The influence of musical
context on tempo rubato
Renee Timmers, Richard Ashley, Peter
Desain, Hank Heijink
‘Music, Mind, Machine’
group, Nijmegen Institute for Cognition and Information, University of
Nijmegen, The Netherlands
School of Music, Northwestern
University, United States
[Timmers,
R., Ashley, R, Desain, P, and Heijink, H. (2000). The influence of musical
context on tempo rubato. Journal of New Music Research 29 (2), 131-158]
Correspondence to:
Renee Timmers
NICI
P.O. 9104
NL-6500 HE Nijmegen
Tel:
+31(0)24-3612768
Fax:
+31(0)24-3616066
Correspondence may
be addressed to any of the other authors at:
E-mail:
r-ashley@nwu.edu, desain@nici.kun.nl, and
heijink@nici.kun.nl
Abstract
Different pieces of music offer different expressive possibilities. Even a single piece of music offers the possibility to be treated in several expressive ways (see “Repp (1998)”). How much of this variety of possible interpretations is exhibited in actual performances of the music? Do pianists make use of the different parameters of the piece to shape their performance? Do variety in performances and variety in musical parameters relate to each other? Previous studies stress the relation between timing variations and musical structure (see “Clarke (1985)”), but provide no clear answer to the freedom that is allowed within this regularity, especially when multiple structural descriptions play a role simultaneously.
In an experiment the melody of Variations
on an Original Theme
(Op. 21, No. 1) by Brahms, is set in different musical contexts derived from
the Theme. Three pianists are asked to perform the melody in the different
settings from a score. They repeat each performance several times. The settings
are 1) the melody without bar-lines, 2) the melody with bar-lines, 3) the
counter-melody, 4) the melody with the counter-melody, 5) the melody with block
chords, 6) the Theme. The Theme contains all material of previous settings (the
melody, counter-melody and block chords). The settings are presented in a fixed
order and the pianists do not know the pieces before hand.
Analysis of the recorded
performances shows that pianists change the onset timing of the melody with
respect to the musical context in which the melody is presented. Aspects of the
context are imbedded in the timing pattern in different ways; for example, the
addition of chords often causes a lengthening of the melody notes with chords,
and the addition of a counter-melody constrains the lengthening of a melodic
ornament. The melody proves to be the primary expressive source, while chords
and a counter melody are good second ones. Both the variety in timing patterns
and the extent of tempo rubato appear to increase with increasing complex
conditions.
I Introduction
I.1 The performer’s
expression of music
When a pianist performs a piece of
notated music (s)he translates this notation into actual tones sounding in
time. Three primary factors are responsible for how the performance will sound:
the composition represented in a score, the expression of the composition
imparted by the performer, and the instrument. A traditional score of a piece
of Western tonal music specifies the pitches and rhythm of the music, and
indicates roughly the articulations, dynamics, and tempi of the performance.
Other “givens” in the composition itself include the musical
texture and, to some extent, the overall character of the music. From the
combinations of pitches and rhythms the performer interprets the harmonic and
melodic function of the notes.
What the performer’s expression adds to the composition is a more closely detailed specification of the execution of notes than is given in the score. In performing the score, the pianist decides on the relative loudness, timing and articulation of simultaneous notes, and also refines the globally indicated dynamics, tempi and articulation marks of the composition. The characteristics of the pianist’s expression will depend on his/her musical interpretation of the piece as well as motor skills, and is in part a function of perceptual constraints (for a detailed description see “Penel & Drake (1999)”).
The pianist’s expression has
an extremely large influence on the way the performance of a piece will sound.
All differences between the ways in which performers produce a sounding piece
of music (good/bad, enlightening/depressing, and the like) are due to
differences in musical expression and, to a minor degree, to the instrument.
This means that most differences between pianists may be seen in expressive
variations in timing, articulation and dynamics.
I.2 Previous literature
on tempo rubato
Tempo rubato (literally, “stolen time”) is the
traditional name for variations in the timing of onsets (beginnings) of subsequent musical
notes. It is a term first used by Tosi in 1723 “Hudson (1994)” in
reference to performers’ alteration of the expected rhythm of a melody
within an underlying base tempo, where time was “stolen” from one
note and “repaid”. Later it was used to refer to the alteration of
the expected tempo of all voices simultaneously “Hudson (1994)”. In
this latter practice time was “stolen”, without the intention of
subsequently restoring it “Donington (1963)”. Nowadays the usual
performance practice of classical and romantic music is to vary all voices
simultaneously, slowing them down or speeding them up in more or less the same
way at the same time (empirical evidence for this is for example provided by
studies of “Repp (1996)”, and “Palmer (1996b)”). In
other musical genres, such as jazz and authentic historical performance
practice, the alteration of the expected rhythm of the melody (to borrow time
as “Donington (1963)” calls it) against a steady beat or base tempo
is still common use “Ashley (in preparation)”.
Sometimes composers or editors
indicate tempo rubato, in the sense of gradual tempo changes, in the score of a
musical work. For example, ritenuto or decelerando indicate a slowing of the tempo, while a
speeding up of the tempo is typically indicated by accelerando. Performers also frequently show a
tendency to speed up and slow down when this is not indicated in the score.
Such modifications of tempo typically occur in relation to phrase structure, as
a way of marking phrase boundaries (see “Palmer (1989)” and
“Repp (1990, 1992a)”).
Alterations of the rhythm which are
not related to tempo per se but which affect more local aspects of temporal relationships between
notes also occur in the course of the performance of a work. Situations in which certain notes are
shortened or lengthened in favour of other surrounding notes are generally not
indicated in a score and are part of what musicians refer to as “performance practice”. An
example of changing the relative timing of notes is the French practice of notes
inégales, as
described from the mid-seventeenth century to the end of the eighteenth
century. In this practice, pairs of equally notated notes are performed with a
slight lengthening of the first note “Hudson (1994)”,
“Donington (1963)”.
In the research literature of music
psychology, music performance has been of particular interest, beginning with
the pioneering work of Seashore in the 1920s (see “Seashore (1967)”
for a summary). The studies are mainly concerned with Western tonal music for
which a score is available. In
this research, the kinds of variations that pianists make in onset timings,
dynamics, and articulations (= offset timing) are described and explanations
for the observed regularities are proposed. One explanation that is quite
appealing involves mapping the expressive variations to the musical structure
of the piece. In this approach the assumption is made that the
performers’ expression of the music serves to highlight the musical
structure “Clarke (1988)”, “Palmer (1996b)”,
“Sloboda (1983)”. In other words, the expressive patterns found in
performances of a piece can be explained with reference to the
performer’s structural interpretation of the piece. In this sense
‘the interpretation of a piece of music acts as a grammar generating the
expressive forms in a performance’ “Shaffer, Clarke & Todd
(1985 p.63)”.
The kinds of relationships between
musical structure and tempo modulations that have been suggested by various
researchers include modulations that clarify the metrical structure of a piece,
modulations in relation to harmonic structure, and phrase final
lengthening. Modulations in
relation to metrical structure are suggested by “Parncutt (1994)”,
and by “Clynes (1995)”, who proposes a “composer-specific
pulse” which is characterised by a special pattern of relatively longer
and shorter beats in each measure. Further, tempo modulations have been
suggested to relate to global patterns of harmonic tension and release, as
found in “Palmer (1996a)”, or more local harmonic phenomena, such
as Sundberg’s “harmonic charge” “Sundberg, Friberg
& Frydén (1991)”.
Phrase final lengthening is a lengthening of notes at the end of musical
groups (or, put another way, a slowing of tempo at the end of a group). It is a
common phenomenon, reported by many researchers in speech and music (among
others, “Shaffer (1980)”, “Todd (1985)”, “Palmer
(1989)”, “Repp (1990, 1992a)”). In the relation between
grouping structure and phrase final lengthening it has been proposed that tempo
rubato in musical performance is used to reflect the hierarchic depth of a
syntactic unit or group by the amount of slowing or pausing at its boundary
“Shaffer (1980)”, “Todd (1985)”. What this means is
that a hierarchic grouping structure (for example, small motivic groups
combining into phrases, combining into themes) is reflected in the amount of
slowing down at group boundaries, with larger-scale boundaries showing more
slowing than smaller-scale boundaries.
Local lengthening or shortening of
notes is often viewed in relation to certain surface characteristics of the
music such as harmonic, rhythmic and melodic contexts of notes. “Sundberg
et al. (1983, 1991)” have developed a generative model of expressive
treatment of notes in relation to such musical characteristics. The model is
based on generalisations of the judgements of a professional musician
“teaching” a computer to play expressively. An example of harmonic
context in Sundberg’s model is harmonic tension or distance of a note to
the tonic, where the duration (and intensity) of a note whose tonal distance is
far from the current tonic is increased. Other factors may also affect
musically expressive performance. For example, contrasts between the duration
of notes is reduced, in such a way that a short note just before a long note is
played longer than a short note in between other short notes. Also, the
duration of notes just after a large leap is lengthened for expressive purposes
in favour of the duration of the note initiating the leap “Sundberg et al.
(1991)”.
