[Desain, P., & Honing, H. (1994). Does expressive timing in music performance scale proportionally with tempo? Psychological Research, 56, 285-292. ]

Does expressive timing in music performance 
scale proportionally with tempo?

Peter Desain & Henkjan Honing

NICI	Computational Linguistics, Faculty of Arts
Nijmegen University	University of Amsterdam
P.O. Box 9104	Spuistraat 134
NL-6500 HE Nijmegen	NL-1012 VB Amsterdam
desain@nici.kun.nl	honing@mars.let.uva.nl

Evidence is presented that expressive timing in music is not relational invariant with respect to global tempo. While Repp’s work on relational invariance (Repp, 1994) is based on Schumann’s Träumerei, our results stem from an analysis of repeated performances of Beethoven’s variations on a Paisiello theme. As in the Träumerei study, recordings were made of two pianists playing the pieces at three tempi. The major differences between the studies are the difference in structure of the pieces used and the larger tempo range of our study.  It is shown that, in contrast with the relational invariance hypothesis, between-tempo correlations are in general lower than within-tempo correlations. Furthermore, in analyses of variance of log-transformed interonset intervals (IOI), there was a significant interaction between tempo and IOI, which is counter evidence for a proportional relation between timing and tempo. Complex but consistent non-proportional patterns were shown in the analyses of the timing of the grace notes in the piece. The analysis suggests that timing aspects of music are closely linked to the musical structure and can only be studied and manipulated respecting this relation - not as a global timing pattern or tempo curve. Finally, the different methodological issues of data collection and analysis that have to be considered for this type of study are investigated. It is shown that without careful consideration these aspects would distort the conclusions. 

INTRODUCTION
In Desain & Honing (1992) we presented a study that criticized the use of tempo profiles as a representation of timing in music performance independent of the musical structure - both in cognitive modeling and technical applications. In these so-called “tempo curves”, measurements of local performance tempo or of note interonset time (IOI) are presented against score time in some form. We studied this representation in two quite different but related pieces of music, that share a harmonic, melodic and metrical structure (a theme and a variation). Many transformations, like scaling the measured tempo curve in global tempo, applying a tempo curve measured in the theme to the variation, or even simply changing the key from major to minor without modifying the tempo curve, produced results that were judged unmusical. This was the basis of our argument against their use. We argued that the various timing aspects of music are linked to the different structural aspects in the music, and therefore should be studied and manipulated respecting this relation and not be seen as a global transformation based on a tempo curve. This research was reported in an informal style, in the form of a Christmas story, and formal analyses were not given. 
Repp (1994), prompted by this work, systematically explored the scaling of expressive aspects of music with regard to global tempo, for performances of Schumann’s Träumerei. In this work, which will be referred to as “Träumerei study”, it was concluded that there was relational invariance to a great extent: expressive timing scaled “approximately” proportionally with tempo.  Consequently a tempo curve for a different tempo can be obtained by plotting the original on a log duration y-axis and shifting it vertically.
To gain further understanding on this issue and the seemingly conflicting results, we decided to publish, in a more formal form, the data and analysis that formed the basis of our original article. We will do that here, striving towards a presentation that works along the lines as set out in the Träumerei study, to allow for easy comparison. We will restrict ourselves to the examination of the patterns of onset timing. The broader scope of this kind of research and its embedding in the literature can be found in the introduction of Repp (1994) and in Desain & Honing (1992; 1993).

