator by means of a revolving perforated disk; and the difficulty of the determinations is increased by the fatigue and deadening of the ear, caused by the beats which enter it from the resonator.

The important applications of this law in the physiology of audition and in the elucidation of the fundamental facts of musical harmony, demanded that I should have my determinations reviewed by ears more highly cultivated than mine in the appreciation of pitch and of musical intervals, and more skilled in the direct oral analysis of composite sounds into their simple tones. Since my publication in October 1874, I have had the good fortune to elicit in Madame Emma Seiler, and in her son Dr. Carl Seiler, a profound interest in my researches. They have spent considerable time in the redetermination of the dura

tions of the residual sonorous sensations, making use, under my directions, of the same apparatus which I employed in my original experiments. Madame Seiler in former years worked much with Helmholtz, assisting him with her fine ear in experiments contained in his renowned work Die Lehre von den Tonempfindungen. This lady unites to educated musical perceptions a knowledge and appreciation of recent advances in physiological acoustics; and hence I have great confidence in the determinations made by her.. These results I give in the following Table, which I desire to take the place of that published in my paper of October 1874.

Column S of this Table contains the simple sounds experimented on; they are designated in the notation stamped by König on his forks. Column N gives the number of vibrations* per second corresponding to the sounds of column S. In column D are the corresponding durations of the residual sensations, expressed in vulgar and in decimal fractions. The reciprocal of the number of beats per second required to produce a continuous sensation by a given sound is taken as the duration of the residual sensation of this soundt. In column L are given the number of wave-lengths contained in the separate impulses into which the sound had been divided in order to produce the continuous sensation.

Although at first sight the apparatus which I have used in this research may appear coarse, yet experience showed that the accuracy of a determination dep ended more on the ear than on

* I here, as always, refer to complete vibrations, i. e. to a motion to and fro, or to what the late Professor De Morgan proposed to call a swing

swang.

That is to say, we take as the duration of the residual sensation the interval during which the impress of a beat has not diminished sufficiently in intensity to cause discontinuity in the sensation. To obtain the duration of the entire sensation, we should have to know the intensity of the sensation at the end of the above interval, and the law giving the rate of diminution of the sensation.

the mechanical appliances of our experiments; for the average difference in the measures of the duration of any one residual sensation did not exceed the 4 of a second. The perforated disk made 3 revolutions to one of the driving-crank; and if the disk has twelve perforations, then the above difference of 4000 of a second is given by the difference between two observations, in one of which the driving-crank made 30 revolutions in ten seconds, and in the other made 29 revolutions in the same time. It is evident that the apparatus readily detects this difference, especially as I often ran it during thirty seconds to obtain the number of beats striking the ear during one second.

Before accepting as final the above determinations, I ascertained that great differences in the intensities of the pulses had little effect on the number of beats required to produce a continuous sensation. When a great increase in intensity was given to the pulses, their number had to be slightly increased to produce the same continuous sensation as that experienced with feebler pulses; but the difference was barely measurable. It is also important to remark that, after the blending of the pulses has been once attained, a further increase in the velocity of the disk does not change the character of the sensation. Extreme velocities, of course, produce such violent agitations at the mouth of the resonator as to render experimenting impossible.

I have projected the above determinations into the accompanying curve (fig. 2), placing on the axis of abscissæ the numbers of vibrations of the various sounds, as designated at the base of the figure, and on the ordinates the corresponding durations of their residual sensations. Thus has been obtained the full-lined curve of the figure. The dotted curve is an equilateral hyperbola, and expresses an assumed law-that the durations of the residual sensations are inversely as the numbers of vibrations producing them. In drawing the latter curve, I took the point corresponding to the ordinate of Ut, as the basis of the assumption.

From the discussion of the curve of the experiments, we find that the law connecting the pitch of a sound with the duration of its residual sensation may be expressed thus,

in which D equals, in fractions of a second, the duration of the residual sonorous sensation corresponding to N number of vibrations per second.

We have already spoken of the difficulty of the determination of the residual sonorous sensations by reason of the complex character of the sound perceived when the vibrations of a tuning

fork are sent intermittently into a resonator by means of a revolving perforated disk. I will now describe the character of the successive sensations experienced when, starting from rest, we gradually increase the velocity of rotation of the disk until the separate beats of the fork blend into a smooth continuous sensation. When the disk is stationary, with one of its openings opposite the mouth of the resonator, it is evident that the ear will

experience a simple sonorous sensation when a tuning-fork is brought near the mouth of the resonator. On revolving the perforated disk, two additional or secondary sounds appear one slightly above, the other slightly below the pitch of the fork. An increasing velocity of rotation causes the two secondary sounds to diverge yet further from the note of the beating fork, until, on reaching a certain velocity, the two secondary sounds become separated from each other by a major sixth, while at the same moment a resultant sound appears, formed by the union of the sound of the fork with the upper and lower of the secondary sounds. This resultant is the lower second octave of the note given by the fork. On further increasing the velocity of rotation of the disk, the two secondary sounds and the resultant disappear, and the ear experiences only the sensation of the simple sound produced by the fork, whose beats at this stage of the experiment have blended into a smooth continuous sensation. These successive and gradual changes, as they happen with an Ut, fork, we have indicated in steps of semitones in the appended musical notation. The sound of the fork is given in the semibreve, while the crotchets represent the secondary sounds and the resultant sound. In the fourth bar the upper note bE proceeds to #D in the fifth bar. This is so because in the natural scale #D is higher than bE.

2. The Determination of the numbers of Beats, throughout the musical scale, which produce the greatest Dissonances.

The determination of the law which shows the connexion existing between the pitch of a sound and the number of its beats which causes the most dissonant sensation, was made with the same apparatus that served for the disco very of the law just discussed. The determination of the number of beats producing the greatest dissonant effect with a given sound is difficult; for the point of maximum dissonance is not sharply marked, and individual judgment and peculiarities come in, so that the range of the determination for any given sound, by different persons, is considerable. But on discussing the determinations reached by any one person, I found that they followed a well-marked law, which, as might have been inferred, is closely connected with the law of the duration of the residual sensation. Indeed we find that any one observer always makes the numbers produ