Global Measures of Tempo Fluctuation

Popular discussions of recorded piano playing (e.g. Dubal, 2004, p. 10) usually argue that performances from the first half of the 20^th century are rhythmically freer than those from the second half. Peres da Costa (2012) makes the following statement:

For the uninitiated, [early] recordings give the general impression of exaggerated temporal waywardness. Yet many important pianists of the era evidently considered such a style to be highly expressive. In contrast, most pianists today adhere more faithfully to the dictates of notation: any modification tends to stay within close proximity of the prevailing tempo. (p. 252)

The present analysis of recorded performances of Chopin's Etude, Op. 25 no. 1, tells a different story. Global measures of tempo fluctuation show no historical trends, increasing or decreasing. The coefficient of variation—i.e. standard deviation of melody IOI divided by average melody IOI—for each performance describes both onset-to-onset changes and changes of tempo between larger sections. 6 If a performer plays a section faster or slower than the prevailing tempo, standard deviation could be high without the local push and pull of rubato. To measure timing deviations between successive melody IOIs, the absolute value of the difference between successive melody IOIs was calculated. These values were then averaged and divided by average melody IOI. Data for both global measures of timing fluctuation are shown in Figure 1, with labels for some of the extreme values. Note the wide range of values for both measures across all time periods and the lack of an overall trend in the data.

Slower performances display a higher absolute quantity, but not a higher relative quantity, of tempo fluctuation. As median tempo increases, we can observe decreases in standard deviation of melody IOIs, r (125) =-.37, p < .001), and onset-to-onset change of melody IOIs, r (125) = -.38, p < .001. Correlations of median tempo with the coefficient of variation and with mean rate of change between MIOI divided by mean MIOI are not significant, p > .05. The role of tempo in the broad changes in expressive timing strategies discussed below is much smaller than the role of the performer's year of birth.

Phrase-Final Lengthening

One of the foundational elements of generative approaches to expressive performance is the concept of phrase-final lengthening—a musical parallel to the widely-observed tendency in human speech of slowing down at the end of a phrase. Todd (1989) states that the degree of slowing is a function of the importance of the boundary. In his description of iconic signification (following Peirce), Clarke (1995) makes the following claim: "A large-scale sectional break will typically be approached with a greater degree of slowing than a small-scale group boundary, an effect which can be heard particularly in performances of nineteenth-century piano music (such as Chopin)." (p. 27) On the large scale, pianists' timing shapes in recorded performances of Chopin's Etude, Op. 25 no. 1, are fairly consistent. Each line in Figure 2 shows the average tempo at each beat of performances grouped by performer's year of birth in 20-year increments; error bars represent the standard error of the mean of each group of performers at each point in time. These average timing profiles display a series of arches that usually correspond to the pianists' interpretations of grouping structure.

A brief analytical discussion of Chopin's Etude, Op. 25 no. 1, is necessary to compare the pianists' performed timing interpretations with the composed phrases. The piece begins with two eight-measure phrases ending in cadences. The section starting at m. 17 is more complicated. Melodically, the grouping structure of mm. 17-24 is 2+2+4, and the composer provides a ritenuto with dashes throughout m. 24. The bass line (^3-^4-^5-^1) and harmony in mm. 21-22 suggests a five-measure unit from mm. 17-22 that completes a harmonic sequence of ascending thirds. Modal mixture intensifies the sense of arrival at m. 22. M. 25 functions as an anacrusis to the sub-phrase beginning in m. 26. This four-measure unit ends with a deceptive resolution, then is repeated with a "concluding expansion" (Burkhart, 1997) in mm. 32-34 before the final cadence in m. 36.

The timing profiles in Figure 2 conform to the composed phrase lengths in mm. 1-16. In the section from mm. 17-36, the average obscures multiple distinct timing interpretations—some privileging certain harmonic events, others focusing on melodic repetition. 7 The ambiguity in grouping structure in this passage leads to a wide variety of possibilities for phrasing demonstrated in performance. A steep phrase-final lengthening occurs next at m. 36. The timing graphs for the post-cadential extension at mm. 36-44 resemble a phrase; note that the dip at 39.3-4 coincides with the high melody note rather than the start of a four-measure group in m. 40.