METHOD
The Music
For our study we used the theme and first variation from the six variations (WoO 70) over the theme “Nel cor più non mi sento” by Giovanni Paisiello composed by Ludwig van Beethoven in 1795 (see Figure 1). These variations belong to the species of melodic variation: they all retain the complete melody (Brendel, 1991). The theme and first variation were chosen because of their relative simplicity and their very similar metrical, harmonic and melodic structure (hereafter we will refer to them as Theme and Variation). Both the Theme and Variation are twenty bars in length. The Theme consists of an eight bar phrase (“Stollen”), followed by a six bar phrase (“Stollen”), and finishes with a six bar phrase (“Abgesang”). This is a, for that time, quite common song structure. The first phrase consists of two sub phrases, that in turn can be divided into two sub phrases each. The next phrase consists of three sub phrases, and the final phrase could be divided in two sub phrases, of which the first could be divided into two again, but other analyses are possible. The topmost level of the phrase structure tree of the Variation follows the phrase structure of the Theme. The Theme has a typical Alberti Bass (broken chord accompaniment), whereas  the Variation has predominantly a chord accompaniment. The Theme has a varied rhythmical structure and several ornaments, whereas the Variation is characterized by runs of sixteenth notes.

Figure 1a. The score of the Theme. Reproduced from Beethoven, Variationen für Klavier, Band I. München: G. Henle Verlag. With kind permission from the publisher.

Figure 1b. The score of the Variation. Reproduced from Beethoven, Variationen für Klavier, Band I. München: G. Henle Verlag. With kind permission from the publisher.

Performers
The two performers were BH, a professionally trained pianist in his early sixties and RB, a professional pianist in his early thirties. Both were familiar with the Beethoven’s Theme and Variation. BH had just studied and performed all six variations.

Recording Procedure
Our recording procedure as described in Desain & Honing (1992) was almost identical to the one reported in Repp (1994). But an important difference was the use of a synthesized harpsichord as to an electronic piano in the Träumerei study. We preferred the harpsichord, since this forced the subjects to use timing as their only expressive means - which works around the difficult problem of the interaction of time and dynamic parameters (e.g., Drake & Palmer, 1993). To make our observations more comparable to Repp’s data, we redid the data collection using a similar instrument and recording procedure as was used in the Träumerei study.

The instrument was a Roland D-5 synthesizer (using only its keyboard and MIDI output), without sustain pedal, connected to an Apple Macintosh Quadra 700 computer running the EZ Vision sequencer software (time resolution 1 ms, within the limits of MIDI transmission speed) under System 7 (with no other programs running). The (MIDI Thru) output of the computer/sequencer was connected to a Yamaha TG-100 sound generator set to a piano sound (“1 GrandPno”), without reverb, played over a SONY SRS-D4 loudspeaker system. 

Each pianist was recorded in a separate session. After some practice on the instrument, he played the full piece at his preferred tempo. Subsequently, each pianist performed the piece at a slower tempo and a faster tempo. These tempi were chosen by the performers to form the end points of a musically sensible range of tempi, where in the Träumerei study they were set by the experimenter. The tempi are given in Table 1. The instruction was to play naturally, as musical as possible. No metronome was used. Both took a small amount of time to get used to the instrument.

Both performers played the Theme and the Variation at each of the three tempi until four performances were recorded that were judged to be good by the performer (this took an average of five recordings per tempo/piece). First the piece was recorded at the normal or medium tempo, followed by the series in a slow and a fast tempo. After a break all good performances where played back to the performer, who was asked to select the three best ones per piece at each of the three tempi. 

The recorded data were stored in standard MIDI files (IMA, 1983). These data files were imported into POCO - a computer environment for research on expression in music (Honing, 1990). For both pieces a score was created from one performance (partly by quantization and partly by hand editing). It was annotated, using POCO tools, with structural information like metrical and phrase structure, chords and ornaments, the different voices and the left- and right-hand parts. The structural information in the score was subsequently transferred to each performance with the POCO performance-score matching facility. The matching dispenses with any further hand annotation of individual performances (a tedious process) and enables sophisticated and easy analysis of information at different structural levels, filtering out certain types of musical objects or voices, and so on. This process is also a secure check that there were no errors in the performance (like accidentally touched keys, missing or wrong notes, etc.). The resulting data were imported into a statistical data analysis program (StatView II) for further processing.

