“Attractor Tempi” in Brahms’s Symphony No. 2/III

“Attractor Tempi” in Brahms’s Symphony No. 2/III Abstract This analysis assesses tempo choice in the third movement of Brahms’s Second Symphony. It is shown that, at key moments of the movement, the average tempi used in a sample of commercial recordings align closely with those suggested by the “attractor” model of cognitively preferable (easiest) tempi (Gotham 2015). Between these stable checkpoints, the movement navigates through a variety of other metrical structures at (formally) the same tempo, each with their own attractor tempi. These necessarily align less well with the tempi used, thus providing a pattern of relative tension/relaxation, which the new model helps to elucidate. 1. INTRODUCTION TO “ATTRACTOR TEMPI” Ever since William James’s pioneering work in the nineteenth century,1 a growing body of evidence in the cognitive sciences has substantiated the intuitive notion that there is a basic, human preference for pulses in a certain tempo range: that we are better able to deal with a range of “moderate” pulse rates than with anything extremely fast or slow. Apart from the extensive music-psychological literature,2 music theorists may be most familiar with these ideas from the notion of metrical “projection,” including observations such as: “Since mensural determinacy is gradually attenuated, evidence of projection or projective potential will, as a rule, become progressively weaker as duration increases.”3 The reader may wish to test the idea of pulse preference by means of a simple experiment. Try tapping along with the second hand of a clock, look away for a few seconds, come back and see how well you kept time. Try the same experiment for an interval of two seconds (without mentally subdividing) and it is likely that you will have fared less well: two seconds is usually found to be a harder duration to project than one. The ramifications that this individual pulse preference might have for tempo preference in (multi-leveled) metrical structures have been alluded to in the literature,4 and my prior work develops a theoretical model of “attractor tempi” for meter-specific tempo preference, systematically deducing those tempi at which different meters may be most easily entrained to and parsed. This modeling is achieved by adopting the view of meter as an interaction between coinciding periodic pulse levels in simple proportions,5 and modeling tempo preference as a trade-off between the competing calls of the various pulse levels involved. Each pulse may “want” to be individually optimized for salience, but the combination of levels and groupings (the meter) is optimized by a tempo, which balances those individual pulses. For instance, where there are just two pulse levels involved, the combined attractor value will split the difference between the individual ones. This can be thought of in terms of a kind of “tempo see-saw” (level and fulcrum): adding “weight” (e.g., metrical levels) to one end of the system moves the “center of mass” (attractor tempo) further in that direction. This manifests in different ways for different metrical structures, but the principle remains: for any meter, the attractor tempo is that which “balances” the combination. Appendix 1 provides a more technical overview of the inner working of the model, including a short summary of the mathematics involved. In short, the model argues that, given any metrical structure, we may be drawn toward the tempo that optimizes salience in that context (the attractor tempo). These attractor tempi are determined by the number of levels represented and their proportional relationships (metrical schemes) in largely simple, categorical ways. To be absolutely clear from the outset, the model does not claim to generate tempi that are “correct,” and which therefore “should” be used. Rather, the attractor tempi represent a kind of notional default to which performers may find they are drawn, and from which composers and performers will very frequently deviate, using deliberately fast or slow tempi to expressive ends. The goal is to engage with a form of relative “tempo dissonance” (divergence from attractor tempi) and its use as a compositional parameter. 1.1. ATTRACTOR TEMPI IN PRACTICE This model invites scrutiny of various kinds. One might assess how successfully it accounts for metrical listening (with cognitive tests) and musical practice (with corpus studies), and there is also plenty of scope for testing the model’s main claim to have explanatory power as an analytical tool. This article tests that claim, using Brahms’s Symphony No. 2, III, as a suitably challenging case study. To do so, it necessarily puts aside questions concerning the fundamentals of the model. It is assumed, for present purposes, that the notion of attractor tempi does speak to an aspect of our musical experience and that the shape of the theoretical model is sufficiently accurate to make meaningful comments about tempo choice. The focus of the article is thus not on proving or disproving the model, though any use of the model should be well informed of its strengths and shortcomings. This section sets out the variables involved in applying the model, engaging some debatable areas; the discussion is then illustrated by a repertoire application. There are three interacting variables in play here: the attractor tempi predicted by the model, the tempi actually used (whether in an individual recording, or in a corpus average), and the metrical structure of the work or passage (including the possible usage of hypermetrical levels). The model itself is new and is based on tempo preference in general—a large, unruly issue, which abstract, context-independent models can only ever hope to approximate. It attempts a balance between accurately corresponding to the data available without “over-fitting” to any individual experiment, and without unduly complicating what can only hope to be an approximate heuristic. The tempi used in a recording may (theoretically) be objectively verified, though summarizing a work or section with a single value presupposes a meaningful “steady state,” or “main tempo,” which may be more or less evident in a given recording. Extensive discussion of average types is beyond the scope of this study; suffice to say that the approach here has been to take mean inter-onset interval (IOI) values for “steady-state” passages deliberately selected to avoid areas of expressive variation in the timing. Further discussion of method follows below, with the introduction of the new corpus data. The metrical structure may seem to be clear and not variable at all; however, the presence or absence of (particularly hyper-) metrical levels is under-determined by the notation and must be deduced by analytical judgment. Reliance on such analytical judgment might be seen as a shortcoming of this analysis. These decisions are indeed readily disputable (as with any analysis), though I have attempted to set out the analysis in such a way that readers with a different view of level usage can still trace a course through the piece in relation to attractors for the levels they consider to be present. To that effect, the two most likely metrical structures for each passage (one usually involving a single additional level to the other) are presented throughout. More limiting is a shortcoming of the attractors model itself: it deals with levels only in terms of presence or absence. There is not yet a mechanism for finer shades of relative usage in a work or performance. 1.2. ATTRACTOR TEMPI IN BACH’S C-MAJOR PRELUDE (BWV 846) A short repertoire example will help clarify how the model works and the effect that these variables have. The first prelude from Bach’s Das Wohltemperierte Klavier is, metrically, much simpler than Brahms’s Symphony No. 2, III. There are no (explicitly notated) tempo or metrical changes in the work, and the piece does not overtly call for a particularly fast or slow tempo. One might, therefore, expect the average approach to be a reasonable reflection of moderate tempo for its metrical structure. The metrical levels self-evidently in use are those of the sixteenth note (the level of the individual notes), as well as the eighth, quarter, and half notes (the last being the length of the repeated, figural pattern). This gives a metrical structure with pulses of ⟨1, 2, 4, and 8⟩ unit lengths and therefore proportions between those four pulse levels of [2, 2, 2], for which the attractor tempo model suggests a quarter note of 70.8 beats per minute (bpm).6 However, chord changes take place at the whole-note (measure) level, which would suggest a metrical structure with an extra level—pulse levels of ⟨1, 2, 4, 8, 16⟩, and proportions of [2, 2, 2, 2]—for which the attractor tempo is quarter note = 100 bpm. The model thus suggests two strategies for “moderate” tempo in this work: 70.8 bpm for metrical levels up to the half-note level (the figural pattern), versus 100 bpm for levels up to the whole-note (measure) level. This highlights a potentially surprising aspect of the model: that the addition of a single metrical level can give rise to a very different attractor