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Tempo and Loudness Analysis of a Continuous 28-Hour Performance of Erik

Satie?s Composition ?Vexations?

Article in Journal of New Music Research · January 2002

DOI: 10.1076/jnmr.32.3.243.16864

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 Journal of New Music Research 0929-8215/02/3101-001$16.00
2003, Vol. 32, No. 3, pp. 243–258 © Swets & Zeitlinger

Tempo and Loudness Analysis of a Continuous 28-Hour

Performance of Erik Satie’s Composition “Vexations”

Reinhard Kopiez1, Marc Bangert1, Werner Goebl2 and Eckart Altenmüller1

1
Hanover University of Music and Drama, Germany and 2Austrian Research Institute for Artificial Intelligence, Vienna

Abstract
This study extends the perspective of music performance no metronome indications; however, the piece is to be per-
research with an examination of a long-term performance. formed “très lent.” “Vexations” remained unknown until it
In a single case study, an uninterrupted recording of Erik came to the attention of the American composer John Cage,
Satie’s “Vexations” performed by one pianist over almost 28 doing research in France in 1949 and was first published
hours is used as a performance of extreme length to explore in the same year (see Bryars, 1983). The most remarkable
new approaches in performance data analysis. The MIDI feature of “Vexations” was Satie’s enigmatic instruction at
and acoustical data are analysed with linear and non-linear the score’s top: “To play this motif 840 times in succession,
methods to describe changes in tempo and loudness. Addi- it would be advisable to prepare oneself beforehand, in the
tionally, the performer’s changing states of consciousness deepest silence, by serious immobilities.”
(alertness, trance, drowsiness) were observed to exert a There is no proof of a performance during Satie’s lifetime
strong influence on tempo and loudness stability. Tempo and and it is not established whether an actual performance
loudness remain stable over the first 14 hours of alertness. A of “Vexations” was intended by Satie at all. For example,
state of trance begins after 15 hours and shows a destabili- Wehmeyer (1998, p. 21) argues that the piece is a parody on
sation of tempo followed by uncontrolled deviations in Satie’s lessons in composition as a student at the Paris Con-
loudness. Time series analysis of loudness changes revealed servatoire. One of the daily exercises at the Conservatoire
periodicities of about 10 minute lengths. Non-linear analy- was the harmonisation of a given bass melody in close and
ses of tempo and loudness changes showed a complex gen- extended position. This compositional technique can be iden-
erator pattern underlying the apparently random fluctua- tified in “Vexations” in its sequence of a theme, followed by
tions throughout the performance. This pattern appears most two variations in double counterpoint (see Fig. 1).
clearly when unfolded in an 18-dimensional embedding Orledge (1992, p. 143) argues that “Vexations” is one of
space. Measures of fractality and chaotic behavior proved to Satie’s “numerous ways to cheat the passage of time” through
be dependent on the states of consciousness. Results are dis- an “absence of any climax or movement towards a goal.” This
cussed in regard to influences of psycho-physiological impression is achieved through the compositional means of
changes (vigilance) on sensorimotor performance and to the a sequence of unresolved diminished and augmented chords.
overall stability of an oscillating psycho-motoric system. The composition joins the tradition of musical works of
extreme duration in European avantgarde music, such as
“The artist does not have the right to take up the audience’s time
Morton Feldman’s Second string quartet (with a duration of
unnecessarily.”
about 5 hours) and John Cage’s ORGAN2/ASLSP (ASLSP
(Satie, 1988, p. 323)
stands for “As slow as possible”). In a recently commenced
In 1893, the eccentric French composer Erik Satie composed undertaking, a performance of Cage’s ORGAN2/ASLSP is
a three-part piece for piano, which he entitled “Vexations.” planned to last at least 639 years, to be played on the organ
The work is the second movement of Pages Mystiques, a of the Burchardi church in Halberstadt, Germany (see Cage,
collection of three short compositions and consists of a 2001). This realisation began on 5 September 2001 with a
sequence of variations on a bass theme of 18 notes. There are one and a half year rest. The first three notes will sound on

Accepted: 1 April, 2003

Correspondence: Reinhard Kopiez, Hanover University of Music and Drama, Emmichplatz 1, 30175 Hannover, Germany. E-mail:
kopiez@hmt-hannover.de
2 Reinhard Kopiez et al.

Fig. 1. Erik Satie’s composition “Vexations” comprises one theme and two variations. One rendition of the score comprises: Theme – Vari-
ation 1 – Theme – Variation 2. It has to be repeated 840 times. Tempo: “très lent.”

5 February 2003 and weights will secure the last combina- gated in an extensive interview with the solo performer of
tion of notes played until the point when the next event is to “Vexations” by Kopiez (1998).
be commenced. The interest in the field of music psychology in “Vexa-
The first public and complete performance of Satie’s tions” can be traced back to 1974. At this time Michon (1974)
“Vexations” was organised by John Cage and took place in was interested in whether note durations in a very slow tempo
New York in 1963 with a total performance duration of 18.40 such as in “Vexations” show a higher deviation in interonset
hours. However, Cage shared the task with 9 other pianists intervals (IOIs) than in medium and fast tempi. With regard
(for a short report see Cage, 1980). As Bryars (1983) dis- to the extremely long performance duration of “Vexations,”
cusses in his report on 23 “Vexations” performances between the author examines how tempo stability is controlled during
1958 and 1977, two points remain contentious: firstly, the the course of a performance. He predicted that an inner clock
question of team vs. solo performance, and, secondly, the model is insufficient as an explanation, due to the consider-
question of tempo. To summarise, there is no decisive argu- ation of only the latest interonset interval. As an alternative
ment in favor of a team performance against a solo one, to serial models of tempo control, Michon suggests a hierar-
and the tempo indication “très lent” can be related to a chical model, which considers the control of durations on five
metronome marking only by comparing the tempo character rhythmic layers, from a beat level of single note events up to
of “Vexations” to other similar compositions by Satie. Ulti- the length of the entire theme. To test his predictions of a
mately, the chosen tempo is decided by the player and thus, multi-layer tempo control system, he recorded 19 hours of
the total performance duration of all 840 renditions varied “Vexations” on several tapes performed by four pianists.
between 5.48 and 24.46 hours (Bryars, 1983). In a recent Results of factor analysis of IOI deviations from grand
publication, deeper insights into the pianist’s coping strate- averaged deviations of IOIs revealed five factors. The author
gies for the specific demands of a performance are investi- interprets this finding as a confirmation of the predicted
 Performance analysis of Saties’s “vexations” 3

hierarchical tempo control system. Despite several technical formance is intended by this paper. Within a broader research
insufficiencies, Michon’s study marks the beginning of framework this aim is of importance for the following
research interest in long-term performances and remains a reasons: firstly, up until now, musical performances with
pioneering work. a duration of one or two hours remained unconsidered;
The second approach was realised by Clarke (1982) who secondly, analysis of long-term performances will allow
analysed (a) the relationship between tempo and grouping conclusions to be made about the general behavior and
of note events and (b) the relationship between tempo and mechanisms of motor stability of the performer as an oscil-
overall tempo drifts. His analyses were based on one-hour lating biological system; thirdly, from the observation of the
recordings of “Vexations” effected through a grand piano performer’s efficiency we hope to draw conclusions about
equipped with photocells under each key attached to a the psycho-motor system’s nature under extreme conditions.
computer, measuring note onset, offset and hammer velocity Although musical repertoire for solo instrument does not
(loudness). Two subjects were instructed to vary tempo contain many works with a duration of more than one hour,
within given limits and to perform repeatedly about one hour we hope that findings will be relevant for the understanding
of “Vexations.” Firstly, the performances showed an overall of other performers, such as conductors who are also con-
tempo drift effect: faster tempi became progressively slower fronted with performances of extreme length. For example,
while slower tempi became progressively faster. Secondly, operas often have a performance duration of between one and
analyses of quarter and eighth note duration distributions five hours (e.g., Wagner’s The Mastersingers of Nuremberg).
showed that tempo control increases when tempo increases. To summarise, this paper is an explorative single-case
Thirdly, the number of note groups increased with slower study that intends to find means for an adequate analysis of
tempo. This finding is interpreted as a tendency for Gestalt an unusual composition. Our focus, which should be the aim
dissociation when group duration exceeds certain limits. of all performance research is to gain more insight into
Thus, their study asserts that there does indeed exist a cor- the complex psycho-physiological system of the musical
relation between tempo and note grouping. performer.