Issues in Measuring Timing
There are a number of methodological issues that are often taken for granted in the study of expressive timing, and that have, as we will show, a significant influence on the results. The issues described below, apply in general to all studies using a global timing pattern or profile. Although lately, more and more caution is taken in the interpretation of these unstructured profiles, still some issues with regard to measurement and representation of timing need closer examination.1

Selecting Onsets in Polyphonic Music
A first issue is the determination of the note onsets that are to be used in calculating the intervals between them (the IOI’s) in a performance. Parallel structures (notes with the same score onset time, for instance notes notated as a chord in the score) and ornamental structures (grace notes) pose the difficulty here. For parallel onsets the basic options are to use only the onsets of the top voice (the highest in pitch), the onsets of melody as resulting from music theoretical analysis (based on the score), or to use the note whose onset happens to be first (see Figure 3). There is no real optimal solution for this issue (apart from restricting oneself to the study of monophonic music). Repp (personal communication) states that the differences in measurement can be neglected, but it is not clear which method was used in the Träumerei study. Because the melodic voice may indeed carry the perceptual tempo, we will use the melody-only approach in the alternative method (described below).

Figure 3. The treatment of parallel structures.

Removing Gracenotes
For a grace note no tempo measurement can be calculated since no reference note duration is given in the score. So, ornamental notes have to be ignored and studied separately (as was, for example, done in Repp, 1994). But the perceptual effect of a short grace note on the perceived timing of the following note can be quite large. This can be easily verified by listening to a performance from which the grace notes have been removed artificially. An explantion for the strange-sounding effect might be a time-shrinking phenomenon described by Nakajima, ten Hoopen, & van der Wilk (1991). Perceptual time-shrinking takes place in a time interval following a short one and was discovered for the short time intervals in the range that is commonly found in musical ornaments. In our case there are so many grace notes in the Theme (in contrast to Schumann’s Träumerei which only has two proper grace notes), that one has to be careful in not letting perceptual effects in the notes following the grace note invalidate the results of the analysis. 
In the method used, to stay on the save side, the onset of the note before the grace note and the onset of note following the grace note are removed (see Figure 4). This will reduce the size of the data set of the Theme by thirty-three data points and the Variation by three data points. Removing this much local detail biases the subsequent analyses towards more global rate or tempo aspects. And because the local structure of grace notes is one example of complex timing that, as we will show, does not generally scale proportionally with tempo - excluding this source of non-proportionality will make the absence of relational invariance in our data a more general conclusion. 

Figure 4. The treatment of grace notes.

Normalized IOI’s Versus Sampled IOI’s
A third issue is the sampling of the IOI’s on a time grid. Sampling, for instance in the way it is used in the Träumerei study, essentially duplicates data points of the tempo measurements of longer notes. These notes with a long nominal (score) duration may then exert more influence on the correlation outcome.2 This method appears to be implicitly based on the notion of a continuous (mental) representation of tempo, that can be accessed, for example, by sampling at regular intervals, even when there are no events - a simplification quite rightly criticized in Gibson’s paper “Events are perceivable but time is not” (Gibson, 1975). Therefore, we will restrict the analysis to IOI’s normalized by score duration

Magnitude Scale of Time Duration
A fourth issue is the magnitude scale to use. A simple duration scale (consisting of raw IOI’s) cannot be used in general to calculate correlations between timing profiles - because different note durations in the score would swamp any timing effects. A beat duration scale does not suffer from that effect. This is the inverse of a tempo or rate scale, another possible scale to calculate correlations from. A choice seems quite arbitrary, but we can fold back Repp’s conclusion (timing is relational invariant with respect to global tempo) to a micro level. Accordingly, timing should be measured on a logarithmic scale. And this makes the difference between duration and tempo (1/duration) measurement essentially irrelevant. Moreover, consistency in performances measured by correlation of log-duration or log-tempo would theoretically coincide better with perceived similarity of timing patterns.3

Averaging Across Performances
A fifth issue is the averaging across performances. Apart from making comparisons of between-tempo and within-tempo correlations impossible, averaging within-tempo IOI’s across performances seems to ignore that - even when performances are highly consistent - higher order control of timing of subsequent events may have a systematic effect on the data that will not average-out as noise. Furthermore, if indeed a log-magnitude scale for IOI’s is most suitable, it can be argued that combining the data before the log-transformation is inappropriate.4 Therefore, we will only report average correlations of individual performances.