tempo. The model seeks to make the levels in use as salient as possible, and adding a level has a strong effect on the best value for this. Think again of the “tempo see-saw”: if you have one pulse level, it is balanced when positioned at the fulcrum, in the center; if you have two, they balance when equidistant from that center. By adding one metrical level, the balanced (centered) position for the continuing level changes markedly. So it is with the metrical options for the Bach: ⟨1, 2, 4, 8, 16⟩ is centered by positioning the four-level (quarter note) in the center at 100 bpm, while ⟨1, 2, 4, 8⟩ keeps its two- and four-levels equidistant from that center, leading to the slower attractor tempo. For illustration, try singing, playing, or just imagining this piece to see what tempi you find “easiest” or most “natural.” First, try emphasizing half-note beats, then shift focus to whole-note beats. The model expects you to find a faster tempo easier when working with the whole-note level. Assuming that performers do (on average) share the view of the work as being in a moderate tempo, then we can look at their tempo choices to see which of the metrical strategies they adopt. Benadon and Zanette (2015) provide a corpus of forty-eight recordings of this work,7 and the first 100 onsets (sixteenth notes, equating to twenty-five quarter-note beats) provide a relatively “steady-state” section in most recordings. The average tempo for this section is 69.3 bpm. This is indeed faster than the average tempo for the whole work (67.1 bpm), and it does correspond well to other steady-state passages (such as the second measure alone, at 69.7 bpm). The performers’ average preference of 69.3 bpm corresponds closely to the attractor tempo option of 70.8 bpm, and thus to a meter with the half note as its highest level. 2. INTRODUCTION TO BRAHMS’S SYMPHONY NO. 2, III The Bach example has provided a short exposition of how the theory works, though to test the full explanatory power of such a model, one needs a more complex work with evident changes of tempo-metrical arrangement, which the model can describe in terms of relative tempo relaxation/tension (or consonance/dissonance) across the whole. The third movement of Brahms’s Symphony No. 2 provides an ideal case study: its metrical disposition is complex and varied, and—partly for that reason—the choice of tempo has been debated at length.8 Two main tempo choices are to be made by performers of this movement: one at the start, and another at the introduction of compound meters in m. 126. This analysis will argue that the choices made by conductors tend toward the attractor tempi for the metrical structures that begin these sections at the expense of the other metrical structures (with different attractor tempi) that follow. Brahms’s numerous changes of meter generate changing attractor tempi; no single tempo can correspond to all of these. This precludes alignment with attractor tempi throughout, and it ensures a process of change in this parameter that plays out as a pattern of relative tempo relaxation/tension across the whole. As such, prior to the close analytical reading itself, a brief overview of the movement’s form is in order to contextualize these changes in terms of a possible compositional strategy that may have a bearing on performers’ strategies for tempo choice. 2.1. FORM IN THIS MOVEMENT (AND IN SOME OTHER, RELEVANT PIECES) The movement appears to divide neatly into five parts, with the Allegretto grazioso (Quasi Andantino) A sections alternating with the Presto ma non assai B sections (ABABA). Example 1 outlines these broad sections and the subsections given by changes to the metrical structure (the most useful and relevant basis of subdivision for the purposes of this tempo-metrical analysis).9 Allegretto grazioso (Quasi Andantino) and Presto ma non assai are designated by ‘A’ and ‘B’ respectively. Subsections serve to identify metrical change. The unit pulse in each meter is the eighth note. Time signatures in inverted commas (“”) serve as a shorthand for the whole structure under discussion, including hypermetrical levels. In each row, two meters are set out, usually distinguished by the absence or presence of one debated hypermetrical level. That debatable hypermetrical level (and its corresponding attractor) is given in soft brackets—“()”—in each relevant column. This does not mean that the version with the hypermetrical level is less preferable (indeed, more often the opposite is true); the notation is merely used to facilitate the reading of this table. Example 2 illustrates each subsection with representative musical material (predominantly melodic). Excerpts are notated in the original time signatures (Brahms’s own); comments above identify the larger metrical units involved in the debated hypermetrical levels (referred to in Ex. 1 and throughout). EXAMPLE 1. View largeDownload slide Form and meter in the movement EXAMPLE 1. View largeDownload slide Form and meter in the movement EXAMPLE 2. View largeDownload slide Representative melodic material and metrical contexts EXAMPLE 2. View largeDownload slide Representative melodic material and metrical contexts However, the form is more equivocal than the neat Allegretto–Presto alternation would suggest. The alternation bears the hallmarks of the classical minuet (or scherzo) and trio form that would be standard fare for a symphonic third movement. Accordingly, the form has been described as a scherzo,10 as ABA′CA″,11 and as ABA′B′A″.12 However, according to that expectation, the intervening trios would ordinarily be slower, and so this movement has also been described as a “back to front scherzo and trio,” or a kind of triple trio with two scherzi (sic, rather than the more usual way around).13 Here a parallel is often drawn with two other works by Brahms that exhibit a similar A–B–A–B–A pattern of slow–fast–slow–fast–slow in their central movements: the String Quintet No. 1, Op. 88, and the Violin Sonata No. 2, Op. 100. The tempo-metrical arrangement of the Op. 88 movement especially bears an uncanny resemblance to the present one: both have A sections in 43 and B sections of which one occurrence is in simple time, and the other is in a compound meter.14 According to a model of symphonic form, this could be viewed as an elision of the second, slower movement (accounting for the anomalously slow A section) and the third (represented by the alternating form and the A section’s triple time).15 However, of these three works, it is the symphony that remains in four movements (retaining a separate slow movement), and the chamber works that have three. Moreover, even the basic, five-part formal division discussed above has been accused of underplaying the “interpenetration” of material between the sections.16 The connection between sections is such that some argue the case for variation form.17 Continuous variation is also intimately linked to that archetypally Brahmsian device, the developing variation.18 Importantly for the present purposes, analysts are keen to note the inclusion of tempo and meter as part of this technique. Walter Frisch notes that it is common to see the “mobile bar line . . . linked to both motivic development and formal articulation” in this way.19 3. TEMPO CHOICE 3.1. CONTEXTUAL, HISTORICAL, AND NOTATIONAL CUES The fact that the main variations in this movement are achieved through rhythmic-metrical transformation motivates the present tempo-metrical analysis and has also led some scholars to draw an alternative formal parallel with the Baroque suite.20 This forms part of a wider range of apparent retrospection in this movement, which Frisch describes as “particularly striking after the Adagio [second movement], one of the least historically retrospective movements in all of Brahms.”21 The formal parallel with Op. 88 may be significant in this light as Pascall has shown the A and second B sections are based on pre-existing movements explicitly identified as Sarabande and Gavotte by the composer.22 The same dance forms may well have been invoked for the symphonic movement in this study. In the A section, the secondary melodic idea introduced in m. 11 could be read as a textbook Sarabande rhythm (see the second system of Ex. 2, noting the 2–3–1 rhythmic pattern and the associated second beat emphasis).23 If the Op. 88 B´ section counts as a Gavotte in Brahms’s reckoning, then the first Presto of the symphonic movement could well be, too—both are marked presto, but can readily be heard in a moderate 2 (indeed, the quintet is notated in cut common time).24 Stylistic allusion could well have an effect on the tempo choices made by Brahms and indeed performers of this work. Unfortunately, however, “very little” can be deduced about specific tempi in Baroque music “either from notation or from contemporary evidence” and the evidence that exists is often extremely inconsistent.25 Instead, this analysis concentrates its efforts on the evidence of contemporary performers’ attitudes to tempi in this movement as expressed in their recordings; no more is said about the possibility of Baroque models.26 As for the compound-time Presto section, this would appear to