Various - Harvard Psychological Studies, Volume 1

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Secondly, in a series of five trochaic measures, the intervals
separating the groups--which in this case follow the unaccented
beat--were successively lengthened by increments identical with those
employed in the preceding set of experiments. The results are
presented in the table below, arranged similarly to the previous one.

TABLE LIX.

Position in Series. Percentage Values.
Ratios. I II III IV I II III IV
1.000 : 1.000 0 0 0 0 0.0 10.0 0.0 0.0
1.000 : 0.874 1 1 3 4 16.5 16.5 50.0 60.0
1.000 : 0.777 4 4 5 6 66.0 66.0 83.0 100.0
1.000 : 0.700 6 6 6 6 100.0 100.0 100.0 100.0
1.000 : 0.636 6 6 6 6 100.0 100.0 100.0 100.0

These results are essentially identical with those of the preceding
section. The sensitiveness to small differences in duration within the
rhythmical series becomes continuously greater as that series
proceeds. The thresholds of perceptible change in terms of the
numerical series of ratios (as in preceding paragraph) are as follows:
I., 4.0; II., 4.0; III., 3.7; IV., 3.6.

Finally, the intensity of the preceding sound was increased as well as
the duration of the interval separating it from the following stroke.
The measure employed was the trochaic, the interval suffering change
was that following the accented beat--in this case, therefore, the
intra-group interval. The relations obtaining among the unchanged
measures were, as to duration of accented and unaccented elements,
1.000:0.714; as to intensity, 0.875:0.250 inch. Instead of a series,
as in the preceding experiments, only one change in each direction
was introduced, namely, an increase in duration of a single accented
element of the series from 1.000 to 1.285, and an increase of the same
element in intensity from 0.875 to 1.875 inch fall. The results are
given in the annexed table:

TABLE LX.

Duration. Stress.
Position Interval Following Louder
in Series. Judged to be Increased Stress.
+ = - Times Noted. Not Noted.
I. 8 per cent. 92 per cent. 0 per cent. 40 per cent. 60 per cent
II. 42 " 50 " 8 " 42 " 58 "
III. 57 " 36 " 7 " 54 " 46 "
IV. 67 " 26 " 7 " 62 " 38 "
V. 30 " 40 " 40 " 60 " 40 "

The figures show that in regard to the discrimination of changes in
duration occurring in intervals internal to the rhythm group, as well
as in the case of intervals separating adjacent groups, there is a
progressive increase in sensibility to variations as the succession of
sounds advances. This increased sensitiveness is here complicated with
another element, the tendency to underestimate the duration of the
interval following a louder sound introduced into a series. The
influence of this second factor cannot be analyzed in detail, since
the amount of underestimation is not recorded unless it be sufficient
to displace the sign of the interval; but if such a quantitative
method be applied as has already been described, the results show a
continuous decrease in the amount of underestimation of this interval
from the first position to the fourth, or penultimate, which presents
the following relative values: 92, 66, 50, 40. A phase of rapid
increase in the amount of underestimation appears in the fifth or
final position, represented on the above scale of relative values by
120. This falling off at the end of the series, which appeared also in
previous experiments, can be attributed only to an interference with
the functions which the several measures bear in the process of
comparison, and indicates that the accuracy of judgment is dependent
on a comparison of the measure or element in question with those which
follow as well as with those which precede it.

The results presented in the preceding section form the statement of
but one half the evidence of higher rhythmical synthesis afforded by
the material of the present investigation. We turn now to the second
set of results. It deals, in general, with the quantitative relations
of rhythmic forms which find expression through finger reactions.
Portions of this evidence have already been presented, through motives
of economy, in connection with the discussion of the phases of
differentiation in intensity and duration which such beaten rhythms
manifest. The burden of it, however, is contained in the results of an
analysis, form by form, of the proportional mean variations which
characterize these types of rhythmic expression. This method has been
applied to a study (_a_) of the characters of the constituent
intervals of the unit, in their relation to accentuation and position;
(_b_) of the simple group which these elements compose; and (_c_) of
the forms of higher synthesis manifested by the variations in
successive groups. The first of these relations concerns, indeed, only
the internal organization of the simple group, and has no direct
bearing on the combination of such groups in higher syntheses; but,
again for the sake of economy, the items are included with the rest of
the material.