The rationale of the present study

Method
Although performance research has made significant
Participant
progress in the last decade, there is a lack of investigations,
which consider musical performance not only from a local The composition was performed by a 40-year-old profes-
(note to note) but from a global perspective (e.g., the long- sional pianist, who had previously performed this piece.
term performances of an entire movement). For example,
as Langner and Kopiez (1996) show, large-scale shaping of
Material
tempo within a time frame of several minutes is characteris-
tic of expert performance. From a global perspective, the The performance was played on a grand piano (Yamaha DS
composition “Vexations” offers a great challenge to perfor- 6 Pro Disklavier) with a built-in MIDI interface. MIDI and
mance research due to its quasi “infinite” duration. The pre- acoustical data were recorded onto hard disc – the audio
condition for an investigation of expressive deviations within recording in CD quality (16 bit, 44.1 kHz sampling rate) –
large time frames is an uninterrupted performance by one using professional microphones (Neumann KMi 84), a
performer and an uninterrupted recording. The first aim and mixing console (Behringer MX 802A) and two PCs (Pentium
main focus of the study is a global examination of tempo and III, 550 MHz) running a LINUX operating system. A sound
loudness in that performance. level limiter (Behringer Composer Pro) between mixing
The second aim is to take advantage of the highly repeti- console and PC avoided digital clipping in the acoustical
tive nature of “Vexations” in an examination of changes on recording. To manage the immense amount of audio data, a
a local level and an analysis of note durations and note loud- researcher-developed software was used to segment the data
ness systematically over time. In other words: are the same stream into separate smaller files of 100 MB each. Addition-
note events always performed in an unchanged manner? It ally, the audio data was backed up on two overlapping DAT
can be hypothesised that systematic variation of expressive tapes (Tascam DA 302). The entire recording procedure is
parameters will determine the organisation of the entire per- displayed in Figure 2.
formance, such as the segmentation and hierarchical group-
ing of musical events into larger units.
Recording procedure
The third aim of the study is to develop and explore ade-
quate methods for long-term performance analysis. Up until The recordings took place in a concert hall in Dresden,
now there has been a significant lack of methods for the study Germany. The entire performance of Erik Satie’s “Vexations”
of entire performances with time-frames of more than a commenced at 5 p.m. and ended at 8.47 p.m. the following
few minutes. The development of adequate methods for the evening. An initial tempo of 52 beats per minute (bpm, eighth
analysis of an extreme example of musical long-term per- notes) was chosen by the performer and was established by
4 Reinhard Kopiez et al.

procedure see Langner, Kopiez, and Feiten (1998) and

Langner (2002). In a third step, the loudness curve was seg-
mented into sections of 30 seconds each for the determina-
tion of note onsets. These small sections were loaded into a
perceptually-based, custom-made onset detection program.
The moment of onset and the corresponding sone-value were
stored into a text file. Errors in onset detection were corrected
manually. The onset detection resulted in a two-column text
file containing loudness values in the first column and onsets
marked by their loudness value (in sone) in the second, each
line representing a time interval of 10 ms. This file contained
9,677,918 lines of text with an amount of more then 170 MB.
From the onset text file, an interonset distribution was cal-
culated. In order to derive tempo values in bpm, we should
know how long a given performed note should have been
Fig. 2. The entire recording procedure including long-term according to the score. A score-to-performance matching
recording of brain activity, MIDI and audio recording. To avoid procedure was not applied, since it would have been too
digital clipping, a limiter was used in the acoustical recording. complex with this enormous amount of data (all existing
However, the limiting threshold was set to the immediate vicinity procedures still include manual correction), and would have
of the maximum digital recording level.
totally failed in the trance section, where the pianist played
anything but correct notes for some time. A simple straight-
forward procedure was used instead. “Vexations” contains
only three different note values: eighth notes, quarter notes,
and dotted quarter notes. All IOIs shorter than 1.58 s were
treated as eighth notes, above that and below 2.9 s as quarter
notes, and above 2.9 s as dotted quarter notes. The values rep-
resent the minima between the three peaks of the IOI distri-
bution. The width of the bins is 59.4 ms (see Fig. 4).
This procedure introduced some noise into the tempo
data, which could have been avoided by meticulous data
correction, but it produced sufficient results for the present
purpose. Not all tempo peaks do represent sudden accelera-
tions, but could result from notes, played accidentally by
the performer. For example, an additional note between two
eighth notes results in two tempo values approximately twice
as large as they should be. To avoid this data noise, the tempo
Fig. 3. Basic post-processing steps of raw data recording for the
curve was smoothed using a rectangular window with 35 data
analysis of timing and loudness.
points on either side of the current value, corresponding to
the 71 tempo values within each of the 840 renditions of the
use of a light-emitting metronome. No specific instructions score.
were given to the performer. Total recording time was 27
hours and 47 minutes (27.78 hours). During the “Theme”
section which is played with one hand, the performer could Calculation of performance trajectories
take minor refreshments, presented on a small table beside
Background
the pianist.
As Langner and Goebl (2002) claim, there is a mismatch
between human perception of expressive music performance
Basic analytical method
and the usual methods in performance research in two points:
The basic analytical approach to the raw data for the analy- (a) performance parameters are not perceived as separate
sis of timing and loudness is displayed in Figure 3. In a first streams of information (e.g., timing and loudness) and (b)
step, audio raw data was converted to standard WAV format. changes in loudness and tempo are perceptually integrated
In a second step, the loudness curve of the entire perfor- over time in human perception so that sudden and very local
mance was calculated in sone by use of a custom-made changes in tempo or loudness do not correspond to the
software, based on the psychoacoustic model by Zwicker impression of an accelerando or a crescendo. The perceptual
and Fastl (1999). Time resolution of the loudness curve was evaluation of this two-dimensional display has yet to be
10 ms. For a detailed description of the loudness analysis validated by listening tests. Thus, a method for performance
 Performance analysis of Saties’s “vexations” 5