RESULTS AND DISCUSSION
Timing Patterns at the Medium Tempo
Figure 2a and 2b show a section of the resulting timing patterns of the three repeated performances in a medium tempo for each pianist of both the Theme and the Variation.

Figure 2a. Timing patterns of eight bars of the Theme by the two pianists at the medium tempo (three repeated performances)

Figure 2b. Timing patterns of four bars of the Variation by the two pianists at the medium tempo (three repeated performances)

In these graphs, a high peak represents a slowing down (a long beat duration). These occur consistently at specific structural boundaries (e.g., in Figure 2a at the end of bar 8). It can be seen that BH employs a deeper rubato (larger tempo excursions) than RB does. 
Now one can use the correlations between the overall profiles of the three performances at the same tempo to measure how consistent and reliable these data are. The average overall correlation between the three individual performances at the medium tempo is for RB 0.77 (Theme) and 0.80  (Variation) and for BH 0.81 (Theme) and 0.95 (Variation). Except for the performances of the Variation by BH, the correlations are lower than the ones reported in the Träumerei study. We think that this can be attributed to the character of the pieces and the measurement method used. The performers differ considerably in the reliability of the replication of performances, BH being the most consistent. But this may be brought about by the fact that BH uses a deeper rubato (comparable to the Träumerei performers), which will raise the intended timing signal higher above the level of timing-noise brought about by, for example, motor control and thus artificially raise the correlations. Correlating between the timing patterns of the two performers (on average 0.56 for the Theme and 0.61 for the Variation, both at medium tempo) indicates indeed that they behave not the same.

Chosen Tempi
The tempi chosen by the performers are given in Table 1. Note that these values are derived from the duration of the piece (from the first note onset to the last note onset). Over repeated performances at the same intended tempo the tempi differed by less then 1.4% from the mean, except for BH’s fast performances of the theme, which had a range of 2.5%. The last column gives the range of tempi used as the ratio’s slow/medium and fast/medium. These ranges constitute a larger range of tempi when compared to the Träumerei  study. In there, the average range is 0.85 - 1.15, though the ranges may not be not completely comparable because the tempi are estimated, not measured, by the performers using a metronome.5

Table 1. Average tempi chosen by the performers and the average tempi range (tempo indicated in M.M. for a dotted quarter note).
		tempo				
piece	performer	fast	medium 	slow	range	
Theme	RB	75.4 	58.1	50.7	0.87 - 1.30	
	BH	65.0	50.6	40.6	0.80 - 1.29	
Variation	RB	59.4	52.5	42.9	0.82 - 1.13	
	BH	59.9	50.7	31.8	0.63 - 1.18	

Consistency of Timing Patterns
In Table 2 the average correlations between performances (between and within tempi) are given, using log-transformed normalized IOI’s of individual performances, derived from the melody, and ignoring grace notes and their context.

Table 2. Average correlations between performances. 
			tempo			
piece	performer	tempo	fast	medium	slow	
Theme	RB	fast	.84	-	-	
		medium	.75	.81	-	
		slow	.69	.80	.89	
	BH	fast	.84	-	-	
		medium	.73	.86	-	
		slow	.56	.81	.87	
Variation 	RB	fast	.80	-	-	
		medium	.65	.76	-	
		slow	.61	.60	.78	
	BH	fast	.94	-	-	
		medium	.85	.93	-	
		slow	.79	.84	.91	

It is useful make some observations on other patterns in Table 2 before turning to the main comparison between performances in different tempi.
Lets first look at the within tempo correlations (the consistency over repeated performances) per piece and performer. One would expect the consistency between performances at the same (high or low) extreme tempi to decrease as compared to the medium tempo, because of the smaller amount of training that the pianists received in these conditions and because of the possibly harder task (considering motor-control and mental time keeping). But in Table 2 it is shown that this only has a very small effect on the correlations. The average correlation for extreme tempi (fast-fast and slow-slow) is 0.86, the average medium-medium correlation is 0.84. One can conclude that the pianists chose a tempo range in which they could master the task of performing consistently and controllably. 

