Chapter XVII: Section II: showed that the shorter filled distances are (13)

Since the progressive variation does not penetrate the whole measure, but affects only a single constituent having a strongly marked functional character, the process of change becomes unlike that of true retardation. In such a case, if the increase in duration be confined to a single element and parallel the changes in a simultaneous variant of a different order, we should regard them as functionally connected, and therefore interpret the successively greater periods of time occupied by the rhythmical measures as constituting no real slowing of the tempo. The measure of relative tempo in such a case consists in the ratios of the successive durations of the rhythmical units after the subtraction of that element of increase due to this extraneous source. Here, since the increase is confined to that member of the group which receives accentual stress, and since the increase of accentuation is typically accompanied by an extension of the following interval, the changes presented do fulfil the conditions of a progressively increased accentuation of the rhythm group, and to this origin I think it is undoubtedly to be attributed. It is to be noted that the final interval also undergoes a slight increase, while the median suffers a similarly slight decrease in duration as the series progresses.

In the amphibrachic form the changes manifested by the constituents of the unit group are more obscure. No progressive retardation of the accented element is apparent. In the initial and final intervals the difference in duration between the first and last members of the series is small and appears early in the process. If we assume the general application of the laws of change presented in the preceding section, there should be here two influences concerned in the determination of the relations presented, the factors, namely, of position and accent. The falling of the accentual stress on the median interval eliminates one of the two factors of progressive reduction in that element and replaces it by a factor of increase, thereby doing away with the curve of change; while at the same time it decreases the changes which occur in the bounding intervals of the group by removing the accent from the first and by the proximate position of its own accent tending to reduce the last interval.

Under this same assumption there should be expected in the anapæstic form of rhythm an exaggeration of the progressive increase in the final interval, together with a further reduction in the duration of the initial; since from the falling of the accent on the final interval two factors of increase combine, while in the initial, which immediately follows the accented interval in the series, a positive factor of reduction appears. This is actually the type of change presented by the quantitative relations, which are given as proportional values in the following table.

TABLE XLI.

Interval. I II III IV V VI VII VIII IX X
First, 1.000 0.950 1.000 0.950 1.000 0.950 1.000 1.000 1.000 1.050
Second, 1.000 1.100 1.000 1.050 1.100 1.000 1.000 1.050 1.100 1.000
Third, 1.000 1.073 1.073 1.024 1.024 1.122 1.098 1.098 1.098 1.146

Between its first and last terms the first interval shows a departure slightly less than that of the previous rhythm from the rate of change which characterizes the dactylic type; but if the average values of the whole series of intervals be taken in each of the three cases, the progressive reduction will be seen clearly to continue in passing from the second to the third form. The figures annexed give these averages as proportions of the first interval in the series.

TABLE XLII.

1st Av. of
Rhythm. Interv. all others.
Dactylic, 1.000 : 1.188
Amphibrachic, 1.000 : 1.019
Anapæstic, 1.000 : 1.000

The relations of the various intervals in the three forms are put together here for comparison:

TABLE XLIII.

Rhythm. 1st Interval. 2d Interval. 3d Interval.
Dactylic, 1.000 : 1.231 1.000 : 1.000 1.000 : 1.066
Amphibrachic, 1.000 : 1.045 1.000 : 1.000 1.000 : 1.054
Anapæstic, 1.000 : 1.050 1.000 : 1.000 1.000 : 1.146

An analysis of the factors of accentual stress and of position in the rhythmical group in isolation from each other, confirms the assumptions already made as to their influence in defining the form of the rhythmic unit. Table XLIV. exhibits the series of temporal changes taking place in accented and unaccented intervals, respectively, for the three forms combined, and therefore independent of position in the group.

TABLE XLIV.

Interval. I II III IV V VI VII VIII IX X
Accented. 1.000 1.064 1.064 1.064 1.064 1.094 1.094 1.064 1.094 1.129
Unaccented, 1.000 1.000 1.000 1.080 1.040 1.040 1.040 1.040 1.040 1.040

Similarly, in Table XLV. are given the proportional values of the series of intervals in order of their position in the group and independent of accentual stress:

TABLE XLV.