diverge from Baroque models in any case, fitting rather with a class of post-Beethoven scherzi commonly used by Brahms. As Komma has it, “Brahms’s Scherzos follow the Beethovenian type: a fast, often staccato 86.” Accordingly, they operate at a “stormy, agitated” tempo—that is, a fast one.27 As mentioned above, Brahms instructs conductors to make two primary tempo choices for this movement, at mm. 1 and 126. The half-note tempo of the first Presto (m. 33) should be equal to the chosen quarter-note tempo for the opening A section.28 Accordingly, the opening notated tempo is, at least theoretically, in operation until the (approach to) the second Presto (section B′) commencing at m. 126. At that point, a second tempo selection is to be made for the compound meter passage’s beginning. That second tempo holds until the retransition to A (Tempo 1), which arrives at m. 194. Brahms initially indicated that the new dottedquarter note for this second tempo should be equal to the quarter note of the A section (and half note of the first Presto), though he later withdrew this instruction.29 Despite the withdrawal, this scheme has found some support,30 and other proportional schemes have also been suggested.31 Brahms does not provide metronome marks for this movement (nor indeed for most of his music—his “blood” did not “go well” with the “mechanical instrument”),32 though the tempo terms he uses may be instructive, as are the changes he made to them.33 Brahms’s many changes to these tempo indications consist largely of minor tinkering with the Italian terminology; however, one more substantial and systematic change appears to present itself. For each of the three tempi provided in Symphony No. 2, III, Brahms added qualifiers suggesting a slower tempo than the original designation: (Quasi Andantino) was added to the A section’s Allegretto grazioso, while ma non assai was added to the two Presto passages (via ma non troppo in the case of the first). This may assume significance in relation to attractor tempi for salience. Yet at the same time, Brahms also suggested the Andante cantabile con moto of Beethoven’s First Symphony to be a good model for the tempo of this movement.34 Unlike Brahms, Beethoven was an enthusiast for the metronome, which was invented during his mature career. Beethoven gives a metronome mark for most of his works; in the case of the Andante cantabile con moto, eighth note = 120, which is quick for any allegretto or andantino. Unfortunately, even if Brahms viewed these two movements as being related in tempo, we do not know whether he had Beethoven’s tempo or something else altogether in mind.35 3.2. TEMPO AS EXPRESSED IN PERFORMANCES OF THIS WORK Brahms’s oblique comments, therefore, join the possible models of Baroque dances in providing interesting (but ultimately conjectural) ideas for tempo selection in this work. More specific and quantifiable evidence is to be had from existing recordings of this work, and it is here that an analysis of attractor tempi is best put to use. This analysis is based on two data sources: Murphy’s corpus of recordings (see Murphy 2009), and a new data set focusing on tempo changes by section (as detailed in Appendix 2). Murphy’s corpus appears to provide a relatively good representation of public, commercial, and professional recordings of this work: the sample size is relatively large (67), and the corpus includes a relatively broad representation of recording years (1928–2005) and nationality of both conductor and orchestra.36 Further analysis could compare the year of recording with the conductor’s birth year, as it has been suggested that the generation in which the conductor came of age may be a more relevant determinant of their approach than that in which the performance is given.37 The corpus provides a useful sense of the range of tempi employed by performers of this movement, and—most usefully for the present purposes—the notion of “average tempi” for the passages in question that can be assessed in relation to the “attractor tempi” of my previous work (or indeed any other models of reference points such as the suggestions of Baroque treatises).38 This analysis focuses on the average tempi partly for practical reasons of space and simplicity, but primarily in the interests of comparing “average” practice with the attractor tempi “defaults.” I reiterate, once again, that there is no intention to prescribe the “correct” tempi for this (or any other work), but neither is there any pejorative connotation intended by the notion of an “average” performance. Indeed, quantitatively averaged forms are often assessed to be the most attractive. This has been shown directly for musical interpretations, as well as for numerous other aesthetic judgments.39 Fittingly, the movement in question appears to lend itself more readily to the notion of average tempo than many of Brahms’s works; Sherman’s data on historical changes in performance practice appear to indicate that the choice of tempo for this movement has been more consistent and less susceptible to the global fashions in tempi that he identifies for other works.40 Example 3 sets out the data for the two tempi used by the performers represented in Murphy’s corpus. The distributions of tempi are selected by performers is shown in the two histograms: the first, simple-time tempo on the left; the second, compound-time tempo on the right. The distribution for the first tempo choice centers on a mean of 91 bpm, and a median of 90 (for the quarter note in the A sections and half note in the first B section). The range is 78–104, and the standard deviation 5.54. The distribution for the second, B´ section’s tempo has a faster beat-level (dotted-quarter-note beat: mean 119, median 118, range 106–136, and standard deviation 7.02). The histogram bars represent spans of two MM values, and a normal distribution of best fit is included in both cases. EXAMPLE 3. View largeDownload slide Histograms showing the distribution of tempi used by performers in Murphy’s corpus: the first tempo selection for the A and simple time B sections (left); the second tempo selection for the compound meter Presto, section B’ (right) EXAMPLE 3. View largeDownload slide Histograms showing the distribution of tempi used by performers in Murphy’s corpus: the first tempo selection for the A and simple time B sections (left); the second tempo selection for the compound meter Presto, section B’ (right) There may only be two distinct tempi in this movement, but there are many changes of meter (cf. Exx. 1 and 2), each with its own attractor. As such, a secondary analysis is needed to assess the extent to which conductors’ tempi change from section to section (despite their being notated at a constant tempo). This requires collecting new data and an opportunity to look at recordings not included in Murphy’s study and to extend the total number addressed. Accordingly, the data set newly collected for this study makes use of six recordings chosen from outside of Murphy’s collection. Of these, three are “modern” (since 2000) and three are “historical.” The results are set out in Example 4 on which there is one data point for each recording (or attractor) and section. Diagonal lines are used to connect those points and lead the eye; they do not imply smooth transitions between these tempi (nor, for that matter, is the x-axis proportional to duration). For the sake of comparison, the tempi are given by tactus beats corresponding to the quarter note (A section), half note (B section), and dotted-quarter note (B’ sections). Thin gridlines are also included for reference: two horizontal lines for the average tempi in Murphy’s corpus, and five vertical lines for the start of each major section (A, B, A, B’, and A). Full details of the recordings are provided in Appendix 2: Examples 7 and 8 contain the tempo data and recording information respectively. EXAMPLE 4. View largeDownload slide Tempo change by section for the six newly studied recordings and the “attractor” tempi for the two most plausible metrical structures EXAMPLE 4. View largeDownload slide Tempo change by section for the six newly studied recordings and the “attractor” tempi for the two most plausible metrical structures As with Murphy’s corpus, the method has been for the author to tap along to the excerpt with basic software that converts tapping into an average tempo count. Several steps have been taken to ensure accuracy and objectivity. I outline the basic method here and provide a summary as part of the data in Example 7. To ensure consistency, three trials were conducted for each section in each recording. The representative ranges used were selected to give the largest continuous part of the section (to provide the most representative averages), while still avoiding any areas with clear rallentandi, or pauses (notated or otherwise).41 Obviously, important exclusions include the first beat of a new start as the onset cannot be reliably predicted. For objectivity, I looked away from the BPM counter during the trial (to avoid attempting to emulate prior results) and conducted a cross-check of my method both with Murphy’s (using a sample of his recordings) and