Syncopated Measures 9.77 8.96 9.61 13.78 Absolute.

Unaccented, 1.000 0.915 0.983 1.165 Relative.

Syncopated Measures 11.57 11.07 11.53 21.50 Absolute.

Accented, 1.000 0.957 0.996 1.858 Relative.

These averages hold for every subject where the syncopated measure receives accentuation, and for two out of three reactors where it is unaccented. The latter individual variation shows a progressive increase in intensity throughout the series.

Here, as in the preceding forms, a well-established type is presented.

Not only when accentuation is consciously introduced, but also when the attempt is made--and in so far as the introspection of the reactor goes, successfully made--to maintain a uniformity among the reactions of the full and syncopated measures, the emphasis on the latter is unconsciously increased. In the accented form, as before, there is a clear discrimination into two grades of intensity (ratio of first three elements to final, 1.000:1.888) while in the unaccented no such broad separation exists (ratio of first three elements to final, 1.000:1.156).

The type of succession in each of these forms of reaction is a transformed dactylic, in which group should now be included the simple four-beat rhythm with final accent, which was found to follow the same curve. The group begins with a minor stress in both of the present forms, this stress being greater in the unaccented than in the accented type. This preponderance I believe to be due to the endeavor to repress the natural accent on the syncopated measure. In both forms the intensive value of the second element is less than that of the third, while the intensity of the initial reaction is greater than that of either of these subsequent beats. This form of succession I have called a _transformed dactylic_. It adheres to the dactylic type in possessing initial accentuation; it departs from the normal dactylic succession in inverting the values of the second and third members of the group. This inversion is not inherent in the rhythmic type. The series of three beats decreasing in intensity represents the natural dactylic; the distortion actually presented is the result of the proximity of each of these groups to a syncopated measure which follows it. This influence I believe to be reducible to more elementary terms. The syncopated measure is used to mark the close of a logical sequence, or to attract the hearer's attention to a striking thought. In both cases it is introduced at significant points in the rhythmical series and represents natural nodes of accentuation. The distortion of adjacent measures is to be attributed to the increase in this elementary factor of stress, rather than to the secondary significance of the syncopation, for apart from any such change in the rhythmical structure we have found that the reactions adjacent to that which receives accentual stress are drawn toward it and increased in relative intensity.

Further quant.i.tative a.n.a.lysis of rhythmical sequences, involving a comparison of the forms of successive measures throughout the higher syntheses of verse, couplet and stanza, will, I believe, confirm this conception of the mutable character of the relations existing between the elements of the rhythmical unit, and the dependence of their quant.i.tative values on fixed points and modes of structural change occurring within the series. An unbroken sequence of dactyls we shall expect to find composed of forms in which a progressive decrease of intensity is presented from beginning to end of the series (unless we should conceive the whole succession of elements in a verse to take shape in dependence on the point of finality toward which it is directed); and when, at any point, a syncopated measure is introduced we shall look for a distortion of this natural form, at least in the case of the immediately preceding measure, by an inversion of the relative values of the second and third elements of the group. This inversion will unquestionably be found to affect the temporal as well as the intensive relations of the unit. We should likewise expect the relations of accented and unaccented elements in the two-beat rhythms to be similarly affected by the occurrence of syncopated measures, and indeed to find that their influence penetrates every order of rhythm and extends to all degrees of synthesis.

To the quant.i.tative a.n.a.lysis of the intensive relations presented by beaten rhythms must be added the evidence afforded by the apprehension of auditory types. When a series of sounds temporally and qualitatively uniform was given by making and breaking an electric circuit in connection with a telephone receiver, the members of a group of six observers without exception rhythmized the stimuli in groups--of two, three and four elements according to rate of succession--having initial accentuation, however frequently the series was repeated. When the series of intervals was temporally differentiated so that every alternate interval, in one case, and every third in another, stood to the remaining interval or intervals in the ratio, 2:1, the members of this same group as uniformly rhythmized the material in measures having final accentuation. In triple groups the amphibrachic form (in regard to temporal relations only, as no accentuation was introduced) was never heard under natural conditions. When the beginning of the series was made to coincide with the initiation of an amphibrachic group, four of those taking part in the investigation succeeded in maintaining this form of apprehension for a time, all but one losing it in the dactylic after a few repet.i.tions; while the remaining two members were unable to hold the amphibrachic form in consciousness at all.

