Chapter XVI: Section II: showed that the shorter filled distances are (12)

The number and the distribution of elements are factors variable at will, and are so treated in both musical and poetical expression. The condition which cannot be transgressed is the maintenance of strict temporal relations in the succession of total groups which constitute the rhythmical sequence. These relations are, indeed, not invariable for either the single interval or the duration of the whole group, but they are fixed functions of the dynamic values of these elements and units. Two identically figured groups (_e.g._, | >q. q q | >q. q q | ), no more possess rhythmically substitutionary values than does the opposition of a single beat to an extended series (_e.g._, | >q. | >q. q q | ), apart from this factor of temporal proportion. Those groups which are identical in figure must also be uniform in duration if they are to enter as substitutionary groups into a rhythmical sequence.[5] When the acatalectic type is alternately departed from and returned to in the course of the rhythmical sequence, the metrical equivalents must present total time-values which, while differing from that of the full measure in direction and degree, in dependence on the whole form of their structure, maintain similar fixed relations to the primary type. The changes which these flexible quantities undergo will here only be indicated. If the substitutionary groups be of different figures, that which comprises the larger number of elements will occupy the greater time, that which contains fewer, the less.

[5] Theoretically and strictly identical; this abstracts from
the coördination of such identical groups as major and minor
components of a higher rhythmical synthesis, which is really
never absent and in virtue of which the temporal values of the
groups are also differentiated.

I do not forget the work of other observers, such as Brücke, who finds that dactyls which appear among trochees are of less duration than the latter, nor do I impugn their results. The rhythmical measure cannot be treated as an isolated unit; it must always be considered in its structural relations to the rhythmical sequence of which it forms a part. Every non-conforming measure is unquestionably affected by the prevailing type of the rhythmical sequence in which it occurs. Brücke points out the converse fact that those trochees and iambs are longest which appear in dactylic or other four-measures; but this ignores the complexity of the conditions on which the character of these intrusive types depends. The time-values of such variants are also dependent on the numerical preponderance of the typical form in the whole series. When a single divergent form appears in the sequence the dynamic relations of the two types is different from that which obtains when the numbers of the two approach equality, and the effect of the prevailing form on it is proportionally greater. Secondly, the character of such variants is dependent on the subordinate configuration of the sequence in which they appear, and on their specific functions within such minor rhythmical figures. The relative value of a single dactyl occurring in an iambic pentameter line cannot be predicated of cases in which the two forms alternate with each other throughout the verse. Not only does each type here approximate the other, but each is affected by its structural relation to the proximately higher group which the two alternating measures compose. Thirdly, the quantitative values of these varying forms is related to their logical significance in the verse and the degree of accentuation which they receive. Importance and emphasis increase the duration of the measure; the lack of either shortens it. In this last factor, I believe, lies the explanation of the extreme brevity of dactyls appearing in three-rhythms. When a specific rhythm type is departed from, for the purpose of giving emphasis to a logically or metrically important measure, the change is characteristically in the direction of syncopation. Such forms, as has been said elsewhere, mark nodes of natural accentuation and emphasis. Hence, the dactyl introduced into an iambic or trochaic verse, which, so far as concerns mere number of elements, tends to be extended, may, in virtue of its characteristic lack of accentuation and significance, be contracted below the value of the prevailing three-rhythm. Conversely the trochee introduced into a dactylic sequence, in consequence of its natural accentuation or importance, may exceed in time-value the typical four-rhythm forms among which it appears. The detailed examination of the relation of temporal variations to numerical predominance in the series, to subordinate structural organization, and to logical accentuation, in our common rhythms, is a matter of importance for the general investigation which remains still to be carried out. In so far as the consideration of these factors entered into the experimental work of the present research, such quantitative time relations are given in the following table, the two types in all cases occurring in simple alternation:

TABLE XXI.

Rhythm. 1st Meas. 2d Meas. Rhythm. 1st Meas. 2d Meas.