The application of such a method, as in all treatment of material by
mean variations, involves much labor,[12] and on that account alone
the lack of its employment to any considerable extent in previous
investigations may be excused; but to this method, as it seems to me,
must the final appeal be made, as an indisputable means by which all
questions concerning the refined features of rhythmical organization,
the definition of units and the determination of the forms in which
they enter into larger rhythmic quantities, are to be settled.

[12] In connection with this work some 48,000 individual measurements
were made (for the transcription of which I am indebted to the patient
assistance of my wife). Half of these were measurements of the
intensity of the successive reactions; the other half, of the
intervals which separated them. The former series has been employed in
obtaining the averages which appear in the section on the distribution
of intensities; the latter in that on the distribution of durations.
The determination of mean variations was made in connection with the
second series only (24,000). These quantities were combined in series
of single groups, and in series of two, four, eight and ten groups,
and for each of these groupings severally the mean variation of the
series was computed.

Of all the possible forms of rhythmic apprehension or expression, the
material for such a statistical inquiry is most readily obtainable in
the form of a series of finger reactions, and to such material the
application of the method in the present investigation has been
restricted.

In the first experiment of this group the reactor was asked to tap out
a series in which temporal, but not intensive variations were
introduced; the strokes were to be of uniform strength but separated
into groups of two beats. No directions as to length of pause between
the successive groups were given, but the whole form of the groups was
to be kept absolutely constant. The reports of the subjects were
uniformly to the effect that no accent had been introduced. At a
cursory examination no intensive grouping was apparent. These records
were the earliest analyzed, when only time relations were in mind, and
no measurements were made of variations in strength. Only the mean
variations of the intervals, therefore, will here be taken up.

A word first as to the relative value of the two intervals and its
significance. The form of a rhythmical series is determined in every
part by subordination to principles of strict temporal arrangement.
Every suppression of elements in such a series, every rest and
syncopated measure has as positive and well-defined a function as have
the successive reactions and their normal intervals. If such a pause
is made as we find introduced in the present case, its value must be a
fixed function of the system of durations of which it forms a part,
whether it replace an element in a rhythmical unit, or a subgroup in a
higher rhythmical quantity. In general, the value of such a rest is
less than the duration of a corresponding full measure or interval.
For example, the syncopated forms | >q % | and | >q % %_| are
demonstrably of shorter average duration than the corresponding
measures| >q q | and | >q q q_|; and the pause occurring at the close
of a syncopated line--such as that in the middle of a catalectic
trochaic tetrameter--should be found of less value than that of the
regular foot.

In the present instance two reactions are made, a pause follows, then
the reactions take place again, and so on. The intervals separating
successive groups of reactions thus result from the coalescence of two
periods, the interval which would regularly follow the reaction and
the additional pause at its close. The value of the latter I interpret
as functionally equivalent to a group of two beats and not to a single
interval; that is, the rhythm beaten out is essentially quadruple, the
second member of each composite group being suppressed, as follows:
>
| q q; % % |.
\______/

To estimate the proper value of such a rest the average relative
duration of first and second intervals was taken in a continuous
series of two-beat measures, in which the first member was accented
sufficiently to define the rhythmical groups. The ratio was
1.000:0.760. In the present instance the values of the simple initial
interval and the composite interval which follows it are, in terms of
the linear measurement, 1.55 mm. and 3.96 mm. Assuming the above ratio
to hold, the duration of a period which included the second
beat-interval and a group-rest should be 1.16 + 1.55 + 1.16 = 3.87 mm.
This is slightly less than the actual value of the period, whereas it
should be greater. It must be remembered, however, that the disparity
between the two intervals increases with initial accentuation, and in
consequence the proportional amounts here added for the second
interval (1.16 to 1.55) should be greater. This interval is not
rhythmically 'dead' or insensitive. The index of mean variation in all
reactors is greater for the first than for the second interval (or
interval + pause) in the ratio 1.000:0.436, that is, the value of the
latter is more clearly defined than that of the former, and the
reactor doubly sensitive to variations occurring within it.