Phrase-final lengthening is the difference between the prevailing tempo and final tempo of a phrase. Larger ritardandi, which sound like deviations from the prevailing tempo, will have less effect on median than average tempo, so the median is a better measure of prevailing tempo. For measuring final tempo, the first two phrases and the climactic phrase ending in m. 36 are unproblematic. In the middle section, most pianists reach a low ebb of tempo in m. 24 or on the downbeat of m. 25. That low point marks the measurement of final tempo for the unit at mm. 17-25. Mm. 36-44 represent a post-cadential tonic expansion. Because of the 2+2+4 melodic structure and the texture change in m. 44, pianists tend to treat this passage as if it were a phrase by lengthening its ending. To compare performances meaningfully, the difference between final tempo and prevailing tempo must be expressed as a proportion. Final lengthening proportion in the following discussion is calculated as the difference between final and prevailing tempo divided by prevailing tempo.

Most performances of segments of the etude show consistent amounts of final lengthening that do not change much over time. Table 1 shows the mean final lengthening proportion for passages where lengthening is not significantly correlated with performer's year of birth. 8

Table 1. Average final lengthening proportion—difference between final tempo and average tempo, divided by average tempo—for segments of the etude.

	Mm. 1-8	Mm. 17-25	Mm. 25-36	Mm. 36-44
Mean Phrase-Final Lengthening Percentage	.35	.28	.71	.30

Figure 3 shows a significant trend over time whereby the final lengthening of the phrase from mm. 9-16, R² = .16, F(1,125) = 27.23, p < .001, as well as a wider variety of timing decisions among earlier pianists. Final lengthening in m. 16 increases, year-by-year (the later the pianist's year of birth, the greater the proportional slowing), by .3% (β = .42, p < .001). This passage—and the section from mm. 36-44 where increases in final lengthening over time barely missed statistical significance—features a harmony that continues into the next group. Instead of a change of chord at the sectional boundary, the C major chord in m. 16 becomes a C 6/5 in m. 17 and the A-flat major chord from m. 43 continues into m. 44, although the melody is replaced by decorative arpeggios. The early performance practice of continuing the same tempo or accelerating over these boundaries has disappeared from more recent performances.

In contrast to the more consistent practice of final lengthening in modern performances, early-recorded pianists use a wide variety of strategies in the transition from m. 16 to m. 17. 9 Some change almost always occurs on the downbeat of m. 17 that clarifies the beginning of a new phrase. Among the performers who choose not to slow down in m. 16, Backhaus (1909, b.1884) makes a dynamic accent and plays in a faster tempo. Planté (1928, b.1839) makes a dramatic change of color by suddenly playing softly and clearing the sustained, pedaled sound abruptly. Koczalski (1938, b.1884) uses dynamic accents, slight desynchronization of the hands and clever pedaling—engaging the right pedal slightly late—to create short articulations in the bass voice at m. 17. The three recordings of this passage can be heard in Audio Example 1. 10 Notably, the two most extreme ritardandi in m. 16 can be found in some of the oldest recordings, for example those of Scharrer (1912, b.1888) and Levitski (1923, b.1898). A few early recordings eschew both approaches; they lack the crafty execution of Backhaus, Planté or Koczalski as well as phrase-final lengthening in m. 16. Nevertheless, the assumption that the communication of grouping structure depends on final lengthening is clearly incorrect in this instance.

The documentary evidence for phrase-final lengthening is mixed. Brown (1999) quotes Crelle's treatise of 1823: "as a rule, the beginning of a musical unit commences powerfully and importantly, the middle carries on in a measured and regular manner and the end increases in speed and decreases in power." (p. 385-386) A decade later, Kalkbrenner advises that "all terminations of cantabile phrases should be retarded." (qtd., in Brown 1999, p. 386) Performance theorists in the end of the 19^th century such as Christiani (1885) and Riemann (1888/1892) were intensely interested in grouping structure. 11 Recordings of Chopin's Etude, Op. 25 no. 1, suggest that early-recorded pianists shared that interest, although the means of expressing their interpretation was dependent on context and highly individualized. Slåttebrekk and Harrison (2010) observed something similar in their project recreating the recordings Grieg made in Paris in 1903: "Grieg seems constantly to avoid any obvious combination of strategies at phrase endings or transitions. A combination of, for example, relaxation, diminution, and retardation with a clear-cut, prepared turning point of the phrase is virtually non-existent in his performances." In his analysis of Schenker's performance commentary on Schubert's Impromptu, Op. 90 no. 3, Cook (2013) writes "although Schenker refers to 'the usual ritardando' it appears that he means not that this cadence [m. 109] should be treated as an exception, but that 'the usual' way of playing is wrong." (p. 72) He then quotes Schenker from 1911: "the requirement that a composition's form not be exposed too nakedly frequently demands considerably quicker playing where the seam occurs." For pianists born before about 1930 it was possible to play the phrase in Chopin's Etude, Op. 25 no. 1, that ends at m. 16, and the section break at m. 44, with little or no ritardando, indeed even with accelerando; more recent performers play as though the ritardando were mandatory.