. > > > > .
q q q; q q % 1.000 1.091 q q %; q q q 1.000 1.140
. > > .
q q q; q q % 1.000 1.159 q q %; q q q 1.000 1.021
. > > .
q q q; q q % 1.000 1.025 q q %; q q q 1.000 1.267
> . . >
q q q; q q % 1.000 0.984 q q %; q q q 1.000 1.112
> . . >
q q q; q q % 1.000 0.766 q q %; q q q 1.000 1.119

As the disparity in numerical constitution increases, so will also the divergence in time-value of the two groups concerned. When differentiation into major and minor phases is present, the duration of the former will be greater than that of the latter. Hence, in consequence of the combination of these two factors--_e.g._, in a syncopated measure of unusual emphasis--the characteristic time-values may be inverted, and the briefer duration attach to that unit which comprises the greater number of elements. Intensive values cannot take the place of temporal values in rhythm; the time form is fundamental. Through all variations its equivalences must be adhered to. Stress makes rhythm only when its recurrence is at regular intervals. The number of subordinate factors which combine with the accented element to make the group is quite indifferent. But whether few or many, or whether that element on which stress falls stands alone (as it may), the total time values of the successive groups must be sensibly equivalent. When a secondary element is absent its place must be supplied by a rest of equivalent time-value. If these proper temporal conditions be not observed no device of intensive accentuation will avail to produce the impression of metrical equivalence among the successive groups.

B. _The Distribution of Elements Within the Group._

(_a_) The Distribution of Intensities.

In the analysis of the internal constitution of the rhythmic unit, as in other parts of this work, the investigation follows two distinct lines, involving the relations of rhythm as apprehended, on the one hand, and the relations of rhythm as expressed, on the other; the results in the two cases will be presented separately. A word as to the method of presentation is necessary. The fact that in connection with each experiment a group of questions was answered gives rise to some difficulty in planning the statement of results. It is a simple matter to describe a particular set of experiments and to tell all the facts which were learned from them; but it is not logical, since one observation may have concerned the number of elements in the rhythmic unit, another their internal distribution, and a third their coalescence in a higher unity. On the other hand, the statement of each of these in its own proper connection would necessitate the repetition of some description, however meager, of the conditions of experimentation in connection with each item. For economy's sake, therefore, a compromise has been made between reporting results according to distribution of material and according to distribution of topics. The evidence of higher grouping, for example, which is afforded by variations in duration and phases of intensity in alternate measures, will be found appended to the sections on these respective classes of material.

In all the following sections the hammer-clang apparatus formed the mechanism of experimentation in sensory rhythms, while in reactive rhythms simple finger-tapping was employed.

In comparing the variations in stress which the rhythmical material presents, the average intensities of reaction for the whole group has been computed, as well as the intensities of the single reactions which compose it. This has been done chiefly in view of the unstable intensive configuration of the group and the small amount of material on which the figures are based. The term is relative; in ascertaining the relations of intensity among the several members of the group, at least ten successive repetitions, and in a large part of the work fifty, have been averaged. This is sufficient to give a clear preponderance in the results to those characteristics which are really permanent tendencies in the rhythmical expression. This is especially true in virtue of the fact that throughout these experiments the subject underwent preliminary training until the series of reactions could be easily carried out, before any record of the process was taken. But when such material is analyzed in larger and smaller series of successive groups the number of reactions on which each average is based becomes reduced by one half, three quarters, and so on. In such a case the prevailing intensive relations are liable to be interfered with and transformed by the following factor of variation. When a wrong intensity has accidentally been given to a particular reaction there is observable a tendency to compensate the error by increasing the intensity of the following reaction or reactions. This indicates, perhaps, the presence of a sense of the intensive value of the whole group as a unity, and an attempt to maintain its proper relations unchanged, in spite of the failure to make exact coördination among the components. But such a process of compensation, the disappearance of which is to be looked for in any long series, may transpose the relative values of the accented elements in two adjacent groups when only a small number of reactions is taken into account, and make that seem to receive the major stress which should theoretically receive the minor, and which, moreover, does actually receive such a minor stress when the value of the whole group is regarded, and not solely that member which receives the formal accentuation.

The quantitative analysis of intensive relations begins with triple rhythms, since its original object was to compare the relative stresses of the unaccented elements of the rhythmic group. These values for the three forms separately are given in Table XXII., in which the value of the accented element in each case is represented by unity.

TABLE XXII.

Rhythm. 1st Beat. 2d Beat. 3d Beat.