Interval. I II III IV V VI VII VIII IX X
First, 1.000 1.043 1.087 1.043 1.087 1.043 1.043 1.121 1.043 1.121
Second, 1.000 1.000 1.000 1.043 1.000 0.956 1.000 0.956 1.000 0.956
Third, 1.000 1.028 1.028 1.055 1.028 1.083 1.083 1.083 1.083 1.083

The former table makes clear the predominance of the increase in the accented element over the average of all unaccented elements of the series; the latter shows the independence of increase in the initial and final, and of decrease in the median interval, of any relation to the position of the accentual stress. Both the intensive accentuation and the demarcation of successive groups thus appear to be factors of definition in the rhythmic unit. Those types which are either marked by a more forcible accent or separated by longer pauses are more distinctly apprehended and more easily held together than those in which the accent is weaker or the pause relatively less. It would follow that the general set of changes which these series of reactions present are factors of a process of definition in the rhythmical treatment of the tapping, and are not due to any progressive change in the elementary time relations of the series.

The figures for measures of four beats are incomplete. They show an increase in the average duration of the group from first to last of the series in three out of the four forms, namely, those having initial, secondary and final stress.

Of the relative amounts contributed by the several elements to the total progressive variation of the measures in the first form, the least marks those intervals which follow unaccented beats, the greatest those which follow accented beats; among the latter, that shows the greater increase which receives the primary accent, that on which falls the secondary, subconscious accent shows the less; and of the two subgroups which contain these accents that in which the major accent occurs contributes much more largely to the progressive change than does that which contains the minor.

When the phases of accented and unaccented elements are compared, irrespective of their position in the rhythmic group, the same functional differences are found to exist as in the case of triple rhythms. Their quantitative relations are given in the following table.

TABLE XLVI.

Phase. I II III IV V VI VII VIII IX X
Accented. 1.000 1.103 1.069 1.172 1.241 1.139 1.206 1.310 1.241 1.310
Unacc., 1.000 1.083 1.128 1.169 1.159 1.208 1.169 1.250 1.169 1.169

The cause of the apparent retardation lies, as before, in a change occurring primarily in the accented elements of the rhythm, and this progressive differentiation, it is inferable from the results cited above, affects adjacent unaccented elements as well, the whole constituting a process more naturally interpretable as a functional accompaniment of progressive definition in the rhythmical treatment of the material than as a mark of primary temporal retardation.

The contribution of the several intervals according to position in the series and irrespective of accentual stress is given in the table following.

TABLE XLVII.

Interval. I II III IV V VI VII VIII IX X
First, 1.000 1.136 1.136 1.182 1.227 1.227 1.227 1.273 1.318 1.318
Second, 1.000 1.042 1.042 1.125 1.166 1.042 1.042 1.083 1.083 1.166
Third, 1.000 1.150 1.250 1.250 1.250 1.250 1.400 1.400 1.450 1.450
Fourth, 1.000 1.059 1.059 1.147 1.179 1.147 1.179 1.294 1.206 1.179

A rhythmical alternation is here presented, the contributions of the first and third elements being far in advance of those of the second and fourth. The values of the minor pair are almost equal; of the major the third exceeds the first. Under the assumption already made this would indicate the existence at these points of nodes of natural accentuation, of which the second marks the maximum reached in the present series.

The determination of relative time-values for accented and unaccented intervals was next sought by indirect experimentation, in which the affective aspect of the experience was eliminated from consideration, and account was taken only of the perception of quantitative variations in the duration of the successive intervals. Proceeding from the well-known observation that if every alternate element of a temporally uniform auditory series receive increased stress, the whole series will coalesce into successive groups of two elements in which the louder sound precedes and the weaker follows, while the interval which succeeds the unaccented sound, and which therefore separates adjacent groups, will appear of greater duration than that which follows the accented element, the investigation sought by employing the method of right and wrong cases with a series of changing time-values for the two intervals to determine the quantitative proportion of the two durations necessary to produce the impression of temporal uniformity in the series.

