TABLE LVI.

TROCHAIC FORM.

Apparent Accentuation Ratio of 1st Second Group Judged to be 2d Group of Second Group.

to 2d Group. + = - Final + = - 1.000:1.000 55.5% 44.4% 100% 71.5% 28.5% 1.000:0.946 83.3 16.6% 100 30.0 70.0 1.000:0.895 66.6 11.1 22.2 100 30.0 60.0 10.0% 1.000:0.846 16.6 41.6 41.6 100 40.0 60.0 1.000:0.800 16.6 41.6 41.6 100 40.0 60.0 1.000:0.756 49.9 24.9 24.9 100 40.0 60.0 1.000:0.714 16.6 41.6 41.6 100 20.0 80.0

TABLE LVII.

DACTYLIC FORM.

Apparent Accentuation Ratio of 1st Second Group Judged to be 2d Group of Second Group.

to 2d Group. + = - Final + = - 1.000:1.000 100.0% 100% 40.0% 60.0% 1.000:0.946 83.3% 16.6% 100 40.0 60.0 1.000:0.895 66.6 33.3 100 20.0 80.0 1.000:0.846 37.5 62.5 100 40.0 60.0 1.000:0.800 100.0 100 40.0 60.0

The syncopated measure, like the redundant, bears to the acatalectic group specific relations of duration, accentual stress, and position in the rhythmical sequence. In position it is final. This relation is independent of the factor of duration, on which the order of elements in the simple measure depends. Even the excessive shortening which occurs in the trochaic form, when the full measure has a duration almost one and one half times as great as the syncopated, brings about no inversion of the order.

In duration the syncopated group is a shortened measure. The amount of reduction necessary to preserve rhythmical proportion with the rest of the sequence is greater in the trochaic than in the dactylic form, as in the relation of accented to unaccented elements in the simple measure it is greater than in the case of the trochaic, a principle of structure which has already been pointed out.

There is similar evidence in beaten rhythms to show that when a full measure is elided, the pause which replaces it is of less value than the duration of a syncopated measure. When trochaic rhythms were beaten out with a distinct pause after each measure, the relative values of the two intervals were 1.000:2.046. Such a pause cannot be equivalent to a suppressed beat and its interval; I regard it as functionally equal to a whole measure. If that value be allowed for the second interval which it possesses in the same rhythm type when no pause is introduced, namely, 1.000:0.920, the first two intervals will have a value--in terms of linear measurement--of 1.93 + 1.77 or 3.70.

The value of the suppressed measure would therefore be 2.15, a ratio of acatalectic to elided group of 1.000:0.581.

Iambic rhythm beaten out without separating pauses presents the following ratio between first and second intervals, 1.000:1.054; on the introduction of a pause between the measures the ratio becomes 1.000:2.131. The a.s.signment of these proportional values gives 1.68 + 1.77, or 3.45, as the duration of the first two intervals, and 1.81 for the pause, a ratio of 1.00:0.524.

In continuous dactylic tapping, the values of the successive intervals are 1.000; 0.756; 0.927; with a separating pause their relations are 1.000; 0.692; 1.346. These being a.n.a.lyzed as before, the elided measure will have the relative value of 0.419. This shows a decline in the proportional duration of the elision as the total value of the measure elided increases. There can be little question that this principle applies also to the value of elisions of higher rhythmic structures as well.

In intensity the syncopated measure is a point of increased accentual stress. This relation is not constantly maintained in the trochaic form, in which at one ratio the accent appears reduced;[10] in the dactylic form divergences are all in the direction of an apparent increase in accentuation. In rhythms beaten out the form of succession > . > > was always prescribed (_e.g._,

q. q; q_%

or

q. %; q. q

, but not ______/ ________/ either at the subjects' preference), so that no material was there afforded for a determination of the primacy of particular figures; but the results must of course show any tendency which exists toward an increased accentuation of the syncopated measure. It needs but a cursory reference to the statements of these results in Pt. III., B, of this paper, to observe how constant and p.r.o.nounced this tendency is.[11]

[10] This result is clearly irregular, and is probably due to the effect of accidental variations on a meager series of judgments. The number of these was three for each observer, making eighteen judgments in all the basis of each percentage in the table.