An analysis of the variations of these intervals separately in series
of four groups reveals a secondary reciprocal rhythm, in which the
changes in value of the mean variation at any moment are in opposite
directions in the two intervals. These values in percentages of the
total duration of the periods are given in the following table.

TABLE LXI.

Interval. 1st Group. 2d. Group. 3d Group. 4th Group.
First, 15.4 per cent. 26.4 per cent. 13.8 per cent. 30.3 per cent.
Second, 12.4 " 7.0 " 9.6 " 7.5 "

Without measurement of their intensive values, interpretation of these
variations is speculative. They indicate that the pairs of beats are
combined in higher groups of four; that the differences of mean
variation in the first interval are functions of an alternating major
and minor accentuation, the former occurring in the second and fourth,
the latter in the first and third; and that the inversely varying
values of the mean variation in the second interval are functions of
the division into minor and major groups, the reduced values of the
second and fourth of these intervals being characteristic of the
greater sensitiveness to variations occurring in the group pause than
to changes occurring within the group.

The fixity of the group is markedly greater than that of the simple
interval. In the one case in which the mean variation of the group is
greater than that of the elementary period the material involved was
meager (five instead of ten repetitions) and the discrepancy therefore
insignificant.

The difference in the mean variation of the first and second intervals
respectively rises to an individual maximum of 3.000:1.000, and
averages for all subjects 2.290:1.000; the fixity, that is to say, of
the inter-group interval in this form of tapping is more than twice as
great as that of the intra-group interval. The fixity of the larger
rhythmical quantities is greater than that of the smaller, whether the
relation be between the elementary interval and the unit group, or
between the synthetic unit and its higher composite. The average mean
variation of the beat intervals exceeds that of the whole group in the
relation of 1.953:1.000. The differentiation of larger and smaller
groups is less clear. When the material is taken in groups of eight
successive beats the mean variation is less in the case of every
subject than when taken in fours, in the ratio 1.000:1.521. The
comparative values for groups of two and four beats is reversed in two
thirds of the cases, yet so that an average for all subjects gives the
ratio 1.000:1.066 between groups of four and two beats. The whole
series of values arranged on the basis of unity for the mean variation
of the beat interval is given in Table LXII.

TABLE LXII.

Proportional. Single Beat. 2-Beat Group. 4-Beat Group. 8-Beat Group.
M.V. 1.000 0.512 0.480 0.320

The persons taking part in the investigation were next required to
make a series of reactions composed of unit groups of two beats, in
each of which the first member received accentuation, a simple
trochaic rhythm. In this type the relation of intra-group to
inter-group interval remains unchanged. In all subjects but one the
mean variation of the first interval exceeds that of the second in the
average ratio 1.722:1.000. The amount of difference is less than in
the preceding type of reaction. In the former there is presented not
an intensively uniform series, but an irregularly rhythmical grouping
of intensities, in dependence on the well-defined parallel types of
temporal differentiation; in the latter such intensive differentiation
is fundamental and constant in its form. Assuming the character of the
second interval to remain unchanged, there is in the intensive fixity
of the initial accented element, on the one hand, and the alternate
assertion of the impulse to accentuation and repression of it in the
attempt to preserve uniformity, on the other, an occasion for the
difference in the relation of the mean variation of this interval to
that of the following in the two cases. It is to be expected that
there should be less irregularity in a series of reactions each of
which represents an attempt to produce a definite and constant
rhythmical accent, than in a series in which such an accent is
spasmodically given and repressed.