1.3 Difficulties of
previous research
Current explanations of tempo rubato
rely heavily on the concept of a tempo curve. This notion refers to a changing tempo
“path” to which the music is synchronised. One implication of
research using the concept of the tempo curve is that these curves have some
independence and can be modulated and transposed without regard to the more
detailed structure of the work, such as rhythmic or harmonic structure. In this
sense a tempo path specifies tempo changes that are natural and general such as
the final retard that is suggested to be an allusion to physical motion
“Kronman & Sundberg (1987)”. “Desain and Honing (1993,
1994)” plead against this notion of an independent tempo path, and claim
that a sense of tempo variation cannot be perceived independently of the events
carrying it.
The focus on the explanation of
expressive timing patterns by generative models, such as “Todd
(1985)” and “Sundberg, Friberg and Frydén (1991)”,
gives another bias, that of providing one expressive pattern for one structural
description. In other words, the rhythmic, harmonic and melodic features of
music are described and coupled with certain expressive patterns of timing or
dynamics. In this account no reference is made to a performer who actually
makes this translation from structure to sound. Therefore, only one expressive
interpretation of the music is provided and differences between performers are
not accounted for. Most of the
time, the modellers do admit this shortcoming. For example, “Sundberg et
al. (1983)” suggest that the rule-system should leave room for variations
in the magnitude of the effects produced by the expressive performance rules to
model the multitude of choices that are available to musicians. They also
suggest that musicians can and should violate one or more of the performance
rules in order to surprise or excite the audience.
I.4 The present study
In the present study we aim to
compensate for at least some of these shortcomings. First of all, we try to get
a grasp on the differences between performances of the same piece. According to
“Repp (1992a)”, performances of the same piece show both
considerable commonality, as well as diversity, in their timing patterns. The
commonality found by Repp existed at a more general structural level than the
diversity; commonality between performances mainly concerned phrase final
lengthening, while the diversity was found in the expression of melodic
gestures of approximately seven notes. The origin of this diversity is uncertain.
On the one hand, differences in interpretation of the musical structure would
lead to different timing patterns, even when the same strategy is used to
translate the structural description into action and sound. On the other hand,
it is possible that performers can express the same structural interpretation
in different ways, which would also lead to different timing patterns.
In the present study we will try to
explain the diversity of performances in relation to the number of possible
structural descriptions of the music. In this explanation we assume first of
all that there is no single structural description of a piece of music which
will be used by all interpreters, but rather that there are multiple
possibilities for making structural descriptions of most pieces of music, at
some appropriate level of detail. The number of descriptions increases with the
complexity or diversity of the music, which leads to smaller similarities
between performers. A number of sophisticated studies in music theory suggest
or show how this diversity of musical structure might be understood; these
include “Meyer (1967)”, “Kunst (1978)”, “Lerdahl
and Jackendoff (1983)”, and “Lewin (1985)”.
Second, we will investigate the
extent of tempo rubato in a more systematic way than has been previously done.
Not much is known about the regularities underlying the ranges of tempo rubato
found in musical performance. A general agreement is that the extent of tempo
rubato changes with the hierarchic depth of the grouping structure.
There are, however, other
regularities that underlie the use of tempo rubato, which have not been
investigated systematically. One of these is that the average extent of tempo
rubato changes with the type of music being performed. For example, Chopin etudes
are frequently performed with a relatively large amount of tempo rubato, while
Bach fugues are often performed in quite strict tempo. Musical style, a
performer’s personal style, and the performer’s conception of the
appropriate way to play a given musical style all play a role in this. We
think, however, that musical structure plays an underestimated, but important
role as well. Evidence for this is found, for example, in the way a theme and
its rhythmic variation--which share many aspects of structure--are performed
with different degrees of rubato (see “Desain & Honing
(1994)”). Even more striking is the difference in rhythmic freedom with
which a prelude and fugue of a harpsichord or organ sonata from the seventeenth
century are generally performed. An example of this is the well-known Toccata
in D minor by J. S. Bach (BWV 565) in which a fugue is interspersed with
sections of free fantasia “Grout & Palisca (1988)”.
To our knowledge it is still unknown
which aspects of the musical structure influence the extent of rubato. We do,
however, think that there are plausible hypotheses. The idea that we pursue in
this study is that the average extent of tempo rubato of a performance depends
on structural characteristics of the music and, in particular, on the richness
of the musical texture. Our main hypothesis is that a rich musical basis is
needed to perform a piece with significant tempo rubato. In this respect we
think a melody alone will not typically be performed with large tempo
variations, while a melody with chords and arpeggios has more potential to be
performed expressively. On the other hand, two melodic lines together could
constrain each other in timing freedom due to the interaction of their
individual timing profiles. Although we think other issues play a role as well
(such as the possible metrical “rigidity” of a piece) we will only
deal with influences of musical texture (inferred from the score by the number
of voices) and interaction of melodic lines on the extent of tempo rubato.
In the present study, we recorded
performances of a melody set in different musical contexts. These settings of
the melody vary in diversity of musical structure as well as in richness of
texture. The influence of these contexts on the performance is examined, in particular
with respect to the variability and the extent of tempo rubato of the
performances.
Specifically, we asked three
professional pianists to perform the melody of the Theme of Brahms’ Variations
on an Original Theme
(D major, Op. 21, No. 1, 1861) in different musical settings. They repeated
each performance several times. The different musical settings are: 1) the
melody without bar-lines, rhythmic beams, dynamics and phrase markings; 2) the
melody with bar-lines, rhythmic beams, dynamics and phrase markings; 3) the
counter-melody alone; 4) the melody with the counter-melody; 5) the melody with
block chords; 6) and the Theme as Brahms originally set it (see figures 1-6).
The Theme combines all the material of the previous settings (the melody,
countermelody and block chords). A more detailed description of the stimuli
follows in the method section.
All of these musical excerpts have a
very similar grouping and metrical structure; this means that differences in
timing of the melody should not be due to differences in these structural
aspects. This provides a way of testing the generality of models that map
timing variations to grouping and metrical structure without taking other
aspects of the music into account such as rhythmic, melodic and harmonic structure.
More commonality between the different settings would implicate grouping and
metrical structure as a strong influence; less commonality would implicate high
influence of other aspects of the musical structures that differ from setting
to setting.
Figure 1
First four measures of condition 1: melody
without bar-lines, rhythmic beams, dynamic and phrase marking.
Figure 2
First four measures of condition 2: melody with
bar-lines, rhythmic beams, dynamic and phrase marking.
Figure 3
First four measures of condition 3:
counter-melody solo.
Figure 4
First four measures of condition 4: melody with
the counter-melody.
Figure 5
First four measures of condition 5: melody with
block chords.
Figure 6
Condition 6: the Theme by
Brahms from which all other stimuli are constructed. The melody is the upper
line of the Theme. The counter-melody is the upper line of the Bass staff. The
chords are the (simplified) chords at each first beat of the measure.
II Method
II.1 Material
The pieces our pianists performed
consisted of a melody in different settings. The melody and the settings were
(for the purpose of the experiment) extracted from the Theme of Variations
on an Original Theme
for piano solo by Brahms. We chose this piece because, although written by a
master composer, it is fairly unknown to most pianists and has interesting
structural features such as rhythmic diversity, rhythmic conflicts,
unconventional phrase lengths (nine measures), and a combination of chords and
several melodic lines. Performers’ general unfamiliarity with the piece
gave us the opportunity to manipulate the music without the risk of the
pianists noticing it and limited the possible confounding effects of prior
knowledge of the music.
The settings of this melody vary in
texture and complexity. The texture is most impoverished for the melody-alone
condition and becomes increasingly rich by combining the melody first with the
counter-melody (two voices), followed by the melody with an accompaniment of
chords (four voices), and finally is in its richest context of all in the
Theme. As an indication of the complexity of a musical context, we take the
number of explicit dimensions of the musical structure. This number is small
for the melody alone condition, greater for the melody with block chords
condition (one melodic line + harmony), great for the melody with
counter-melody condition (two melodic lines + harmony) and greatest for the
Theme (two to three melodic lines + harmony + arpeggio chords).
To these five settings of the
melody, one stimulus is added that does not contain the melody, but consists of
the counter-melody on its own. This brings the total of conditions to 6, each
containing three repetitions of one stimulus.
In more detail, the conditions can
be described as follows.
· In Condition 1, the player is
confronted with the melody of the Theme with no explicit metric, phrasing or
dynamic information given. The stimulus contains, in other words, only the
“raw ingredients” of melody: pitch and rhythm (the first four
measures of stimulus 1 are given in figure 1). In this condition there are
still a variety of sources of information upon which the performer can draw,
however. For grouping structure,
there are parallelisms of motivic structure caused by the repeat of the first
and second “halves” of the Theme, which indicate large-level
grouping. There are individual notes, such as the d’’ and
g’’ in figure 1, which are emphasised by being the local maxima of
the melodic contour, and there are notes that receive “agogic” or
durational stresses, such as longer notes surrounded by shorter ones. However, note that in these cases some
of the stresses so indicated go against the actual meter of the melody as
notated in the original score. For example, in the first few measures all of
the notes that would receive stresses by virtue of being the goal of a melodic
leap (e.g. the third note in the melody) occur not on the downbeat but
internally to the measure. Such rhythmic and metric conflicts are a typical
feature of Brahms’ music.