Two rhythm forms only were tested, the trochaic and dactylic, since without an actual prolongation of considerable value in the interval following the louder sound, at the outset, no apprehension of the series as iambic or anapæstic could be brought about. The stimuli were given by mechanism number 4, the distance of fall being 2/8 and 7/8 inch respectively for unaccented and accented sounds. The series of changes included extreme proportional values of 0.714 and 1.769 in duration of the two intervals. Six persons took part in the investigation. In the following table is given the percentage of cases in which the interval following the unaccented element was judged respectively greater than, equal to, or less than that which followed the accented element, for each of the series of ratios presented by the time-values of the intervals in trochaic rhythm.

TABLE XLIX.

Ration of Unaccented to Unaccented Interval Judged to be
Accented Interval. + = -
1.000 : 1.769 0.0 per cent. 100.0 per cent 0.0 per cent.
1.000 : 1.571 12.5 " 50.0 " 37.5 "
1.000 : 1.400 22.0 " 56.0 " 22.0 "
1.000 : 1.222 16.0 " 84.0 "
1.000 : 1.118 26.0 " 74.0 "
1.000 : 1.000 61.6 " 38.4 "
1.000 : 0.895 100.0 "
1.000 : 0.800 88.8 " 11.2 "
1.000 : 0.714 100.0 "

The anomalous percentage which appears in the first horizontal row needs explanation. The limit of possible differentiation in the time-values of accented and unaccented intervals in a rhythmical group is characteristically manifested, not by the rise of a perception of the greater duration of the interval following the accented element, but through an inversion of the rhythmical figure, the original trochee disappearing and giving place to an iambic form of grouping, the dactyl being replaced by an anapæst. In the case in question the inversion had taken place for all subjects but one, in whom the original trochaic form, together with its typical distribution of intervals, remained unchanged even with such a great actual disparity as is here involved.

For this group of observers and for the series of intensities taken account of in the present experiment, the distribution of time-values necessary to support psychological uniformity lies near to the ratio 1.400:1.000 for accented and unaccented intervals respectively, since here the distribution of errors in judgment is arranged symmetrically about the indifference point. Overestimation of the interval following the louder sound appears by no means invariable. Under conditions of objective uniformity the judgment of equality was given in 38.4 per cent, of all cases. This cannot be baldly interpreted as a persistence of the capacity for correct estimation of the time values of the two intervals in the presence of an appreciation of the series as a rhythmical group. The rhythmic integration of the stimuli is weakest when the intervals separating them are uniform, and since the question asked of the observer was invariably as to the apparent relative duration of the two intervals, it may well be conceived that the hearers lapsed from a rhythmical apprehension of the stimuli in these cases, and regarded the successive intervals in isolation from one another. The illusions of judgment which appear in these experiences are essentially dependent on an apprehension of the series of sounds in the form of rhythmical groups. So long as that attitude obtains it is absolutely impossible to make impartial comparison of the duration of successive intervals. The group is a unit which cannot be analyzed while it continues to be apprehended as part of a rhythmical sequence. We should expect to find, were observation possible, a solution of continuity in the rhythmical apprehension in every case in which these distortions of the normal rhythm form are forced on the attention. This solution appears tardily. If the observer be required to estimate critically the values of the successive intervals, the attention from the outset is turned away from the rhythmical grouping and directed on each interval as it appears. When this attitude prevails very small differences in duration are recognized (_e.g._, those of 1.000:1.118, and 1.000:0.895). But when this is not the case, the changes of relative duration, if not too great for the limits of adaptation, are absorbed by the rhythmical formula and pass unobserved, while variations which overstep these limits appear in consciousness only as the emergence of a new rhythmic figure. Such inversions are not wholly restricted by the necessity of maintaining the coincidence of accentuation with objective stress. With the relatively great differences involved in the present set of experiments, the rhythmical forms which appeared ignored often the objective accentuation of single groups and of longer series. Thus, if the second interval of a dactyl were lengthened the unaccented element which preceded it received accentuation, while the actual stress on the first sound of the group passed unobserved; and in a complex series of twelve hammer-strokes the whole system of accentuation might be transposed in the hearer's consciousness by variations in the duration of certain intervals, or even by simple increase or decrease in the rate of succession.[6]

[6] Bolton found one subject apperceiving in four-beat groups a
series of sounds in which increased stress fell only on every
sixth.