analysis that considers human perception should display Kohlmetz, Kopiez, & Altenmüller, 2003). Trance is often
the changes in tempo and loudness simultaneously over time. referred to as a distinct level of consciousness, characterised
Additionally, an option for averaging data with adjustable by a restful yet fully alert state of mind with a heightened
time-frames (according to musical units such as one perception. Thus, in trance one may experience conflicting
measure) should be included. Such an integrated approach perceptions and time shortening (Travis & Pearson, 1999).
would result in a graph (a so-called “trajectory”) which The characteristic features of this meditative state, being the
displays the course of tempo and loudness simultaneously loss of the external frameworks (time, space, and bodily sen-
over time. Samples for the application of this method to the sation) and mental content (inner and outer perception), are
analysis of a Chopin Etude are presented in Langner and often interpreted as the result of a dominant right hemi-
Goebl (2002; in press). sphere. In practitioners of transcendental meditation, EEG
recordings showed a distinct pattern of electrocortical activ-
ity (Dunn, Hartigan, & Mikulas, 1999; West, 1980), includ-
Analytical method
ing synchronisation of the alpha spectrum (Jevning, Wallace,
Tempo data was extracted from the two-column interonset & Beidebach, 1992) and an increase in the relative power
file. The loudness information (perceptual measurement of of theta 2 (6.0–7.5 Hz) and alpha 1 (8.0–10.0 Hz) activity
loudness in sone according to Zwicker & Fastl, 1999) was (Alexander, Davies, & Dixon, 1990; Mason et al., 1997). In
derived from the corresponding acoustical recording of accordance with these findings, the significant increase of
“Vexations.” The smoothed tempo and loudness data was alpha 1 activity was also used as aphysiological indicator of
resampled to a time frame of 0.25 seconds and displayed in the state of trance in our study. Within the trance state, there
a two-dimensional space of tempo and loudness. were two “blackout” episodes during which the performance
was completely suspended for about a minute each (proto-
col: “microsleep”). These short blackout episodes were
Time series analyses of note durations and loudness
included in the analysis as well but yielded no consistent
A 28-hour uninterrupted performance of a complex repeti- results in terms of the target parameters and are therefore not
tive motor pattern with a cycle duration of approximately 2 presented here.
minutes represents an ideal data source for linear as well as
nonlinear analytical computations. The complete recording
may exhibit either transient properties, periodic properties, Linear time series analyses of note durations
quasi-periodic properties, or properties of deterministic and loudness
chaos. Of importance to this study is the question of whether Autocorrelation
distinctive features within the performance determine subse-
quent or future structures. The periodic structure of the piece The autocorrelation function gives a measure of how far one
itself would – if performed by a machine – produce a per- has to shift a given signal, compared to a copy of itself, to
fectly periodic time series. If performed by a human being, make the time signals appear similar again. In other words,
one would expect some sort of quasi-periodic deviation. time lags where the autocorrelation value is high, point to
However, this quasi-periodicity might be superimposed by putative periodicities within the signal.
deterministic processes on a larger time scale. Therefore, The series of transient keystrokes, each regarded as a
further analyses are applicable to clarify whether the time singular event, resemble the succession of so-called’ action
course of the performance is non-deterministic (i.e., noise), potentials’ generated by nerve fibers. Neuroscientists cus-
deterministic with a convergent, stable behaviour, or even tomarily analyse the series of onsets of action potentials only.
deterministic with “unpredictable,” divergent behavior – These series’ are known as “spike trains.” In our analysis,
namely, chaos. spike train temporal autocorrelations were computed for a
modified time series, where only the timepoint of detected
note onsets were kept and all intermittent data (sone-values)
Analytical method were set to zero. The resulting data set can be considered a
The linear and non-linear time series analyses were calcu- linear sum of delta pulses at the times of the note onsets.
lated with the software bundle TISEAN (Hegger, Kantz, Information about the performance loudness is still present
& Schreiber, 1999) and plotted using MATLABTM (Math- in the data, as the delta pulses have the sone amplitude of the
works). During the performance a spectator recorded a respective onset.
protocol of events. Additionally, the pianist recorded a
retrospective protocol of events after the performance.
Nonlinear time series analyses of note durations
According to this protocol we divided sections of the
and loudness
performance into three different states and three data sub-
sets were extracted: alertness (0.10–2.10 hr), trance (14.10– The use of non-linear methods was motivated through the
19.09 hr) and drowsiness (19.20–21.00 hr). Behavioural and immense amount of performance data that had to be reduced
EEG data of all three states were compared (for details see without the risk of a loss of information buried in the
6 Reinhard Kopiez et al.

complex data. The application of tools from non-linear time the piece again and again. But is the parameter constant at
series analysis seemed to be the most promising way to fulfill the instance of each recurrence? Is there, in addition to sys-
(a) the condition of data reduction and (b) of information tematic global shifts, an underlying system predicting a
maintenance. parameter in the following keystroke, phrase, or whole cycle,
on the basis of one or several of the preceding keystrokes,
phrases, or cycles? Usually (with dissipative dynamical
Multidimensional embedding of the time series
systems), trajectories are confined to lower dimensional
In the first step in non-linear analysis approaches, it is a subsets of the phase space. This simply means that the tra-
common practice to “embed” a given time series. This means, jectory is an object with fewer dimensions than we have to
one has to build up an appropriate multidimensional space (m use to “embed” it (not unlike the example that a wildly
dimensions), in which each dimension contains a measured twisted wire is embedded in 3-dimensional space but has not
value of the time series temporally separated from each other more than one dimension inherently). These subsets can be
by a specific amount of time (“delay” d). For example, let us extremely complicated, and they frequently possess a fractal
consider a three-dimensional embedding space (m = 3) with structure, meaning that they are self-similar in a nontrivial
a delay of d = 1 sec. For each sample of the time series, one way. Generalised dimensions are one class of quantity which
plots the value of a sample on the first axis, the value of a characterise this fractality. The Hausdorff dimension, from
sample one second later on the second axis, and the value of the mathematical point of view, is the most natural concept
a sample two seconds later on the third axis. Thus, each point to characterise fractal sets (Eckmann & Ruelle, 1985),
of the resulting pattern (or trajectory) contains information whereby the information dimension takes the relative visita-
not only about a given sample, but also about the future devel- tion frequencies into account and is therefore more attractive
opment of the time series. for physical systems. Ultimately, other similar concepts, like
It is obvious that the shape of the resulting trajectory is the correlation dimension, are more useful for the character-
very sensitive to the choice of the embedding dimension and isation of measured data. Dimensions are invariant under
delay. An appropriate embedding dimension can be estimated smooth transformations and are thus again computable in
by plotting the “correlation dimension” for increasing values time delay embedding spaces.
of m (see the section “correlation dimension” below). This is The correlation dimension was chosen to analyse the data
an estimator of the embedding dimension. in “Vexations.” Generalised dimensions are promising for
With respect to the time axis of the non-linear embedding this kind of explorative approach because the ever-repeating
procedure, two approaches were tested. Firstly, the complete cycle of the piece suggests a self-similar or fractal structure
time series with 100 samples per second was processed. Sec- of the performance, and because every order of magnifica-
ondly, as the acoustical data between two keystrokes cannot tion from local to global features (single note, phrase, cycle,
be controlled by the pianist, a relative time measure was entire piece) remains included as long as there is no reason
introduced with each note onset being a distinct timestep, to exclude any possibility in the first place.
regardless of whether the played note was of eighth, quarter,
or dotted quarter note length. Thus, the embedding procedure
Correlation dimension
could take the predictability of the loudness/interonset-
interval of a specific tone into account, depending on the Correlation dimension is a measure of the structural com-
history of the preceding tones. The second approach proved plexity of an attractor. Roughly speaking, the idea behind
to provide more robust results, which will be presented later certain quantifiers of dimensions is that the weight p(e) of a
in this paper. typical e-ball covering part of the invariant set scales with its
We estimated the maximum Lyapunov coefficient and the radius like p(e) ª eD (where the value for D depends also on
correlation dimension. For the embedding of the time series, the precise way one defines the weight). Using the square
various parameters were tested in order to determine the of the probability pi to find a point of the set inside the ball,
optimum embedding situation. The delay d with the value of the dimension is called the “correlation dimension” D2,
1 was chosen for the embedding, the Theiler window w was which is computed most efficiently by the correlation sum
set to 100 in order to exclude repetitions of one cycle of the (Grassberger & Procaccia, 1983):
Satie piece (1 cycle = 72 notes), and the embedding dimen-
N
sion was varied from 1 to 72. 1
C (m,e ) =
N pair s
  Q(e - s j - sk ) (1)
j=m k< j-w

Generalized dimensions and self-similarity

where si are m-dimensional delay vectors, Npairs = (N - m -
The “perpetually” repetitive character of the piece offers an 1)(N - m - w + 1)/2 the number of pairs of points covered
opportunity to apply the tool of generalised dimensions to by the sums, Q is the Heaviside step function, and w is the
the performance data set. A “trajectory” of virtually any para- so-called Theiler window (Theiler, 1990).
meter recorded throughout the performance of this particu- If the correlation dimension converges with increasing
lar piece is expected to re-enter similar recurring points of embedding dimension m to a fixed value, one can consider
 Performance analysis of Saties’s “vexations” 7

the embedding dimension at which saturation is reached.