with an automatic beat counter: the “tempo and beat tracker” plug-in of the Sonic Visualiser software package.42 Sections were once again set at a consistent length before I proceeded to run the software, manually check the results for “octave” and phase errors (still relatively common in automatic beat extraction), export the annotation, and take the average. In short, the new results match exactly with the semiobjective measurement of Sonic Visualiser. They also match Murphy’s results with the singular exception of Sanderling’s second (B´ section) tempo choice. The summarized data is provided in Appendix 2 (Ex. 9), with the anomalous value identified by an asterisk. The resulting tempi are discussed in the course of the text in relation to possible attractors. In the meantime, Example 4’s illustration enables a broad summary. As shown by the roughly horizontal lines across the first seven sections (an A–B–A unit), the recordings do broadly preserve the notated equivalence across the A and first Presto (B) section. The tactus of the second, compound time B´ section—from B´(iv)—is faster in all recordings, and again the tempo equivalence within this section is broadly preserved in most recordings. Knappertsbusch’s recording very clearly does not preserve consistent within-section tempi. This recording is the most variable in tempo from section to section and often follows the vicissitudes of the attractor tempi more closely than do the others, as will be discussed below. 4. ANALYSIS: METER AND TEMPO, SECTION BY SECTION The primary changes of metrical structure in this movement appear in the summarizing Examples 1 and 2, above. This analysis focuses on the three-way relationship between those metrical structures, the attractor tempi suggested for them, and the tempi used in the recordings studied (with a primary focus on Murphy’s averages). In total, the analysis traces a process of (non-)alignment that stands to comment on the performers’ view of this movement’s structure. It outlines each of the metrical forms encountered and discusses what the attractor tempo model has to say about these selections. Professional performers (such as those represented in the corpus) are almost certain to select tempi on the basis of a consideration of the whole movement, or even multi-movement work in question. Most trivially, this involves choosing a speed at which the fastest passages are feasible, but it applies as well to preferred tempi for phrasing and other factors. This analysis engages some elements of the interpretative form of that process, including whether and which sections may be optimized for salience and, relatedly, the presence or absence of hypermetrical levels as part of the meter. Level usage is debated throughout in reference to perceptual limits on the upper length of what can be considered “metrical” (cf. discussion in Gotham 2015 after London 2012) and to all relevant musical parameters including grouping boundaries delimited by changes in motif, phrase, harmony, and orchestration. 4.1. THE A SECTIONS As Example 1 shows, the minimum number of metrical levels that are unequivocally represented by the notation of the A section are those of the eighth note, quarter note, and dotted-half note ( 43 measure): a metrical structure in the ratio ⟨1, 2, 6⟩.43 The “attractor tempo” for this three-level metrical scheme is given at quarter note = 85.2 bpm, and is shown on the lower attractors line on Example 4. This is relatively close to the average of the conductor corpus (mean 90, median 91 bpm) already, though it is not the whole picture—we must consider the possible inclusion of one or more hypermetrical level. There is a strong case for the presence of a two-measure level here. Binary pairing would be the normal assumption for classical phrasing of this kind, and the relevant pulse length for this two-measure level is under 4 seconds, thus falling within the acceptable limits for pulse projection. Finally and most importantly, the two-measure level is strongly borne out by the musical design throughout the prototypical first A section, and almost consistently in the reprises thereof. Unsurprisingly, Brahms makes artful changes to this hypermetrical structure later in the movement,44 but these alternations are sufficiently few and occur sufficiently late that they must be viewed as deviations from a well-established two-measure norm. Thus the two-measure level is strongly represented. By contrast, additional hypermetrical levels are equivocal both analytically and perceptually in terms of projecting very slow pulses: the possibility of a consistent four-measure level is quickly ruled out (especially at mm. 11 and 23), and the prospects for projecting a pulse of ca. 8 seconds are remote. Murphy’s analysis accords with this view; he includes the two-measure level but none higher.45 Adding the two-measure hypermetrical level into the consideration gives a metrical structure of ⟨1, 2, 6, 12⟩ and an attractor tempo of quarter note = 90.4 bpm, which aligns exactly with the average of the conductor corpus (mean 90, median 91). This is shown on Example 4 by the alignment of the upper attractors line (Attractors 2) with the lower of the two thin, solid gridlines (the corpus average), and is highlighted by the first large arrow. The spread of individual recordings in Example 4 gives a sense of the range in the new set. For this opening A section, three of the recordings align very closely, one is slower (Knappertsbusch), and two are faster (Gardiner, Jurowski). Averages for this group are discussed below. 4.2. B(i) SECTIONS: MM. 33–49; 63–90 The B section begins (mm. 33–49) and is reprised (mm. 63–90) with simple binary meters that are strongly articulated with pulse levels from the eighth note right up to the four-measure unit. However, whatever the number of levels in operation here, there is no sense of an alignment between the average tempi (90–91 bpm). For an even number of binary metrical levels (in this case ⟨1, 2, 4, 8⟩ without the four-measure level) the attractor for the half-note level is 70.8 bpm, while for an odd number (⟨1, 2, 4, 8, 16⟩ with that four-measure level) it is 100 bpm.46 On average, then, a tempo has been selected, which is close to an attractor for the A section(s), and far from those of the first Presto (B) section, at least at the start.47 This establishes a tension between the average tempo used in this Presto section and the attractor for its metrical structure. This tension will play out in a most interesting way as the metrical groupings change, specifically as levels of three-groupings are introduced and migrate from the remotest to the most salient levels. In this situation, where the tempo is supposed to remain constant but no longer aligns with any possible attractor, any tempo changes that do take place in performance may assume significance. In the new sample shown on Example 4, four of the six recordings opt for a clearly slower tempo in this B section. If either of the possible attractors is having an effect here, it is possible that the slower, 70.8 bpm (Attractor 1) may be drawing conductors toward a slower tempo. Particularly notable is the strict alignment of the slowest recording (Knappertsbusch) with the Attractors 1 line across the first four sections (A[i], B[i], B[ii], and B[i]). 4.3. FRAGMENTATION AND TRANSITION. B(ii) AND B(iii); MM. 49FF. AND 91FF The ends of those B(i) sections (mm. 49–51; 91–100) fragment the four-measure level into a two-measure form as part of a transition into new metrical structures. In the first instance, this leads to a regrouping of the two-measure fragment in 3 seconds at m. 51 (Rehearsal A) to give a highest (possible) level of six measures, with the hypermetrical structure 13.48 This is designated B(ii) and set out on the fourth system of Example 2. That highest level (grouping two-measure pulses into sets of 3 rather than 2) marks the first deviation from simple binary grouping in the B section. At the average tempo, this six-measure level equates to a plausible period of approximately 4 seconds, though it is musically equivocal. The formal division is clear—and analytically important, in view of what follows—but those six-measure units are not necessarily a strong candidate for metrical pulse projection. The reader may wish to listen to a sample of different recordings of this passage and consider whether (or how often) they (a) do in fact project a six-measure pulse, even at its second iteration (mm. 57–62); (b) take the change of texture after four measures as indicative of a new start which is cut short (a division of 4 + 2 rather than 6);49 or (c) do not bother with metrically projecting any grouping at a higher level than the two-measure pulse. Assuming option (a), the metrical structure is ⟨1, 2, 4, 8, 24⟩,50 optimized by half note = 73.5 bpm; option (b) presents the nearest attractor tempo to the average tempo used (half note = 100), but it is thwarted as a metrical strategy by the six-measure units; option (c) rejects periods above the two-measure level, resulting in the binary structure ⟨1, 2, 4, 8⟩ much as at the start of the Presto (half note = 70.8). Both viable attractors (73.5, 70.8) are far from the average used (90–91), while the attractor at 100, which is closer to the average is invalidated by the six-measure units. Whatever the listening strategy, either the tempo or the irregular phrasing perpetuates the dissonance here. The second, equivalent two-measure fragmentation (m. 91) leads to another new grouping. As discussed above, m. 63 returns to the binary metrical grouping with which the Presto began. The parallel is strengthened by the reprised material at Rehearsal B (m. 79). Once again, the four-measure level is initially clear, but is then subtly fragmented to two-measure exchanges of a one-measure idea (mm. 93–100). The single measures are then regrouped in 3 s as a retransition to the A section: mm. 101–3, 104–6. This last transitional passage presents a new structure of 42 measures with three-measure hypermeter, which is set out on the fifth system of Example 2 described by the time signature 23, and designated B(iii). According to Brahms’s instructions, this 23 tempo matches exactly the 43 of the A section that follows, and—sure enough—there is also a thematic retransition in the wind and cello parts: indeed, m. 101 can easily be heard as a stronger formal division than 107, contrary to the visual appearance of the notation. As Epstein observes, “Where Brahms changes tempi through a true alteration of beat duration, he invariably sets up some rhythmic example prior to the change, which serves as the reference for the changed pulse itself. The change is thus composed into the music, so to speak.”51 This resonates with the technique of developing variation (briefly mentioned above) and, indeed, to supposedly “later” compositional developments such as metric modulation. The three-measure pulse here is just over 3 seconds at the average speed: an eminently feasible duration for projection, though a much weaker one than the alternative (two-measure) grouping (as at the previous incarnation of this material at mm. 41ff.). The six-measure grouping must be excluded. It is both at the limits of pulse length, and without a second iteration there is little chance to engage it as a meter. Here, we have a new attractor tempo to consider for the grouping (of ⟨1, 2, 4, 12⟩, excluding 24) that equates to a half note = 54.2 (quarter note = 108). This is even further from the average (90–91) than were the binary groupings with which the B section began (attractor half note = 70.8 or 100 depending on the level usage). Thus the tension briefly increases before returning to the neatly attractor-aligned A section at m. 107. 4.4. REPRISE OF A(i) AND OBSERVATIONS FROM THE NEW DATA SET The newly studied recordings suggest that the tempo fluctuates less within the B section, than from A to B (though once again, Knappertbusch’s tempo dives obligingly in the direction of the extremely slow attractors for the B[iii] section just discussed). Returning to the A(i) section at the reprise (m. 107), the new data set suggests a markedly slower tempo for many of the recordings at the reprise (average 87.6—slightly below the attractor of 90.4) than was the case at the opening (average 94.3—slightly above it). While some conductors (Knappertbusch, Jurowski) bring the tempo back up for the A reprise (following a slower B section), even these come “back” to tempi slower than those with which they began. In this A section, reprise is also an area of conspicuously slower tempo in most recordings (not notated). As such, the new data set introduces an additional A(ii) section for mm. 114ff. (Reh. C, and shown on the sixth system of Ex. 2), whose average is 81.7 bpm. As can be seen in Examples 4 and 7, this A(ii) section is slower than any of the tempo-equivalent sections (within A and B) in all but one of the recordings (Knappertbusch being the exception once again). This is a fascinating (and complicated) area in terms of a possible motivation from attractor tempi. The most conspicuous changes to the metrical structure here appear at the level below the tactus, with the addition of triplets (strings, leading to what could be notated as 89), and then dotted rhythms (winds, superimposed on ongoing triplets in the strings). The psychological literature suggests that “there is no such thing as [a perceptual mechanism for] polymeter,”52 but rather that listeners tend to hear one of the subdivision classes as a rhythm in the context of the other as a meter. Accordingly, the two attractor tempi put forward by the model for this moment correspond respectively to a meter of 89 with three levels (tactus = 100, attractors 2, the higher line on Ex. 4) and a binary meter with four levels (tactus = 54.2, attractors 1, lower line). Of course, some conductors may have felt it beneficial to adopt a slower tempo for other reasons. For instance, a slower tempo would allow the simultaneous triplets and dotted rhythms to be more clearly distinguished; at anything other than a very slow tempo, they tend to collapse into a single metrical category. 4.5. B´(IV): MM. 126FF.—THE PIVOTAL MOMENT After the A section reprise, the Presto returns at m. 126, but in 83—that is, with three-unit grouping at the level of the dotted-quarter-note beat. This section is labeled as an altogether new C section in some accounts, as discussed above. It is also the moment at which a new tempo is to be chosen, the mean and median tempo for the conductor corpus being 119 and 118 respectively. In Example 4, this tempo (118.5) is given as the upper (other) horizontal gridline, and the extent to which the new recordings fit with this can be seen at a glance (their average comes in at the slightly lower 117, thanks largely to Knappertsbusch’s leisurely interpretation of “presto”). As well as being a significant moment in the traditional formal sense, it is also pivotal in the ongoing process of metrical change in the movement. So far, the single level of ternary grouping (hereafter “three-level”) has migrated from a 13 grouping to a 23, and, via the A section’s 43, now arrives at 83. Simultaneously, this moment begins the same process for a second level of ternary grouping as those 83 measures are grouped in six-measure units exactly as at m. 51.53 The equivalence between the material is made especially clear at m. 132 (refer again to Ex. 2, comparing the seventh and fourth systems—B´[iv] and B[ii]). This generates an “ 818” (to match the 13 above), which will undergo the same process as the three-level migrates to neighbor the beat level as part of a (notated) 89 at m. 188. The proportional scheme for this whole process of three-level migration in the B section can be seen in the “Proportion” column of Example 1 and is clarified diagrammatically in Example 5. In Example 5, numbers are given at the bottom, while 2 seconds and 3 seconds represent proportions between consecutive levels, and arrows indicate the migration described. Levels are aligned according to their exact or nearest durational equivalent, with the various note values, and notated “measure”-levels included for ease of reference. EXAMPLE 5. View largeDownload slide Migration of three-level grouping across the two scherzi EXAMPLE 5. View largeDownload slide Migration of three-level grouping across the two scherzi It may also be significant that this pivot occurs at the midpoint of the movement (m. 126 of 240; ca. 2’40” of ca. 5’20”). This would appear to be center of something significant; as such, the moment could perhaps be read in terms of Agawu’s observation that, “Deep into each of Brahms’s sonata-form movements (and sometimes others as well) is a significant turning point, a moment of reversal that announces closure.”54 Perhaps “reversal” is not quite indicative of the process here, but there is certainly a strong sense of “pivot” and of an extreme boundary point in the metrical expansion, with the migrating three-level occupying both extremes of the structure. Crucially then, what is the attractor tempo for this pivotal moment? The shorter measures make the six-measure level more feasible as a metrical pulse for projection than was the equivalent in the first Presto (m. 51), though similarly unattractive from the perspective of musical phrasing. Including the six-measure level yields a structure of ⟨1, 3, 6, 18⟩ (“ 818”), which is optimized by a measure-level dotted-quarter-note pulse of 142 bpm—faster than all of the performances studied and so clearly not operating as an “attractor” for them (see the upper “Attractors 2” line of Ex. 4). The other viable metrical option (corresponding to “option c” at m. 51) involves neglecting levels above that of the two-measure period. This yields a structure of ⟨1, 3, 6⟩ ( 86) for which the attractor tempo is dotted-quarter note = 118, aligning precisely with the averages of the conductor corpus and the new data set (as shown by the second arrow in Ex. 4). This is the first and only time in the many and varied meters of the B sections that there is a possible structural representation that even comes close to the tempo used. Against that unpromising background, the structural representation in question is analytically strong, and the tempo alignment is as exact as was the alignment between the first tempo choice and the attractor for the A section’s meter. In summary, it would appear that: The initial tempo aligned well with the attractor tempo for a version of the A with two-measure level (only) leading to a divergence from the attractors tempi for the various meters in the tempo-equivalent first Presto (B). The new tempo at the second Presto (B´) also aligns with the attractor for its meter with two-measure level only, leaving higher levels to the domain of form. Thus, the attractor tempo model suggests a possible motivation for the value of the average tempi used, as well as a view of the metrical level usage, beyond that which is self-evident from the notation, which that tempo suggests. This study does not comment on proportional tempo schemes, but the result is compatible with Murphy’s proposition of a 3:4 ratio. 