(_b_) The Distribution of Durations.

The inquiry concerning this topic took the direction, first, of a series of experiments on the influence which the introduction of a louder sound into a series otherwise intensively uniform exerts on the apparent form of the series within which it occurs. Such a group of experiments forms the natural preliminary to an investigation of the relation of accentuation to the form of the rhythm group. The apparatus employed was the fourth in the series already described. The sounds which composed the series were six in number; of these, five were produced by the fall of the hammer through a distance of 2/8 inch; the sixth, louder sound, by a fall through 7/8 inch. In those cases in which the intensity of this louder sound was itself varied there was added a third height of fall of two inches. The succession of sounds was given, in different experiments, at rates of 2.5, 2.2, and 1.8 sec. for the whole series. The durations of the intervals following and (in one or two cases) preceding the louder sound were changed; all the others remained constant. A longer interval intervened between the close and beginning of the series than between pairs of successive sounds. After hearing the series the subject reported the relations which appeared to him to obtain among its successive elements. As a single hearing very commonly produced but a confused impression, due to what was reported as a condition of unpreparedness which made it impossible for the hearer to form any distinct judgment of such relations, and so defeated the object of the experiment, the method adopted was to repeat each series before asking for judgment. The first succession of sounds then formed both a signal for the appearance of the second repet.i.tion and a reinforcement of the apperception of its material.

In order to define the direction of attention on the part of the observer it was made known that the factors to be compared were the durations of the intervals adjacent to the louder sound in relation to the remaining intervals of the series, and that all other temporal and intensive values were maintained unchanged from experiment to experiment. In no instance, on the other hand, did any subject know the direction or nature of the variation in those quant.i.ties concerning which he was to give judgment. In all, five subjects shared in the investigation, C., E., F., H. and N. Of these C only had musical training. In the tables and diagrams the interval preceding the louder sound is indicated by the letter B, that following it by the letter A. Totals--judgment or errors--are indicated by the letter T, and errors by the letter E. The sign '+' indicates that the interval against which it stands is judged to be greater than the remaining intervals of the series, the sign '=' that it is judged equal, and the sign '-' that it is judged less.

The first series of changes consisted in the introduction of variations in the duration of the interval following the loud sound, in the form of successive increments. This loud sound was at the third position in the series. All intensive relations and the duration of the interval preceding the louder sound remained unchanged. The results of the experiment are presented in the following table.

TABLE XXVIII.

Ratio of A to B A Errors Total Per cent.

Other Intervals. + = - + = - B A T judgts. of errors

1.000 : 0.625 2 2 2 4 2 0 4 2 6 12 50 1.000 : 0.666 4 2 0 1 3 2 4 5 9 12 75 1.009 : 0.714 5 3 0 2 2 4 5 6 11 16 69 1.000 : 0.770 5 4 0 1 1 7 5 8 13 18 72 1.000 : 0.833 1 5 0 0 0 6 1 6 7 12 50

Totals, 17 16 2 8 8 19 19 27 46 70

The value of the interval following the louder sound is correctly reported eight times out of thirty; that preceding it is correctly reported sixteen times out of thirty. The influence which such a change in intensive value introduced at a single point in a series of sounds exerts on the apparent relation of its adjacent intervals to those of the remainder of the series is not equally distributed between that which precedes and that which follows it, but affects the latter more frequently than the former in a ratio (allowing lat.i.tude for future correction) of 2:1. In the case of interval A the error is one of underestimation in twenty-seven cases; in none is it an error of overestimation. In the case of interval B the error is one of overestimation in seventeen instances, of underestimation in two. The influence of the introduction of such a louder sound, therefore, is to cause a decrease in the apparent duration of the interval which follows it, and an increase in that of the interval which precedes it.