In the experiments on dactylic rhythm the changes introduced affected the initial and final intervals only, the one being diminished in proportion as the other was increased, so that the total duration of the group remained constant. The figures, arranged as in the preceding table, are given in Table L.

The percentage given in the case of the highest ratio is based on the reports of two subjects only, one of them the exceptional observer commented on in connection with two-beat rhythms; for all other participants the anapæstic form had already replaced the dactylic. The distribution of values which supports psychological uniformity in this rhythmic figure lies between the ratios 1.166, 1.000, 0.800, and 1.250, 1.000, 0.755, since in this region the proportion of errors in judgment on either side becomes inverted. The two rhythmic forms, therefore, present no important differences[7] in the relations which support psychological uniformity. A comparison in detail of the distribution of judgments in the two cases reveals a higher percentage of plus and minus, and a lower percentage of equality judgments throughout the changes of relation in the dactylic form than in the trochaic. This appears to indicate a greater rhythmical integration in the former case than in the latter. On the one hand, the illusion of isolation from adjacent groups is greater at every point at which the intervening interval is actually reduced below the value of either of the internal intervals in the dactylic than in the trochaic rhythm; and on the other, the sensitiveness to differences in the whole series is less in the case of the trochee than in that of the dactyl, if we may take the higher percentage of cases in which no discrimination has been made in the former rhythm as a negative index of such sensibility.

[7] The ratios of initial to final intervals in the two cases
are, for trochaic measures, 1.400:1.000, and for dactylic,
1.400(to 1.666):1.000.

TABLE L.

Ration of Unaccented Unaccented Interval Judged to be
to Accented Interval. + = -
1.000 : 2.428 100.0 per cent
1.000 : 2.000 20.0 per cent. 33.3 per cent 46.7 "
1.000 : 1.666 33.2 " 23.9 " 42.9 "
1.000 : 1.400 39.0 " 46.0 " 15.0 "
1.000 : 1.182 60.0 " 37.2 " 2.8 "
1.000 : 1.000 85.4 " 12.2 " 2.4 "
1.000 : 0.846 89.2 " 10.8 "
1.000 : 0.714 100.0 "
1.000 : 0.660 96.0 " 4.0 "

The increase in the number of inverted forms which occur is coördinated percentually in the following table with the successive increments of difference between the accented and unaccented intervals of the group:

TABLE LI.

Rhythm. 2.428 2.000 1.769 1.666 1.571 1.400 1.222 1.182 1.118 1.000
Trochaic, 93.7 74.0 44.2 25.0 25.0 2.9
Datylic, 93.6 54.0 39.4 18.4

These figures are corroborative of the preceding conclusions. The dactylic figure is maintained in the presence of much greater differences in the relative durations of accented and unaccented intervals than is the trochaic. In the latter, inversions not only appear earlier in the series, but become the (practically) exclusive mode of apprehension at a point where not fifty per cent, of the dactyls have suffered transformation. At a certain definite stage in the process the tendencies toward the two forms of apprehension balance each other, so that with the slightest change in direction of attention the rhythmical figure inverts and reverts to the original form indifferently. These points are defined, in the case of the two rhythms here reported on, by the following (or intermediate) ratios: Trochaic-Iambic, (1.400-1.571): 1.000; Dactylic-Anapæstic, (1.666-2.000): 1.000.

The temporal conditions of such equilibrium are a strict function of the degree of accentuation which the rhythm group presents. The location of the indifference point must, therefore be independently determined for each intensive value through which the accented element may pass. Its changes are given for five such increments in the following table, in which the values of the various intervals are represented as proportions of the absolute magnitudes which appear in the first, or undifferentiated series.

TABLE LII.

