Various - Harvard Psychological Studies, Volume 1

V >> Various >> Harvard Psychological Studies, Volume 1

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On the other hand, a rhythm is already presented in the sounds of the
metronome, occasioned by the qualitative differentiation of the
members of each pair of ticks, a variation which it was impossible to
eliminate and which must be borne in mind in estimating the following
results.

Five reactors took part in the experiment, the results of which are
tabulated in the following pages. The figures are based on series of
one hundred reactions for each subject, fifty accompaniments to each
swing and return of the metronome pendulum. When taken in series of
ten successive pairs of reactions, five repetitions of the series will
be given as the basis of each average. The quantitative results are
stated in Tables VII.-XIV., which present the proportional values of
the time intervals elapsing between the successive reactions of an
accompaniment to the strokes of a metronome beating at the rates of
60, 90 and 120 per minute.

TABLE VII.

I. AVERAGES ACCORDING TO REACTORS OF ALL RATES FOR BOTH PHASES.

(_a_) In Series of Ten Successive Pairs of Beats.

Subject. I II III IV V VI VII VIII IX X

J. 1.000 1.005 1.022 1.053 1.044 1.116 1.058 1.061 1.055 1.052
K. 1.000 1.027 1.057 1.111 1.093 1.086 1.074 1.096 1.093 1.071
N. 1.000 1.032 1.062 0.990 1.009 0.980 1.019 1.040 1.067 1.040

Aver. 1.000 1.021 1.047 1.051 1.049 1.061 1.050 1.066 1.072 1.054

TABLE VIII.

(_b_) First and Second Halves of the Preceding Combined in Series of
Five.

Subject. I II III IV V
J. 1.058 1.031 1.041 1.054 1.048
K. 1.043 1.050 1.076 1.102 1.082
N. 0.990 1.025 1.051 1.028 1.024

Aver. 1.030 1.035 1.056 1.061 1.051

TABLE IX.

AVERAGES OF ALL RATES AND SUBJECTS ACCORDING TO PHASES OF METRONOME.

(_a_) In Series of Ten Successive Reactions in Accompaniment of Each
Phase.

Phase. I II III IV V VI VII VIII IX X
First, 1.000 1.055 1.102 1.097 1.082 1.066 1.053 1.123 1.120 1.074
Second, 1.000 0.988 0.992 1.007 1.016 1.055 1.015 1.009 1.024 1.001

TABLE X.

(_b_) First and Second Halves of the Preceding Combined in Series of
Five.

Phase. I II III IV V
First, 1.033 1.054 1.112 1.108 1.078
Second, 1.027 1.001 1.000 1.015 1.008

TABLE XI.

AVERAGES OF ALL SUBJECTS ACCORDING TO RATES AND PHASES OF METRONOME.

(_a_) First Phase, Series of Ten Successive Reactions.

Rate. _I II III IV V VI VII VIII IX X_
60 1.000 1.168 1.239 1.269 1.237 1.209 1.265 1.243 1.237 1.229
90 1.000 1.048 1.063 1.095 1.086 1.069 1.102 1.127 1.168 1.095
120 1.000 1.004 0.942 1.043 1.057 0.978 0.949 1.065 1.065 0.967

TABLE XII.

(_b_) Second Phase, Series of Ten Successive Reactions.

Rate. I II III IV V VI VII VIII IX X
60 1.000 0.963 0.942 0.947 1.009 0.695 0.993 0.995 1.023 0.996
90 1.000 0.893 0.987 1.018 1.036 1.005 0.995 1.000 0.977 1.000
120 1.000 1.000 0.990 1.048 1.040 1.007 0.986 1.030 1.037 0.962

TABLE XIII.

AVERAGES OF ALL SUBJECTS AND BOTH PHASES OF METRONOME ACCORDING TO
RATES.

(_a_) In Series of Ten.

Rate. I II III IV V VI VII VIII IX X
60 1.000 1.065 1.140 1.108 1.123 0.952 1.129 1.119 1.130 1.112
90 1.000 0.970 1.025 1.056 1.061 1.037 1.048 1.063 1.072 1.047
120 1.000 1.000 0.990 1.048 1.040 1.007 0.986 1.030 1.037 0.962

TABLE XIV.

(_b_) Above Combined in Series of Five.