In comparing the Theme with the Variation, they are played with a similar consistency. Both pieces show an average correlation between performances of  0.79. This might be surpising considering the more elaborate rhythmical structure and performance demands of the Theme. 

In comparing the two performers, the impression is confirmed that BH replicates his performances more reliably. The average correlation between performances of BH is 0.83, and the average correlation between performances of RB is 0.75.

Now we arrive at the main question: do the expressive timing profiles of the performances at different tempi change? For that we have to compare the average correlations between the slow-fast tempo extremes (on average 0.66) with the correlations between same tempo (fast-fast, medium-medium, slow-slow) performances (on average 0.85). For testing the significance of this difference it is easier to consider the individual correlations - not their average. For the Variation all individual correlations of performances between the tempo extremes were significantly lower (p< .05, N = 221, Fisher Z transform) than any correlation between performances of the same piece by the same performer at the same tempo. Thus one may conclude here that a change in global tempo affects the shape of the timing profile - which is counter evidence for relational invariance. For the Theme not all comparisons were significant, which may be induced by the small amount of data points (N = 48).

Analysis of Variance 
A high between-tempo correlation is evidence for a linear relation between expressive timing profiles at different tempi. But this measure does not differentiate between an additive or multiplicative model. This means that a high correlation does not refute a “multiplicative rate parameter hypothesis”, which postulates a timing profile as a motor program with a yet to be determined parameter which sets the global playback speed.
But high between-tempo correlation provides no evidence for its choice over a model with an additive component, for instance, the purely additive event-shift model (Bilmes, 1993) in which timing is absolutely invariant with tempo. For that, the intercept of the regression line should be zero. This drawback of using only correlations is acknowledged by Repp (personal communication) and he therefore proceeds with an analysis of the variance on log-transformed IOI’s. It is reasoned that if the log-transformed timing profiles for different tempi are parallel, then the IOI by tempo interaction should be non-significant. Separate ANOVA’s per piece and per performer were conducted to test this hypothesis. In our study, each ANOVA has IOI number (113 levels for the Theme, 223 for the Variation) and tempo (3 levels) as fixed factors and performance within tempo as random factor. The IOI by tempo interactions from ANOVA’s of the performances are significant for the Theme of RB [F(224,678) = 2.122, p < .0001] and BH [F(224,678) = 4.128, p < .0001], and for the Variation of RB [F(444,1338) = 3.022, p < .0001] and BH [F(444,1338) = 4.444, p < .0001]. Because of the significant interactions between tempo and IOI, one can state that expressive timing in the performances of the Theme and Variation is not relational invariant. This differs from the findings reported in the Träumerei study using the same method. We will leave a possible explanation to the general conclusion. 