· The second condition is the same
melody, this time presented in its entire form (for the first four measures,
see figure 2): bar-lines, time signature, phrasing, dynamics, rhythmic grouping
(beams), ornamented eighth notes (in measure 10 and 12 (that is, the first and
third measures of the second half). The addition of metric information gives
the player much more clarity as to the musical structure of the melodic line.
In particular, the location of downbeats--relatively strong beats--are now
disambiguated. However, a number
of new decisions are now made available, as well: for example, how should the
relative weight of the downbeats and the notes emphasised by leaps be handled? There are multiple implications for
performance, coming from different aspects of the musical structure, such as
meter, phrase and melodic contour, giving the player an interesting set of
factors to balance in producing a performance.
· Condition 3 (see figure 3) differs
from all others in that it does not include any version of the melody. Instead, it consists only of the
counter-melody taken from the Theme.
Our intention here is to determine the way in which the counter-melody
would be played expressively if taken by itself, to better understand its
contribution to the performance of the Theme as a whole. In fact, there are noticeable
differences between the construction of the counter-melody and that of the
melody. For example, the most
notable “goal” of melodic direction in the counter-melody is on the
downbeat of measure 3 (the turning point of the melodic contour)--a point not
emphasised in any significant way in the melody. If a performer should focus on this aspect of the music,
there might be some timing changes produced with the goal of defining this point
in time as significant (for example, slowing down in the vicinity of this
beat).
· Condition 4 consists of a two-voice
texture, containing the melody and counter-melody played together (see figure
4). It is in this condition that
truly interacting sets of musical possibilities begin to emerge, as the related
but individual structures of the two lines are presented together. Further, the
harmonic intervals are new in this condition. Up to this point any sense of
chord, consonance, or dissonance in the music was something that had to be
purely inferred by the performer, as there were no simultaneously sounding
tones. However, now certain
melodic tones--for example, the downbeats of mm. 8, 11, and 17--are mildly dissonant
in the new context, affording new possibilities for expression (in this case,
perhaps lengthening the tones so as to emphasise the tension of the
dissonance).
· Condition 5 is somewhat like
condition 4, in that the melody is now placed in the context of other voices.
However, this time the melody is presented with block-chords (see figure 5),
which is a harmonic differentiation of the melody, with chords and chord
functions now quite explicit rather than implicit. This condition also brings a
change in texture, because of the fuller sound of the chords. In condition 5
the more dissonant tones are the downbeats of measures 3 and 6, neither of
which were dissonant in condition 4.
For this reason, different melodic pitches may be emphasised in the harmonic-block
chord context of this condition than in the more purely contrapuntal condition
4.
· The last condition is the full Theme
(figure 6), which contains the melody, the counter-melody and the chords in
full context. This is the most complex piece of music and it has the richest
texture. It is the only stimulus composed for musical purposes by a well-known
composer and not for experimental purposes. In this condition we hope to see
the way in which the performer chooses from among the different possibilities
inherent in the composition, which should have been highlighted in the previous
conditions, in order to produce a rich and interesting performance.
The Theme is in D major, in
three-eight meter and starts poco forte. The first measures of the piece emphasise D
major, with the tonic chord in the first measure, sub-dominant chords in
measures 2-4 and dominant chords in measures 5-6. From the tonic of D, the
music modulates to A major at the end of the first half. The second half starts
in d minor, modulates to the relative major tonality of F (mm. 10-11) and
returns via the dominant of G (mm. 14-15) to D major (last three measures). The
piece begins and ends with a pedal on D and parallel melodic lines in eighth
notes. In between the movement is primarily in chords of eighth note duration.
Dissonances regularly occur as passing notes and changing notes. Strong
dissonant chords occur, for example, at the start of measure 5 and at the last
beat of measure 8.
The musical structures of the
stimuli are quite similar in several respects. All stimuli have a common
metrical and phrase structure, except for the melody without bar-lines
(stimulus 1), which has no explicit metric structure. Stimuli 2-6 are in
three-eight meter, with an eighth note metrical level and a larger dotted
quarter note level; sometimes the eighth note metrical level is subdivided in a
sixteenth notes. The eighth notes which are ornamented with a main-note or
five-note turn (for a definition see “Donington (1963)”) in
measures 6, 10 and 12 occur irregularly, in that they do not fall into a
metrical framework other than the eighth note and bar levels.
Stimulus 1 lacks a metrical
indication. It is therefore unclear what metrical interpretation the performer
will make. The performer may choose between a binary and a ternary meter, which
are the most common meters in western classical music, or, possibly, he will
change the meter within the piece. The piece could either start on the
downbeat, with an upbeat or with upbeats. Cognitive rules like the metrical
rules of “Lerdahl and Jackendoff (1983)” provide plausible downbeat
markers, such as relative early notes within groups, the note after a leap, a
relative long note, a relative low note. For stimulus 1 these rules do not
unambiguously show one solution. For example downbeat markers within the frame
of the first eight notes would fall on the first note (early note within a
group), third note (note after a leap) and sixths note (note after a leap).
These last two accented notes preferably receive parallel metrical structure.
The first long note is spaced fourteen eighth notes away from the accented note
at the sixth eighth note beat. This distribution of accented notes cannot be
combined in one optimal way with the metrical well-formedness rules that state
that strong beats should be spaced either two or three beats apart and each
metrical level must consist of equally spaced beats (see “Lerdahl &
Jackendoff (1983)”).
All stimuli are divided in two
halves (A and B) both of which are repeated in the performance. Each section is
nine measures long. Both sections contain two sub-phrases of which the first
sub-phrase is built out of a 2 + 2 measure structure, whereas the second
sub-phrase consists of a 2 + 3 measure structure. This periodicity is found in
both the melodic and harmonic structure of the piece. In section A, both
sub-phrases (mm. 1-4 and 5-9) show a rising and falling melodic movement. The
start of the second sub-phrase of section A (in measure 5) is marked by a
variation of the beginning measures of the piece. In section B, the melody of
the first sub-phrase rises towards the start of the second sub-phrase (in
measure 14). This second sub-phrase mainly contains a falling movement. In the
first sub-phrase of section B the chord progression is in eighth notes, while
in the second sub-phrase the pedal on D dominates.
Because all stimuli are derived from
the same Theme the global underlying harmonic progressions of the stimuli are
the same. It is, however, unclear whether a performer will interpret the
underlying harmony in the same way for each stimulus. In the first 4 stimuli
only one or two melodic lines are given, which is not enough to present the
performer with an unambiguous harmonic context. Slightly different harmonic
interpretations of the stimuli cannot therefore be excluded.
II.2 Subjects
Three professional pianists
participated in the experiment. They have all completed their undergraduate
conservatory-level studies and are presently active as performing musicians.
Subject 1 (S1), age mid-twenties, is continuing his studies at an advanced
level. Subject 2 (S2,) age mid-thirties, is working as a professional
accompanist. Subject 3 (S3), age late-twenties, is a professor of piano at a
conservatory. The pianists were paid an appropriate fee for their services.
The analysis of the performances of the pieces by only three
pianists provided the opportunity to obtain detailed insights into the
treatment of the different contexts by each pianist, without the danger of an
overload of data. It also gave the opportunity for (thorough) comparison
between the performances. Insight into the diversity between performances of
the same piece is limited, because of the small number of subjects. We accept this limitation in favour of
a more detailed comprehension of each single performance.
II.3 Procedure
Each pianist participated in a
separate recording sessions of one hour including two short breaks. Each
recording session had the following set-up. Upon arrival, the pianist was given
time to familiarise himself with the laboratory arrangement and to warm up. The
first musical fragment was presented to him on a score. The pianist was given
time to examine the piece and to study it. A metronome was provided during the
practise to help the performer to set the right global tempo of the eighth notes
at approximately 60 beats per minute (BPM). During the recordings no tempo
indication was given. The instructions were to examine the piece and to play it
four times with approximately the same expression. The first time through was
meant as a trial, with the other repeats being the actual recordings. The
pianist was also asked to play as musically and naturally as possible.
The tempo of the eighth notes at 60
BPM was chosen in accordance with a CD recording of a performance of the Theme
by Idil Biret (Naxos CD 8550509). The tempo was appropriate for the Theme, but
a bit slow for the simpler first four conditions. By asking the pianists to
perform all conditions in approximately this slow tempo, we assured that the
tempo contrast between the first four conditions and the last two conditions
would be limited. In the experimental design we favoured expressive freedom for
the pianists above experimental constraints. We wanted the pianists to play as
freely and expressively as possible. The result of this procedure was that they
were given a metronome to use during practice and establish a reference tempo,
but the metronome was not used during the recording sessions. This made tempo
differences between performers and conditions unavoidable, as, without a steady
mechanical beat given as a “click track”, all performers will tend
to drift up and down in tempo over the course of time. The actual average tempi
therefore show some differences between conditions and between performers (see
table 1). Our reason for being concerned with base tempo is that previous
research has shown that timing patterns change when a piece is performed in
different tempi “Desain & Honing (1994)”, “Repp
(1994)”. This means that we cannot generalise over tempi, and that tempo
can be a factor in determining the timing pattern of a piece.
|
|
Condition 1 |
Condition 2 |
Condition 4 |
Condition 5 |
Condition 6 |
Subject 1 |
82.7 |
84.7 |
75.4 |
70.1 |
54.0 |
|
Subject 2 |
97.5 |
98.0 |
110.5 |
103.3 |
87.1 |
|
Subject 3 |
79.5 |
77.6 |
67.8 |
59.9 |
48.1 |
Table 1
Mean tempo (in beats per minute) per performer per condition.