4.6. AFTER THE PIVOT: MM. 156–END. B(v) and (vi); A The rest of the scherzo plays out the process of a “migrating three-level” described above (cf. Ex. 5). During the completion of that process, the new metrical structures are associated with attractor tempi that diverge from the tempo used. At m. 156, the two-measure level continues to operate, but the higher level moves from three-grouping to two-grouping. In addition to the motivic parallel connecting m. 156 with m. 132 and m. 51 (as shown in Ex. 2), this continuity at the two-measure level (but no higher) provides another possible motivation for the ⟨1, 3, 6⟩ (without 12) representation of mm. 126–55. This would unite the best part of the section (the two-measure level operates from mm. 126–88), while an emphasis on the change of higher-level structure would divide the section in two at m. 156. However, the four-measure level at m. 156 is an eminently salient pulse and is strongly articulated in the music, especially by changes in harmony and orchestration. Such is the strength of the musical articulation that incorporating this four-measure ( 812) metrical level is almost inevitable, forcing a divergence of metrical structures from attractor tempi for the passage. The attractor without the four-measure level ⟨1, 3, 6⟩ continues to be 118 bpm, while the equivalent including it ⟨1, 3, 6, 12⟩ would be the rather extreme 185 bpm. This is even further than the attractor for ⟨1, 3, 6, 18⟩ (a possibility for mm. 126–55). It seems that in this case, a form of tempo-metrical tension is (somewhat paradoxically) the path of least resistance here. As previously, the newly studied recordings do broadly honor the notated tempo consistency across this B´ section, though once again there is a noticeable tendency to slow down (see the averages of Example 7, and the recordings by Jansons and Jurowski in particular). This is consistent with the very slow attractors for B´(vi), at which point four of the recordings adopt a slow tempo.55 This moment (B´[vi], m. 188) is where the three-level migration reaches fulfillment, with the two three-unit levels finally adjacent, as shown in the final column of Example 5. Finally, a metrical retransition even more brilliant than that at m. 101 occurs at m. 190. The final system of Example 2 shows how the motivic content in the upper strings continues across the divide from 89 to 43. This is invariably performed such that the compound beat of m. 189 = the duple beat of 190 (though nowhere is that stated).56 Nevertheless, the motivic parallel is suggestive of a kind of hemiola structure on the 43 versus 86 level. This has been hinted at in the preceding music (see the entry of voices in mm. 180–87) and is also coupled with a much clearer hemiola on the 23 versus 46 level as lower strings and wind parts have onsets on alternate quarter- note beats.57 There is therefore a bewildering range of possible metrical structures to represent, though 23 and 46 are the most probable along with sub-structures omitting those highest levels ( 42 and 43, respectively). The attractor tempi for these meters are all far wide of the average ( 23 coming closest at quarter note = 108), and the rallentando obviously complicates matters further. In short, this section provides a final bout of divergence from the attractors before the final tempo-aligned return to the A section at m. 194. It should be noted that the start of this final A section (m. 194) is a melodic reprise only and not a tonal one, being as it is in the remote key of F♯ major. The tonal reprise of the same material only comes later, at m. 219 (Rehearsal F). One might suppose that tempo would be used to comment on this non-tonic reprise, but the new data set suggests otherwise, with recordings tending to remain constant between the two (excepting Szell and Knappertsbusch). Perhaps tempo is being used to enhance the sense of “false” reprise and the power of the subsequent tonal movement. 5. SUMMARY AND CONCLUSIONS This analysis has examined a three-way interaction between metrical structures (including a close focus on level usage), the set of “attractor tempi” suggested for them by the model advanced in Gotham (2015), and the tempi used in two samples of commercial recordings. The primary observations concern how the average tempi used by performers in a large, preexisting corpus align with the attractors associated with particular representations of important metrical structures in the movement. This story is contextualized and qualified with a new corpus of recordings, with tempo data for each internal section. Formally, just two tempi are to be selected for this movement: one for the Allegretto sections (A) along with the Presto passage, which is proportionally related to them (B); and another for the compound-time Presto section, which occurs later (B´). The average tempo is such that the opening A section aligns with its attractors, and so the various meters in the proportionally related Presto that follows are necessarily divergent from their attractors. The metrical structure of the first Presto changes several times, systematically introducing levels of ternary grouping at the extremes of hypermeter and then bringing them into focus at the most salient levels. The introduction of the second Presto and second tempo at m. 126 marks the midpoint of this long-range metrical process, as well as the middle of the movement. At this pivotal point, there occurs a metrical structure, which completes this process for the first three-unit level and begins this process for the second. Here, the corpus average takes the opportunity of a new tempo to realign this new passage with the attractor associated with its metrical structure, once again leading to divergence from attractors in the metrical structures that follow. Therefore, of all the many metrical structures in the Presto sections, m. 126 is the only one whose attractor tempo aligns with the average tempo of the conductor corpus. The specific alignments that emerge here may stand to illuminate the extent of metrical-level usage by performers of those passages. In both cases, the tempo aligns best with a representation that includes a single hypermetrical level. In the A section ( 43), this marks the highest (almost) consistently tenable level (generating a ⟨1, 3, 6, 12⟩ total scheme), while in the compound-time scherzo, higher levels are possible, but the focus is apparently on just three levels: ⟨1, 3, 6⟩. Motivic parallels with earlier forms of this material (at m. 51) provide a possible explanation for this, as does a possible attraction to continuity between successive metrical structures, to unite large formal sections. Brahms’s variation of the metrical structure ensures that the tempo cannot be aligned with attractors throughout: the meters (and thus attractors) change, while the tempo is expected to stay constant, and so there emerges a pattern of changing tension-relaxation for the tempo-metrical alignment independent of the two “primary” tempi selected. This study assesses the extent to which six recordings do indeed remain constant across these passages and suggests possible attractor-based reasons for why they might diverge as they do in specific sections. In so far as performers do observe the formal tempo continuities in this work, we can interpret their strategy of tempo choice in a number of ways. The fact that the average tempi selected optimize the metrical structure at section beginnings (m. 1, m. 126) leads to the possibility of the cynical interpretation that the performers optimize tempi at the moment of selection for that moment only, without regard for what is coming in later sections. This hardly seems likely to be the case for the commercial recordings studied, which involve professionals at the highest level. A more convincing alternative interpretation is that the A section is optimized at the expense of the first Presto (B) in order to enhance the leisurely, grazioso feel of the A sections, and the more unsettled quality of B (which is fast and changes meter more frequently). The optimization of the ⟨1, 3, 6⟩ structure for the second scherzo may be attractive in that that meter could be representative of a large part of the section (rejecting most hypermetrical possibilities, as discussed). Apart from interpreting other performers’ choices, we might