Grace Note Timing Patterns
One of the original motivations for the choice of the Beethoven variations was the number of grace notes in the Theme. They are all notated in the same way in the score (see Figure 1a, in which the grace notes are numbered in square boxes). The grace notes occur in five different rhythmical contexts: I (grace note number 2 and 10), II (number 3 and 11), III (4 and 5), IV (1, 6 and 9), and V (7 and 8). The grace notes in context I can be characterized as Appogiatura (“time-taking” grace notes), because of their pitch interval with the next note and the derivation of the singing style of the Theme (comparable to the grace notes notated as small eighth-notes in the Träumerei score). The grace notes in the other contexts can be characterized as Acciaccaturra (“time-less” grace notes). 
We looked in detail to the individual grace notes in the rhythmical contexts I, II and III for BH (Figure 5, 6 and 7). In each figure the data of the 9 performances of one grace note is depicted. The horizontal axis shows the average beat duration of the performance. The vertical axis depicts the duration of the specific grace note in that performance. The duration of a grace note is defined to be the IOI between the onset of the grace note and the onset of the note it is attached to. Note that each figure has a different vertical scale. The predicted tempo scaling with respect to the average beat duration in the medium tempo is plotted as a reference line. If a grace note IOI scales proportionally with global tempo, then it is close to this line. A quick inspection indicates that replications are quite accurate: same-tempo grace notes are about the same length. But relational invariance is violated consistently: in almost all cases low and high tempo grace notes are not close to the reference line.
In Figure 5 the timing patterns of the grace notes 2 and 10 are plotted. Both of them seem to scale proportionally from the medium to the fast tempo. But from the medium to the slow tempo BH exaggerates the timing of the grace note with respect to the global tempo. 
Grace notes 3 and 11 (see Figure 6) are clearly different. From the medium to fast tempo they stay invariant. From the medium to slow tempo less so. The maximum deviation over all tempi in this figure is 10 ms.6
Finally, Figure 7 shows the timing of grace notes 4 and 5. Most remarkable is grace note 5 that seems to scale inversely with respect to the tempo. 
These observations are evidence for the fact that there is more to the timing of grace notes than mere proportional scaling. But there is of course more data needed to come to a precise description and explanation of these different behaviors of grace notes.

Figure 5. Timing of grace notes 2 and 10 for three repeated performances in the three tempi. 

Figure 6. Timing of grace notes 3 and 11 for three repeated performances in the three tempi.

Figure 7. Timing of grace notes 4 and 5 for three repeated performances in the three tempi. 

These observations were confirmed by a significant interaction between IOI and tempo for ANOVA’s over all log-transformed grace note IOI’s in the theme for both performers. The IOI by tempo interactions from ANOVA’s of the grace notes of the Theme were significant for both RB [F(20,66) = 4.352, p < .0001] and BH [F(20,66) = 3.242, p < .0002].

GENERAL DISCUSSION AND CONCLUSION
In this study we analyzed in more detail the informal observations made in Desain & Honing (1992) on the implications of a “tempo curve” as a representation of time in music cognition research. This implicit notion that is present in several models of timing and analysis methods (e.g., Clynes, 1987; Feldman, Epstein, & Richards, 1992; Sundberg & Verillo, 1980; Todd, 1989) seems to have little cognitive, perceptual and musical reality. The analysis of the expressive timing profiles of performances of two works from Beethoven’s Variations (WoO 70), as were presented in this study, showed that timing, in general, does not scale proportionally with respect to global tempo, which is counter evidence for the usefulness of the notion of an overall tempo curve. 

The reasons for the contrasting results found in the Träumerei study now have to be investigated. There are different sources that may have contributed to the differences. A first source is the distinct musical structure of the pieces used in the studies. Schumann’s Träumerei is a slow, expressive piece of the Romantic period, with a prominent harmonic and melodic structure, but rhythmically vague. Of the Beethoven pieces, the Theme has a very clear rhythmical structure and a number of characteristic ornaments. However, one would expect similar results for the Variation and for Träumerei because, although the tempo of the Variation is higher, both consist mainly of sixteenth notes and have little or no rhythmic variation or ornaments. Nonetheless, even here we found significant differences in expressive timing between different global tempi. So the piece under study may not be that important.

A second source for the contrasting results between the studies could be the range of tempi chosen by the subjects. While for Träumerei the extreme tempi differed on average by 15% from the medium tempo, in the Beethoven pieces the range was double that size. By measuring consistency of repeated performances it was shown that the performances at more extreme tempi were still under control of the performers, and our measurements might indicate a systematic breakdown of relational invariance at these more extreme tempi. 

Finally, some methodological issues of data collection and analysis were examined. The method of deriving IOI’s from the measured data (note onsets) has an influence on the data analyses that cannot be ignored. While these issues may be relatively unimportant for Repp’s results, without careful consideration they would have distorted the conclusions in our case.