Dactylic, 1.000 0.436 0.349
Amphibrachic, 0.488 1.000 0.549
Anapæstic, 0.479 0.484 1.000

The dactylic form is characterized by a progressive decline in intensity throughout the series of elements which constitute the group. The rate of decrease, however, is not continuous. There is a marked separation into two grades of intensity, the element receiving accentual stress standing alone, those which possess no accent falling together in a single natural group, as shown in the following ratios: first interval to third, 1.000:0.349; second interval to third, 1.000:0.879. One cannot say, therefore, that in such a rhythmic form there are two quantities present, an accented element and two undifferentiated elements which are unaccented. For the average is not based on a confused series of individual records, but is consistently represented by three out of four subjects, the fourth reversing the relations of the second and third elements, but approximating more closely to equivalence than any other reactor (the proportional values for this subject are 1.000; 0.443; 0.461). Moreover, this reactor was the only musically trained subject of the group, and one in whom the capacity for adhering to the logical instructions of the experiment appears decidedly highest.

In the amphibrachic form the average again shows three degrees of intensity, three out of four subjects conforming to the same type, while the fourth reverses the relative values of the first and third intervals. The initial element is the weakest of the group, and the final of median intensity, the relation for all subjects being in the ratio, 1.000:1.124. The amphibrachic measure begins weakly and ends strongly, and thus approximates, we may say, to the iambic type.

In the anapæstic form the three degrees of intensity are still maintained, three out of four subjects giving consistent results; and the order of relative values is the simple converse of the dactylic. There is presented in each case a single curve; the dactyl moves continuously away from an initial accent in an unbroken decrescendo, the anapæst moves continuously toward a final accent in an unbroken crescendo. But in the anapæstic form as well as in the dactylic there is a clear duality in the arrangement of elements within the group, since the two unaccented beats fall, as before, into one natural group, while the accented element is set apart by its widely differentiated magnitude. The ratios follow: first interval to second, 1.000:1.009; first interval to third, 1.000:2.084.

The values of the three elements when considered irrespective of accentual stress are as follows: First, 1.000; second, 1.001; third, 0.995. No characteristic preponderance due to primacy of position appears as in the case of relative duration. The maximum value is reached in the second element. This is due to the coöperation of two factors, namely, the proximity of the accentual stress, which in no case is separated from this median position by an unaccented element, and the relative difficulty in giving expression to amphibrachic rhythms. The absolute values of the reactions in the three forms is of significance in this connection. Their comparison is rendered possible by the fact that no change in the apparatus was made in the course of the experiments. They have the following values: Dactylic, 10.25; amphibrachic, 12.84; anapæstic, 12.45. The constant tendency, when any difficulty in coördination is met with, is to increase the force of the reactions, in the endeavor to control the formal relations of the successive beats. If such a method of discriminating types be applied to the present material, then the most easily coördinated--the most natural--form is the dactyl; the anapæst stands next; the amphibrach is the most unnatural and difficult to coördinate.

The same method of analysis was next applied to four-beat rhythms. The proportional intensive values of the successive reactions for the series of possible accentual positions are given in the following table:

TABLE XXIII.

Stress. 1st Beat. 2d Beat. 3d Beat. 4th Beat.

Initial, 1.000 0.575 0.407 0.432
Secondary, 0.530 1.000 0.546 0.439
Tertiary, 0.470 0.407 1.000 0.453
Final, 0.492 0.445 0.467 1.000

The first and fourth forms follow similar courses, each marked by initial and final stress; but while this is true throughout in the fourth form, it results in the first form from the preponderance of the final interval in a single individual's record, and therefore cannot be considered typical. The second and third forms are preserved throughout the individual averages. The second form shows a maximum from which the curve descends continuously in either direction; in the third a division of the whole group into pairs is presented, a minor initial accent occurring symmetrically with the primary accent on the third element. This division of the third form into subgroups appears also in its duration aspect. Several inferences may be drawn from this group of relations. The first and second forms only are composed of singly accented groups; in the third and fourth forms there is presented a double accent and hence a composite grouping. This indicates that the position in which the accent falls is an important element in the coördination of the rhythmical unit. When the accent is initial, or occurs early in the group, a larger number of elements can be held together in a simple rhythmic structure than can be coördinated if the accent be final or come late in the series. In this sense the initial position of the accent is the natural one. The first two of these four-beat forms are dactylic in structure, the former with a postscript note added, the latter with a grace note prefixed. In the third and fourth forms the difficulty in coördinating the unaccented initial elements has resulted in the substitution of a dipodic division for the anapæstic structure of triple rhythms with final accent.

The presence of a tendency toward initial accentuation appears when the average intensities of the four reactions are considered irrespective of accentual position. Their proportional values are as follows: First, 1.000; second, 0.999; third, 1.005; fourth, 0.981. Underlying all changes in accentuation there thus appears a resolution of the rhythmic structure into units of two beats, which are primitively trochaic in form.

