[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 coordinated 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-Anapaestic, (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 pa.s.s. 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 const.i.tuent 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 quant.i.tative 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 quant.i.tative 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 ant.i.thesis of rhythmical quant.i.ties 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 const.i.tuents, 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 a.n.a.lyzed, 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 quant.i.tative results are given in the following tables, in each of which the relations of duration, position and stress are included.