The stimuli were presented in a
fixed order (in the order described above). The stimuli could not be presented
in random order because they are transformed versions of a theme and we wanted
the pianists to perform each stimulus in as “unprejudiced” a manner
as possible. That is to say, the
first condition had to be the first stimulus to be performed, as otherwise the
pianists would have known the metrical structure of the piece. Likewise, the Theme had to be the last
stimulus presented. It was also crucial that the pianists did not know the
Theme beforehand (as noted earlier, one of the reasons why this piece was
chosen). By presenting the melody with increasingly more information, we
ensured that each fragment had its own identity.
The recordings were made in the
‘Music, Mind, Machine’ laboratory on a Yamaha Disklavier MIDI grand
piano. This instrument detects key velocities and pedal movements optically and
converts this information to standard MIDI messages “MMA (1996)”.
II.4 Analyses
The analysis of the three recordings
of the six stimuli from each pianist involved several steps. The MIDI files
with the performance data were imported in POCO, a computer environment for
research into expression in music (see appendix and “Honing
(1990)”). A performance-score-matching facility in POCO was used to link
notes in the performance to their corresponding score representation and to
extract the timing data from the recordings for the melody of each condition
and the counter-melody of condition 3 and 4 “Desain, Honing and Heijink
(1997)”, “Heijink & Desain (in press)”. For further
processing and statistical analysis of the data the statistical software
package JMP 3.2.2 was used.
The timing data consist of scaled
and normalised inter-onset-intervals (IOI’s) at the eighth-note metrical
level. First the melody eighth note IOI’s are constructed, by calculating
the duration between succeeding onset times of melody eighth notes. When there
is no melody note onset that coincides with the eighth note beat the interval
to the next melody note is interpolated. This eighth note IOI pattern is than
scaled to the slowest tempo of all performances (i.e., 54 eighth notes per
minute). We divided the IOI’s of a single performance by its mean eighth
note IOI and multiplied this by the new, standard mean eighth note IOI (1132
ms). The scaled IOI patterns are finally normalised by subtracting the mean IOI
from each eighth note IOI. The result is a timing pattern that indicates for
each melody eighth note how much its duration deviates from the mean eighth
note IOI. This deviation is given in milliseconds (see figure 7). The same is
done for the counter-melody in condition 3 and 4.
The reason for this normalisation
and scaling of tempi is that we wanted to have a uniform scale in which all
tempo variations can be compared directly with each other, independent from
mean tempo of the performances. Both the scaling and the normalisation did not
influence the correlation measurements on which the results of the first
“Result and Discussion” section are based. They do, however, effect
the extent of tempo rubato measurements. This influence is further discussed in
section III.2.
We take this approach aware of its
possible limitations. In particular, this procedure of scaling the performances
seems to imply that one can generalise over tempo, and this is clearly not the
statement we want to make. The only reason for scaling is the uniform scale for
tempo variations, as far as the explanation of the differences is concerned,
tempo is not excluded as a factor (see also III.1.f)
Figure 7
Above: Score condition 5, melody with block
chords, measures 5 – 9.
Below: Timing profile condition 5: S1, 5mel, measures 5-9 The duration between succeeding melody eighth note onsets (above) is calculated resulting in an eighth note IOI pattern. This pattern is normalised by scaling the mean tempo to the slowest tempo of the performances. An eighth note timing pattern (below) is calculated by subtracting the mean eighth note IOI from the normalised eighth note IOI pattern. The deviation from the mean in milliseconds is then depicted. When no melody note falls at the eighth note beat, the duration till the next melody onset at an eighth note beat is calculated and divided by the nominal duration of the interval.
We focused on the interpretation of
the tempo rubato pattern at the eighth-note level of the melody. By doing this
we made two assumptions: first, that tempo rubato is expressed by variations in
the length of beats, and second, that beats coincide with the onset of melody
notes. In other words, we measure tempo rubato by calculating the duration
between onset-times of successive eighth notes. Measuring onset-onset intervals
is the most accurate way of measuring tempo variations available.
The choice of the eighth note level
is made, because, on the one hand, this level is small enough to contain
detailed tempo variations, which provides the opportunity for performer
differences to come through, and the other hand, it is the “tactus”
level--that from which the performer is feeling the ongoing “pulse”
of the music. By focusing on the tactus level we expect that inconsistencies
due to lack of motor and conceptual control play a minor role.
III Results &
Discussion
The following section is split in
two. First we will report the
results of the study in relation to the timing contour of the melody in the
different conditions for the different subjects, and then we will report the
results as far as they are concerned with the extent of tempo rubato.
We use two kinds of abbreviations:
1) S1, S2 and S3 for respectively subject 1, 2 and 3; and 2) a combination of a
number and “mel” or “counter-mel” refers to the timing
data of either the melody or the counter-melody from a certain condition. For
example, 4mel refers to the timing of the melody in condition 4, while
4counter-mel refers to the timing data of the counter-melody in the same
condition.
The focus is on the analysis of the
onset timing of the melody in the different conditions. At one point in the
analysis, however, we will also consider the onset timing of the
counter-melody. In these analyses only the performances of S2 and S3 are taken
into consideration. We leave out the analysis of the counter-melody by S1, since
the recording of the performance of condition 4 by that subject showed some
missing notes in the counter-melody due to very soft playing, which the
Disklavier did not register.
Remember that all timing patterns
are scaled to the lowest tempo. This results in relatively large tempo
variations. The actual tempo variations can be reconstructed with the aid of
table 1. The effect of the scaling is only noticeable in the figures that show
timing deviations in ms. The correlation measurements are the same for raw
IOI’s as for scaled and normalised IOI’s.
III.1 Changing timing
contour
In general we found that the
variability in timing contour is largest between performers, moderately large
between conditions within performers, and smallest within conditions and within
performers. The average correlation between timing profiles (= eighth note IOI)
of repeated performances is 0.81, which means that the effect of repetition on
the tempo rubato patterns is small (for further detail see table 2). Generally
the correlations between eighth note IOI’s of the melody in the different
conditions were lower than the correlation between eighth note IOI’s of
repeated performances. Significant differences between correlations are for
example the low correlation between performances of 2mel and 4mel by S3 in
contrast to the consistency with which 2mel and 4mel are timed over repetitions[1].
Taken as a whole, these results indicate that the contexts effect the rubato
patterns more than the repeats do. In other words, subjects do change the
timing of the melody with changes in context. This change occurs, however, only
to a limited degree (the correlation between tempo patterns are all
significantly greater than 0 (p < 0.05); for an overview see table 3).
|
Performer |
Condition |
Average
correlation between repetitions |
S1 |
1mel 2mel 4mel 5mel 6mel |
0.864 0.756 0.834 0.859 0.788 |
|
S2 |
1mel 2mel 4mel 5mel 6mel |
0.890 0.793 0.888 0.850 0.882 |
|
S3 |
1mel 2mel 4mel 5mel 6mel |
0.546 0.750 0.741 0.713 0.836 |
Table 2
Average correlations between repetitions split
by performer and condition. The correlations are the same for scaled,
normalised and raw eighth note IOI patterns.
|
Variable S1
1mel S1 2mel S1 4mel S1 5mel |
|
|
|
Variable S2
1mel S2 2mel S2 4mel S2 5mel |
|
S2
2mel 0.546 |
|
|
|
Variable S3
1mel S3 2mel S3 4mel S3 5mel |
Table 3
Correlation between averaged timing profiles of conditions (averaged over repetitions) split by performer. The correlations are the same for scaled, normalised and raw eighth note IOI patterns.
The correlations between melody
eighth note IOI’s within conditions and between subjects were all
significantly greater than 0 and generally significantly smaller than
correlations of repeated performances (p < 0.05). In other words, the timing profiles
of the melody of a single condition by different performers were related but
showed clear differences as well. The correlations between performers (within
conditions) were, on average, lower than the correlations between conditions
and within performers (the correlation between performers is further discussed
in section III.1.c and in figure 10). This means that in this study the timing
patterns found are in general more typical for performers than they are for
conditions[2].