The influence exerted by the accented element on adjacent members of the group is manifested in these forms more clearly than heretofore when the values of the several elements are arranged in order of their proximity to that accent and irrespective of their positions in the group. Their proportional values are as follows:

TABLE XXIV.

2d Remove. 1st Remove. Accent. 1st Remove. 2d Remove.
0.442 0.526 1.000 0.514 0.442

This reinforcing influence is greater--according to the figures just given--in the case of the element preceding the accent than in that of the reaction which follows it. It may be, therefore, that the position of maximal stress in the preceding table is due to the close average relation in which the third position stands to the accented element. This proximity it of course shares with the second reaction of the group, but the underlying trochaic tendency depreciates the value of the second reaction while it exaggerates that of the third. This reception of the primitive accent the third element of the group indeed shares with the first, and one might on this basis alone have expected the maximal value to be reached in the initial position, were it not for the influence of the accentual stress on adjacent members of the group, which affects the value of the third reaction to an extent greater than the first, in the ratio 1.000:0.571.

The average intensity of the reactions in each of the four forms--all subjects and positions combined--is worthy of note.

TABLE XXV.

Stress. Initial. Secondary. Tertiary. Final.
Value, 1.000 1.211 1.119 1.151

The first and third forms, which involve initial accents--in the relation of the secondary as well as primary accent to the subgroups--are both of lower average value than the remaining types, in which the accents are final, a relation which indicates, on the assumption already made, a greater ease and naturalness in the former types. Further, the second form, which according to the subjective reports was found the most difficult of the group to execute--in so far as difficulty may be said to be inherent in forms of motor reaction which were all relatively easy to manipulate--is that which presents the highest intensive value of the whole series.

In the next group of experiments, the subject was required to execute a series of reactions in groups of alternating content, the first to contain two uniform beats, the second to consist of a single reaction. This second beat with the interval following it constitutes a measure which was to be made rhythmically equivalent to the two-beat group with which it alternated. The time-relations of the series were therefore left to the adjustment of the reactor. The intensive relations were separated into two groups; in the first the final reaction was to be kept uniform in strength with those of the preceding group, in the second it was to be accented.

The absolute and relative intensive values for the two forms are given in the following table:

TABLE XXVI.

Rhythm. 1st Beat. 2d Beat. 3d Beat. Value.

Syncopated Measures 13.00 15.12 16.50 Absolute.
Unaccented, 1.000 1.175 1.269 Relative.

Syncopated Measures 10.95 11.82 16.11 Absolute.
Accented, 1.000 1.079 1.471 Relative.

These averages hold for every individual record, and therefore represent a thoroughly established type. In both forms the reaction of the syncopated measure receives the greatest stress. In the first form, while the stress is relatively less than in the second, it is at the same time absolutely greater. The whole set of values is raised (the ratio of average intensities in the two forms being 1.147:1.000), as it has already been found to be raised in other forms difficult to execute. To this cause the preponderance is undoubtedly to be attributed, as the reports of every subject describe this form as unnatural, in consequence of the restraint it imposes on an impulse to accent the final reaction, _i.e._, the syncopated measure.

In the next set of experiments the series of reactions involved the alternation of a syncopated measure consisting of a single beat with a full measure of three beats. The same discrimination into accented and unaccented forms in the final measure was made as in the preceding group. The series of absolute and relative values are given in the following table.

TABLE XXVII.

Rhythm. 1st Beat. 2d Beat. 3rd Beat. 4th Beat. Value.

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 quantitative analysis 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 quantitative 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 quantitative analysis 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 repetitions; 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 repetition 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 quantities 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 latitude 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 pronounced 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 quantitative 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 (quantitative) 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 quantitative 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 XXX.

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) quantitatively.

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 XXXI.

Interval B.¹ 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%

¹Interval 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 passing 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 XXXII.

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".

These results are converted into percentages of the total number of judgments in the following table:

TABLE XXXIII.

Rate of B A
Success. + = - Errors. + = - Errors.
2.5 secs 10 85 5 15 49 44 7 51
2.2 " 36 59 5 41 33 34 33 67
1.8 " 43 53 4 47 38 38 24 62

In the case of interval A the direction of the curve of error changes in passing from Rate II. to Rate III. In the case of interval B the increase is continuous.

This increase in the percentage of error is, further, distinctly in the direction of an accentuation of the overestimation of the interval B, as is shown in the percentage of cases in which this interval appeared greater than the rest of the series for each of the three rates.