Rate. I II III IV V
60 0.976 1.097 1.129 1.119 1.117
90 1.018 1.009 1.044 1.059 1.054
120 1.003 0.993 1.010 1.042 1.001

In the following table (XV.) is presented the average proportional
duration of the intervals separating the successive reactions of these
subjects to the stimulations given by the alternate swing and return
of the pendulum.

TABLE XV.

Subject. Rate: 60. Rate: 90. Rate: 120.
B. 0.744 : 1.000 0.870 : 1.000 0.773 : 1.000
J. 0.730 : 1.000 0.737 : 1.000 0.748 : 1.000
K. 0.696 : 1.000 0.728 : 1.000 0.737 : 1.000
N. 0.526 : 1.000 0.844 : 1.000 0.893 : 1.000

The corresponding intensive values, as measured by the excursion of
the recording pen, are as follows:

TABLE XVI.

Subject. Rate: 60. Rate: 90. Rate: 120.
B. (1.066 : 1.000) 0.918 : 1.000 (1.010 : 1.000)
J. 0.938 : 1.000 0.943 : 1.000 0.946 : 1.000
K. 0.970 : 1.000 0.949 : 1.000 (1.034 : 1.000)
N. 0.883 : 1.000 0.900 : 1.000 0.950 : 1.000

These figures present a double process of rhythmic differentiation,
intensively into stronger and weaker beats, and temporally into
longer and shorter intervals. The accentuation of alternate elements
has an objective provocative in the qualitative unlikeness of the
ticks given by the swing and return of the pendulum. This phase is,
however, neither so clearly marked nor so constant as the temporal
grouping of the reactions. In three cases the accent swings over to
the shorter interval, which, according to the report of the subjects,
formed the initial member of the group when such grouping came to
subjective notice. This latter tendency appears most pronounced at the
fastest rate of reaction, and perhaps indicates a tendency at rapid
tempos to prefer trochaic forms of rhythm. In temporal grouping the
cooerdination of results with the succession of rates presents an
exception only in the case of one subject (XV. B, Rate 120), and the
various observers form a series in which the rhythmizing tendency
becomes more and more pronounced.

Combining the reactions of the various subjects, the average for all
shows an accentuation of the longer interval, as follows:

TABLE XVII.

Rate. Temp. Diff. Intens. Diff.
60 0.674 : 1.000 0.714 : 1.000
90 0.795 : 1.000 0.927 : 1.000
120 0.788 : 1.000 0.985 : 1.000

The rhythmical differentiation of phases is greatest at the slowest
tempo included in the series, namely, one beat per second, and it
declines as the rate of succession increases. It is impossible from
this curve to say, however, that the subjective rhythmization of
uniform material becomes more pronounced in proportion as the
intervals between the successive stimulations increase. Below a
certain rapidity the series of sounds fails wholly to provoke the
rhythmizing tendency; and it is conceivable that a change in the
direction of the curve may occur at a point beyond the limits included
within these data.

The introduction from time to time of a single extra tap, with the
effect of transposing the relations of the motor accompaniment to the
phases of the metronome, has been here interpreted as arising from a
periodically recurring adjustment of the reaction process to the
auditory series which it accompanies, and from which it has gradually
diverged. The departure is in the form of a slow retardation, the
return is a swift acceleration. The retardation does not always
continue until a point is reached at which a beat is dropped from, or
an extra one introduced into, the series. In the course of a set of
reactions which presents no interpolation of extra-serial beats
periodic retardation and acceleration of the tapping take place. This
tertiary rhythm, superimposed on the differentiation of simple phases,
has, as regards the forms involved in the present experiments, a
period of ten single beats or five measures.

From the fact that this rhythm recurs again and again without the
introduction of an extra-serial beat it is possible to infer the
relation of its alternate phases to the actual rate of the metronome.
Since the most rapid succession included was two beats per second, it
is hardly conceivable that the reactor lost count of the beats in the
course of his tapping. If, therefore, the motor series in general
parallels the auditory, the retardations below the actual metronome
rate must be compensated by periods of acceleration above it. Regarded
in this light it becomes questionable if what has been called the
process of readjustment really represents an effort to restore an
equilibrium between motor and auditory processes after an involuntary
divergence. I believe the contrasting phases are fundamental, and that
the changes represent a free, rhythmical accompaniment of the
objective periods, which themselves involve no such recurrent
differentiation.