This is suitable for complete embedding and unfolding of a
possibly underlying attractor by which the dynamics of the
system can be described. The m value for which the curves
converge is an estimator of the embedding dimension. If
there is no underlying deterministic process but noise only,
the respective curves will not saturate and will not gather
around the correlation dimension.

Lyapunov exponents
Chaos arises from the exponential growth of infinitesimal
perturbations, together with global folding mechanisms to
guarantee boundedness of the solutions. This exponential
instability is characterised by the spectrum of Lyapunov
exponents (Eckmann & Ruelle, 1985). If one assumes a local
decomposition of the phase space into directions with dif-
ferent stretching or contraction rates, then the spectrum of Fig. 4. Overall distribution of IOIs over the entire performance
exponents is the proper average of these local rates over the duration of about 28 hours. Left peak = quarter notes, middle =
whole invariant set, and thus consists of as many exponents eighth notes, right = dotted quarter notes (last syncopated note of
staff). (Note: about 20 outliers are located between IOIs from 5 to
as there are space directions.
30 seconds which cannot be displayed here due to the chosen
resolution of the x-axis.)
The maximal exponent
The maximal Lyapunov exponent can be determined without
the explicit construction of a model for the time series. Since mean note duration for all dotted quarter notes (IOIs > 2.9 s)
the convergence of the correlation dimension with increas- was 3.6 s (SD = 1.2 s). This means that eighth notes varied
ing embedding dimension has been previously checked (see with 10%, quarter notes with 9% and dotted quarter notes
the “correlation dimension” section), we applied an embed- with 33% of their value. The higher variability of the dotted
ding dimension of m = 18 with a delay of d = 1 to compute note might be due to the fact that it is the final note of the
theme.
Ê 1 ˆ From this, the development of tempo over the entire per-
S (e, m, t ) = lnÁ
Ë Un  S n+t - S n¢+t ˜
¯
(2)
formance duration of 27.78 hours was analysed. As shown in
sn , ŒU n n
Figure 5, the mean tempo remained surprisingly stable over
We used the very similar algorithm of Rosenstein, Collins, the first 15 hours of performance although the light-emitting
and De Luca (1993) where only the closest neighbor is fol- metronome was only used for the initial fixing of tempo (mean
lowed for each reference point. Also, the Euclidean norm is tempo for 00.00–14.00 hours = 54.7 bpm, SD = 5.6 bpm). The
used. If S(e, m, t) exhibits a linear increase with identical mean tempo curve in Figure 5 was calculated by averaging the
slope for a reasonably large m (in this case, m = 18), then current tempo over 35 data points on either side of the current
this slope can be taken as an estimate of the maximal expo- IOI-value within a rectangular window (corresponding to 71
nent l. For the IOI data, no satisfying linear slope for values within each of the 840 renditions of the score). No
S(e, m, t) was detected in any of the conditions; alert, drowsy, general trend in tempo change could be observed in this first
or trance. section. Commencing with the transition to the trance section
at t = 14.00 hours, a slight increase in tempo and tempo insta-
bility can be observed (mean tempo 14.00–16.08 hours =
Results 58.5 bpm, SD = 7.7). However, according to the different
states of consciousness as reported in the pianist’s retrospec-
Analysis of tempo
tive protocol (see Kohlmetz et al., in press), we have to bear
From the onset text file, an interonset distribution was cal- in mind that the pianist was in a deep trance between 14.10
culated (see Fig. 4). We should bear in mind that tempo peaks and 19.09 hours. This state of consciousness seems to have a
artefacts are caused by the straight-forward tempo analysis. strong influence on the average tempo stability (mean tempo
Analysis of overall interonset durations by categories of 14.00–19.09 hours = 58.7 bpm, SD = 10.3 bpm). After the end
note lengths (see Fig. 4) revealed a surprisingly high stabil- of trance (after about 19.00 hours) the initial tempo stability
ity: mean note duration for all eighth notes (IOIs < 1.58 s) could not be re-established and shows higher deviations com-
was 1.0 s (SD = 0.1 s), mean note duration for all quarter pared to the beginning (mean tempo from 19.09 hours until
notes (IOIs > 1.59 and < 2.9 s) was 2.2 s (SD = 0.2 s) and the end = 54.2 bpm, SD = 7.1 bpm).
8 Reinhard Kopiez et al.

Fig. 5. Development of tempo changes over the entire performance duration of about 28 hours. The bright line represents the smoothed
mean tempo curve. The box indicates the state of trance.

Fig. 6. Development of changes in loudness over the entire performance duration of about 28 hours. The bright line represents the smoothed
mean loudness curve. Box indicates state of trance.

Analysis of loudness Table 1. Mean loudness values (and their standard deviations) in
sone during three different states of consciousness.
The most obvious result of the analysis of the overall loud-
ness curve (see Fig. 6, smoothed curve calculated as in Fig. 0.00–17.80 hrs 17.80 hrs–19.30 hrs 19.30 hrs–end
5) is the segmentation of the loudness curve into three parts: (alert) (trance) (drowsy)
a first part which is characterised by an overall and continu-
ous decline of loudness over roughly the first 18 hours with Mean 14.7 18.97 13.7
a mean loudness of 14.7 sone, a second part which is char- SD 3.6 7.7 3.4
acterised by a higher degree of instability and more sudden
increase in loudness (mean: 18.97 sone), and a third part that
shows more dynamic instability than the first part and less
extreme changes than the second (mean: 13.7 sone; for ness curve shows, the beginning of trance at 14.00 hours does
statistical details see Table 1).We have to remember that the not seem to influence the stability of the general decline. The
second part corresponds to the end of the trance section sudden loudness burst at 19.00 hours corresponds to the end
(about 14.00–19.00 hours). However, as the averaged loud- of the trance section. Surprisingly, the beginning of tempo
 J

Performance analysis of Saties’s “vexations” 9

Fig. 8. Twenty-eight hours tempo-loudness trajectory of Satie’s

“Vexations” (x-axis: tempo in bpm, y-axis: loudness in sone). The
individual points are 250 ms apart.