If the three rates be combined in the one set of results, the difference in the effects produced on the interval following the louder sound and on that which precedes it becomes again apparent. This is done in the table below.

TABLE XXXIV.

B A B A
Ratio + = - + = - T.E. T.J. % T.E. T.J. %
I. 2 20 2 0 12 12 2 24 8.5 12 24 50.0
II. 5 18 2 4 16 5 5 25 20.0 21 25 84.4
III. 10 22 0 9 19 4 10 32 31.0 23 32 72.0
IV. 13 18 2 20 9 8 13 33 39.0 17 37 46.0
V. 8 8 0 12 0 4 8 16 50.0 4 16 25.0
VI. 7 9 1 13 1 3 7 17 41.0 4 17 24.0

The overestimation of the interval before the louder sound also tends to increase in extent with the actual increase in duration of the interval following that sound over the other intervals of the series.

Thus, the form which the sensible time-relations of such a limited series of sounds present is found to be intimately dependent on the intensive preponderance of certain elements within it, on the degree of increased stress which such elements receive, on their local position in the series, and on the rate at which the stimulations succeed one another. The knowledge of these facts prepares us for the whole series of relations manifested in the special quantitative investigations reported in the sections which follow. In the first of these is presented the time-relations obtaining among the successive reactions of the various rhythm types discussed in the preceding division of this part, the section, namely, on the distribution of intensities.

In the first group of reactions the series was not to be consciously accented, nor to be divided into groups by the introduction of pauses. The reactor was required only to conceive it as a succession of two-beat groups continuously repeated, the way in which the groups should be defined, whether by counting or otherwise, being left to his own discretion. The experimental group was composed of five subjects.

The following table presents the quantitative results of an analysis of the material in series of ten successive pairs of reactions, upon the basis of unity as the value of the first element.

TABLE XXXV.

Quantities. I II III IV V VI VII VIII IX X
Whole Meas., 1.000 0.894 1.035 0.912 1.000 0.877 1.070 0.877 1.070 0.841
First Inter., 1.000 1.142 1.071 1.142 1.000 1.285 1.000 1.214 1.000 1.214
Second Inter., 1.000 0.837 1.023 0.860 1.000 0.744 1.093 0.767 1.093 0.790

Within the limits of the calculation no progressive change appears, either of acceleration or of retardation, whether in general or on the part of individual reactors. In narrower ranges the inconstancy of the periods is very marked, and their variations of clearly defined rhythmical character. The duration of the total measures of two beats is throughout alternately longer and shorter, the average of their values presenting a ratio of 1.000:0.847. The order of this arrangement, namely, that the longer period precedes the shorter in the larger group, is drawn from the fact that measurements consistently began with the initial reaction of the series.

An analysis of the constituent intervals of the unit group, as shown in the second and third lines of the table, reveals the existence of a complex subordinate rhythm. The two components of the rhythmical group do not increase and decrease concomitantly in temporal value in composing the alternate long and short measures of the fluent rhythm. The movement involves a double compensating rhythmical change, in which the two elements are simultaneously in opposite phases to each other. A measure which presents a major first interval contains always a minor second; one introduced by a minor first concludes with a major second. The ratios of these two series of periodic variations must themselves manifestly be different. Their values are, for the first interval of the measure, 1.000:1.214; and for the second interval, 1.000:0.764. The greater rhythmical differentiation marks the second of the two intervals; on the variations of this second interval, therefore, depends the appearance of that larger rhythm which characterizes the series. The ratios of these primary intervals are less consistently maintained than are those of the rhythmical measures built out of them. It will be noted that in both intervals there is a tendency for the value of the difference between those of alternate groups to increase as the tapping progresses. This change I have interpreted as indicative of a progressive definition in the process of rhythmization, depending on an increase in coördination and differentiation of the reactions as the series advances.

A simple stress on alternate elements was next introduced in the series, forming a simple trochaic measure repeated without interruption. The quantitative results follow, arranged as in the preceding experiment.

TABLE XXXVI.