Of the existence of higher rhythmic forms evidence will be afforded by
a comparison of the total durations of the first and second
five-groups included in the decimal series. Difference of some kind is
of course to be looked for; equivalence between the groups would only
be accidental, and inequality, apart from amount and constancy, is
insignificant. In the results here presented the differentiation is,
in the first place, of considerable value, the average duration of the
first of these groups bearing to the second the relation of
1.000:1.028.

Secondly, this differentiation in the time-values of the respective
groups is constant for all the subjects participating. The ratios in
their several cases are annexed:

TABLE XVIII.

Subject. Ratio.
J. 1.000:1.042
K. 1.000:1.025
N. 1.000:1.010

It is perhaps significant that the extent of this differentiation--and
inferably the definition of rhythmical synthesis--corresponds to the
reported musical aptitudes of the subjects; J. is musically trained,
K. is fond of music but little trained, N. is without musical
inclination.

The relations of these larger rhythmical series repeat those of their
constituent groups--the first is shorter, the second longer. The two
sets of ratios are brought together for comparison in the annexed
table:

TABLE XIX.

Subject. Unit-Groups. Five Groups.
J. 1.000:1.354 1.000:1.042
K. 1.000:1.388 1.000:1.025
N. 1.000:1.326 1.000:1.010

It is to be noted here, as in the case of beating out specific
rhythms, that the index of differentiation is greater in simple than
in complex groups, the ratios for all subjects being, in simple
groups, 1.000:1.356, and in series of five, 1.000:1.026.

There is thus present in the process of mechanically accompanying a
series of regularly recurring auditory stimuli a complex rhythmization
in the forms, first, of a differentiation of alternate intervals, and
secondly, of a synthesis of these in larger structures, a process here
traced to the third degree, but which may very well extend to the
composition of still more comprehensive groups. The process of
reaction is permeated through and through by rhythmical
differentiation of phases, in which the feeling for unity and
equivalence must hold fast through really vast periods as the long
slow phases swing back and forth, upon which takes place a swift and
yet swifter oscillation of rhythmical values as the unit groups become
more limited, until the opposition of single elements is reached.

III. THE CHARACTERISTICS OF THE RHYTHMICAL UNIT.

A. _The Number of Elements in the Group and its Limits._

The number of elements which the rhythmical group contains is related,
in the first place, to the rate of succession among the elements of
the sequence. This connection has already been discussed in so far as
it bears on the forms of grouping which appear in an undifferentiated
series of sounds in consequence of variations in the absolute
magnitude of the intervals which separate the successive stimuli. In
such a case the number of elements which enter into the unit depends
solely on the rate of succession. The unit presents a continuous
series of changes from the lowest to the highest number of
constituents which the simple group can possibly contain, and the
synthesis of elements itself changes from a succession of simple forms
to structures involving complex subordination of the third and even
fourth degree, without other change in the objective series than
variations in tempo.

When objectively defined rhythm types are presented, or expression is
given to a rhythm subjectively defined by ideal forms, these simple
relations no longer hold. Acceleration or retardation of speed does
not unconditionally affect the number of elements which the rhythm
group contains. In the rhythmization of an undifferentiated series the
recurrence of accentuation depends solely on subjective conditions,
the temporal relations of which can be displaced only within the
limits of single intervals; for example, if a trochaic rhythm
characterizes a given tempo, the rhythm type persists under conditions
of progressive acceleration only in so far as the total duration of
the two intervals composing the unit approximates more closely to the
subjective rhythm period than does that of three such intervals. When,
in consequence of the continued reduction of the separating intervals,
the latter duration presents the closer approximation, the previous
rhythm form is overthrown, accentuation attaches to every third
instead of to alternate elements, and a dactylic rhythm replaces the
trochaic.

