Various - Harvard Psychological Studies, Volume 1

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The presence of this tendency to break up the four-rhythm into
subgroups of two beats explains a variety of peculiarities in the
records of this investigation. The four-beat rhythm with final accent
is found most pleasant at the close of a rhythmical sequence. The
possibility of including it in a continuous series depends on having
the final interval of 'just the right length.' If one keeps in mind
that a secondary initial accent characterizes this rhythm form, the
value required in this final interval is explained by the resolution
of the whole group into two units of three beats each, the latter of
the two being syncopated. The pause is of 'just the right length' when
it is functionally equal to two unaccented elements with their
succeeding intervals, as follows: | .q. q q; .q % % |.

Likewise in four-rhythms characterized by initial stress there appears
a tendency to accent the final beat of the group, as well as that to
accent the third. Such a series of four may therefore break up in
either of two ways, into | >q. q; .q q | on a basis of two-beat units,
or into | .q. q q; >q % %| on a basis of three-beat units.

The persistence of these simple equivalences appears also in the
treatment of syncopated measures and of supplementary or displaced
accents. Of the form | >q. q >q. | one reactor says, and his
description may stand for all, "This deliberate introduction of a
third accent on the last beat is almost impossible for me to keep. The
single group is easy enough and rather agreeable, but in a succession
of groups the secondarily accented third beat comes against the first
of the next group with a very disagreeable effect." This is the case
where no pause intervenes between the groups, in which case the rhythm
is destroyed by the suppression, in each alternate simple group, of
the unaccented phase; thus, | >q. q >q. | alone is pleasant, because
it becomes | .q. q; >q % |, but in combination with preceding and
succeeding groups it is disagreeable, because it becomes in reality
| >q. q; .q % |, etc. A long pause between the groups destroys this
disagreeableness, since the lacking phase of the second subgroup is
then restored and the rhythm follows its normal course.

The amphibrachic form, | >q q. q |, is more difficult to maintain than
either the dactylic or the trochaic, and in a continuous series tends
to pass over into one of these, usually the former. 'With sufficient
pause,' the reactors report, 'to allow the attitude to die away,' it
is easily got. The same inability to maintain this form in
consciousness appears when a continuous series of clicks is given,
every third of which is louder than the rest. Even when the beginning
of the series is made coincident with the initial phase of the
amphibrachic group the rhythmic type slips over into the dactylic, in
spite of effort. In this, as in the preceding type of reaction, if the
interval separating adjacent groups be lengthened, the rhythm is
maintained without trouble. The 'dying away' of the attitude lies
really in such an arrangement of the intervals as will formally
complete a phrase made up of simple two-beat units.

The positive evidence which this investigation affords, points to the
existence of factors of composition in all rhythms of more than three
beats; and a variety of peculiarities which the results present can be
explained--and in my estimation explained only--on the basis of such
an assumption. I conclude, therefore, that strictly stated the
numerical limit of simple rhythm groups is very soon reached; that
only two rhythmical units exist, of two and three beats respectively;
that in all longer series a resolution into factors of one of these
types takes place; and, finally, that the subordination of higher
rhythmical quantities of every grade involves these simple relations,
of which, as the scope of the synthesis increases, the opposition of
simple alternate phases tends more and more to predominate over
triplicated structures.

Variation in the number of elements which enter into the rhythmic
unit does not affect the sense of equivalence between successive
groups, so long as the numerical increase does not reach a point at
which it lessens the definiteness of the unit itself. For the purpose
of testing this relation the reactors beat out a series of rhythm
forms from 'one-beat' rhythms to those in which the group consisted of
seven, eight and nine elements, and in which the units were either
identical with one another or were made up of alternately larger and
smaller numbers of elements. Two questions were to be answered in each
case; the manner in which these various changes affected the sense of
rhythmical equivalence in the alternate groups, and the variations in
affective quality which these changes introduced into the experience.
With the former of these problems we are here concerned. From
'one-beat' to four-beat rhythms the increase in number of constituents
in no way affects the sense of rhythmical equivalence. Beyond this
point there is a distinct falling off. 'The first part of the rhythm
begins to fade away before the end of the second,' says one; and
another: 'The series then reverts to a monotonous succession without
feeling of rhythm.' This decline marks those groups composed of an odd
number of elements much earlier and more strongly than those which
contain an even number. The sense of equivalence has fallen off at
five and practically disappears at seven beats, while groups of six
and eight retain a fairly definite value as units in a rhythmical
sequence. This peculiar relation must be due to the subconscious
resolution of the larger symmetrical groups into smaller units of
three and four constituents respectively.

Likewise the introduction of variations in the figure of the
group--that is, in the number of elements which enter into the groups
to be compared, the distribution of time values within them, the
position of accents, rests, and the like--does not in any way affect
the sense of equivalence between the unlike units. Against a group of
two, three, four, or even five elements may be balanced a syncopated
measure which contains but one constituent, with the sense of full
rhythmical equivalence in the functional values of the two types.
Indeed, in the case of five-beat rhythms the definition of values is
greater when such opposition finds place than when the five-beat
group is continuously repeated. This is to be explained doubtlessly by
the more definite integration into a higher rhythmical unity which is
afforded under the former conditions.

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 cooerdination 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 Bruecke, 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. Bruecke
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 cooerdination 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
Anapaestic, 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 anapaestic 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 anapaest moves continuously toward a final accent in an unbroken
crescendo. But in the anapaestic 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 cooeperation 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; anapaestic, 12.45. The constant tendency, when any
difficulty in cooerdination 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 cooerdinated--the most
natural--form is the dactyl; the anapaest stands next; the amphibrach
is the most unnatural and difficult to cooerdinate.

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 cooerdination 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
cooerdinated 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 cooerdinating the
unaccented initial elements has resulted in the substitution of a
dipodic division for the anapaestic 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