Tempo Arching

Tempo arching—a slow beginning, followed by accelerando and final lengthening—has been part of the performance practice of the etude since the earliest recordings. Later in the 20^th century it became a primary means of delineating grouping structure. Studies such as those by Todd (1985), Repp (1997) and Windsor and Clarke (1997) model tempo arches, viewing the expressive execution of the music as historically static. Close examination of recordings of the etude suggests differences between historically conditioned norms and more absolute rules, at least as regards the last 100 years or so, and possibly earlier. 12

The following discussion is indebted to Cook's (2013, pp. 176-223) important account of the rise of phrase arching in the post-war period. Cook compared performances to an integrated model of hierarchically-nested tempo and dynamic arches. This model is based on Todd's (1992) theory of expression. Instead of reducing a performance to a single measurement (even a highly descriptive one, as in Cook's study), the present study describes the timing only of specific passages and how pianists' readings of them have changed over time. Note that this analysis does not attempt to find a curve of best fit for each segment of each performance, but rather to determine the presence or absence of an arch in a particular segment of music.

First, sections where the pianists employed tempo arching were identified, limiting the passages investigated to those starting and ending on the first beat of a measure and respecting the most obvious elements of Chopin's grouping structure. Pearson product-moment coefficients (r-values) were then calculated for the correlation between tempo and a quadratic function with negative sign, having x-intercepts at the origin and the value corresponding to the length of the musical segment. A quadratic function has been used in previous research (Todd, 1985; Cook, 2013) because it describes physical phenomena such as the path of a thrown object. In this case, r represents the smoothness or predictability of a certain tempo arch in a given performance; the "height" of the pianists' tempo arch has no impact on the correlation. Pianists' tempo shapes were compared with prototype arches rather than fitting the curve to each particular performance for several reasons. 13 The sign of the curve should be negative because tempo arches do not start fast and climax slowly. To prevent a regression that fits only one side of the arch, the curve must contain an inflection point. To avoid confusing a six-measure segment of an eight-measure arch with a true eight-measure arch, for example, the inflection point should come at the middle of the musical segment. Therefore, this analysis does not describe how closely quadratic arches model actual performances—interpreted aurally, perhaps, as overall smoothness—but whether a symmetrical arch that starts and ends slowly is present in a given segment of a performance.

The threshold for counting a segment as an arch was set at r = .60. Foundational research on the validity of this measure in the broader population has yet to be conducted. Video Example 1 shows tempo data plotted against the prototype arch for an eight-measure segment (mm. 1-8) performed by three pianists: Wild (1947, b.1915) at r = .54, Ohlssohn (1998, b.1948) at r = .63, and Rubinstein (1964, b.1887) at r = .70. The same trends shown on Figures 4-7 for r > .60 can be observed in the data using r = .70 as a threshold, indicating that the number of two- and four-measure arches had doubled in performances by pianists born in 2000 compared with performances by pianists born in 1880.

Figures 4-7 show the number of segments of a given length for which correlation with the prototype arch produces r > .60, plotted against the pianist's year of birth. Residuals from the regression analyses of numbers of arches are significantly non-normal. This is most likely caused by a threshold measurement (r > .60) to count the presence of an arch; a segment where r = .59 will not be counted while a segment of r = .60 will, although they are very similar. The statistical models are nevertheless robust; large sample size and low p values mitigate this violation of the assumptions of regression analysis. Figure 4 shows that the most dramatic increase of tempo arching over time is in groupings of two-measure units, R² = .17, F(1,125) = 25.46, p < .001. Four-measure units also increase significantly over time (see Figure 5), but the effect size is smaller, R² = .05, F(1,125) = 7.20, p < .01. Figure 6, for eight-measure arches, has no trend line because the change over time is not significant. Irregular groupings—three-, five-, six- and seven-measure segments—are charted together in Figure 7 because each type is uncommon; together they show a significant increase over time, R² = .06, F(1,125) = 7.21, p < .01. Clarke (1985) suggests that "at slower tempi, the upper limit of the perceptual present puts a constraint on overall group size and requires a performer to subdivide large groups and to establish the boundaries of smaller groups more convincingly" (p. 217); see also Clarke (1982). Both year of birth and median tempo predict the number of two-measure arches in a performance, Adj. R² = .20, F(2,124) = 15.16, p < .001, but performer's year of birth (β = .37, p < .001) is a much stronger predictor than tempo (β = -.17, p = .04). 14 Median tempo is not correlated with the number of any other length arches. A negative correlation between median tempo and number of eight-measure arches was found to be marginally significant: r(125) = -.17, p = .051.