use this heuristic as a part of the basis for planning a performance. To do so, one must address a fundamental question: is it best to employ a tempo that compromises between the various meters involved, or one that prioritizes some meters (as stable forms) and differentiates them from the other (less stable) meters? I would suggest that the answer depends entirely on the piece, but that differentiation is the better strategy in this dynamic, discursive work. This brings us back to an important comment strongly emphasized in the text: “attractor” tempi are not about dictating “correct tempi” for performers, but are, rather, about observing and engaging with the presence of “default” or “easiest” tempi that exist for classes of metrical structures. In this case, the analysis has observed a correlation between those attractors and the averages used. In future work, the attractor tempi model could be used to assess other performance matters, such as why certain tempi are easier to maintain in a given passage than others. These lines of enquiry are important to pursue as temporal matters (of rhythm, meter and tempo) affect the performer “more than any other compositional parameter.”58 In terms of future developments, it would be useful to include in the “attractor tempi” model a quantification of the relative usage of metrical levels, rather than simply asserting that they are present or absent. Volk (2008) provides a context-specific model of metrical weightings through quantification of each metrical position’s usage. Volk’s model is important and encouraging, though it is currently limited to note onsets. Developments to include other considerations would be welcome, especially with respect to parameters such as harmony and orchestration, both of which would appear to be significant articulators of metrical structure in this piece. A connected performance analysis incorporating all of these parameters systematically would take this line of enquiry to the next level. Footnotes 1 See, for instance, James (1890), The Principles of Psychology, Vol. 2. 2 See especially London (2012) for a summary. 3 Hasty (1997, 183). 4 See London (2012) once again, as well as London (2002) and Parncutt (1994), for instance. 5 Lerdahl and Jackendoff (1983). 6 See Appendix 1 for a more technical explanation of how these values are reached from the “Attractor Tempos” model (Gotham 2015). The bracketing convention of angle brackets for sets of pulse lengths, and square brackets for proportions is also adopted from Gotham (2015) and used here throughout. 7 I am grateful to Fernando Benadon for sharing this data. 8 See, for instance, Epstein (1995) and Murphy (2009) as discussed in the main text. 9 Please note that Ex.1 is a point of reference for much of the discussion which follows and, thus, includes a great deal of other detail that will become relevant in due course. See Murphy (2009, 18ff.) for another specifically metrical analysis of this movement, especially his Fig. 9 for comparison with Ex. 1. 10 Williams (1909, 226). 11 Komma (1967, 449). 12 McClelland (2010, 267). Others identify the “intermezzo” as an alternative kind of norm with which Brahms “replaced the symphonic scherzo” (Swafford 1998, 440). For Pascall, “Stylized dance-types are brought into an overall rondo form” (2013, 39). 13 Brinkmann 1995, 164. Note that the change of tempo in this context would ordinarily be given by a lower metronome value for a shared tactus level, whereas Brahms achieves the same result by the arrangement of metrical levels. This speaks to a difficulty we have in discussing even such basic concepts as “fast” and “slow”—a difficulty that the attractor tempi help to address. 14 In the quintet: A ( 43), B ( 86), A ( 43), B′ (cut-common time), A ( 43); while in the symphony: A ( 43), B ( 42), A ( 43), B′ ( 83), A ( 43). By contrast, the Op. 100 Sonata movement alternates 42 Andante with 43 Vivace: Andante–Vivace–Andante–Vivace di più–Andante–Vivace (short). 15 Pascall observes Brahms’s “interest in radical middle-movement forms” (2013, 39). 16 Frisch (2003, 80). 17 The movement is “thoroughly infused with variation technique” (McClelland [2010, 267]). Compare, for instance, mm. 1–2 with 33–34 and 176–77. The same motive is reprised in the three different metrical contexts ( 43– 42– 83). Indeed, this motive is an inversion of that which opens the symphony and even dominates the Finale of preceding Symphony No. 1. 18 Schoenberg’s origination of the term is often associated with his iconic “Brahms the Progressive” (1951, 398–441), but see Severine Neff’s 1994 translation of Schoenberg’s 1917 Coherence, Counterpoint, Instrumentation, and Instruction in Form for a much earlier appearance of this term (Schoenberg 1917 [1994], 39–43]. 19 Frisch (1990, 155). 20 See comments from Frisch (2003, 80), Komma (1967, 448−49; quoting Willi Lahl), and Pascall (2013, 39), for instance. 21 Frisch (2003, 80). 22 See Pascall (1976) for a short article assessing the sources, and including an edition of both pieces, but see especially Pascall (2013) for a more complete exposition of Brahms’s re-workings of these neo-Baroque pieces into not only the Op. 88 Quintet, but also the Second String Sextet and the Clarinet Quintet. 23 That said, the section has also been likened to the minuet and indeed the Ländler. See Brinkmann (1995, 160), for instance. For Max Kalbeck (an important contemporary and biographer of Brahms himself), the A section is a minuet-like Ländler (“menuettartiger Ländler”), and the two B sections are best described as a “Galopp,” and “tinglingly fast waltz” (“prickelnder Geschwind-Walzer”), respectively (1913, 171). 24 Incidentally, the notational custom of beginning Gavottes on the half measure is apparently no impediment here: in the quintet (where the model is known to have been a Gavotte), the custom is ignored; in the symphony, the use of 42 avoids commitment to any hypermetrical grouping. Incidentally, Kirnberger (a contemporary Baroque musician and theorist) complains about music notated in incorrect metrical displacements. For a modern, translated edition, see Kirnberger (1982). 25 Donington (1973, 243). 26 For an introduction to possible tempo choices in Baroque dances, see Donington’s (1963) translation of Johann Joachim Quantz’s seminal treatise, or the E. R. Reilly 1966 translation for a modern edition of the original commentary; and the summary of treatises by L’Affilard, Lachapelle, Onzembray, and Choquel in Epstein (1995). 27 “Brahms’s Scherzi folgen dem Beethoven’schen Typus, stehen in schneller, oft staccartierter 86” (Komma 1967, 448); stürmisch bewegte (ibid.). 28 Murphy’s corpus (discussed below) assumes that conductors honor these equivalence relationships. (I am grateful to the author for private correspondence on this matter.) The new data set introduced here assesses whether and to what extent that is the case. 29 See Pascall and Struck (2004) or Murphy (2009, 23) for a discussion and list of sources. 30 See Epstein (1995), for instance. 31 Murphy (2009) proposes a 3:4 tempo ratio, asserting that this is followed by 3167 recordings (46 percent) in his corpus (discussed below) to an accuracy within the bounds of a 5 percent “just noticeable distance” (JND). See his Fig. 8, p. 24). That said, the JND is not designed for assessments of this kind, but rather experimental contexts in which stimuli are presented side by side. 32 Sherman observes that Brahms himself “made it clear that he did not believe that there is one ideal tempo for a work” (1997, quoted on 469) and—furthermore—that “if Brahms ever said a kind word about the metronome, we have no record of it” (2003, 99). Brahms’s disinclination is associated with the incompatibility of performance with the cold, mechanical objectivism of the metronome. The clearest extant expression of Brahms’s opinion comes in a letter to George Henschel, dated February 1880: “I myself have never believed that my blood and a mechanical instrument go well together” (quoted in Avins [1997, 559]). 33 See Pascall and Struck (2001) edition of the Symphony in the Neue Ausgabe sämtlicher Werke for full details of designations and sources (270–74 for this movement). 34 See Pascall and Struck (2004, viii). Presumably, Brahms was comparing his own quarter-note tempo (in 43) to Beethoven’s eighth note (in 83). 35 What is more, Beethoven’s notated tempi have been the subject of heated contention since he first committed them to paper. For a considered defense of Beethoven’s markings, see Kolisch’s famous essay (retranslated and reprinted as Kolisch 1993a, b) along with the commentary in Levin (1993). This also includes a rumination of the role of the metronome in a more general sense. 36 Nationalities, within the major European and American centers for Classical music, that is. There is little representation of artists from outside of Europe, and (among conductors at least) even the few exceptions such as Ozawa tend to have been trained and made their name in the West. That said, this may be broadly reflective of the production of public, commercial, professional recordings of this work, at least during the temporal span that this corpus covers. 