[11] The subjective notes of the observers frequently refer to this as an explicitly conscious process, the nature of the rhythmical sequence requiring a greater stress at that point than elsewhere. Extracts are appended:

_Trochaic Syncopation._--"There is almost a necessity for an accent on the last beat." "... an almost imperative tendency to emphasize the final syllable beyond the rest." "The two taps were followed by a pause and then a tap with increased pressure." "This was not satisfactory with any adjustment of time relations so long as the stress of all three beats was the same. In attempting to make them all equal I almost involuntarily fell into the habit of emphasizing the final one."

_Dactylic Syncopation._--"In this series it was easy to lay stress on the last (beat) ... this is the natural grouping; I unconsciously make such." "... of these the heavy one (accented syncopation) was much more satisfactory." "It was constantly my tendency to increase the strength of the last tap." "In this it is natural for me to make the final stroke heavy. To make the second group balance the first by equalizing the time alone is less satisfactory than by introducing elements of both time and force." "I felt that the latter part of the rhythm (unaccented syncopation) was lacking in force.

Something seemed continually to be dropped at the end of each group."

The reactors frequently repeated the full measure several times before introducing the syncopated measure, which thus brought a series to its close. It will probably be found that in the actual construction of poetic measures the syncopated or partially syncopated foot is systematically introduced coincidently with points of rhythmical or logical pause.

Conclusive evidence of the integration of simple rhythm forms in higher structures is presented by the process of increasing definition which every rhythmical sequence manifests between its inception and its close. This process is manifested equally in the facts of sensory apprehension and those of motor reproduction of rhythm forms. On the one hand, there is a progressive refinement in the discrimination of variations from temporal uniformity as the series of stimulations advances; and correspondingly, the sequence of motor reactions presents a clearly marked increase in coordination taking place parallel with its progress. A rhythmical form is thus given to the whole succession of simple measures which are included within the limits of the larger series, a form which is no less definite than that exhibited by the intensive and temporal relations of the rhythmical unit, and which, there can be little doubt, is even more important than the latter in determining the character of the rhythm experience as a whole.

The presentation of experimental results bearing on this point will follow the lines already laid down. Only that part of the material which is derived from the apprehension of sensory rhythm forms can be applied to the determination of this formal curve for the ordinary metrical types and their complications. The facts of progressive coordination presented by beaten rhythms are based on the repet.i.tion of simple forms only. The completion of the evidence requires a quant.i.tative a.n.a.lysis of the temporal relations presented by the whole sequence of integrated measures which compose the common verse forms: dimeter, trimeter, etc. This matter was not taken up in the present investigation.

The perception of variations in the measures of an iambic pentameter line was first taken up. The series of sounds was produced by the fall of hammer, the distances traversed being, for the accented elements 0.875 inch, and for the unaccented, 0.250 inch. The series was followed by a pause equal to one and a half measures, and was repeated before judgment was made. The time occupied by the series of sounds was 2.62 seconds. The intervals between the successive sounds were adjusted on the basis of previous experimentation concerning the most acceptable relations between the durations of accented and unaccented intervals. Their values were in the ratio 1.000:0.714 for accented and unaccented respectively. The variations were introduced in a single element, namely, the interval following the accented beat of the group, which, in this form of rhythm, is also the inter-group interval. This interval was changed by successive increments of one seventh its original value, or one twelfth the duration of the whole measure. Four such additions were made, the final value of the interval standing to its original duration in the ratio 1.000:0.636.

The same series of changes in the duration of the accented interval was made successively in each measure of the pentameter series. In all these experiments the subjects were in ignorance of the character and position of the changes introduced. The results appear in the annexed table.

TABLE LVIII.

Position in Series. Percentage Values.