Comparing tempo in discrete segments of music with arches shows that the way pianists realize certain compositional elements in the score has changed over time. Smoother arches, described by higher r-values, are characterized by predictable speeding-up at the beginning, slowing at the end, and a lack of agogic accentuation in the middle of the arch. They often give a strong sense of the pianist's interpretation of grouping structure. The tempo shape of almost every segment of the etude is becoming, on average, more arch-like over time. Dramatic increases in arching can be observed in the two-measure arches starting in mm. 3, 11, 26, 28 and 30, perhaps responding to two-measure melodic units and their repetitions. Some of the most significant increases seem to be driven by slowing in cadential progressions: the five-measure arch from mm. 17-22, the four-measure arch from mm. 32-36, and the two-measure arch at mm. 15-16. Pianists increasingly bind m. 21 to m. 20; Figure 8 shows the Pearson correlation (r) between pianists' tempo in a five-measure arch beginning in m. 17 with their year of birth.

Note the wider variety of values among earlier performers. There is a significant trend such that smooth tempo arches can be found in more recent recordings of this passage, R² = .21, F(1,125) = 32.68, p < .001. Performer's year of birth is also correlated with a three-measure arch beginning in m. 19, r (125) = .37, p < .001. The stronger correlations between pianists' timing shapes and quadratic arches in more recent performances reflect the growing tendency to mark the cadence in m. 16 and the V7-I (with modal mixture) in m. 22 with slowing, as well as a generally smoother, less agogically-marked style of playing.

Tempo arching was an interpretive possibility throughout the recorded history of Chopin's Etude, Op. 25 no. 1. Nearly all the tempo arches that occur in the later 20^th century can be found in the early recordings as well, although two-measure units—especially in mm. 1-16—are extremely rare. The Polish-American pianist and Busoni pupil Michael Zadora (1922, b.1882) seems to use tempo arching as a guiding principle of his interpretation. Zadora plays six four-measure arches (r > .6), a surprising number of arches for a musician born in 1882 (see Audio Example 2). Where he might have learned it is unclear. The recording made by his contemporary and fellow Busoni pupil Egon Petri (1951, b.1881) shows much less arching. Zadora and Paderewski (1912, b.1860) both play a five-measure arch from mm. 17-22, which, according to Figure 7, is a modern idea. The three-measure arch from 19-22 can be found in the recording by Broman (1919, b.1887). The six-measure arch from mm. 30-36, always a popular practice, has among its earliest exponents Godowsky (1924, b.1870), Ganz (1919, b.1877) and Cortot (1925, b.1877). A seven-measure arch from 25-32 appears first in Petri's recording (1951, b.1881). Eight-measure arching (mm. 1-8, 9-16, 17-24) goes all the way back to Paderewski's performance. In his study of phrase arching in Chopin's Mazurka, Op. 63 no. 3, Cook (2013) writes that "the development of a new approach to phrase arching did not extinguish previous performance options." (p. 206) The number of possible interpretations expanded "during the period up to the 1950s." In the analysis of the present etude, the data for individual phrasing decisions sometimes show a shift either from an eight-measure arch towards two-measure arches in mm. 1-8, or from an eight-measure arch towards a 2+2+4 arching structure in mm.36-44. In other passages the data show the field of interpretive possibilities narrowing, as a smooth four-measure arch in mm. 32-36 or a five-measure arch in mm. 17-22 becomes more common.

Cook (2009) describes a dichotomy between "rhetorical" and "structuralist" performance styles, and tempo arching data for Chopin's Etude, Op. 25 no. 1, do suggest that structuralist performances are increasingly ascendant. However, as shown above in the discussion of early practices of phrase-final lengthening in mm. 16-17, the quantitative data need a rich context to be understood. Backhaus, Planté and Koczalski clearly mark the phrase boundary in m. 17, but not by using tempo. Figure 9 shows arching according to performer's year of birth in another such passage: mm. 32-36; pianists' tempo curves in this section have become significantly more arch-shaped over time, R² = .09, F(1,125) = 11.78, p < .01.