of the dot clouds does not correlate with duration. However,

during the performer’s different states of consciousness, the
elliptical shape of the loudness distribution increasingly
disintegrates from alertness to trance. The distortion of the
Fig. 7. Loudness of an onset (in sone, y-axis) versus the per- shape also means that timing becomes progressively more
formed interonset interval (IOI in seconds, x-axis), for each single unstable while the performed loudness remains controlled.
note (keystroke) during 10-minute excerpts from different stages of This homogeneity of disintegration of tempo control affects
the performance. Displayed are alertness (top), drowsiness (mid), all note durations and is the most striking feature of the
and two excerpts from the trance episode (bottom panels). For the trance section.
trance stages the data displayed in the upper graph was selected
from the state in which loudness minimum was reached (cf. right
half of the respective box in Fig. 6), while those displayed in the The performance as tempo-loudness trajectory
lower graph was characterised by greater psychomotoric instability Figure 8 shows the development of loudness and tempo over
of performance (final stage of the trance, compare the tempo vari-
the entire performance duration of about 28 hours. The tra-
ability in Figs. 5 and 8).
jectory has the form of a wool ball with more transparent
threads in the periphery. The black spot marking a tempo
vicinity of 55 bpm and 15 sone seems to be a kind of “gravity
instability at 14.00 hours (see Fig. 5) does not coincide with center” for the performance. In total, the entire performance
the onset of dynamic instability at 18.00 hours. Although the shows that tempo and loudness vary independently: faster
increasing instability in tempo and loudness seem to be two does not mean louder. Over most of the performance loud-
independent processes, we can observe a synchronisation ness varies between 10 and 15 sone, except a loudness peak
between tempo and loudness instability at about 19.00 hours up to 35 sone, and tempo varies between 45 and 65 bpm.
after the end of the trance section. To summarise, we can say that the trajectory shown in
Analysis of loudness changes on a more local level Figure 8 corresponds well to the perceived overall impres-
revealed an interesting pattern. Figure 7 displays the loud- sion of “Vexations” as an inexpressive composition with no
ness of notes in three duration categories (eighth notes, clear climax. The repetition, which is its main feature, is rep-
quarter notes, dotted quarter notes) over the three states of resented in the small variation of loudness and a higher vari-
consciousness. The idea, not unlike the tempo-loudness tra- ation in tempo. In the trajectory, these findings correspond
jectory of Figure 8, is to define the x and y coordinates of a to the small surface covered by the trajectory’s trace.
single keystroke by two putatively independent parameters of However, we would like to emphasise that the interpretation
musical behavior. The figure shows that performed loudness of the trajectory in terms of “smallness” is only of descrip-
is independent of the note duration, as the vertical extension tive value at the current state of research.
10 Reinhard Kopiez et al.

Fig. 9. Autocorrelation function of the loudness time series, for Fig. 10. Larger scale image of Figure 9. In addition to the fine
the alert state (0.00–9.00 hours). Periodical similarities of the structure, a slow periodicity with a period of 600 seconds (10
pattern can be seen at multiples of about 60 and 120 seconds, which minutes or 5 full cycles, respectively) becomes evident.
resembles one half and one full cycle of the piece at the actual
performance speed. The hyperfine structure is produced by the
sequence of the singular notes of each cycle. dimension increases from 5.5 to 14 with decreasing vigilance
(see Fig. 12).

Linear time series analyses of note durations

Lyapunov exponents
and loudness
However, the sone note onset data revealed a considerable
Figure 9 depicts the autocorrelation function of the loudness
linear part, which is shown in Figure 13. A Lyapunov
time series (in sone) for the state of alertness reported by the
coefficient could be estimated for all conditions. As with the
pianist (verified by the EEG data, see Kohlmetz et al., in
correlation dimension, a steady trend can be observed that
press). The picket-fence-like hyperfine structure is produced
correlates with the overall state of alertness (cf. protocol and
by the repetition of eighth notes throughout the performance
EEG data). l increases from an initial value of 0.025 to
creating a period length of 1.15 sec (the reciprocal of
0.036 as vigilance declines from alertness to deep trance.
52 bpm). Mid-scale periodical similarities of the pattern can
A summary of the nonlinear estimation is given in Table 2.
be seen at multiples of about 60 and 120 seconds, which
unsurprisingly resembles one half and one full cycle of the
piece at the actual performance speed. However, the larger
time scale shown in Figure 10 reveals a slow periodicity with
Discussion
a period of 600 seconds (10 minutes or 5 full cycles, respec- Straight-forward procedure of IOI analysis
tively). This slow periodicity in the loudness time series
The application of a simple straight-forward procedure for
might be due to physiological ultradian oscillations and will
the analysis of note durations was only a pragmatic method
be discussed below in detail.
developed for the specific features of the “Vexations” data.
An exact matching of score to performance events would
Non-linear time series analyses of note durations have been possible. However, due to omnipresent variation
and loudness in data (such as omitted notes, variation of tempo, wrong
notes etc.) this matching would always need manual correc-
Correlation dimension
tion and thus would be extremely time-consuming. This also
In Figures 11 and 12, the correlation dimension is plotted means that the IOI categorisation should only be applied to
versus e for different embedding dimensions m ranging from the analysis of more complex scores, if the number of dif-
1 to 72. In each case, asymptotic behavior reaches saturation ferent note durations is not too high. Otherwise IOIs with
for m ª 18. only a small difference would fall through the net of this IOI
The graphs do not exhibit a clear plateau or saddle; there- categorisation procedure.
fore the actual value of the correlation dimension for m > 18
can only be estimated. For the IOI data, the correlation
Tempo and loudness analysis
dimension increases from 0.3 to 1.2 with decreasing vigi-
lance (alert -> drowsy > trance, see Fig. 11). For the loud- Data analysis started with the analysis of tempo changes over
ness data, a similar trend can be observed: the correlation the entire performance duration. The main finding of this first
 Performance analysis of Saties’s “vexations” 11

A
A

B
B

C
C
Fig. 12a–c. Correlation dimension versus e, based on the loudness
Fig. 11a–c. Correlation dimension versus e, based on the IOI data data (sone) of different states of consciousness. Correlation dimen-
of different states of consciousness. Correlation dimension e is a sion e is a measure of the structural complexity of an attractor. The
measure of the structural complexity of an attractor. The set of set of curves is produced by varying the embedding dimension m
curves is produced by varying the embedding dimension m from 1 from 1 to 72. Alert (upper panel): asymptotic behavior reaches satu-
to 72. Alert (upper panel): asymptotic behavior reaches saturation ration for m > 18, yielding a correlation dimension of 5.5; drowsy
for m > 18, yielding a correlation dimension of 0.3; drowsy (middle (middle panel): asymptotic behavior reaches saturation for m > 18,
panel): asymptotic behavior reaches saturation for m > 18, yielding yielding a correlation dimension of 12; trance (lower panel): asymp-
a correlation dimension of 1.0; trance (lower panel): asymptotic totic behavior reaches saturation for m > 18, yielding a correlation
behavior reaches saturation for m > 18, yielding a correlation dimension of 14.
dimension of 1.2.
12 Reinhard Kopiez et al.