Ratios. I II III IV I II III IV 1.000 : 1.000 0 0 0 0 0 0 0 0 1.000 : 0.874 4 4 4 7 40 40 40 70 1.000 : 0.777 6 6 8 10 60 60 80 100 1.000 : 0.700 6 6 10 10 60 60 100 100 1.000 : 0.636 6 6 10 10 60 60 100 100

In the five horizontal rows on the left of the table are set down the number of times, out of a total of ten judgments, the interval in question was perceived to be greater than the like interval in other groups, under the original relation of uniformity and for the four successive increments. On the right these numbers are given as percentages of the whole number of judgments. These figures show an increase of discriminative sensibility for such changes as the series advances. The percentage of correct discrimination, as it stands in the table, is the same for the first and second positions in the line, but this coincidence is to be attributed to accident, in consequence of the relatively small number of judgments on which the results are based, rather than to a functional indifference in the two positions. I conclude that fuller experiments would show a curve of continuous increase in the number of correct judgments for the whole series of measures here included. If we number the series of ratios given above from one to five, the thresholds of perceptible change for this series of positions, expressed in terms of this numerical series, would be: I., 4.1; II., 4.1; III., 3.9; IV., 3.6.

Secondly, in a series of five trochaic measures, the intervals separating the groups--which in this case follow the unaccented beat--were successively lengthened by increments identical with those employed in the preceding set of experiments. The results are presented in the table below, arranged similarly to the previous one.

TABLE LIX.

Position in Series. Percentage Values.

Ratios. I II III IV I II III IV 1.000 : 1.000 0 0 0 0 0.0 10.0 0.0 0.0 1.000 : 0.874 1 1 3 4 16.5 16.5 50.0 60.0 1.000 : 0.777 4 4 5 6 66.0 66.0 83.0 100.0 1.000 : 0.700 6 6 6 6 100.0 100.0 100.0 100.0 1.000 : 0.636 6 6 6 6 100.0 100.0 100.0 100.0

These results are essentially identical with those of the preceding section. The sensitiveness to small differences in duration within the rhythmical series becomes continuously greater as that series proceeds. The thresholds of perceptible change in terms of the numerical series of ratios (as in preceding paragraph) are as follows: I., 4.0; II., 4.0; III., 3.7; IV., 3.6.

Finally, the intensity of the preceding sound was increased as well as the duration of the interval separating it from the following stroke.

The measure employed was the trochaic, the interval suffering change was that following the accented beat--in this case, therefore, the intra-group interval. The relations obtaining among the unchanged measures were, as to duration of accented and unaccented elements, 1.000:0.714; as to intensity, 0.875:0.250 inch. Instead of a series, as in the preceding experiments, only one change in each direction was introduced, namely, an increase in duration of a single accented element of the series from 1.000 to 1.285, and an increase of the same element in intensity from 0.875 to 1.875 inch fall. The results are given in the annexed table:

TABLE LX.

Duration. Stress.

Position Interval Following Louder in Series. Judged to be Increased Stress.

+ = - Times Noted. Not Noted.

I. 8 per cent. 92 per cent. 0 per cent. 40 per cent. 60 per cent II. 42 " 50 " 8 " 42 " 58 "

III. 57 " 36 " 7 " 54 " 46 "

IV. 67 " 26 " 7 " 62 " 38 "

V. 30 " 40 " 40 " 60 " 40 "

The figures show that in regard to the discrimination of changes in duration occurring in intervals internal to the rhythm group, as well as in the case of intervals separating adjacent groups, there is a progressive increase in sensibility to variations as the succession of sounds advances. This increased sensitiveness is here complicated with another element, the tendency to underestimate the duration of the interval following a louder sound introduced into a series. The influence of this second factor cannot be a.n.a.lyzed in detail, since the amount of underestimation is not recorded unless it be sufficient to displace the sign of the interval; but if such a quant.i.tative method be applied as has already been described, the results show a continuous decrease in the amount of underestimation of this interval from the first position to the fourth, or penultimate, which presents the following relative values: 92, 66, 50, 40. A phase of rapid increase in the amount of underestimation appears in the fifth or final position, represented on the above scale of relative values by 120. This falling off at the end of the series, which appeared also in previous experiments, can be attributed only to an interference with the functions which the several measures bear in the process of comparison, and indicates that the accuracy of judgment is dependent on a comparison of the measure or element in question with those which follow as well as with those which precede it.