The illusion is more p.r.o.nounced and invariable in the case of the interval following the louder sound than of that preceding it, the proportion of such characteristic misinterpretations to the whole number of judgments in the two cases being, for A, 77 per cent.; for B, 54 per cent. The effect on interval A is very strong. In the second group, where the ratio of this interval to the others of the series is 3:2, it is still judged to be equal to these others in 50 per cent. of the cases, and less in 35 per cent. Further, these figures do not give exhaustive expression to the whole number of errors which may be represented in the judgments recorded, since no account is taken of greater and less but only of change of sign; and an interval might be underestimated and still be reported greater than the remaining intervals of the series in a group of experiments in which the relation of the interval in question to these remaining intervals ranged from the neighborhood of equivalent values to that in which one was double the other. If in a rough way a quant.i.tative valuation of errors be introduced by making a transference from any one sign to that adjacent to it (_e.g._, - to =, or = to +) equal to _one_, and that from one extreme sign to the other equal to _two_, the difference in the influence exerted on the two intervals will become still more evident, since the errors will then have the total (quant.i.tative) values of A 46, and B 19, or ratio of 1.000:0.413.

Next, the position of the louder sound in the series of six was changed, all other conditions being maintained uniform throughout the set of experiments. The series of intervals bore the following relative values: A, 0.900; B, 1.100; all other intervals, 1.000. The louder sound was produced by a fall of 0.875 inch; all others by a fall of 0.250 inch. The louder sound occurred successively in the first, second, third, fourth and fifth positions of the series. In the first of these forms it must of course be remembered that no interval B exists. The results of the experiment are shown in the following table:

TABLE XXIX.

Position Apparent Values. Errors. % of Errors Ditto in B A B A T in tot. judg. quant.

Series + = - + = - B A B A 1 2 6 6 0 12 12 85.7 85.7 2 2 8 2 1 7 4 10 11 21 83.3 91.6 73.3 91.6 3 1 9 3 1 8 3 10 11 21 76.9 91.6 71.9 91.6 4 1 8 4 2 6 5 9 11 20 69.2 84.6 52.8 84.6 5 0 12 0 0 4 8 12 12 24 100.0 100.0 60.0 100.0 Totals, 4 37 9 6 31 26 41 57 98 82.3 90.7 64.5 90.7

Total judgments, 113; Errors (B = 31), A = 57.

The relatively meager results set forth in the preceding section are corroborated in the present set of experiments. That such a variation of intensity introduced into an otherwise undifferentiated auditory series, while it affects the time-values of both preceding and following intervals, has a much greater influence on the latter than on the former, is as apparent here as in the previous test. The number of errors, irrespective of extent, for the two intervals are: B, 82.3 per cent, of total judgments; A, 90.7 per cent. When the mean and extreme sign displacements are estimated on the quant.i.tative basis given above these percentages become B, 64.5; A, 90.7, respectively--a ratio of 0.711:1.000.

The direction of error, likewise, is the same as in the preceding section. Since the actual values of the two intervals here are throughout of extreme sign--one always greater, the other always less--only errors which lie in a single direction are discriminable.

Illusions lying in this direction will be clearly exhibited, since the differences of interval introduced are in every case above the threshold of discrimination when the disturbing element of variations in intensity has been removed and the series of sounds made intensively uniform. In case of a tendency to underestimate B or overestimate A, errors would not be shown. This problem, however, is not to be met here, as the results show; for there is recorded a proportion of 82.3 per cent. of errors in judgment of interval B, and of 90.7 per cent. in judgment of interval A, all the former being errors of overestimation, all of the latter of underestimation.

The influence of position in the series on the effect exerted by such a change of intensity in a single member can be stated only tentatively. The number of experiments with the louder sound in position five was smaller than in the other cases, and the relation which there appears cannot be absolutely maintained. It may be also that the number of intervals following that concerning which judgment is to be given, and with which that interval may be compared, has an influence on the accuracy of the judgment made. If we abstract from this last set of results, the tendency which appears is toward an increase in accuracy of perception of comparative durations from the beginning to the end of the series, a tendency which appears more markedly in the relations of the interval preceding the louder sound than in those of the interval which follows it. This conclusion is based on the succession of values which the proportion of errors to total judgments presents, as in the annexed table.

TABLE x.x.x.

Percentage of Errors for Each Position.