Quantity. I II III IV V VI VII VIII IX X
Measure, 1.000 1.035 1.070 1.035 1.087 1.070 1.071 1.052 1.070 1.070
1st Int., 1.000 1.000 1.111 1.000 1.055 1.111 1.166 1.111 1.111 1.111
2d Int., 1.000 1.025 1.051 1.051 1.102 1.051 1.025 1.025 1.051 1.051

Here again there is no progressive acceleration or retardation. The rhythmical differentiation of alternate measures is very slight--the average ratio of the first to the second being 1.000:0.993--but is of the same type as in the preceding. The excess in the amount of this differentiation presented by the first type of reaction over the second may be due to the presence of a tendency to impart rhythmical character to such a series of reactions, which, prohibited in one form--the intensive accent--finds expression through the substitution for this of a temporal form of differentiation.

In this trochaic rhythm the phases of variation in the constituent intervals of the measure are concomitant, and their indices of differentiation almost identical with each other. Their values are, for the first, 1.000:0.979; and for the second, 1.000:0.995. The higher index is that of the first interval, that, namely, which follows the accented beat of the measure, and indicates that the rhythmical change is due chiefly to a differentiation in the element which receives the stress.

In iambic measures similarly beaten out there is likewise no acceleration nor retardation apparent in the progress of the tapping. The temporal differentiation of alternate measures is of the same extent as in the preceding group, namely, 1.000:0.991. the proportional quantitative values of the measure and its constituent intervals, taken in series of ten successive repetitions, are as follow:

TABLE XXXVII.

Quantity I II III IV V VI VII VIII IX X
Measure, 1.000 0.979 1.000 0.979 1.020 0.979 0.979 1.020 0.979 0.979
1st Int., 1.000 0.941 0.941 1.000 1.000 0.941 8.082 0.941 0.941 0.941
2d Int., 1.000 1.000 1.032 0.967 1.032 1.000 1.000 1.032 1.000 0.967

The alternation of greater and less duration in the rhythm groups is due to a variation in the time-value of the second interval only, the index of average change in the first member being zero. That is, the greater index of instability again attaches to that element which receives the stress. Though this holds true throughout these experiments, the amount of difference here is misleading, since on account of the smaller absolute value of the first interval the proportional amount of change within it which passes unrecorded is greater than in the case of the second interval.

In general, the larger temporal variations of the trochaic and iambic rhythm forms are too slight to be significant when taken individually. The evidence of rhythmical treatment in such a series of reactions, which is strongly marked in the unaccented form, nevertheless receives reinforcement from these inconsiderable but harmonious results.

The proportional values of the variations in alternate measures for accented and unaccented elements are given in the following table, in which the figures for the trochaic and iambic forms are combined:

TABLE XXXVIII.

Interval I II III IV V VI VII VIII IX X
Accented, 1.000 1.000 1.083 1.000 1.041 1.000 1.083 1.000 1.041 1.000
Unacc. 1.000 1.000 1.000 1.035 1.071 1.000 0.964 1.000 1.000 1.000

It is perhaps worthy of note that in this table a still higher rhythmical synthesis of regular form appears in the accented elements if the figures be taken in series of four consecutive pairs of reactions.

In the group of triple rhythms next taken up--the dactylic, the amphibrachic and the anapæstic--each type presents an increase in the duration of the unit group between the beginning and end of the series, but without any regular curve connecting these terms. Neither the average results nor those of the individual subjects show anywhere a decrease of duration in the progress of the tapping. The proportional results for each of the three rhythm forms, and their averages, are given in the following table.

TABLE XXXIX.

Rhythm. I II III IV V VI VII VIII IX X
Datyl., 1.000 1.062 1.062 1.087 1.087 1.075 1.125 1.112 1.125 1.112
Amphib., 1.000 1.000 1.000 1.069 1.085 1.046 1.046 1.046 1.046 1.035
Anapæs., 1.000 1.012 1.023 1.012 1.037 1.037 1.023 1.059 1.023 1.084

Average, 1.000 1.024 1.036 1.060 1.060 1.060 1.072 1.072 1.072 1.084

When all types and subjects are thus combined the summation of these inconstant retardations presents sharply differentiated terms and a curve uninverted at any point.

A separate analysis of the components of the rhythmical group shows, for the dactylic form, an important increase in duration in only one of the three intervals, namely, that following the element which receives accentual stress. The proportional values for these intervals follow.

TABLE XL.

Interval. I II III IV V VI VII VIII IX X First, 1.000 1.153 1.153 1.153 1.153 1.231 1.193 1.193 1.231 1.231 Second, 1.000 0.917 0.917 1.000 0.917 0.917 0.917 0.917 0.917 0.917 Third 1.000 1.000 1.033 1.066 1.055 1.066 1.133 1.066 1.066 1.066