For a like reason, the difference in value between the mean variations
of the elementary interval and the unit group should be less in the
case of the positive rhythm form than in that of a series which
combines a definite temporal segregation with an attempt to maintain
intensive uniformity. The mean variation of the interval is still of
greater value than that of the unit group, but stands to it in the
reduced ratio 1.000:0.969.

The relations of higher groups present certain departures from the
preceding type. In three cases out of five the unit has a greater
> .
fixity than its immediate compound ( | q. q; q q |), with an average
\_______/
ratio of 0.969:1.072. The original relation, however, is reestablished
in the case of the next higher multiple, the eight-beat group, the
whole series of values, arranged on the basis of unity for the simple
interval, being as follows:

TABLE LXIII.

Proportional Single Beat 2-Beat Group 4-Beat Group 8-Beat Group
M.V. 1.000 0.969 1.072 0.859

An analysis of the material in successive pairs of two-beat groups
revealed a pronounced rhythm in the values of the mean variations of
the first and second members of the pair respectively, the fixity of
the second group being much greater than that of the first, the mean
variation having a ratio for all subjects of 0.801:1.000. The
interpretation of this rhythmical variation, as in the preceding
reaction series, must be speculative in the absence of quantitative
measurement of intensive changes, but is still not left in doubt. The
rhythmic material is combined in larger syntheses than the groups of
two beats, alternately accented and unaccented, which were avowedly in
mind. This secondary grouping appears in at least a measure of four
beats, into which the unit group enters as the elementary interval
entered into the composition of that unit. In this larger group the
initial period, or element of stress, is characterized by a greater
mean variation than the unaccented period which follows it. There are
present in this first interval two factors of instability: the factor
of accent, that element which receives the stress, being in general
characterized by a greater mean variation than the unaccented; and the
factor of position, the initial member of a rhythmical group,
independent of accentuation, being marked by a like excess of mean
variation over those which follow it. The interpretation of the latter
fact lies in the direction of a development of uniformity in the motor
habit, which is partially interrupted and reestablished with the
ending and beginning of each successive group, large or small, in the
series of reactions.

Further, when the material is arranged with four unit groups in each
series, the same relation is found to hold between the first period
composed of two unit groups and the second like period, as obtained
within these pairs themselves. The mean variation of the first period
of four beats is greater than that of the second in the case of all
subjects but one, with an average ratio for all subjects of
1.000:0.745. The analysis was not carried further; there is, however,
nothing which points to a limitation of the process of synthesis to
groups of this magnitude; rather, to judge from the close
approximation in definition of the two orders manifested here, there
is suggested the probability that it is carried into still higher
groupings.

In the next rhythmical type analyzed--the iambic form--that relation
of the first to the second interval holds which was found to obtain in
the preceding forms. The excess of mean variation in the former over
the latter presents the ratio 1.274: 1.000. In amount it is less than
in either of the previous types (2.290:1.000 and 1.722:1.000). For
here, though both elements have constant relations as accented or
unaccented members of the group, the factor of stress has been
transferred from the initial to the final beat. Instead, therefore, of
combining in a single member, the factors of inconstancy due to stress
and to position are distributed between the two elements, and tend to
neutralize each other. That the preponderance of irregularity is still
with the initial interval leads to the inference that position is a
greater factor of inconstancy than accentuation.

Also, the group presents here, as in the preceding forms, a greater
fixity than does the individual interval. This relation holds for all
subjects but one, the average mean variations of the simple interval
and of the unit group having the ratio 1.000:0.824.

In larger groupings irregularities in the relations of higher and
lower again occur, and again the greater constancy obtains between the
first and second orders of higher grouping (in which for only one
subject has the lower group a greater fixity than the higher, and the
averages for all subjects in the two cases are in the ratio
1.149:0.951), and the lesser constancy between the unit group and the
first higher (in which two subjects manifested like relations with
those just given, while three present inverted relations). The whole
series of relations, on the basis of unity for the mean variation of
the simple interval, is given in Table LXIV.