III.1.a Ground pattern
The timing contour of the melody is
different for the different musical settings of the melody; still, there is
considerable consistency and similarity between the profiles of different
conditions within subjects and even between subjects. To get a first general
impression of the way in which the performers time the melody we discuss the
main characteristics of the grand average of all performances. Such a grand
average highlights the aspects shared by all performances and diminishes the
variable aspects. In this way the grand average can be seen as an indication of
a “ground pattern” shared by the different performances
The grand average timing profile is
shown in figure 8. There are a couple of aspects that characterise this timing
pattern. First of all, the repeats of the first and second half have a very
similar timing profile, with only the last measures of the second half being
timed differently the first and second times through (this is because the
phrase final lengthening at the end of the piece does not occur the first time
through). Second, the large peaks at 17 and 44 coincide with a greatly
lengthened ornamented eighth note of measure 6. The smaller peaks in 56 and 83
coincide with a lengthened ornamented eighth note in measure 10. Third, the
first half and second half have different timing profiles. The first half
speeds up towards the second beat of measure 3 (score-time 10 and 37 in figure
8) and slows down towards the end of the phrase in measure 9 (score-time 27 and
54 in figure 8), with the locally lengthened ornamented eighth notes in the
middle. The second half has less clear accelerations of tempo. It does contain
clearly lengthened notes, which occur more frequently compared to the first
half. Phrase final lengthening occurs in the second half two times as much as
in the first half, due to the sub-phrase ending at measure 13 that is clearly
marked by a lengthening, as can be seen in the figure at score-time 66 and 93
(repeat).
Figure 8
Grand average timing profile, calculated by averaging over all (scaled and normalised) timing profiles of the melody. The timing profile contains for each eighth note its deviation in milliseconds from the mean eighth note IOI. The dashed lines indicate phrase boundaries.
III.1.b Influence of
contexts on timing contour
The grand average highlighted the
common elements within the timing profile of the melodies. In this section, we
examine the relative impact or influence of the contexts on the timing contour
of the melody. The analysis is done in two ways. First, a factor analysis is
used to obtain insight into the relatedness between the melodies of the
different conditions. Second, partial correlations highlight the degree to
which pairs of conditions are uniquely related with respect to the timing of
the melody in the other conditions.
In the factor analysis, the data
within one subject was satisfactorily explained by two orthogonal common
factors (eigenvalue > 1), calculated by taking the rank order of the timing
patterns of the melody as variables. The choice to use the rank orders was made
to cancel out the possibility that one of the factors would represent the
degree of (lack of) normal distribution. The disadvantage of this method is
that the interpretation of the factors is less easily done on the ground of
their own characteristics, but has to be reconstructed from the correlation
measurements. The two latent factors explain on average 66.8% of the data. The
largest factor contributes four to seven times as much to this explanation as
does the second factor. This large factor can be seen as the basic form of a
fundamental ground pattern that explains much of the timing variability. For S1
and S3 this ground pattern is most closely related to the melody of condition
6, while for S2 condition 2 correlates most highly with the main factor. The second factor is orthogonal to the
first factor. The conditions that most highly correlate with this second factor
(and correlate only to a minor degree with the main factor) are: for S1, the
melodies of condition 4; for S2, the melodies of condition 6; and, for S3, the
melodies of condition 2 (see figure 9).
Figure 9
Loading of each
condition on the two factors calculated in a two-way factor analysis. Variables are the (scaled and normalised) timing patterns
of the melody of single performances. Number combination: indication of
condition – indication of repetition
On the ground of these correlations
some characteristics of the two factors were reconstructed. The main factor of
S1 contains, in the first half of the piece, a clear periodic pattern of
accelerations and decelerations. The second half, however, is better
characterised by a high density of large local lengthenings and shortenings,
such as the lengthening (and compensation of the lengthenings) of the
ornamented eighth note and of the third beats of measures 12 and 15. The second
factor that is needed to explain the performance of S1 primarily consists of a
global, constant tempo, with small local decelerations. The second half of this
factor has a greatly lessened density of timing changes, compared to the second
half of the main factor.
The main factor of S2 consists of a
global constant tempo with local lengthening of the ornamented eighth note and
of the upbeats to measures 8, 9, 15 and 16. The second factor contains a global
speeding up of the tempo. It also contains clearly pronounced
acceleration-deceleration patterns. Lengthenings primarily occur in the first
half, such as at the first beat of measure 3 and 5 and the upbeat to the repeat
of the first half. In the second half accelerations occur towards the second
beat of measure 15 and the end of measure 16, the middle of the last phrase.
The two factors that explain a
considerable amount of the data of S3 are less easy to characterise and
separate from each other than the factors used to explain the data of S1 and
S2. The main difference between the factors of S3 is the point of direction to
which accelerations and decelerations occur. In the main factor gradual
accelerations occur within a phrase, while the tempo decelerates at the end of
phrases quite suddenly and quickly. In the second factor, however, this is
reversed, in that sudden accelerations are followed by gradual slowing of the
tempo.
For each pianist a particular timing
strategy becomes clear. For S1 condition 4 and condition 6 represent two
opposed ways of timing the melody, while conditions 2 and 5 are timed in ways
that are quite similar to each other as well to the other conditions (see
figure 9, upper graph). For S2 conditions 2, 4 and 5 are all very similar to
each other. S2 is very consistent over repetitions as well as over conditions,
with only conditions 1 and 6 timed in a different way. Condition 6 correlates
highly with the second factor explaining the similarity between the
performances, but hardly relates to the main factor, which makes it clearly
different from the other conditions (see figure 9, middle graph).
The data of S3 imply a strategy in
which each condition gets its own individual timing characteristic. From the
factor analysis, it becomes clear that for S3 each condition has its own
characteristic profile, being both related and differentiated from the other
conditions. Only 1mel does not have such a clear and distinct profile. For S3,
the timing of 4, 5 and 6mel are most closely related to each other. Within
these three conditions 5mel is most closely related to 2mel. It seems as if
6mel is a prototypical performance of the melody, which 4mel and 5mel approach.
2mel is different from the other conditions by its high correlation to factor 2
and low correlation to factor 1.
For all subjects, the performance of
1mel is least consistent and least characteristic of all performances,
correlating only to a minor degree to both factors. It seems as if the melody
without metrical and phrasing information provides limited or confusing
information about the musical identity of the melody, which may account for its
low “representative” status as a performance of the melody.
A different way of looking at the
impact of contexts on the timing of the melody is by calculating, for each
performer, the partial correlations between averaged timing profiles (averaged
over repetitions) of the different conditions. In table 4 partial correlations
between conditions are shown separately for each performer. Partial
correlations show the remaining correlation between conditions when the
features common to all conditions have been corrected for. In other words, the
partial correlations give insight into the amount that certain conditions
relate to each other more than average. The benefit of partial correlations
beyond simple correlations is that the similarity between conditions is not due
to agreement between all conditions, but rather is specific for the pair of
conditions. This is of importance since all conditions consist of the same
melody and thereby share perforce many similarities. For the decision between
high and low partial correlation we take a partial correlation of 0.27 as a
border, since below 0.27 the correlation would not be significantly greater
than 0 (p <
0.05).
|
Variable S1
1mel S1 2mel S1 4mel S1 5mel |
|
|
|
Variable S2
1mel S2 2mel S2 4mel S2 5mel |
|
S2
2mel 0.349 |
|
|
|
Variable S3
1mel S3 2mel S3 4mel S3 5mel |
Table 4
Partial correlations between averaged timing profiles (averaged over repetitions) of four conditions, split per subject. The correlations are the same for scaled, normalised and raw eighth note IOI patterns.
From the partial correlations the
influence of each musical setting of the melody seems to be as follows.
Condition 1, the melody without bar-lines and other performance indications,
shows very low partial correlations with all other conditions, with the
exception of condition 2. This
confirms that this condition is performed in an atypical way. However, it is
worth noting that, for conditions 1 and 2, the correlations between melody with
and without bar-lines are small, but the partial correlations are relatively
large. This means that the common features between condition 1 and 2 are not
very large, but they are very specific to these two conditions.
The addition of the counter-melody
to the melody is interpreted differently by the different subjects. For S1 and
S2 the effect of this combination is limited. They time the melody in
combination with the counter-melody in much the same way as they timed the
melody solo. This is not because the timing contour of the counter-melody solo
and the melody solo are so much alike (they are quite different instead--the
correlation between 2mel and 3counter-mel is, on average, 0.23). It is, rather,
a result of what we take to be these performers’ primary focus on the
melody. For S3, however, the addition of the counter-melody changes the timing
of the melody considerably. The correlation between the melody only and the melody
with counter-melody is for S3 the lowest correlation between conditions,
together with the correlation between the melody only and the melody without
bar-lines (r =
in both cases 0.55) (see table 3).
Our intuition is that these low correlations are due to S3’s
greater sensitivity to the details and specifics of each musical context.
S2 and S3 time the melody with the
block chords in a manner most like condition 2, the melody solo. For S2 this is
not a surprise, as he does not change the timing of the melody much in
conditions 2, 4 and 5 in any event. For S3, however, this is a bit more
surprising. S3 times the melody in condition 5 much more in the way he timed
the melody solo than was the case with the melody and counter-melody combined.
For S3 the addition of the harmony of the block chords does not seem to make a
large difference with respect to the melody solo.