With outliers excluded (un-arched performances by Godowsky and Backhaus) this trend is still significant, R² = .05, F(1,122) = 6.11, p = .02. Arching has become more consistent over time, and tempo shapes showing very low correlation with an arch (either flat or marked agogically) have disappeared from recent performances. Irrespective of historical period, pianists' tempo curves in this section—from the start of a phrase expansion leading to final confirmation of the tonic key—are among the most consistently arch-shaped segments of the etude. Phrase-final lengthening in m. 36 is unsurprisingly common to almost all performances, accounting for a significant portion of the correlation. Multiple late-19^th-century sources prescribe a tempo arch for such an arch-shaped melody. 15 However the earliest performers are the least likely to heed this advice. Koczalski (1938, b.1884), Planté (1928, b.1839), Backhaus (1909, 1928, 1951, b.1884) and Gillé (1950, b.1884) all clearly mark the start of the expansion with a faster tempo, then either play in a sharply rhetorical way with less predictable rubato or use a four-measure ritardando. Audio Example 3 consists of recordings of performances by Planté, Backhaus (1928) and Gillé. The faster tempo at m. 32 is a common idea; 83 of the 127 performances increase the tempo by at least 5%. Some early-recorded pianists, such as Broman (1919, b.1887), Novaes (1958, b.1895), Petri (1951, b.1881) and Scharrer (1912, b.1888), create very smooth tempo arches in mm. 32-36. Audio Example 4 contains mm. 30-36 played by Petri (r = .94 for arching in mm. 32-36), Broman (r = .83), and Novaes (r = .85). The pianists who create a tempo arch and those who do not both seem equally aware of m. 32's status as a deviation from the expected grouping structure. Some pianists born after 1980 have shied away from the smooth arch. This may represent a backlash against the style of the previous generation.

A pianist's use of a tempo arch does not necessarily represent an interpretation of grouping structure. In the examples of mm. 32-36 provided above, early-recorded pianists often communicate grouping ideas without using tempo arches. Likewise, a clear tempo arch does not necessarily indicate an interpretation of phrase structure. Scharrer (1912, b.1888), for example, creates an arch in mm. 2-3 (Audio Example 5) by accenting the dissonances agogically and accelerating through the change of harmony in m. 3.1. In general, increasing numbers of short arches crowd out this kind of rhetorical use of rubato.

Performers' increasing tendency to use tempo arching for smaller groups of measures creates a sense of rhythmically smooth predictability as to where tempo fluctuation will occur; acceleration becomes gradually more likely to be followed shortly by deceleration. This is part of the story about how, while the total amount of tempo fluctuation has not changed, its location and the priorities that undergird its use have. Rubato has become more periodic, as the simple sweep of directed motion allows us to visualize each phrase as though it were the progress of an ocean wave, or feel it as though we were breathing slowly. 16 Cook (2007) describes how Paderewski's performance of a Chopin Mazurka comes from the salon tradition, whereas modern performances (he gives the example of Fou Ts'ong) are mediated by the long distances and rich acoustics of the concert hall. An architectural or diachronic projection of musical time thus replaced the element of surprise, the synchronic or immediate act of listening. 17

Metrical Patterning

Tempo arching and phrase-final lengthening show how performance practice has changed on the large scale. A more granular approach reveals even more about where pianists employ rubato and how their tendencies to employ it in different kinds of locations has shifted. Metrical patterning in each recording was measured by averaging melody IOIs for each beat of the measure in mm. 1-48 and then dividing them by the average length of the measure. To negate the impact of final lengthening and large-scale tempo arching on metrical patterning, average lengths of beats were calculated using measures that do not include the beginnings and ends of phrase: mm. 1, 8, 9, 16, 17, 35, 36 and 29-49 were thus excluded. Figure 10 shows average melody IOI for each beat of the measure, expressed as a proportion of the average measure's length, plotted against the year of performance.

The broad hierarchy is clear, although individual performances show many exceptions. The fourth beat is longest, followed by first beats, third beats and second beats. 18 Trend lines illustrate the growing tendency to lengthen fourth beats at the expense of second and third beats; statistics for these trend lines are shown in Table 2. The divergence of beat proportions shows that lengthening is happening more and more consistently on beats one and four in later performances. 19

Table 2. Linear regression statistics: Year of birth with proportional lengths of beats two, three and four

Linear regression of Year of Birth with Duration of:	R2	F(1,125)	p	B	β
2nd Beats	.14	20.79	<.001	-.000086	-.38
3rd Beats	.21	33.26	<.001	-.000095	-.46
4th Beats	.32	57.53	<.001	-.000188	.56

For pianists born before the turn of the 20^th century, fourth beats were almost never the longest of the four on average; for pianists born since 1960, they always are. At the measure level, these graphs show a significant change in the kind of rubato pianists employ. Slower tempi in the later recordings are unlikely to be responsible for the increasing length of fourth beats. Controlling for year of birth, median tempo is not correlated with beat proportions (p > .05 in all cases).