Interval. I II III IV V B. 83.3 76.9 69.2 (100) Irrespective A. 85.7 91.6 91.6 84.6 (100) of extent.

B. 73.3 71.9 53.8 (60) Estimated A. 85.7 91.6 91.6 84.6 (100) quant.i.tatively.

Next, the relation of the amount of increase in intensity introduced at a single position in such a series to the amount of error thereby occasioned in the apprehension of the adjacent intervals was taken up.

Two sets of experiments were carried out, in each of which five of the sounds were of equal intensity, while one, occurring in the midst of the series, was louder; but in one of the sets this louder sound was occasioned by a fall of the hammer through a distance of 0.875 inch, while in the other the distance traversed was 2.00 inches. In both cases the extent of fall in the remaining hammers was uniformly 0.25 inch. The results are given in the following table:

TABLE x.x.xI.

Interval B. Interval A.

Ratio of Interval 0.875 in. 2.00 in. 0.875 in. 2.00 in.

B to Interval A. + = - + = - + = - + = - 1.000 : 1.000 0 6 0 0 4 2 0 5 1 0 0 6 0.909 : 1.000 2 4 0 0 4 2 0 2 4 2 2 2 0.833 : 1.000 0 6 0 0 4 2 4 0 2 1 3 2 0.770 : 1.000 0 6 0 2 2 2 2 4 0 4 0 2 0.714 : 1.000 0 6 0 1 5 0 6 0 0 2 2 2 Totals, 2 28 3 19 8 12 11 7 9 7 14 T.E., T.J., 2 30 11 30 13 30 21 30 and per cent., 6.6% 36.6% 60.0% 70.0%

Interval B in these experiments is of the same duration as all others but that following the louder sound; hence, judgments in the second column are correct.

Again the markedly greater influence of increased intensity on the interval following than on that preceding it appears, the percentage of errors being, for B (both intensities), 21.6 per cent.; for A, 56.6 per cent. Also, in these latter experiments the direction of error is more definite in the case of interval A than in that of interval B.

The influence of changes in intensity on the amount of error produced is striking. Two intensities only were used for comparison, but the results of subsequent work in various other aspects of the general investigation show that this correlation holds for all ranges of intensities tested, and that the amount of underestimation of the interval following a louder sound introduced into an otherwise uniform series is a function of the excess of the former over the latter. The law holds, but not with equal rigor, of the interval preceding the louder sound. So far as these records go, the influence of such an increase of intensity is more marked in the case of interval B than in that of interval A. It is to be noted, however, that the absolute percentage of errors in the case of A is several times greater than in that of B. I conclude that A is much more sensitive than B to such influences, and that there is here presented, in pa.s.sing from intensity I. to intensity II., the rise of conditions under which the influence of the louder sound on B is first distinctly felt--that is, the appearance of a threshold--and that the rate of change manifested might not hold for higher intensities.

Lastly, the rate at which the sounds of the series succeeded one another was varied, in order to determine the relation which the amount of influence exerted bore to the absolute value of the intervals which it affected. Three rates were adopted, the whole series of sounds occupying respectively 2.50 secs., 2.20 secs, and 1.80 secs. The results are summed in the following table:

TABLE x.x.xII.

Rate: 2.5 secs. Rate: 2.2 secs. Rate: 1.8 secs.

Ratio of Interval B B A B A B A to Interval A. + = - + = - + = - + = - + = - + = -

1.000 : 1.000 2 8 0 0 8 2 0 8 2 0 2 8 0 4 0 0 2 2 0.917 : 1.000 0 8 2 4 6 0 3 8 0 0 8 3 2 2 0 0 2 2 0.846 : 1.000 1 9 0 5 4 1 3 8 0 3 7 1 6 5 0 1 8 2 0.786 : 1.000 1 10 0 11 0 0 6 6 0 7 3 4 6 2 2 2 6 2 0.733 : 1.000 4 2 0 4 0 2 4 6 0 8 0 2 0.687 : 1.000 5 3 1 6 1 2 2 6 0 7 0 1

Totals 4 35 2 20 18 3 21 35 3 20 21 20 20 25 2 18 18 11*

*Transcriber's Note: Original "1".