In objective rhythms, on the other hand, the determination of specific
points of increased stress makes it impossible thus to shift the
accentuation back and forth by increments of single intervals. The
unit of displacement becomes the whole period intervening between any
two adjacent points of accentuation. The rhythm form in such cases is
displaced, not by those of proximately greater units, but only by such
as present multiples of its own simple groups. Acceleration of the
speed at which a simple trochaic succession is presented results thus,
first, in a more rapid trochaic tempo, until the duration of two
rhythm groups approaches more nearly to the period of subjective
rhythmization, when--the fundamental trochaism persisting--the
previous simple succession is replaced by a dipodic structure in which
the phases of major and minor accentuation correspond to the
elementary opposition of accented and unaccented phases. In the same
way a triplicated structure replaces the dipodic as the acceleration
still continues; and likewise of the dactylic forms.

We may say, then, that the relations of rate to complexity of
structure present the same fundamental phenomena in subjective
rhythmization and objectively determined types, the unit of change
only differing characteristically in the two cases. The wider range of
subjective adjustment in the latter over the former experience is due
to the increased positive incentive to a rhythmical organic
accompaniment afforded by the periodic reinforcement of the objective
stimulus.

An investigation of the limits of simple rhythmical groups is not
concerned with the solution of the question as to the extent to which
a reactor can carry the process of prolonging the series of elements
integrated through subordination to a single dominant accentuation.
The nature of such limits is not to be determined by the introspective
results of experiments in which the observer has endeavored to hold
together the largest possible number of elements in a simple group.
When such an attempt is made a wholly artificial set of conditions,
and presumably of mechanisms, is introduced, which makes the
experiment valueless in solving the present problem. Both the
direction and the form of attention are adverse to the detection of
rhythmical complications under such conditions. Attention is directed
away from the observation of secondary accents and toward the
realization of a rhythm form having but two simple phases, the first
of which is composed of a single element, while within the latter
fall all the rest of the group. Such conditions are the worst possible
for the determination of the limits of simple rhythm groups; for the
observer is predisposed from the outset to regard the whole group of
elements lying within the second phase as undifferentiated. Thus the
conditions are such as to postpone the recognition of secondary
accents far beyond the point at which they naturally arise.

But further, such an attempt to extend the numerical scope of simple
rhythm groups also tends to transform and disguise the mechanism by
which secondary stresses are produced, and thereby to create the
illusion of an extended simple series which does not exist. For we
have no right to assume that the process of periodic accentuation in
such a series, identical in function though it be, involves always the
same form of differentiation in the rhythmical material. If the
primary accentuation be given through a finger reaction, the fixating
of that specific form of change will predispose toward an overlooking
of secondary emphases depending on minor motor reactions of a
different sort. The variety of such substitutional mechanisms is very
great, and includes variations in the local relations of the finger
reaction, movements of the head, eyes, jaws, throat, tongue, etc.,
local strains produced by simultaneous innervation of flexor and
extensor muscles, counting processes, visual images, and changes in
ideal significance and relation of the various members of the group.
Any one of these may be seized upon to mediate the synthesis of
elements and thus become an unperceived secondary accentuation.

Our problem is to determine at what point formal complication of the
rhythmical unit tends naturally to arise. How large may such a group
become and still remain fundamentally simple, without reduplication of
accentual or temporal differentiation? The determination of such
limits must be made on the basis of quantitative comparison of the
reactions which enter into larger and smaller rhythmical series, on
the one hand, and, on the other, of the types of structure which
appear in subjective rhythmization and the apprehension of objective
rhythms the forms of which are antecedently unknown to the hearer. The
evidence from subjective rhythms is inconclusive. The prevailing
types are of two and three beats. Higher forms appear which are
introspectively simple, but introspection is absolutely unable to
solve the problem as to the possible composite nature of these
extended series. The fact that they are confined to even numbers, the
multiples of two, and to such odd-numbered series as are multiples of
three, without the appearance of the higher primes, indicates the
existence in all these groups of secondary accentuation, and the
resolution of their forms into structures which are fundamentally
complications of units of two and three elements only. The process of
positive accentuation which appears in every higher rhythmical series,
and underlying its secondary changes exhibits the same reduction of
their elementary structure to double and triple groups, has been
described elsewhere in this report. Here it is in place to point out
certain indirect evidence of the same process of resolution as
manifested in the treatment of longer series of elements.