For S1 it is interesting to see that
condition 5 is, of all conditions, timed most like condition 6. In other words,
for this subject the melody with block chords in condition 5 approximates the
final version of the melody to a large extent. It seems as if the chords and
the melody in condition 6 have a relative large impact on the timing contour
for S1, in contrast to the melody in relation to the counter-melody which is
less well represented in condition 6.
S2 performs condition 6 in a way
that is quite different from the other conditions. Apparently his conception
and performance of the melody changes considerably with the addition of the
full context in the Theme. For S3 the melody in condition 6 correlates almost
equally well with the melodies in conditions 2, 4 and 5. The partial
correlations, however, show a larger similarity between the timing of the
melody in condition 4 and 6 than between condition 2, 5 and 6. This implies
that the melody in condition 6 approaches the melody in condition 4, which
suggests that S3 takes the counter-melody in condition 6 into account in the
timing of the melody. A regression analysis substantiates this suggestion by
indicating that the timing profile of the counter-melody in condition 4 is able
to explain 20% of the timing variation of 6mel and 30% of the timing variation
of the counter-melody in condition 6 (6counter-mel).
III.1.c
Similarity between performers
From the outset of this study we
were interested in the way in which pieces offer different possibilities for
performing the piece with tempo rubato. The idea was that certain pieces are
interpreted in a highly consistent way while other pieces will tend to give
rise to multiple timing patterns. In this study we found that the agreement in
timing profile between the performers is not equally great for the different
conditions. The correlations between averaged timing profiles (averaged over repetitions)
of different performers is for the melody without bar-lines and (partly) for
the melody alone significantly higher (p < 0.05) than for the Theme (see figure 10).
Figure 10a
Correlation between
averaged timing patterns (averaged over repetitions) of performers split by
condition. All pairs of correlations in condition 1 are significantly greater
than the pairs of correlations of conditions 4 and 6 (p < 0.05). The
mean correlation between performers in condition 1 is also significantly
greater than the mean correlation between performers of condition 5 (p < 0.05). The
correlation between the timing profiles of S2 and S3 of 2mel is significantly
greater than the correlation of 6mel (p < 0.05). The correlations
are the same for scaled, normalised and raw eighth note IOI patterns.
Figure
10b
Example of the (scaled and normalised) timing profiles of the melody in condition 2 and condition 6 performed by S2 and S3. Score-time 56-72 refers to the upbeat of measure 10 to the first beat of measure 15. The differences between the timing profiles of S2 and S3 are in condition 6 (below) clearly greater than in condition 2 (above).
The high correlation between the
timing profiles of the different performers in condition 1 seems due to the
relative large lengthening of the ornamented eighth note in contrast to the
generally small timing variations. For the descending trend of the correlations
over conditions 2, 5, 4, and 6, however, we would like to posit an
interpretation that needs further testing in future research. In figure 11 the
correlation between subjects (taken as a measure of similarity between
performers) is plotted for conditions with increasing diversity/complexity of
musical structure. We see that the similarity between performers decreases (the
variability between performers increases) with increasing diversity of musical
structure. In this graph the least “diverse” musical structure is
the melody alone, which is timed in a quite uniform way (mean correlation
between the performers = 0.64). The second-least diverse structure is the
melody with block chords (having one melodic line and harmony) which is still
timed with some substantial agreement. The second most diverse structure is the
melody with the counter-melody (having two melodic lines and harmony). For this
condition the correlations are all a bit lower (r = 0.52). The most diverse structure is the full Theme (having two to
three melodic lines, harmony, and arpeggio chords); the melody within this
complete Theme is timed with least agreement (mean correlation between the
performers = 0.48).
Figure 11
Correlation between averaged timing patterns (averaged over repetitions) of performers plotted by conditions with increasingly divers musical structure. A = 2mel, B = 5mel, C = 4mel, D = 6mel. The correlations are the same for scaled, normalised and raw eighth note IOI patterns.
A reason for the similarity between
performers’ timing of the conditions could be an agreement in
interpretation caused by a common performing practice or by limited
interpretative possibilities of the music. A reason for the decrease of
similarity between the timing profiles of the different subjects could be that,
when there is more musical diversity, more choices and interpretations have to
be made by the pianists, which they make in increasingly different ways. In
this view a complex musical structure provides several interpretative
possibilities, while a simple and plain musical piece provides less room for
alternatives. Supplemental support for this hypothesis is found in the changing
timing profile of the melody in different conditions. Among conditions 4, 5 and
6, the Theme (condition 6) differs the most from condition 2, which is probably
due to the great number of new elements in the musical context of the melody
within the Theme. In other words, the timing of the melody within performers
changes the most with diversity of musical structure, as was also found between
performers, within conditions.
The decrease of similarity between
performers with diversity of musical structure is, however, not entirely
self-evident. An arguable and contrasting effect that might be found would be
an increase of similarity between performers with increasing musical
complexity, because of constraints or direction provided by the structure. A
melody on its own provides limited cues for its interpretation while a melody
with contexts in which harmony, meter, and grouping are more strongly defined
provides many more cues to be heeded. With increasing complexity of the musical
context, these effects could become more pronounced. For example, a Bach four-voice fugue is likely to be played
in a rather strict manner with respect to timing variations, due to the complex
way in which the voices interact with one another. In that case it seems unlikely that increasing complexity of
structure would yield greater potential for flexibility of timing.
III.1.d Diversity between
conditions
Having introduced a hypothetical
ground pattern, it would be interesting to know what, exactly, is timed
consistently over different conditions and what changes. Although a detailed
discussion of this falls beyond the scope of this article, we do want to point
out some of the common features and some of the differences. In general what we
found is in substantial agreement with the explanations of expressive timing
variations reported in the literature (cf. “Palmer (1996a)”,
“Sloboda (1983)”, and “Sundberg et al. (1983, 1991)”);
for example, all performers generally ended the piece with a phrase final
lengthening. Consistent patterns were some rhythmic and metrical figures, such
as lengthened upbeats (also found by “Desain & Honing (1994)”),
shortened second beats (see composers pulse in triple meter “Clynes
(1983)” “Repp 1989”) and a highly prolonged ornamented note.
Differences between conditions or
between repetitions were often due to local changes in stress patterns at
ambiguous places. For example, S2 timed the first few measures of condition 1
differently in each repetition (see figure 12). In repetition 2 of 1mel, S2
clearly lengthened the high notes in the melody and plays first beats
relatively short. In 2mel, however, he instead lengthened the first beat of the
second measure and favours metrical stress, indicated by bar-lines, over the
stress of high notes in the melodic contour (possibly effecting a shift of
downbeat by one eighth note).
Figure
12
(Scaled and normalised) timing profile of the three repetitions of the first two measures of the melody without bar-lines performed by S2. From the second beat of the first measure on no clear timing pattern exists.
Globally consistent changes were
minor. For S3, condition 5 shows a major change in timing in relation to other
conditions in that the first beat of each measure is, on average, lengthened more
than other beats in the measure. This regular pattern does not occur in the
other conditions, and we interpret the lengthening as a result of the block
chords occurring at the first beat of each measure in this condition. S2
treated condition 5 in a special way by varying the “chord
spread”--the length of time between the onset of the first note in the
chord and the last note in the same chord--expressively. In this condition, S2
performs some chords nearly simultaneously, while he breaks others noticeably.
For an impression of the asynchrony in condition 5 for S2 see figure 13.
Figure
13
Example of S2 treating the asynchrony between voices as an expressive device in the condition of the melody with block chords. The asynchrony between voices is calculated by subtracting onset times of the first from the last note of a chord. In this graph only the first 9 measures are given.
Not all of our findings were in
keeping with the implications of previous research. One finding that is not in agreement with previous research
on this topic is the tendency of one subject to sometimes lengthen the notes in
the middle of the phrase and to speed up towards the end of the phrase. This
was the case for S2, who, in condition 5, lengthened either the third beat of measure
3, or the first or even second beat of measure 4, where after he speeded up
towards the down beat of measure 5. In other words, he lengthens the notes in
the middle of
the phrase and speeds up towards the start of the new phrase (see figure 14), the opposite
of the tempo curve given by the tendency toward phrase final lengthening.
Figure 14
(Scaled and normalised) timing profile of measures 3 and 4 by S2. S2 speeds up towards the end of the sub-phrase in measure 5. The short note at the second beat of measure 3 consists of two sixteenth notes that are played very fast by all subjects, consistent over all conditions.
Another inconsistency with the
literature is that the decelerations at the end of musical groups (phrases) are
not the largest lengthenings of notes that occur. Instead, the ornamented
eighth notes in measures 6, 10 and 12 are generally lengthened more than the
notes at the end of a phrase; only the final phrase lengthening, at the end of
the piece, has the degree of the lengthening of the ornamented eighth note (see
figure 15). Still, the effect of the phrase final lengthening is greater than
the effect of the ornament lengthening, since the lengthening of the ornamented
eighth note is compensated by a fast preceding eighth note. The phrase final lengthening is, on the
contrary, approached by a gradual slowing down of the tempo (see also figure
15). The lengthening of the ornament has therefor less impact on the duration
of higher level units (e.g. a two-measure unit) than the phrase final
lengthening has.