Intensive Form. 1st Interval. 2d Interval. 3d Interval.
1/8 1/8 1/8 1.000 1.000 1.000
3/8 1/8 1/8 1.042 1.010 0.948
7/8 1/8 1/8 1.142 1.021 0.862
15/8 1/8 1/8 1.146 1.042 0.808
24/8 1/8 1/8 1.291 1.000 0.708

IV. THE COMBINATION OF RHYTHMICAL GROUPS IN HIGHER SYNTHESES AND THEIR EQUIVALENCES.

In the elaboration of higher rhythmical forms the combination of formally identical groups is rather the rule than the exception, since in poetical structures the definition of the metrical form and the maintenance of its proper relations depend on a clear preponderance of its own particular unit-type over local variants. In the experimental investigation of composite rhythm forms the temporal relations of structures presenting such likeness in their constituent groups were first taken up. In the conduct of the research those differences of intensity which are actually expressed and apprehended in the utterance of a rhythmic sequence were uniformly employed. While there is no doubt that a succession of perfectly identical forms would, under the requisite temporal conditions, be apprehended as presenting major and minor phases of accentuation, yet in the expression of rhythmic relations the subordination of accents is consistently observed, and all our ordinary apprehension of rhythm, therefore, is supported by an objective configuration which fulfils already the form of our own subjective interpretation.

The temporal relations of these major and minor phases cannot be considered apart from the index of their respective accentuations. As the distribution of elements within the simple group fluctuates with the changes in intensive accentuation, so does the form of temporal succession in larger structures depend on the relations of intensity in their primary and secondary accentuations. The quantitative values hereafter given apply, therefore, only to those specific intensities involved in the experiment. Two types were chosen, the trochee and the dactyl. The series of sounds was given by successive hammer-falls of 7/8 and 1/8 inch for the major, and 3/8 and 1/8 inch for the minor phase. The distribution of time-values within each group was made on the basis of previous experimentation to determine those relations which support psychological uniformity. These internal relations were maintained unchanged throughout the series of ratios which the durations of the two groups presented. Four subjects took part in the experiment. The quantitative results in the composition of trochaic forms are given in the following tables (LIII., LIV.), the figures of which present, in the form of percentages of total judgments, the apprehension of sensible equality or disparity in the two groups.

In the earlier set of experiments the series of ratios diverged in both directions from unity; in the later it departed in one only, since every divergence in the opposite direction had, in the previous experiments, been remarked at once by the observer. In this second set the series of differences is more finely graded than in the former; otherwise the two sets of figures may be considered identical. Using the equilibrium of errors as an index of sensible equality, the two trochaic groups are perceptually uniform when the temporal ratio of major and minor lies between 1.000:0.757 and 1.000:0.779.

TABLE LIII.

Ratio of Duration 2d Group Judged to be
of 1st Group to 2d. + = -
1.000 : 1.250 100 per cent.
1.000 : 1.116 100 "
1.000 : 1.057 100 "
1.000 : 1.000 100 "
1.000 : 0.895 68 " 22 per cent.
1.000 : 0.800 25 " 75 "
1.000 : 0.714 100 per cent.

TABLE LIV.

Ratio of Duration 2d Group Judged to be
of 1st Group to 2d. + = -
1.000 : 1.000 100.0 per cent.
1.000 : 0.973 87.5 " 12.5 per cent.
1.000 : 0.870 66.6 " 33.3 "
1.000 : 0.823 33.3 " 22.2 " 44.4 per cent.
1.000 : 0.777 50.0 " 50.0 "
1.000 : 0.735 33.3 " 33.3 " 33.3 "
1.000 : 0.694 33.3 " 66.6 "

In the dactylic form, as in the second trochaic series, ratios varying from unity in one direction only were employed. The results follow:

TABLE LV.

Ratio of Duration Second Group Judged to be
of 1st Group to 2d. + = -
1.000 : 1.000 100.0 per cent.
1.000 : 0.946 62.5 " 37.5 per cent.
1.000 : 0.915 33.3 " 66.6 "
1.000 : 0.895 8.3 " 33.3 " 58.3 per cent.
1.000 : 0.800 40.0 " 60.0 "

As in the preceding case, when relations of equality obtained between the two subgroups, the secondary period in every instance appeared longer than the primary. This prolongation was uniformly reported as displeasing. The distribution of values which here support psychological uniformity lies between 1.000:0.915 and 1.000:0.895, that is to say, the difference of phases is less marked than in the case of the simpler trochaic composite. This is a structural principle which penetrates all rhythmical forms. The difference in the case of both of these composites is less than in the opposition of phases within the simple group, in which for identical intensities and (practically) the same group of observers these presented the ratio 1.000:0.714. It is evident that the relative differentiation of accented and unaccented intervals due to specific variations in intensity is greater than is that of successive groups characterized by similar differences of accentual stress; and if still more extensive groups were compared it would unquestionably be found that a further approximation to equality had taken place.