analytical step was the high degree of mean tempo stability tinuous slowing down in slow tempo) occurred in the first 14
over a long period of time (about 14 hours). As Figure 4 hours of performance, as would have been expected accord-
shows, at the end of this phase of alertness, instability ing to findings in continuation experiments with tapping. Up
increased by an acceleration of tempo and remained un- until now it is inconclusive as to whether highly trained musi-
stable during the phase of trance. The initial tempo stability cians are able to compensate for error in timing, and can thus
could not be re-established by the player in the third avoid general tempo drifts instead of error cumulation. For
“drowsy” section. As an overall tendency, note durations example, Clynes and Walker (1982) found that tempo drift
remained surprisingly constant with a standard deviation of in the tapping of isochronous intervals disappeared when
less then 10% for eighth notes and quarter notes. Only dotted tapping to the pulse of an imagined Mozart piano concerto.
quarter notes with a duration longer than 2.9 s were charac- The subject showed a remarkable mean IOI duration of
terised by a standard deviation of 33%. This finding is in 0.511 s with a SD of 0.0026 s during 4000 taps. Additionally,
accordance with results from experiments on isochronous results from the measurement of repeated performances of
serial interval production. As Madison (2000) reports, a stan- Bach’s Goldberg Variations by one pianist over more than one
dard deviation between 3 and 6% of the IOI is typical for decade (Clynes & Walker, 1982) showed a surprisingly high
IOIs up to two seconds. Even with extensive training, musi- degree of duration stability for the single variations. Follow-
cians show an IOI deviation of 2.8% for IOIs of 300 ms. This ing on from this, it cannot be excluded that the tempo
tempo variability is determined by central nervous processes, stability of a musical performance is influenced by psy-
such as the individual tempo discrimination threshold, as well chomotoric “noise” and drift on a more local level. However,
as the peripheral motor delay of fingers and hands (Wing at a global level, the duration of an entire movement is stored
& Kristofferson, 1973). Generally, tempo stability depends and coded in a different memory system and could work as
on IOI duration and decreases with increasing IOIs, corre- an overall error compensation mechanism.
sponding with decreasing tempo (for an overview see The analysis of the loudness curve over the entire perfor-
Madison, 2000). With this in mind, the performer played with mance showed a surprising result: over more than 18 hours
an extraordinary stability as regards mean tempo. It remains a continuous decline of loudness can be observed. Technical
uncertain as to why no general tempo drift (such as a con- reasons for this decline can be excluded.1 This tendency is
resumed after the sudden eruption of loudness at the end of
the trance section at t = 19 hours, and continues for the rest
of the performance. Up until now there has been no report
on a decrescendo over such a long period of time. The com-
parison between the courses of loudness and tempo reveals
an interesting result: during the trance section instability
occurs firstly in tempo. This phenomenon is interpreted as an
asynchrony of parameter control in musical performance.
Tempo seems to be more sensitive to a loss of control (caused
by the state of trance) than loudness, and as Figure 7 reveals,
control of loudness can be stabilised even when the
performer is in a state of extreme drowsiness. This desyn-
chronisation of performance parameter stability has been
described here for the first time. The relative independence
of changes in tempo and loudness can also be observed in

1
Note: Data sheets of microphones and mixing console gave no
Fig. 13. Estimation of the maximum Lyapunov coefficient (based indication of instability of phantom power or electrical charge over
on event succession of onsets) by regression of the linear part of the time. Temperature-dependency of microphone sound pressure sen-
S(e, m, t)-plot and determining the slope of the fit. Embedding sitivity can also be excluded. High degree of electrical stability was
dimension: m = 18, delay d = 1. confirmed by the manufacturer.

Table 2. Summary of embedding dimension, correlation dimension, and Lyapunov coefficient.

Subject’s state d [hours] Embedding Correlation Correlation Lyapunov

dimension dimension (IOI) dimension (sone) coefficient (sone)

Alertness 0.00–9.00 18 0.3 (5.5) +0.025

Drowsiness 19.00–21.00 18 1.0 (12) +0.034
Trance 14.00–19.00 18 1.2 (14) +0.036
 Performance analysis of Saties’s “vexations” 13

the trajectory of Figure 8. An assumed coupling of perfor- positive. Together with the saturation of the correlation
mance parameters (such as “the faster, the louder”) would dimension (suggesting complex deterministic behavior), this
result in a diagonal trajectory. However, Figure 8 clearly dis- finding points towards the presence of a “performance
plays that tempo and loudness are controlled independently. generator” which exhibits characteristics of a deterministic
chaos.
In our case, as customary with measured data, the picture
Time series analysis
seems to be somewhat ambiguous, suggesting that noise as
The evaluation of the autocorrelation function, in addition to well as a trace of a high-dimensional chaotic component
revealing note repetition periodicities and phrase repetition contributes to the shaping of tempo and loudness in the
periodicities, shed light on performance oscillations at larger performance of “Vexations.”
time scales. These oscillations have possibly physiological
sources and are superimposed onto the piano performance
Vigilance, periodic changes and endogenous rhythms
with rather feeble amplitudes. To highlight these weak
performance oscillations, a very large number of coherent The most remarkable feature of tempo and loudness analy-
samples (over a long time series) is needed. The performance sis is a de-synchronised and increasing degree of instability
of “Vexations” provides a unique model that allows for such over the first 19 hours of performance. An initial explanation
novel large-scale performance analyses. for this drift could be made by referring to changes in vigi-
The goal of the non-linear approach was to address the lance as a confounding variable. However, a research of the
question as to whether those large-scale performance fluc- literature of the influence of vigilance on the long-term esti-
tuations are basically random, or rather highly complex yet mation of loudness or time duration showed no results. We
deterministic. The embedding procedure clearly suggests that have serious doubts as to whether there can be a direct influ-
the correlation dimension does not increase beyond a certain ence of psychophysiological activation on performance at all.
value as the embedding dimension is increased. From this, An indirect effect on performance data could be assumed by
an underlying structure of the performance can be inferred, a shift of perceptual thresholds through changes in vigilance.
which is complex enough to require an unfolding of at least Even the assumption of indirect effects of vigilance on
18 dimensions. The dimensionality of n = 18 fits perfectly performance implies an insolvable methodological problem
into the properties of the piece, since a single repetition of in the determination of perceptual thresholds: as such
the phrase (theme, or one of the variations, respectively) thresholds can only be measured by a subject’s reaction or
consists of 18 notes. In other words, if one assigns an inde- performance, it is impossible to separate performance from
pendent coordinate of an 18-dimensional space to each indi- compensational neuropsychological mechanisms. This view
vidual event of the theme – i.e., one single point in that space is supported by findings from experiments in occupational
contains all the information on the performance of one full medicine. As Galley (1998) demonstrates in an experiment
theme/variation – then a complex trajectory unfolds on the relationship between saccadic activation and perfor-
that cannot be sufficiently displayed in less than those 18 mance, subjects show no decrease in ability in an eye-
dimensions. tracking task when activation was reduced after intake of
The correlation dimension gives an idea as to how benzodiazepine. Due to compensational neuropsychological
complex a possible attractor should be. If the correlation mechanisms of task adaptation, performance seems to be
dimension does not converge, the irregularity is produced by practically independent of changing activation.
noise. However, the mere fact that the correlation dimension In addition to the assumption of a “performance genera-
converges, suggests that the underlying dynamical process is tor,” the underlying psychophysiological foundations of
a deterministic chaos, rather than just noise. With regard to periodic changes in tempo and loudness should be examined.
this chaos, the notion of the 18-dimensional trajectory as a Our finding of periodic changes in tempo and loudness (see
possible attractor should be addressed. Naturally, the perfor- Figs. 9 and 10), within time frames of up to 30 minutes,
mance of the piece is highly constrained by the composition remains a challenge for performance research. Let us bear in
itself, and at least with respect to timing (IOIs), subsequent mind: up until now, there has been no performance research
repetitions of the theme force the player to perform similar that would support the assumption of a conscious and inten-
motor movements over and over again. The composition does tional process as an explanation for the observed shaping of
not allow the performer to leave this fixed frame. Yet, the esti- tempo and loudness within those large time frames. Thus, it
mation of the Lyapunov exponent suggests a rather different would be speculative to interpret these changes in terms of
phenomenon. A positive value of the Lyapunov coefficient an intentional or voluntary musical interpretation. As an
means that for very similar initial conditions, the resembling alternative explanation for the observed periodic changes in
future phase states tend to drift apart; the larger l, the faster loudness and tempo, we will look at underlying endogenous
the tempo. Thus, a positive l is a sign of irregularity of the periodic processes, which could influence motor perfor-
time series. As mentioned above, the constraints of the com- mance in the following section. This explanation is based on
position suggest a highly converging performance over time. the fundamental assumption of Birbaumer and Schmidt
However, the estimated Lyapunov coefficients are small but (1999) who postulate that all body functions are shaped by
14 Reinhard Kopiez et al.