Figure 15
(Scaled and normalised) timing profile of one repetition by S1 of 2mel, rep 3. The phrase final lengthenings are not the greatest decelerations; instead the lengthening of the ornamented eighth note is often larger. The numbers indicate an ornamented eighth note in the score, the dashed lines mark phrase boundaries. Only the phrase final lengthening at the end of the piece is larger.
III.1.e Explanation of
the changing timing profile of the melody; different voices, one timing profile
When we take the performance of the
melody solo in condition 2 as point of reference, we notice that the timing
contour of the melody changes when it is performed in different musical
settings. The reason for this changing timing contour seems twofold:
1) Different
voices are timed in a uniform way.
2) In
the common timing pattern, different musical characteristics are highlighted,
related to
·
each
voice, and to
·
characteristics
of the music due to the combination of the voices (such as harmony).
In other words, the timing contour
of the melody does not remain the same when other voices are added, because
supplemental characteristics of the other voices are taken into account by the
pianist when timing the melody.
With reference to 1) the fourth
condition provides our most provocative evidence for the statement. In this
condition the asynchrony between the two voices is small (it lies between 0 and
55 ms, averaging 17 ms), which means that both voices are timed practically
simultaneously (for a comparison see the chord spread S2 uses in condition 5,
figure 13, which in average exceeds 50 ms). Further, the timing patterns of the
two voices correlate highly (average correlation between the counter-melody and
melody line of one repetition = 0.81), which suggests that the voices share a
common timing pattern (see also figure 16). In fact, a correlation between the
voices of 0.81 is as high as the correlation between melodies of repeated
performances within a single condition. It is, further, much higher than the
correlation between 2mel and 3counter-mel for example, which is in average 0.28
(0.33 for S2 and 0.22 for S3). We see this as an indication that the different
voices of condition 4 are timed in a uniform way. (Remember: the analysis of
condition 4 is only based on S2 and S3, because of missing notes in the
counter-melody of S1.)
Figure 16
(Scaled and normalised) timing profile of the melody and counter-melody in measures 1-9 of condition 4 performed by S3. The timing profiles correlate highly.
With respect to 2) The timing
patterns of condition 4 also provide insight into the relationship between the
timing characteristics of the melodic lines alone and of the lines together
when they are compared with the timing patterns of condition 2 and 3 provide.
The timing patterns of condition 4 correlate significantly (p < 0.05) with both the melody solo
and the counter-melody solo timing patterns (average correlation of condition 4
with 2mel = 0.67 and with 3counter-mel = 0.36). This means that aspects of both
the melody and the counter-melody solo are present in condition 4.
A multiple regression analysis shows
that the average timing patterns (averaged over repeats) of condition 4 can be
explained fairly well on the ground of averaged timing patterns of condition 2
and 3. For S2 82.6% of 4mel and 56.8% of 4counter-mel is explained by 2mel and
3ten. For S3 the numbers are 36.8% for 4mel and 40.8% for 4counter-mel (for
more detail see table 5). The significant contribution of both 2mel and
3counter-mel to the explanation of 4mel and 4counter-mel confirms that aspects
of both the melody and the counter-melody solo are present in condition 4[3].
|
Performer |
Explained performance |
by |
Proportion
of variance explained* |
|
S2 |
4mel |
2mel |
0.76 |
|
3ten |
0.28 |
||
|
2mel & 3ten |
0.83 |
||
|
4ten |
2mel |
0.56 |
|
|
3ten |
0.11 |
||
|
2mel & 3ten |
0.57** |
||
|
S3 |
4mel |
2mel |
0.31 |
|
3ten |
0.13 |
||
|
2mel & 3ten |
0.37 |
||
|
4ten |
2mel |
0.27 |
|
|
3ten |
0.12 |
||
|
2mel & 3ten |
0.41 |
* All contributions to the
explanation of the explained performance are significant (p < 0.05) except
**.
* * The
contribution of 3ten to the explanation of 4ten is not significant, 2mel does
significantly contribute to the explanation of 4ten.
Regression analysis between averaged timing profiles (averaged over repetitions) of 2mel, 3ten, 4mel and 4ten. The (scaled and normalised) timing patterns of condition 4 are explained on the ground of 2mel and 3ten separate and of 2mel and 3ten together.
Although this explanation covers the
data fairly well, the timing patterns of condition 4 cannot be fully explained
by the linear combination of the melody solo and counter-melody solo. The
unexplained part, which is quite large for S3, suggests that the combination of
the counter-melody and melody lines in condition 4 produces a case of the whole
being more than the sum of the parts. Supplemental musical characteristics are
introduced by this combination of voices: for example, harmonic intervals that
are explicit as well as implied, as well as a new composite rhythmic texture
and an interweaving of melodic goals between the two voices.
For S2 the contribution of 2mel and
3counter-mel to the explanation of condition 4 is asymmetrical. S2 seems to
have the melody as his major focus in two ways: 1) The performance of both the
melody and the counter-melody in condition 4 are very well explained by 2mel
and to a lesser degree by 3ten, and 2) 2mel and 3counter-mel explain more of
the performance of the melody in condition 4 than they do of the performance of
the counter-melody in this condition. We interpret this as an indication of the
melody in condition 4 carrying more intentional variation than the counter-melody in this
condition. That is to say, the melody line continues to be the focus of
intentional variation on the part of the player, while the counter-melody is
less carefully modulated for expressive purposes. Rather than having a truly independent life of its own, the
counter-melody seems to be modulated by the changes in the melody-- the melody
seems to be leading the expressive variations of the counter-melody.
III.1.f Tempo and motor
factors
The influence of some other factors,
other than musical structure, on the timing of the melody were not cancelled
out in this study. For example, the full Theme (condition 6) was harder to
perform than the melody or counter-melody solo. This resulted mainly in a
change in mean tempo, with the Theme being performed on average more slowly
than the other conditions. The timing profile itself would be affected largely
in the area of difficult chords vs. easy chords to perform. That is to say,
difficult chords might need extra preparation, which would probably lead to a
lengthening of the previous note. We think, however, that the influence of this
preparation is limited. This is suggested by the slower mean tempo (probably
taken by the performers to allow for increased difficulty of the material) and
by the high consistency over repetitions of the timing profiles for the Theme
(condition 6) and melody with block-chords (condition 5). The consistency over
repetitions of the last conditions was as high as (or even higher than) the
consistency within other conditions. This means that the timing variations
found are in large measure intentional. From hearing the performers at work we
are confident that S2 and S3 are skilled enough to perform the Theme in the way
they intended. For S1, however, we are not entirely confident about this. The
hand size of S1 could have been a constraining factor in his performance of the
Theme[4].
The influence of tempo as a factor
altering the timing contour was not cancelled out either. We tried to limit its
influence by indicating the ideal average tempo to the subjects. They were
however, not able to restrict themselves to the indicated tempo and changed the
tempo in accordance with the various conditions (see table 1). It is very well
possible that some of the differences in timing pattern between conditions are
partially a result of tempo differences.
The musical context of the melody,
however, is the primary source of the differences in timing patterns, since it
is the main cause of the motor and tempo differences and since the differences
in the timing patterns between conditions is demonstrated to be the result of
new aspects in the context for several cases.
III.2 Extent of tempo
rubato
In this section of the study, the
extent of tempo rubato and its relationship to musical context will be
considered. In order to be able to
compare the extent of tempo rubato of performances in different tempi, we
scaled all performances to one tempo (the slowest tempo = 54 eighth notes per
minute). We divided the IOI’s of a single performance by its mean eighth
note IOI and multiplied this by the new, standard mean eighth note IOI (which
is 1132 ms). We did this to create a uniform scale for tempo variations in
which rubato extents within different tempi can be compared to each other. All
observations about the rubato extent are made relative, not absolute. When no
such scaling would be made, the differences in rubato extent would be
considerably larger than reported below, due to global tempo differences. By
scaling all performances to the same average tempo, the differences in the
extent of rubato are underestimated. We can therefore be sure that significant
differences in the extent of tempo rubato between performances are indeed
significant. In other words, we wanted to be sure that an increase in
variability is an increase in relative variability, that is, relative to the
mean tempo. For example, a small lengthening or shortening of notes in a slow
tempo is (when taken in absolute terms) relatively large in a fast tempo.
III.2.a Performer
characteristics
When the performances are scaled it
becomes clear that the three subjects use approximately the same amount of
tempo rubato. Only S3 uses tempo changes that are a bit smaller than those
produced by the other subjects. On average the standard deviation of the eighth
note IOI is 108 ms, which is 9.54% of the mean eighth note duration (average
eighth note duration of the scaled performances = 1132 ms).
This similarity in extent of timing
variability was a surprising result for us. The performances did not sound like
as if they had a similar extent of timing variations. Apparently, more aspects
of the tempo changes than the extent of tempo rubato alone influence the
impression of the amount of tempo rubato used by performers. The kind of tempo
rubato used could be the significant factor, or perhaps the density of tempo
changes could play a role. S2 and S3 showed very different kind of tempo
rubato. While S3 made global and gradual tempo changes, the timing
characteristics of the performances of S2 are better described as give and take
and are more local.