TABLE LXIV.

Proportional. Single Beat. 2-Beat Group. 4-Beat Group. 8-Beat Group
M.V. 1.000 0.824 1.149 0.951

There is also presented here, as in the preceding forms, a synthesis
of the material into groups of four and eight beats, with similar
differences in the fixity of the first and last periods in each. A
single subject, in the case of each order of grouping, diverges from
the type. The ratio of difference in the mean variations of the first
and second members of the groups is, for series of four beats,
1.000:0.657, and for series of eight beats, 1.000:0.770. This
indicates a diminishing definition of rhythmical quantities as the
synthesis proceeds, but a diminution which follows too gradual a curve
to indicate the disappearance of synthesis at the proximate step in
the process.

Three-beat rhythms were next taken up and the same method of analysis
carried out in connection with each of the three accentual forms,
initial, median, and final stress. In these types of rhythm the
intra-group intervals are more than one in number; for the purpose of
comparison with the final, or inter-group interval, the average of the
first and second intervals has been taken in each case.

The results agree with those of the preceding types. The mean
variation of the interval separating the groups is less throughout
than that of the average group-interval. The ratios for the various
rhythm types are as follows:

TABLE LXV.

Rhythm Form. Initial Stress. Median Stress. Final Stress.
Ratios, 1.000 : 0.758 1.000 : 0.527 1.000 : 0.658

This relation, true of the average intra-group interval, is also true
of each interval separately. Among these ratios the greatest departure
from unity appears in the second form which all subjects found most
difficult to reproduce, and in which the tendency to revert to the
first form constantly reasserts itself. The difference in value of the
mean variations is least in the first form, that with initial accent,
and of intermediate magnitude in the third form when the accent is
final. The contrary might be expected, since in the first form--as in
the second also--the factors of stress and initial position are both
represented in the average of the first two intervals, while in the
third form the factor of stress affects the final interval and should,
on the assumption already made concerning its significance as a
disturbing element, tend to increase the mean variation of that
interval, and, therefore, to reduce to its lowest degree the index of
difference between the two phases. That it does so tend is evident
from a comparison of the proportional mean variations of this interval
in the three forms, which are in order: initial stress, 4.65 per
cent.; median stress, 4.70 per cent., and final stress, 7.15 per cent.
That the consequent reduction also follows is shown by the individual
records, of which, out of four, three give an average value for this
relation, in forms having final stress, of 1.000:0.968, the least of
the group of three; while the fourth subject departs from this type in
having the mean variation of the initial interval very great, while
that of the final interval is reduced to zero.

If, as has been assumed, the magnitude of the average mean variation
may be taken as an index of the fixity or definition of the rhythm
form, the first of these three types, the ordinary dactylic is the
most clearly defined; the second, or amphibrachic, stands next, and
the third, the anapaestic, has least fixity; for in regard to the final
interval, to the average of the first and second and also to each of
these earlier intervals separately, the amount of mean variation
increases in the order of the accents as follows:

TABLE LXVI.

Interval. Initial Stress. Median Stress. Final Stress.
First, 5.82 per cent. 9.95 per cent. 11.95 per cent.
Second, 6.45 " 7.87 " 9.77 "
Third, 4.65 " 4.70 " 7.15 "

In these triple rhythms, as in the two-beat forms, the simple interval
is more variable than the unit group, and the lower group likewise
more unstable than the higher. The series of proportional values for
the three forms is given in the table annexed:

TABLE LXVII.

Rhythm Form. Single Interval. 3-Beat Group. 6-Beat Group.
Initial Stress, 1.000 1.214 1.037
Median " 1.000 0.422 0.319
Final " 1.000 0.686 0.524