Various - Harvard Psychological Studies, Volume 1

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[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 anapaestic 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
cooerdinated 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 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 cooerdination 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
cooerdination 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.