Using Linear Mixed-Effects Models to Analyze Historical Trends in Performance Strategies

In this commentary, I would like to expand on Majid Motavasseli’s article “Interpretation of Cyclic Form in Bach’s ‘Goldberg Variations’ through Performance History” by applying Motavasseli’s data to further address issues related to the performance history of the “Goldberg Variations.” More specifically, I will use the tempo measurement database built in the context of the PETAL research project[1] to explore the historical changes in performance tempo in seventy-six selected recordings (from sixty-two different performers) of the “Goldberg Variations” spanning a period from 1928 to 2020.[2] The database includes the name of the performer and year of the recording, a “main tempo” for each of the thirty-two pieces comprising the set of variations (with two measurements for the two different sections of Variation 16, yielding a total of thirty-three tempo values per recording),[3] the instrument employed for the recording (piano or harpsichord), the mode (major/minor), and finally the type of variation (character, virtuoso, or canonic).[4]

Unlike Motavasseli, I will not discuss the tempo relations between pieces in the context of their position within the cycle of variations. Rather, I aim to uncover broader trends over time in the interpretative choices made by performers recording the “Goldberg” cycle. Here, I will limit myself to the analysis of tempo choices, which are the main focus of the PETAL database. More specifically, I will explore the interconnections between tempo and choice of instrument, type of variation, as well as mode.

Another aim of the present article is to familiarize the music-theoretical community with more advanced statistical methods that may help uncover statistical trends that remain difficult to detect using only basic tools such as means, standard deviations, and correlations. In particular, the current article will make use of linear mixed-effects models (LMM), a powerful class of statistical models that are increasingly employed in other fields such as psychology or ecology.[5] LMMs allow for the modeling of both fixed effects (typically, variables of interest that are selected by the researcher to include all experimental conditions [or parameter levels] in the study) and random effects (typically, a classification parameter used to group together measurements obtained on the same individual [or cluster of individuals] which are randomly sampled from a larger population). In the example at hand, year of recording, instrument employed for the recording, mode, and type of variation are treated as fixed effects, whereas performer and piece are identified as random effects.[6] As this example suggests, LMMs are especially interesting for performance analyses aiming to disentangle statistical effects corresponding to performers’ individual styles from piece-specific effects, something which is difficult to do using more traditional modeling approaches such as analysis of variance (ANOVA).[7]

All the analyses presented here are based on the PETAL database as described above. LMM models were built using the lmer function from package lme4[8] in the R programming environment (version 3.6.1),[9] and the package sjPlot was used to create the graph shown in Figure 1.[10] As indicated above, the goal was to model the main tempo (the dependent variable) using both fixed-effects and random-effects predictors. The initial model included the following predictors: instrument, year of recording, mode of the variation, and type of the variation as fixed effects, as well as all possible interactions between these parameters, and performer and variation number (using a number to identify each individual variation) as random effects. The step function of the lmerTest package was used to simplify this initial model by automatically selecting only parameters (including higher-level interactions) that significantly improve the fit of the model to the data.[11]

After applying the step function, a final model was obtained, which included two significant higher-level interactions, as determined with a likelihood ratio test: a two-way interaction between instrument and mode (χ²(1) = 5.5985, p < 0.05), and a three-way interaction between instrument, type, and year of recording (χ²(3) = 15.666, p < 0.01). The two-way interaction is relatively easy to interpret: variations in the minor mode recorded on the piano are noticeably slower than when recorded on the harpsichord,[12] whereas the mean tempo is practically the same for major-mode variations whether they are recorded on the harpsichord or on the piano. This is probably because, while minor-mode variations are played at a slower mean tempo regardless of the instrument, choosing an extremely slow tempo on the harpsichord may lead to a “disjointed” impression given the quick decay associated with this instrument, a particularity which is not as pronounced on the piano.

The three-way interaction between instrument, type, and year of recording is slightly more complex to interpret but can still be visualized fairly easily (Fig. 1). Essentially, the mean tempo varies according to the year of the recording, but not in the same manner according to the instrument and the variation type. More specifically, Figure 1 shows that, for both instruments, virtuoso variations (in purple) are played at a slower tempo in more recent recordings. The same trend is visible for the Aria (in red), although much more pronounced for piano recordings than for harpsichord recordings (the Aria is generally played at a slow tempo and, as mentioned above, extremely slow tempi may not be appropriate for the harpsichord). For the canonic variations (in blue), the tempo also slows down in more recent recordings, but only very slightly (the trend is similar for both piano and harpsichord recordings). Finally, and interestingly, characteristic variations (in green) tend to be played faster in more recent harpsichord recordings, while they are played slower on the piano recordings. To summarize, there is a general tendency to play the “Goldberg” cycle slower in more recent recordings, but this tendency is much more pronounced in piano recordings than in harpsichord ones, and also more pronounced for the virtuoso variations and for the Aria than for the other variation types (the exception being the characteristic variations which are actually played faster in recent harpsichord recordings).

It is worth noting that the PETAL dataset includes performances with some extreme or otherwise atypical tempo values, such as a few exceptionally fast piano-roll recordings by Rudolf Serkin (1928). To evaluate the potential impact of these outliers, I used the influence.ME package in R, which identified seventy-two influential recordings out of 2504 with a Cook’s distance larger than 4/n (with n being the total number of observations, 2504 in this case).[13] I then repeated the analysis on a subset of the PETAL dataset excluding these seventy-two “extreme” recordings (most of them from Glenn Gould, Ralph Kirkpatrick, Wanda Landowska, Rudolf Serkin, and Yūji Takahashi). I obtained the same LMM model as described above, with only minor differences in the p-values and coefficient estimates. This suggests that the statistical trends reported here are not driven by a few atypical recordings but are indeed generally representative of the entire PETAL dataset for the “Goldberg” cycle.

Figure 1: Visualization of the three-way interaction between instrument, year of recording, and variation type. Aria: opening and closing Arias; Chara: characteristic variations;
Canon: canonic variations; Virtu: virtuoso variations; lighter bands represent the 95% confidence intervals for each estimate.

It would be beyond the scope of this commentary to speculate on the reasons and motives behind these changes in performance strategies over time, as well as on the complex interplay between tempo, choice of instrument, and variation type (with an additional interaction between instrument and mode). Nevertheless, the observation that these statistical trends exist might, in itself, be a starting point for further musicological and performance-based analysis along the lines of Motavasseli’s article. On a related note, it should be pointed out that the analysis presented here was not based on a dataset that was selected for the express purpose of confirming a preconceived hypothesis. Rather, the findings reported here are entirely the products of an objective, principle-based statistical approach, and emerged serendipitously, as it were, from an analysis of the PETAL database.

As mentioned at the outset of this commentary, the analysis reported here does not address temporal relationships between individual variations, at least not beyond general trends related to variation type and mode. However, although such an analysis falls outside the purview of this essay, it is also possible to apply more sophisticated statistical models that take into account the order of the variations within the cycle as well as other possible interconnections within the various pieces. For instance, autoregressive models, notably time-series analyses, have been fruitfully applied to analyze tempo variations within a single piece, taking into account the order in which the notes are played, as well as the relationship between tempo and other factors.[14] While these models are somewhat more complex than LMM models, they would allow a statistically principled analysis of the temporal relationships explored in Motavasseli’s article. It is to be hoped that the brief outline presented here will inspire researchers in the field of performance-based analysis to further familiarize themselves with these statistical approaches that can help reveal rich interconnections between various factors that would otherwise remain concealed.