Various - Harvard Psychological Studies, Volume 1

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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 cooerdination 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 anapaestic--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
Anapaes., 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

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 anapaestic
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
Anapaestic, 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
Anapaestic, 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 anapaestic 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 anapaest. 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]