The different timing strategies of Ganz (1919, b.1877) and Bonnecaze (1999, b.1965) typify the broad changes illustrated by diverging metrical proportions. The two pianists play with almost identical amounts of beat-to-beat tempo change, although Ganz plays a bit quicker overall. Ganz's average measure is more even than Bonnecaze because his choices of beats to lengthen are more varied. Video Example 2 shows melody IOIs for the first sixteen measures of each performance. In twelve of the sixteen measures, Bonnecaze's fourth beat is longest. Eleven of these have the same basic shape—long first beats, shorter second and third beats, longest fourth beats. Ganz also lengthens fourth beats, but usually as a prelude to an even longer first beat. Ganz favors first beats for his agogic climaxes; they are longest in nine of Ganz's measures. More importantly, Ganz's measures do not have a repeating basic shape, making his agogic nuances less predictable. 20 Lengthening dissonances on first beats gives the impression of clinging to the expressive moment as it flees, while lengthening a fourth beat before a dissonance serves as preparation or warning.

Agogic Accents

Empirical analyses of rubato have tended not to examine the role of changes of harmony and dissonance at the surface level. Palmer's (1996) tension-relaxation model combines surface dissonance and harmony distance, based on the circle of fifths, into a single measure. Her discussion of the correlation between this model and the actual performance being analyzed does not include granular detail about the kinds of events that motivate expressive timing strategies. Todd (1995) starts from a model that ignores the role of dissonance and harmony. Benadon and Zanette (2015), Cook (1987) and Dodson (2011a) focus on interactions between expressive timing and harmonic structure. Milsom and Peres da Costa (2014) call for closer examination of "special moments of deliberate and enhanced agogic accentuation in modeling expressive performance," (p. 83) noting that the rhetorical style espoused by 19^th-century theorists contradicts the phrase-arch model of performance. Elements of both styles can co-exist in the same performance. The low points of tempo arches, particularly in shorter units, often create agogic accents, and pianists do not apply arching equally across the entire performance.

Analysis of recordings of Chopin's Etude, Op. 25 no. 1, shows that strategies of agogic accentuation have changed significantly since the early 20^th century. Early 20^th-century performers tended to use agogic lengthening for accented dissonances, changes of harmony and melodic inflections. Modern performers increasingly delay the arrival of those events, such that lengthening the melody IOI of an important event is no longer the default option for the most recent generation.

Friberg and Sundberg's (1995) research, the findings of which are consistent with those of previous experiments, suggests that the minimum threshold for perceiving timing deviation is about 5% of the tempo-defining IOI. 21 Ignoring the sextuplets' role in defining rhythm, I therefore used this measure as a conservative estimate of the difference between the intentional and unintentional timing deviations made by the highly skilled performers whose recordings were analysed in the present study. Therefore, a melody IOI was considered an agogic accent if one of the following conditions is true: it is longer than both the preceding and following IOIs by 5% of the prevailing tempo; it is longer than the preceding IOI by 10% of the prevailing tempo and longer than the following IOI; it is longer than the following IOI by 10% of the prevailing tempo and longer than the preceding IOI. Prevailing tempo is defined as the average of the nearest five melody IOIs (that is, the accent IOI, the two preceding and the two following IOIs). In absolute terms, the threshold ranges from about 25 milliseconds on both sides of an accent IOI or 50 ms on one side in the fastest recording (Koczalski [1938, b.1884], median tempo=120) to 42/84 ms in the slowest (Fialkowska [1997, b.1951], median tempo=71). For this analysis all the measures with a melody were included (mm. 1-44), while the coda's harmonically static arpeggios were excluded. Video Example 3 shows how this agogic definition works in practice. The first eight measures of two performances—one fast, one slow—are marked with melody IOIs and the ">" sign for agogic lengthening.

There is no correlation between the total number of agogic accents in a performance and the year of performance or performer's year of birth. However, the location of those accents within the measure has not remained constant over time. Figures 11 and 12 show a strong and significant trend over time whereby the proportion of agogic accents that fall on the third beat to total agogic accents in a performance declines, R² = .25, F(1,125) = 41.51, p < .001, and the proportion of agogic accents that fall on the fourth beat increases, R² = .32, F(1,125) = 57.93, p < .001. The proportion of agogic accents on fourth beats to total agogic accents is more strongly a function of year of birth (β = .53, p < .001) than of median tempo (β = -.17, p = .02), Adj. R² = .34, F(2,124) = 32.70, p < .001. The proportion of first beat agogic accents to total agogic accents also decreases by year of birth, but at much slower rate, R² = .06, F(1,125) = 8.22, p < .01. The proportion of agogic accents on the second beat of the measure has remained constant over time.