periodic changes. Mental performance (such as in memory with a periodicity of 80 minutes for the verbal task, and of
tasks) is synchronised with body temperature amongst other 96 minutes for the spatial task. Luteinizing hormone had a
factors. The authors show that the suprachiasmatic nucleus period of 120 minutes. The authors conclude that cognitive
plays the role of a central ultradian pacemaker which co- task performance is associated with endogenous neurochemi-
ordinates numerous oscillators with different periodicities cal systems. However, Neubauer and Freudenthaler (1995)
such as body temperature, level of growth hormone, level of investigated ultradian rhythms in cognitive performance and
cortisol, or pain threshold. found no significant 90-min periodicity. The authors are criti-
Since the discovery of the “basic rest-activity cycle” cal of the earlier BRAC studies due to a lack of conservatism
(BRAC) by Kleitman (1967), an influence of ultradian in statistical methods.
rhythms (with periodicities smaller then 20 hours) on basic
vital functions, such as motor performance and sensory
2.5 to 20 hours
acuity has become widely accepted. BRAC means a change
of intensitiy in low frequency brain waves during sleep with Studies of long-term performances are rare. However, two
a periodicity of 90 minutes. In a review study representing studies reveal interesting insights into the role of endogenous
the state of BRAC research of the late 1970s, Kleitman rhythms on human performance. Miller (1995) collected
(1982) found much evidence for the existence of endogenous 10,000 hours of EEG and behavioral data from commercial
rhythms. A short review of the current state of research on truck drivers, driving runs of 10 to 13 hours. Typed into a
the relationship between ultradian rhythms and psychomotor database this immense collection is now available for all
performance can be sorted by the size of time frame affected: researchers interested in long-term psychophysiological and
performance data. Up until now there has been nothing
comparable in music performance. In a longitudinal study
40 seconds to 30 minutes
Nakano et al. (2000) measured the psychomotor performance
Makeig and Inlow (1993) measured the coherence of slow and physiological data during 19 hour sleep deprivation.
mean variations in EEG power and in local error rate in Periodicities of physiological indicators (heart rate, body
an acoustic target detection task. A significant coherence temperature, etc.) differed significantly and influenced the
between many EEG frequencies and task performance with degree of errors.
cycles of 4 min and 90 s is shown. Conte, Ferlazzo, and Renzi
(1995) investigated whether reaction time to an acoustic
stimulus is influenced by individual performance rhythms.
The authors showed that attention capacity changed with
General discussion
periods ranging from 5 to 30 minutes. In summary, we can conclude that up until now, very little
has been established about the influence of circadian rhythms
on musical performance. However, the analysis of tempo and
0.5 to 2.5 hours
loudness fluctuations shows that an influence of endogenous
Investigations of periodicity within a time frame of 90 min processes on the long-term shaping of expressive parameters
BRAC show an ambiguous picture. In order to study the cannot be denied. The recording of “Vexations” is a first step
relationship between body temperature and reaction time, in the collection of long-term performance data, however, the
Almirall, Ferrer, and Sanchez-Turet (1988) measured the “Vexations” project investigates an extreme case of musical
performance in a reaction time task over a period of 5 hours. performance. In a more normal case the main interest of
Time series analysis did not show an ultradian cyclicity of performance research focuses on performance durations of
90 min. Hayashi, Sato, and Hori (1994) studied ultradian between five minutes and one hour. With this background
rhythms in task performance and EEG activity over a period it is clear that “Vexations” is an exploration into a thus far
of 9 hours. Spectral analysis of behavioral data revealed a unknown territory. However, the analysis demonstrates that
cyclicity of 2 hours, and analysis of EEG data showed a it is possible to develop and apply linear as well as non-linear
slower component (with a periodicity of 4 hours) and a faster analytical tools for the analysis of extremely long perfor-
component (with a periodicity of 1.3 to 2.4 hours). Grau et mances. It becomes clear that these tools must fit the specific
al. (1995) investigated ultradian rhythms in human gross needs of the performed composition. This calls for further
motor activity and recorded the frequency of motor activity development of adequate analytical tools in future research.
in a monotonous environment over 5 hours. Rhythmometric Another interesting examination would be the comparison of
analysis showed activity rhythms with periods between 0.5 the composition played by different performers in order to
and 2.5 hours. Due to the analysis of ultradian rhythms in reveal the similarities as well as differences among the artists.
cognitive functions, the study by Gordon, Stoffer, and Lee For example, would other performers keep the same steady
(1995) is of special interest. The authors measured the per- tempo over 13 hours? The pianist who took part in our
formance of cognitive tasks (such as verbal, spatial and per- project was well prepared and experienced in long-term
ceptual speed tests) over a period of 8 hours. Additionally, performances. Thus, we cannot be sure whether his psycho-
blood samples were taken. Results showed multiple cycles physiological behavior is unique. Other performers could
 Performance analysis of Saties’s “vexations” 15