While in the Träumerei study the set of pieces in which relational invariance holds is characterized as those “with expressive gestures on an even rhythmical flow”, in both studies counter evidence is presented with respect to the structural aspects of the music (chord spread, note overlap and certain types of ornaments). This indicates that the separate analysis of the timing contributions of these structural aspects of music is one promising route to take in future research. This is already done for grace notes in this study, but is still an open question for concepts like meter, phrase structure and surface structure. It is not unrealistic to assume that these components behave differently when scaled in global tempo.

ACKNOWLEDGMENTS
Many thanks to Bruno Repp for stimulating discussions and making available his data and results in an early stage, and to RB and BH for their performances and patience. Valuable comments on earlier versions of this paper were made by René van Egmond and Dirkjan Povel.
The research of Dr. Desain and Dr. Honing has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (KNAW).
Part of this study was done while visiting New York University on a kind invitation by Robert Rowe and was made possible by a grant from the Netherlands Organization for Scientific Research (NWO). The NYU Music Department, the NICI institute in Nijmegen, and the Computational Linguistics department in Amsterdam formed the stimulating environment in which our work could evolve.

REFERENCES
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Brendel, A. (1991). Theme and Variations II. Schumann - Beethoven. Liner Notes for CD (Philips 432 093-2).
Clynes, M. (1987). What can a musician learn about music performance from newly discovered microstructure principles (PM and PAS)? In Action and Perception in Rhythm and Music, edited by A. Gabrielsson. Royal Swedish Academy of Music, No. 55, 201-233.
Desain, P. & Honing, H. (1992). Music, Mind and Machine, Studies in Computer Music, Music Cognition and Artificial Intelligence. Amsterdam: Thesis Publishers.
Desain, P. & Honing, H. (1993). Tempo curves considered harmful. In Time in Contemporary Music Thought, edited by J. D. Kramer. Contemporary Music Review. 7(2), 123-138. London: Harwood Press. Preprinted in Desain & Honing (1992).
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Feldman, J. , Epstein, D. & Richards, W. (1992). Force Dynamics of Tempo Change in Music. Music Perception, 10(2), 185-204.
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Honing, H. (1990). POCO: An Environment for Analysing, Modifying, and Generating Expression in Music. In Proceedings of the 1990 International Computer Music Conference. 364-368. San Francisco: Computer Music Association. Reprinted in Desain & Honing (1992).
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FOOTNOTES
1In the Träumerei study the following method was used to preprocess the timing data. Of all the note onsets, the onsets of grace notes are removed because their timing cannot be related to a notated score duration - no tempo can be calculated. The onsets of notes that form part of a chord are removed as well, except for the one note (it is unclear whether this is the highest note or the melody note). Then the IOI’s are calculated by taking the difference of two subsequent onsets. These are related to their duration in the score, by dividing the IOI of a longer note into several equal intervals (i.e., having multiple data points for IOI’s longer than one eight-note; see Figure 2 in Repp, 1994). This basically transforms the data to a 1/tempo measurement based on a regular time grid in the score. This data is then log-transformed. This kind of data is still referred to as “IOI profile”, but “log beat duration profile” would be a better term. 

2The correlation measures become higher when using the sampling method (with a maximum increase of 0.09 and an average of 0.02 compared to non-sampled data) - which would bias the results in our study towards relational invariance. In Repp (1994; footnote 4) it is stated that the effect of duplicated data points is neglectable for the correlations, but he removes these data points for the analysis of variance. 

3Using a linear scale would raise the correlations (with a maximum increase of 0.1 and an average of 0.03) compared to (sampled) log data - biasing results towards relational invariance. Repp initially did use linear IOI’s in calculating correlations. Comparing the correlations based on raw IOI’s from his draft manuscript with the log-transformed IOI’s in the final version, a similar effect can be observed.

4Using average performances would raise all correlations systematically (with a maximum of 0.12 and an avarage of around 0.08) compared to (sampled, linear) individual performance data - again biasing results towards relational invariance. 

5The determination of the global tempo is an acknowledged problem in itself (see, e.g., Repp, submitted). 

6The short size of this time interval indicates that the time resolution of 5 ms used in the Träumerei study is too coarse to be able to measure and determine possible relational invariance here.