The main motivators of agogic accents are changes of harmony, metrically-accented dissonances in the melody, and changes of melodic direction from ascending to descending motion (that is, peaks). The taxonomy presented by Parncutt (2003), implemented in a computer model by Bisesi et al. (2011), also contains grouping accents and metrical accents. 22 In the present analysis, the role of grouping accents is captured mainly by tempo arching. Metrical accents universally coincide with changes of harmony, dissonances and melodic inflections. With changes of harmony, dissonance or chromatic tones introduced over the same bass note are included if there was no dissonance or chromaticism on the previous quarter note. For example, m. 4.3 is a change of harmony, but m. 6.2 is not. Thus in the present system dissonances in the harmony are described as changes of harmony, while metrically-accented dissonances in the melody are given their own category. This reflects the dominant role of melody in expressive timing. The term "delaying accents" will be used to describe agogic accents that occur one beat before changes of harmony, metrically-accented melodic dissonances and changes of melodic direction. The proportion of delaying accents to total agogic accents has increased significantly over time, R² = .22, F(1,125) = 35.76, p < .001. The later the year of the pianist's birth, the larger the proportion of delaying accents in their performance, by an additive increase of .2% each year, β = .47, p < .001. 23

Because so many beats in Chopin's Etude, Op. 25 no. 1, feature one or more of the events listed above, the data are broken down into various types. Figures 13 and 14 show the difference between agogic accents that occur on the melody IOI of the accented dissonance or change of harmony and those that occur on the previous melody IOI; negative values represent a preponderance of delaying accents. 24 The unstandardized coefficients (B) in Table 3 show that the proportion of delaying accents has increased more quickly for changes of harmony than for accented dissonances or melodic inflections. Pianists who favor delaying accents tend to do so for all three types of events. Percentage of delaying accents on changes of harmony is correlated with percentage of delaying accents on melodic inflections (r = .57, p < .001) and accented dissonances (r = .68, p < .001); percentage of delaying accents on melodic inflections is correlated even more strongly with percentage of delaying accents on accented dissonances (r = .80, p < .001).

Table 3: Linear regression statistics: Year of birth with difference between on-beat and delaying accents for accented dissonances, changes of harmony and melodic inflections.

Linear regression of Year of Birth with Difference Between On-Beat and Delaying Accents for:	R2	F(1,125)	p	B	β
Accented Dissonances	.20	31.75	<.001	-.088	-.45
Changes of Harmony	.16	24.51	<.001	-.111	-.41
Melodic Inflections	.06	8.10	<.01	-.063	-.25

Delaying the arrival of a change of harmony was never an uncommon practice; note that many earlier pianists appear below the line indicating 0 in Figure 13, whereas very few early pianists consistently delayed accented dissonances (Figure 14).

The delayed change of harmony has recently become a strong default option. Melodic inflections show a slower trend towards delaying accents, and year of birth is a weaker predictor than for accented dissonances and changes of harmony. 25 Controlling for year of birth, there is no correlation between each type of event and median tempo, p > .05.

Accented dissonances, changes of harmony and melodic inflections very often occur simultaneously, most often on the first beat. In fact, the first beat of every measure in the etude features one or more of these events; in a piece of smooth, flowing and uniform texture, they are Chopin's primary means of delineating meter. Two hypotheses can be proposed to explain the lengthening of fourth beats over time and the greater prominence of agogic stress on the fourth beat. First, modern pianists lengthen fourth beats irrespective of the musical context. They have a characteristic way of playing a measure in 4/4 time that tends to lengthen the fourth beat more than earlier performers. Second, modern pianists tend to lengthen the melody IOI before an important musical event, as opposed to earlier pianists who tended to lengthen the IOI of the event itself. These hypotheses can be tested by examining dissonance, change of harmony and inflection on third beats. The difference between on-beat and delaying accents for third beats shows the same trend as those described in Table 3: R² = .12, F(1,125) = 16.61, p < .001, B = -.056, β = -.34. The small, although statistically significant, increase over time of lengthening a second beat to highlight an event on the following third beat is not a function of metrical patterning. Figure 10 above shows second beats decreasing in length. Increasing numbers of agogic accents on fourth beats are not merely the consequence of groove. The evidence points to a gradual shift towards greater acceptance of delaying an important event instead of lengthening after it occurs.