experience different sequences in their altering states of of human development. In: C.N. Alexander, & E.J. Langer
consciousness. (Eds.), Higher stages of human development: perspective on
Our findings also contribute to the understanding of adult growth (pp. 286–341). New York: Oxford University
complex artistic processes. We could clearly show that the Press.
highly repetitive and simple structure of “Vexations” does Almirall, H., Ferrer, R., & Sanchez-Turet, M. (1988). External
not result in a corresponding interpretation of low complex- auditory canal temperature and reaction time relationship
ity. Non-linear methods revealed that changes in loudness during long performances. International Journal of Psy-
and tempo are of a highly complex nature, and both para- chophysiology, 6, 215–220.
meters unfold in an 18-dimensional space. This has never Birbaumer, N., & Schmidt, R.F. (1999). Biologische Psycholo-
before been demonstrated in performance research. Although gie (4th ed.). Berlin: Springer.
we did not expect to generalise our analytical tools, we would Bryars, G. (1983). “Vexations” and its performers. Contact, 26,
like to state that our method of dimensional analysis is inde- 12–20.
pendent from the performance duration and thus can also be Cage, J. (2001). ORGAN/ASLSP – John Cage in
applied to much shorter compositions. The use of tempo- Halberstadt. Retrieved, from the World Wide Web:
loudness trajectories has been a further promising way for http://www.john-cage.halberstadt.de
the analysis of performance features over the unfolding of Clarke, E.F. (1982). Timing in the performance or Erik Satie’s
time within a performance. The advantage of this method is “Vexations.” Acta Psychologica, 50, 1–19.
that it reveals the complex interaction between the per- Clynes, M., & Walker, J. (1982). Neurobiologic functions of
former’s shaping of tempo and loudness and is also inde- rhythm, time, and pulse in music. In: M. Clynes (Ed.), Music,
pendent of the performance’s duration. We would also like to mind, and brain. The neuropsychology of music. New York:
emphasise that this integration of performance parameters Plenum Press.
into clearly structured visualisations corresponds to the lis- Conte, S., Ferlazzo, F., & Renzi, P. (1995). Ultradian rhythms of
teners’ subjectively perceived auditory sensation of a perfor- reaction times in performance in vigilance tasks. Biological
Psychology, 39, 159–172.
mance, and we predict that the visualisation of performance
Dunn, B.R., Hartigan, J.A., & Mikulas, W.L. (1999). Con-
data will be a central point in further progress of performance
centration and mindfulness meditations: Unique forms of
research.
consciousness? Applied Psychophysiolog and Biofeedback,
In conclusion: linear and non-linear methods, the integra-
24, 147–165.
tion of performance data in easily understandable visualisa-
Eckmann, J.-P., & Ruelle, D. (1985). Ergodic theory of chaos
tions and the comprehension of the role of underlying
and strange attractors. Review of Modern Physics, 57, 617–
psycho-physiological processes are predicted as being the
656.
most promising ways for an adequate understanding of a
Galley, N. (1998). An enquiry into the relationship between
musical performance’s idiosyncrasies. In this respect, the
activation and performance using saccadic eye movement
“Vexations” project is a first step in a new direction.
parameters. Ergonomics, 41, 698–720.
Gordon, H.W., Stoffer, D.S., & Lee, P.A. (1995). Ultradian
rhythms in performance on tests of specialized cognitive
Acknowledgements function. International Journal of Neuroscience, 83, 199–
This project could not have been realised without the help of 211.
numerous people. We are indebted to the support of the Grassberger, P., & Procaccia, I. (1983). Measuring the strange-
following people: Armin Fuchs (Würzburg), Christine ness of strange attractors. Physica D, 9, 189–208.
Kohlmetz, Christian Wolf, Michael Großbach, Timo Grau, C., Escara, C., Cilveti, R., Garcia, M., Mojon, A.,
Weggen, Wolfgang Trappe, Matthias Harnitz, Annika Fernandez, J.R., & Hermida, R.C. (1995). Ultradian rhythms
Hempel (Hanover), Simon Dixon (Vienna), and Jörg Langner in gross motor activity of adult humans. Physiological
(Berlin). The project was funded by the Hanover University Behavior, 57, 411–419.
of Music and Drama, Yamaha Europe Corp. and the Cultural Hayashi, M., Sato, K., & Hori, T. (1994). Ultradian rhythms in
Foundation of Saxony. The recording took place in the task performance, self-evaluation, and EEG activity. Percep-
Societaetstheater, Dresden, Germany, on May 20 and 21, tual & Motor Skills, 79, 791–800.
2000. The pianist was Armin Fuchs. Hegger, R., Kantz, H., & Schreiber, T. (1999). Practical imple-
Performance data and software used for data analysis can mentation of nonlinear time series methods: The TISEAN
be obtained from the website package. Chaos, 9, 413–435.
http://musicweb.hmt-hannover.de/satie. Jevning, R., Wallace, R.K., & Beidebach, M. (1992). The physi-
ology of meditation: A review. A wakeful hypometabolic
integrated response. Neuroscience & Biobehavioral Reviews,
16, 415–424.
References
Kleitman, N. (1967). The basic rest-activity cycle and physio-
Alexander, C.N., Davies, J.L., & Dixon, C.A. (1990). Growth of logical correlates of dreaming. Experimental Neurology,
higher states of consciousness: Maharishi’s Vedic psychology Suppl 4, 2–3.
16 Reinhard Kopiez et al.

Kleitman, N. (1982). Basic rest-activity cycle-22 years later. troencephalography & Clinical Neurophysiology, 86, 23–
Sleep: Journal of Sleep Research & Sleep Medicine, 5, 311– 35.
317. Mason, L.I., Alexander, C.N., Travis, F.T., Marsh, G., Orme-
Kohlmetz, C., Kopiez, R., & Altenmüller, E. (2003). Stability of Johnson, D.W., Gackenbach, J., Mason, D.C., Rainforth, M.,
motor programs during a state of meditation: Electrocortical & Walton, K.G. (1997). Electrophysiological correlates of
activity in a pianist playing “Vexations” by Erik Satie con- higher states of consciousness during sleep in long-term
tinuously for 28 hours. Psychology of Music, 31, 173–186. practitioners of the Transcendental Meditation program.
Kopiez, R. (1998). Die Performance von Erik Saties Vexations Sleep: Journal of Sleep Research & Sleep Medicine, 20,
aus Pianistensicht. In: R. Kopiez, B. Barthelmes, H. Gembris, 102–110.
J. Kloppenburg, H. von Loesch, H. Neuhoff, G. Rötter, & Michon, J.A. (1974). Programs and “programs” for sequential
C.M. Schmidt (Eds.), Musikwissenschaft zwischen Kunst, patterns in motor behavior. Brain Research, 71, 413–424.
Ästhetik und Experiment. Festschrift Helga de la Motte- Miller, J.C. (1995). Batch processing of 10000 h of truck driver
Haber zum 60. Geburtstag (pp. 303–311). Würzburg: EEG data. Biological Psychology, 40, 209–222.
Königshausen & Neumann. Nakano, T., Araki, K., Michimori, A., Inbe, H., Hagiwara, H., &
Langner, J. (2002). Musikalischer Rhythmus und Oszillation. Koyama, E. (2000). Temporal order of sleepiness, perfor-
Eine theoretische und empirische Erkundung. Frankfurt a. mance and physiological indices during 19-h sleep depriva-
M.: Peter Lang. tion. Psychiatry and Clinical Neuroscience, 54, 280–282.
Langner, J., & Goebl, W. (2002). Representing expressive Neubauer, A., & Freudenthaler, H.H. (1995). Ultradian rhythms
performance in tempo-loudness space. Proceedings of the in cognitive performance: No evidence for a 1.5-h rhythm.
10th Anniversary conference of the European Society for the Biological Psychology, 40, 281–298.
Cognitive Sciences of Music. Liège, 5–8 April, CD-ROM. Orledge, R. (1992). Satie the composer. Cambridge: Cambridge
Langner, J., & Goebl, W. (in press). Visualizing expressive University Press.
performance in tempo-loudness space. Computer Music Rosenstein, M.T., Collins, J.J., & De Luca, C.J. (1993). A prac-
Journal. tical method for calculating largest Lyapunov exponents from
Langner, J., & Kopiez, R. (1996). Entwurf einer neuen Methode small data sets. Physica D, 65, 117–134.
der Performanceanalyse auf Grundlage einer Theorie oszil- Satie, E. (1988). Schriften. Hofheim: Wolke.
lierender Systeme. In: K.-E. Behne, H. de la Motte-Haber, Theiler, J. (1990). Estimating fractal dimension. Journal of the
& G. Kleinen (Eds.), Jahrbuch Musikpsychologie (12, Optometric Society of America A, 7, 1055.
pp. 9–27). Wilhelmshaven: Noetzel. Travis, F., & Pearson, C. (1999). Pure consciousness: distinct
Langner, J., Kopiez, R., & Feiten, B. (1998). Perception and phenomenological and physiological correlates of “con-
representation of multiple tempo hierarchies in musical per- sciousness itself.” International Journal of Neuroscience,
formance and composition. In: R. Kopiez, & W. Auhagen 100, 77–89.
(Eds.), Controlling creative processes in music (pp. 13–35). Wehmeyer, G. (1998). Erik Satie. Reinbek: Rowohlt.
Frankfurt: Peter Lang. West, M.A. (1980). Meditation and the EEG. Psychological
Madison, G. (2000). On the nature of variability in isochronous Medicine, 10, 369–375.
serial interval production. In: P. Desain, & L. Windsor (Eds.), Wing, A.M., & Kristofferson, A. (1973). Response delays
Ryhthm. Perception and production (pp. 95–113). Lisse: and the timing of discrete motor responses. Perception &
Swets & Zeitlinger. Psychophysics, 13, 455–460.
Makeig, S., & Inlow, M. (1993). Lapses in alertness: Coherence Zwicker, E., & Fastl, H. (1999). Psychoacoustics. Facts and
of fluctuations in performance and EEG spectrum. Elec- models (2nd ed.). Berlin: Springer.

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