In the integration of rhythmical groups this subordination of the intensive accents which characterize them is not the sole mechanism of higher synthesis with which we are presented. Another mode is the antithesis of rhythmical quantities through verse catalepsis. Such variation of the rhythmical figure can take place in two directions and in two only: by an increase in the number of constituents, giving what may be called _redundancy_ to the measure, and by a decrease in their number, or _syncopation_. Each of these forms of departure from the typical figure fulfils a specific rhythmic function which determines its temporal and intensive characters, and its local position in the rhythmical sequence.

(_a_) _Redundant Measures._--The position of such a measure is uniformly initial. On rare occasions individual observers reported an inversion of this order in the earlier portion of the series,[8] but in no case were subjectively formulated series concluded in this way; and when the objective succession ended with the redundant measure the experience was rhythmically displeasing. In accentual stress the redundant measure is of secondary rank, the chief intensity falling upon the shorter, typical groups. Variation from the type does not, therefore, unconditionally indicate a point of accentual stress, though the two are commonly connected.

[8] This was probably due to beginning the series of
stimulations with the typical measure. Such beginning was
always made by chance.

In regard to the relative duration of the redundant measure the subjective reports indicate a large variability. The dactylic form appears to be slightly longer than the trochaics among which it appears; but not infrequently it is shorter.[9] These variations are probably connected with differences in stress due to the relation which the measure bears to the accentual initiation of the whole series; for this accent apparently may fall either within the redundant measure itself or on the first element of the succeeding ___ _____ >/ \ > | | > > group, thus: | q q q; q q; |, or | e e e q q; q q |. \_/

[9] The only form taken up was the occurrence of dactylic
measures in trochaic series.

Two rhythm forms were analyzed, the trochaic and the dactylic, the series of sounds being given by hammer-falls of 7/8 and 1/8 inch for accented and unaccented elements respectively. In each experiment full and syncopated measures alternated regularly with each other in continuous succession, giving the forms

> > > >
| q. q; q % | and | q. q q; q. % % |.
\_____/ \____________/

The initiation of the series was in every case determined by chance. Six observers took part in the work with trochaic forms, five in that with dactylic. The quantitative results are given in the following tables, in each of which the relations of duration, position and stress are included.

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 assignment 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 analyzed 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 pronounced 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 coördination 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 coördination presented by beaten rhythms are based on the repetition of simple forms only. The completion of the evidence requires a quantitative analysis 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 analyzed 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 quantitative 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 quantitative 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 analysis, 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 constituent 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 quantities, 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 assistance 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 quantities 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 analyzed, 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 quantity. 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; % % |. \______/

To estimate the proper value of such a rest the average relative duration of first and second intervals was taken in a continuous series of two-beat measures, in which the first member was accented sufficiently to define the rhythmical groups. The ratio was 1.000:0.760. In the present instance the values of the simple initial interval and the composite interval which follows it are, in terms of the linear measurement, 1.55 mm. and 3.96 mm. Assuming the above ratio to hold, the duration of a period which included the second beat-interval and a group-rest should be 1.16 + 1.55 + 1.16 = 3.87 mm. This is slightly less than the actual value of the period, whereas it should be greater. It must be remembered, however, that the disparity between the two intervals increases with initial accentuation, and in consequence the proportional amounts here added for the second interval (1.16 to 1.55) should be greater. This interval is not rhythmically 'dead' or insensitive. The index of mean variation in all reactors is greater for the first than for the second interval (or interval + pause) in the ratio 1.000:0.436, that is, the value of the latter is more clearly defined than that of the former, and the reactor doubly sensitive to variations occurring within it.