The almost imperceptibly slow shifts illustrated in Figures 11-14 show how Slåttebrekk and Harrison (2010) could arrive at the view that the difference between Grieg's playing style and modern interpretations is of crevasse-like proportions:

Another element, never found in Grieg's performances, is the usual procedure today of emphasising a single note or a chord by adding time immediately before them. This 'holding' or waiting in order to heighten the effect of an accentuation, intensifying the expression or adding general weight to the performance, is very alien to Grieg. Quite the contrary, Grieg's impulse to a new phrase or the placing of the climax of a phrase, is almost inevitably slightly before the projected beat, a projected beat being where we would expect the beat to appear.

Perhaps surprisingly, the documentary evidence from the late 19^th century predicts pianists' use of agogic accents in the early recordings. Riemann (1888/1892) makes clear exactly where the agogic emphasis should fall: "the usual dynamic signs [cresc. and decresc.] apply also to the 'agogics', and the chief lengthening (the agogic accent) falls on the most accented note; therefore in the limits of the bar the one directly following the bar-line." (p. 64) 26 In light of this and early recorded practice, when Lussy (1874/1896) advises performers to slow down "on expressive notes" (p. 167) he most likely means the note itself, not the beat before.

The "'holding' or waiting" described by Slåttebrekk and Harrison, above, is common in the phrasing of 21^st-century pianists, but rarely referred to explicitly in writing. Boris Berman (2000), cautions:

against a bad habit that creeps into the playing of too many performers: hesitation before the downbeat, which can become a constant feature of one's playing… At the root of this habit is probably a desire to emphasize the importance of the downbeat, but the result is terribly unnatural and often irritating. This agogic accent is often better achieved by getting to the important musical point without any hesitation and lingering on it slightly… I myself suffered from this ailment at one time and had difficulty purging my playing of it. (pp. 95-96)

One of the only descriptions of the metrical patterning and delaying accents noted in this study thus comes from a performer swimming against the tide. Slåttebrekk's and Harrison's description arises out of a recognition of stylistic difference as well.

Using linear regression to describe trends in performance practice obscures some details of how style changes. The real correlation between year of birth and tendency to use delaying accents may, in fact, be stronger than suggested above. For example, Darré (1946, b.1905) and Lympany (1995, b.1916), two older pianists who tend to lengthen second beats in preparation for an event on the following third beat, are not obvious examples of performers whose agogic accentuation delays the arrival of important events. Both play certain passages in an unusual lilting way, lengthening beats two and four, creating a 2/4 feel. Metrical patterning is difficult to distinguish from agogic accentuation in these recordings; Audio Example 6 shows an extreme passage from Darré's recording.

The best example of the style that accentuates events by lengthening the beat before is in Ashkenazy's recording (1975, b.1937). Video Example 4 shows the first eight measures with agogic accents on melody IOIs marked; note that the small peaks at 1.4 and 2.4 fall below the significance threshold used in this study. The progressively increasing size of agogic accents on the fourth beat of the measure creates a hierarchically organized phrase. The accents guide our listening to the important events on the first beat of the measure. Ashkenazy is not the only pianist to utilize this style, although consistent application of the delaying accent is unusual. Gavrilov (1985, b.1955), Zayas (1983, b.1940), Szabó (1971, b.1942) and Kempf (2003, b.1977) show the same tendency to highlight both dissonances and changes of harmony by stretching the beat before. At the same time, we can witness the development of a style that emphasizes a smooth and gentle rocking motion rather than the use of agogic accents to emphasize musical events in the score. The lower-rightmost point in Figure 14 represents Albright's performance (2010, b.1989). Video Example 5 shows the first eight measures of this recording. A regular metrical pattern persists irrespective of changes of harmony or melodic events.

At the other end of the spectrum is the recording made by Novaes (1958, b.1895), in which melodic and harmonic events are highly correlated with agogic accents. As shown in Video Example 6, Novaes stretches beats with dissonances and changes of harmony. Of the two fourth-beat agogic accents, the first emphasizes the dissonant leap in the bass (m. 6.4) rather than the downbeat that follows, and the second ends the phrase (m. 8.4). Performances in which the first beat tends to be emphasized are less likely to give an impression of metrical patterning than those of Albright (2010) and Bonnecaze (1999). Varsi (1982, b.1939) is the pianist who places the most agogic accents on the first beat (32), and Bonnecaze plays the most on the fourth (32), closely followed by Albright (30). Varsi (1982, b.1939), Mozzati (1984, b.1917) and Perahia (2001, b.1947) play the longest first beats as a proportion of the measure. Of these only Varsi provides a sense of groove: see Video Example 7. Most pianists take care not to be predictable in the measure-to-measure application of rubato.