III.2.b Conditions
All subjects change the rubato
extent with respect to the performed music. For S1 the eighth note IOI variance
is in 2mel and 4mel rather small, in 1mel it is moderately large and for 5mel
and 6mel it is fairly large (see figure 17). Significant differences between
the mean standard deviation per condition are: 1) 6mel is timed with greater
tempo variance (duration variance) than all melodies in other conditions (p < 0.01), and 2) 5mel and 1mel are timed with greater
tempo variance (duration variance) than 2mel and 4mel (p < 0.05)[5].
Figure 17
Average standard deviations in ms (averaged over repetitions) of the scaled timing profiles split by performer and conditions. The letters A-D indicate conditions with increasingly rich musical texture. A = 2mel, B = 4mel, C = 5mel, D = 6mel
For S2 the variance of eighth note
IOI of 1mel is greater than in the other conditions (except 5mel). This tempo
variance is mainly due to a very large lengthening of the ornamented eighth
note. The tempo variance of 2mel, 3ten and 4mel are the smallest and the
variance of 5mel and 6mel are in between (see figure 17). The tempo variance of
1mel is significantly greater than the tempo variance of 2mel, 3ten, 4mel and
6mel (p <
0.05). The tempo variance of 5mel is significantly greater than of 2mel, 3ten
and 4mel (p <
0.025). The tempo variance of 6mel is greater than of 3ten (p < 0.05).
The large rubato extent of condition
5 is probably a result of the asynchrony with which S2 performs the
block-chords. In condition 5, S2
uses asynchrony as an expressive device; he sometimes plays the chords broken
and sometimes simultaneously.
S3 uses the least tempo variation
for 3counter-mel and 4mel. For 1mel, 2mel and 5mel it becomes increasingly
greater and for 6mel it is the highest (see figure 17). S3 times the melody in
condition 6, 1, 5 and 2 with significantly larger tempo variation than the
melody in condition 4. The timing variance of the melody in condition 1, 5 and
6 are greater than the timing variability of 3ten. In other words, the timing
variation of conditions 3 and 4 are significantly smaller than the timing
variations in all other conditions.
III.2.c Texture
When
only the melody of condition 2, 4, 5 and 6 are considered, a general trend
occurs: the extent of rubato appears to increase with richness of musical
texture. In figure 17 this relation is depicted. The extent of tempo rubato is
plotted by conditions with increasingly rich texture. The mean extent of rubato
of a performance is represented by the standard deviation of the eighth note
IOI’s. The texture is indicated by numbers (1 to 4) which represent
conditions with increasingly rich texture. Condition 2 (texture 1) has a plain
texture (one voice), while condition 4 (texture 2) has the next plain texture
(2 voices). Condition 5 (texture 3) has the next-richest texture (three to four
voices), while condition 6 (texture 4) has the richest texture of all (four to
five voices).
It is interesting to see that the
diversity between performers and the amount of tempo rubato used do not behave
in the same way. While in condition 4 the performers differ quite a lot in
their timing of the melody, they all perform this condition with small tempo
rubato; In other words the melody with counter-melody condition provides an
expressively differentiated or varied source, but it does not lend itself for
large tempo rubato. On the other hand all performers time condition 5 with deep
tempo rubato which, however, does not mean that the diversity in tempo rubato
patterns increases in this condition.
III.2.d Confounding
factors
The design of this study did account
for enough degrees of richness of sound to be confident about these factors
influencing the extent of tempo rubato. Nevertheless, in musical situations of
this complexity, one should be aware of possible conflicting factors that were
not accounted for in this study. A possible influence on the extent of tempo
rubato was, for example, the constraining influence of the counter-melody on
the melody in the performances by S3. The musical texture is richer in
condition 4 than in condition 2 (melody alone). Still, S3 performed condition 4
with a smaller extent of tempo rubato than condition 2, which is probably due
to the counter-melody being preferably timed with small tempo rubato (see
figure 17).
The rich texture in condition 5 and
6 is a complex factor containing influences of various kinds. Condition 5 and 6
are not only characterised by a richer sound, but also by broken chords, wide
chords and dense musical passages which are motorically demanding (for example
the ornamented eighth note in measure 6 and the sixteenth notes in measures 7
and 8). These local characteristics are related to the richer musical texture,
but have different origins. They cause local lengthenings, which increase the
globally measured extent of tempo rubato. Rather than condemn these local
delays we think they probably belong to the compositional intent of Brahms, who
was a pianist himself and who should have been aware by the time delay caused
by arpeggio chords and by the separation of bass and chord notes by more than
an octave (see figure 6).
V Conclusion
Three pianists performed a melody in
five different settings. These settings theoretically provide different
expressive possibilities. These possibilities were translated by the pianists
into differences in timing contour, extent of tempo rubato and use of
expressive devices (for example, asynchrony was used expressively in the
conditions with chords). The pianists were generally consistent in their timing
of the melody within a single condition.
Differences in timing contour
between conditions consisted of the lengthening of beats to which chords are
added, shifts in points towards which is accelerated or decelerated, shifts in
locally lengthened (and thereby accented) beats. The condition in which the
melody was given without bar-lines, rhythmic beams and phrase indications was
timed in the least idiomatic manner. The impact of the contexts on the timing
of the melody depended on the context and the performer. One subject
considerably changed the timing pattern only with the addition of the full
context. Another subject was especially sensitive to the presence of a
counter-melody. Explanations of the changes in timing contour were based on the
ground of two observations: 1) Different voices are timed in a uniform way. 2)
In the common timing pattern, musical aspects are highlighted that relate to
structural aspects of both the individual voices and the combination of voices.
The extent of tempo rubato appeared
to be modulated in a much more systematic way than could be expected from the
lack of interest into this aspect of tempo rubato in previous studies. The
extent of tempo rubato was found to increase with richness of musical texture.
The melody, when played alone and in the context of the counter-melody, was
timed very much in tempo, while the fuller textures of condition 5 and 6
resulted in deep tempo rubato patterns. Locally, changes found consisted of an
increasing use of accelerations and periodic tempo variations in later
conditions with richer musical structure. Both the amount of the final retard
and the amount of lengthening of the ornamented eighth note changed in
accordance with the globally found extent of tempo rubato.
In addition to the findings
mentioned above, the study has shown to be fruitful in a number of ways. First,
a relationship between diversity of performances and musical structure has been
found. The diversity of performances increased with the number of explicit
musical dimensions in the score (such as melodic lines and chords). This is a
relationship not often reported in the literature, and seems to be related to
the complexity of musical structure, or at least to the presence of multiple
cues in the musical texture. Second, a beginning has been made in investigating
the scope of musical elements that pianists take into account when performing
structurally diverse music. This scope appeared to be different for different
pianists. One subject focussed strongly on one aspect of the music (the
melody), while another took multiple aspects of the music into account. The
degree of sensitivity to multiple structural descriptions is possibly related
to musical experience. This hypothesis has also been stated by “Clarke
(1988)”, but needs further investigation. Last, but not least, this study
has contributed to the discussion of ‘the tempo curve considered
harmful’ “Desain & Honing (1993)”. Timing patterns appear
to depend heavily on the musical events or musical structure in which they are
originally produced, meaning that they have limited generality over different
pieces.
We conclude that pianists take
different parameters of the music into account when deciding on the timing of
the melody of a piece. This is shown by the changing interpretation of the
melody in the different conditions, but also by the contribution of both the
melody solo and the counter-melody solo to the melody and counter-melody in
combination.
We are cautious, however, about
generalising the results of this study. Its generality is limited by the scope
of the study, which included only different versions of one piece, and only
diversity of performances between three pianists. In this set-up, the influence
of each performer is quite large, which means for example that if one pianist
times a certain melody very different, it lowers the correlations considerably.
Further investigation is needed in
which the musical factors influencing the diversity and extent of tempo rubato
patterns are separated and tested. The scope of the study should be broadened
to include other expressive devices such as dynamics, articulation, chord
spread. The purpose of such a study is to further clarify the constraints on or
freedom of performing the piece expressively.
Acknowledgements
This work was undertaken as part of
the ‘Music, Mind Machine’ project at the Nijmegen Institute for
Cognition and Information (NICI) which is funded through a Pionier grant from
the Netherlands Organisation for Scientific Research. The second author’s
work in Nijmegen was supported by a Fulbright Senior Scholar grant from the
Netherlands America Commission for Educational Exchange and the Council for
International Exchange of Scholars.
The authors would like to express thanks to Dick Willems and Jules Ellis
(Mathematical Psychology, NICI) for their assistance with some of the
statistical issues in this study. Especial thanks are due to Rob Broek, Jeroen
Malaise and Bart van de Roer, for their patience and excellent playing.
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Appendix
POCO
POCO is a workbench
for analysing, modifying and generating expression in music, to be used in a
research context. POCO contains a consistent and flexible representation of
musical objects and structure. The integration of existing models of expression
makes it possible to compare and combine models using the same performance and
score data. New tools are developed for specific “micro surgery” on
expression. A lot of attention is given to the openness, integration, and
extendibility of the system “Honing (1990)”.