The breaking up of such series into subgroups may not be an explicitly
conscious process, while yet its presence is indispensable in giving
rhythmical form to the material. One indication of such
undiscriminated rhythmical modification is the need of making or
avoiding pauses between adjacent rhythmical groups according as the
number of their constituents varies. Thus, in rhythms having units of
five, seven, and nine beats such a pause was imperative to preserve
the rhythmical form, and the attempt to eliminate it was followed by
confusion in the series; while in the case of rhythms having units of
six, eight, and ten beats such a pause was inadmissible. This is the
consistent report of the subjects engaged in the present
investigation; it is corroborated by the results of a quantitative
comparison of the intervals presented by the various series of
reactions. The values of the intervals separating adjacent groups for
a series of such higher rhythms are given in Table XX. as proportions
of those following the initial, accented reaction.

TABLE XX.

Rhythm. Initial Interval. Final Interval
Five-Beat, 1.000 1.386
Six " 1.000 0.919
Seven " 1.000 1.422
Eight " 1.000 1.000
Nine " 1.000 1.732
Ten " 1.000 1.014

The alternate rhythms of this series fall into two distinct groups in
virtue of the sharply contrasted values of their final intervals or
group pauses. The increased length of this interval in the
odd-numbered rhythms is unquestionably due to a subdivision of the
so-called unit into two parts, the first of which is formally
complete, while the latter is syncopated. In the case of five-beat
rhythms, this subdivision is into threes, the first three of the five
beats which compose the so-called unit forming the primary subgroup,
while the final two beats, together with a pause functionally
equivalent to an additional beat and interval, make up the second, the
system being such as is expressed in the following notation:
| .q. q q; >q. q % |. The pause at the close of the group is
indispensable, because on its presence depends the maintenance of
equivalence between the successive three-groups. On the other hand,
the introduction of a similar pause at the close of a six-beat group
is inadmissible, because the subdivision is into three-beat groups,
each of which is complete, so that the addition of a final pause would
utterly unbalance the first and second members of the composite group,
which would then be represented by the following notation:
| >q. q q; .q. q q % |; that is, a three-group would alternate with a
four-group, the elements of which present the same simple time
relations, and the rhythm, in consequence, would be destroyed. The
same conditions require or prevent the introduction of a final pause
in the case of the remaining rhythm forms.

The progressive increase in the value of the final interval, which
will be observed in both the odd-and even-numbered rhythms, is
probably to be attributed to a gradual decline in the integration of
the successive groups into a well-defined rhythmical sequence.

This subdivision of material into two simple phases penetrates all
rhythmical structuring. The fundamental fact in the constitution of
the rhythmical unit is the antithesis of two phases which we call the
accented and the unaccented. In the three-beat group as in the
two-beat, and in all more complex grouping, the primary analysis of
material is into these two phases. The number of discriminable
elements which enter each phase depends on the whole constitution of
the group, for this duality of aspect is carried onward from its point
of origin in the primary rhythm group throughout the most complex
combination of elements, in which the accented phase may comprise an
indefinitely great number of simple elements, thus:

______ __________ ______________
/ \ / \ / \
> . > . >> .
| q q ; q q |, | q q q; q q q |, | q q q q; q q q q |, etc.
\_/
>

An indication of this process of differentiation into major and minor
phases appears in the form of rhythm groups containing upwards of four
elements. In these the tendency is, as one observer expresses it, 'to
consider the first two beats as a group by themselves, with the others
trailing off in a monotonous row behind.' As the series of elements
thus bound up as a unit is extended, the number of beats which are
crowded into the primary subgroup also increases. When the attempt was
made to unite eleven or twelve reactions in a single group, the first
four beats were thus taken together, with the rest trailing off as
before. It is evident that the lowest groups with which attention
concerned itself here were composed of four beats, and that the actual
form of the (nominally) unitary series of eleven beats was as follows:

_______________________
/ \
>> > >
| q q q q; q q q q; q q q q |.
. . .

The subscripts are added in the notation given above because it is to
be doubted if a strictly simple four-beat rhythm is ever met with. Of
the four types producible in such rhythm forms by variation in the
accentual position, three have been found, in the course of the
present investigation, to present a fundamental dichotomy into units
of two beats. Only one, that characterized by secondary accentuation,
has no such discriminable quality of phases. Of this form two things
are to be noted: first, that it is unstable and tends constantly to
revert to that with initial stress, with consequent appearance of
secondary accentuation; and second, that as a permanent form it
presents the relations of a triple rhythm with a grace note prefixed.