The results presented in the preceding section form the statement of but one half the evidence of higher rhythmical synthesis afforded by the material of the present investigation. We turn now to the second set of results. It deals, in general, with the quant.i.tative relations of rhythmic forms which find expression through finger reactions.

Portions of this evidence have already been presented, through motives of economy, in connection with the discussion of the phases of differentiation in intensity and duration which such beaten rhythms manifest. The burden of it, however, is contained in the results of an a.n.a.lysis, form by form, of the proportional mean variations which characterize these types of rhythmic expression. This method has been applied to a study (_a_) of the characters of the const.i.tuent intervals of the unit, in their relation to accentuation and position; (_b_) of the simple group which these elements compose; and (_c_) of the forms of higher synthesis manifested by the variations in successive groups. The first of these relations concerns, indeed, only the internal organization of the simple group, and has no direct bearing on the combination of such groups in higher syntheses; but, again for the sake of economy, the items are included with the rest of the material.

The application of such a method, as in all treatment of material by mean variations, involves much labor,[12] and on that account alone the lack of its employment to any considerable extent in previous investigations may be excused; but to this method, as it seems to me, must the final appeal be made, as an indisputable means by which all questions concerning the refined features of rhythmical organization, the definition of units and the determination of the forms in which they enter into larger rhythmic quant.i.ties, are to be settled.

[12] In connection with this work some 48,000 individual measurements were made (for the transcription of which I am indebted to the patient a.s.sistance of my wife). Half of these were measurements of the intensity of the successive reactions; the other half, of the intervals which separated them. The former series has been employed in obtaining the averages which appear in the section on the distribution of intensities; the latter in that on the distribution of durations.

The determination of mean variations was made in connection with the second series only (24,000). These quant.i.ties were combined in series of single groups, and in series of two, four, eight and ten groups, and for each of these groupings severally the mean variation of the series was computed.

Of all the possible forms of rhythmic apprehension or expression, the material for such a statistical inquiry is most readily obtainable in the form of a series of finger reactions, and to such material the application of the method in the present investigation has been restricted.

In the first experiment of this group the reactor was asked to tap out a series in which temporal, but not intensive variations were introduced; the strokes were to be of uniform strength but separated into groups of two beats. No directions as to length of pause between the successive groups were given, but the whole form of the groups was to be kept absolutely constant. The reports of the subjects were uniformly to the effect that no accent had been introduced. At a cursory examination no intensive grouping was apparent. These records were the earliest a.n.a.lyzed, when only time relations were in mind, and no measurements were made of variations in strength. Only the mean variations of the intervals, therefore, will here be taken up.

A word first as to the relative value of the two intervals and its significance. The form of a rhythmical series is determined in every part by subordination to principles of strict temporal arrangement.

Every suppression of elements in such a series, every rest and syncopated measure has as positive and well-defined a function as have the successive reactions and their normal intervals. If such a pause is made as we find introduced in the present case, its value must be a fixed function of the system of durations of which it forms a part, whether it replace an element in a rhythmical unit, or a subgroup in a higher rhythmical quant.i.ty. In general, the value of such a rest is less than the duration of a corresponding full measure or interval.

For example, the syncopated forms

>q %

and

>q % %_

are demonstrably of shorter average duration than the corresponding measures

>q q

and

>q q q_

; and the pause occurring at the close of a syncopated line--such as that in the middle of a catalectic trochaic tetrameter--should be found of less value than that of the regular foot.

In the present instance two reactions are made, a pause follows, then the reactions take place again, and so on. The intervals separating successive groups of reactions thus result from the coalescence of two periods, the interval which would regularly follow the reaction and the additional pause at its close. The value of the latter I interpret as functionally equivalent to a group of two beats and not to a single interval; that is, the rhythm beaten out is essentially quadruple, the second member of each composite group being suppressed, as follows: >

q q; % %

.