An analysis of the variations of these intervals separately in series of four groups reveals a secondary reciprocal rhythm, in which the changes in value of the mean variation at any moment are in opposite directions in the two intervals. These values in percentages of the total duration of the periods are given in the following table.

TABLE LXI.

Interval. 1st Group. 2d. Group. 3d Group. 4th Group.
First, 15.4 per cent. 26.4 per cent. 13.8 per cent. 30.3 per cent.
Second, 12.4 " 7.0 " 9.6 " 7.5 "

Without measurement of their intensive values, interpretation of these variations is speculative. They indicate that the pairs of beats are combined in higher groups of four; that the differences of mean variation in the first interval are functions of an alternating major and minor accentuation, the former occurring in the second and fourth, the latter in the first and third; and that the inversely varying values of the mean variation in the second interval are functions of the division into minor and major groups, the reduced values of the second and fourth of these intervals being characteristic of the greater sensitiveness to variations occurring in the group pause than to changes occurring within the group.

The fixity of the group is markedly greater than that of the simple interval. In the one case in which the mean variation of the group is greater than that of the elementary period the material involved was meager (five instead of ten repetitions) and the discrepancy therefore insignificant.

The difference in the mean variation of the first and second intervals respectively rises to an individual maximum of 3.000:1.000, and averages for all subjects 2.290:1.000; the fixity, that is to say, of the inter-group interval in this form of tapping is more than twice as great as that of the intra-group interval. The fixity of the larger rhythmical quantities is greater than that of the smaller, whether the relation be between the elementary interval and the unit group, or between the synthetic unit and its higher composite. The average mean variation of the beat intervals exceeds that of the whole group in the relation of 1.953:1.000. The differentiation of larger and smaller groups is less clear. When the material is taken in groups of eight successive beats the mean variation is less in the case of every subject than when taken in fours, in the ratio 1.000:1.521. The comparative values for groups of two and four beats is reversed in two thirds of the cases, yet so that an average for all subjects gives the ratio 1.000:1.066 between groups of four and two beats. The whole series of values arranged on the basis of unity for the mean variation of the beat interval is given in Table LXII.

TABLE LXII.

Proportional. Single Beat. 2-Beat Group. 4-Beat Group. 8-Beat Group.
M.V. 1.000 0.512 0.480 0.320

The persons taking part in the investigation were next required to make a series of reactions composed of unit groups of two beats, in each of which the first member received accentuation, a simple trochaic rhythm. In this type the relation of intra-group to inter-group interval remains unchanged. In all subjects but one the mean variation of the first interval exceeds that of the second in the average ratio 1.722:1.000. The amount of difference is less than in the preceding type of reaction. In the former there is presented not an intensively uniform series, but an irregularly rhythmical grouping of intensities, in dependence on the well-defined parallel types of temporal differentiation; in the latter such intensive differentiation is fundamental and constant in its form. Assuming the character of the second interval to remain unchanged, there is in the intensive fixity of the initial accented element, on the one hand, and the alternate assertion of the impulse to accentuation and repression of it in the attempt to preserve uniformity, on the other, an occasion for the difference in the relation of the mean variation of this interval to that of the following in the two cases. It is to be expected that there should be less irregularity in a series of reactions each of which represents an attempt to produce a definite and constant rhythmical accent, than in a series in which such an accent is spasmodically given and repressed.

For a like reason, the difference in value between the mean variations of the elementary interval and the unit group should be less in the case of the positive rhythm form than in that of a series which combines a definite temporal segregation with an attempt to maintain intensive uniformity. The mean variation of the interval is still of greater value than that of the unit group, but stands to it in the reduced ratio 1.000:0.969.

The relations of higher groups present certain departures from the preceding type. In three cases out of five the unit has a greater > . fixity than its immediate compound ( | q. q; q q |), with an average \_______/ ratio of 0.969:1.072. The original relation, however, is reëstablished in the case of the next higher multiple, the eight-beat group, the whole series of values, arranged on the basis of unity for the simple interval, being as follows:

TABLE LXIII.

Proportional Single Beat 2-Beat Group 4-Beat Group 8-Beat Group
M.V. 1.000 0.969 1.072 0.859