37 See Leech-Wilkinson, in which the author suggests that “most recorded musicians for whom we have a lifetime’s output seem to have developed a personal style early in their career and to have stuck with it fairly closely for the rest of their lives” (2009, 250), and “that being so, it is more important to know a performer’s birth year than the date of the recording” (250n14). This would move the average back several decades. Assuming very roughly that conductors come of age at twenty, and that the mean age at the time of recording is fifty-five (half way between forty and seventy), would generate a new “conductor coming of age” mean at approximately 1939. 38 Gotham (2015). One brief disclaimer must be passed from Murphy’s analysis (25n21) to the present one. Deviations in tempo are included in the construction of those tempi used. Given the numerous possibilities for local ritardandi, this may make some of the tempi given slower than is accurate. This therefore equates to the first of Gabrielsson’s (1988) types: the “abstract mean tempo.” The second type—“main tempo”—would have been preferable. This should be borne in mind, but does not affect the substance of the claims made in this analysis. 39 “Empirical evidence confirm[s] that morphed (quantitatively averaged) human faces, musical interpretations, and human voices are preferred over most individual ones” (Wöllner et al. 2012, 1390). The paper from which this quote is taken includes a useful summary of relevant literature, but is primarily concerned with demonstrating a new, practical application: that prototypical conductor patterns are more easily followed than any individual’s. 40 This material can be found in Sherman (1997, see especially the table on p. 467; 2003). 41 The shortest sections (with the fewest taps) often gave the most consistent averages. 42 Cannam et al. (2010). I am grateful to Scott Murphy for providing the exact extracts he used. 43 Here, I exclude the fleeting melodic triplet eighth notes from a metrical consideration, though its increased use at Rehearsal C, m. 114 may be considered grounds for investigating a 98 meter there. 44 These are prepared by potential ambiguities in the initial material. The original m. 4 is used as a new phrase beginning at Rehearsal C, preceded by a 4 + 3 measure phrase structure from the start of the A section reprise (mm. 107–13). Similarly, the mm. 8–9 register transfer serves just as well as m. 4 for such a trick, and it, too, is used to ensure that m. 207 (in the second reprise) is unequivocally phrase-commencing. That connection is enhanced in readings that emphasize the m. 207 theme (which first occurs at m. 11—see the second system of Ex. 2) as a parallel of that which commences the sections at m. 51, and 132. See, for instance, McClelland (2010, 268 including 268n3); and Musgrave (1994, 217). 45 See Murphy (2009), summarizing Fig. 9 on p. 27. 46 These were the meters and attractors discussed in relation to the Bach example. 47 That said, these “attractor” values of 70.8 ad 100 broadly delimit the extremes of the tempi used in the corpus (78–104). It is, therefore, possible that those “extreme” conductors are aligning with attractors for this B section rather than the A section, thus creating an alternative pattern of tension-relaxation for this movement in those individual cases. 48 As in Ex. 1, inverted commas are used here to indicate a time signature that would express the full metrical structure for the note values used but that is not actually notated in the score. 49 This is the view of Williams (1909), for instance. 50 Eighth notes continue to form the faster metrical period involved. Sixteenth notes are only weakly used, and (at the average tempo) are shorter than 0.1 seconds, ruling them out as a metrically useful pulse level. 51 Epstein (1990, 206–10). 52 London (2012, 50). See Polak (2010, nn9–10) for further references rejecting polymeter, particularly in the context of Jembe music. 53 This process could be conceived as an intra-structural-level form of the latent-emergent-manifest principle that Rink invokes in the context of a related Brahms analysis (1995, 273). 54 Agawu (1999, 136). 55 The exceptions are Stokowski who remains broadly constant, and Szell who moves slightly faster. 56 For Epstein, this is “implicit” (1990, see the diagram on p. 208), and no more or less problematic than the (highly dubious) beat equivalence of m. 126, which he marks in the same way. 57 The bass part of this hemiola is also included in Ex. 2. 58 Rink (1995, 25). APPENDIX 1: ATTRACTOR TEMPI This appendix summarizes the mathematical structure of the Attractor Tempi model. Please refer to Gotham (2015) for full details, illustrative diagrams, and a discussion of how the model attempts a balance between accurately corresponding to the data available without “over-fitting” to any individual experiment, and without unduly complicating what is necessarily designed as a heuristic. The model is based on cognitive literature that suggests that: (a) there is a preference for pulses around 100 beats per minute (which equates to an inter-onset interval [IOI] of 0.6 seconds); (b) pulses shorter than 0.1 seconds cease to be metrically useful; and (c) the upper limit for what can be parsed as metrical unit is approximately 6 seconds. To model this salience (S), we need a smoothly continuous curve that accounts for all x-values, peaks at 0.6 seconds, and returns negligibly low values for very fast (x < 0.1) and very slow (x > 6.0) pulses. This is achieved with the following logarithmic Gaussian function:   S=exp⁡-(log⁡(x/0.6))20.18. This gives us a metric for the salience of an individual pulse. Attractor tempi are based on combining these values to give a salience value for the whole meter (“metrical salience,” M), which is operationally defined by the combined saliences of all the pulse levels present in the metrical structure. If we represent the fastest pulse as x, and all higher levels by the relevant multipliers (px, qx, rx…), then this salience value is given by   M=∑n=1,p,q,r,…exp⁡-(log⁡(nx/0.6))20.18. Each metrical structure has an “attractor tempo” given by the x-value that maximizes its metrical salience (M). Example 6 sets out these values for some of the simplest meters, with the attractor value given both as an IOI in seconds for the fastest level, x (in the third column), and as a tempo in beats per minute (“bpm”—60/nx, the last column) for a central level near 100 bpm that might act as the tactus. EXAMPLE 6. View largeDownload slide Some common meters and their attractor values EXAMPLE 6. View largeDownload slide Some common meters and their attractor values APPENDIX 2: TEMPO DATA This appendix provides data for the additional recordings used in this study (such as the commercial release information) as well as methodological matters concerning the collection of that new data and a formal test of compatibility among the data collecting methods (as discussed in the main text). EXAMPLE 7. View largeDownload slide Tempo data (and methodological information) for the recordings added in this study EXAMPLE 7. View largeDownload slide Tempo data (and methodological information) for the recordings added in this study EXAMPLE 8. View largeDownload slide Information about the newly studied recordings EXAMPLE 8. View largeDownload slide Information about the newly studied recordings EXAMPLE 9. View largeDownload slide Summarized data for the methodological cross-check of Murphy and Gotham’s tempo judgments against the results from Sonic Visualiser EXAMPLE 9. View largeDownload slide Summarized data for the methodological cross-check of Murphy and Gotham’s tempo judgments against the results from Sonic Visualiser NEWLY COLLECTED DATA FOR THIS STUDY Works Cited Abdy Williams C. F. 1909. The Rhythm of Modern Music . London: Macmillan and Co. Agawu Kofi. 1999. “Formal Perspectives on the Symphonies.” In Cambridge Companion to Brahms . Ed. Musgrave Michael. 133– 55. Cambridge: Cambridge University Press. Google Scholar CrossRef Search ADS   Avins Styra. 1997. Johannes Brahms: Life and Letters . Oxford: Oxford University Press. Benadon Fernando, Zanette Damian. 2015. “ A Corpus Analysis of Rubato in Bach’s C Major Prelude, WTC I.” Music Performance Research  7 ( 1): 1– 26. Brinkmann Reinhold. 1995. Late Idyll: The Second Symphony of Johannes Brahms . Cambridge, MA: Harvard University Press. Cannam Chris, Landone Christian, Sandler Mark. 2010. “Sonic Visualiser: An Open Source Application for Viewing, Analysing, and Annotating Music Audio Files.” In Proceedings of the ACM Multimedia 2010 International Conference. 1467–68. New York: ACM. Donington Robert. 1963. The Interpretation of Early Music.  London: Faber and Faber. ———. 1973. A Performer’s Guide to Baroque Music . London: Faber and Faber. Epstein David. 1990. “Brahms and the Mechanisms of Motion: The Composition of Performance.” In Brahms Studies: Analytical and Historical Perspectives . Ed. Bozarth George S. 191– 228. Oxford: Clarendon Press. ———. 1995. Shaping Time: Music, the Brain, and Performance . New York: Schirmer. Frisch Walter. 1990. “The Shifting Bar Line: Metrical Displacement in Brahms.” In Brahms Studies: Analytical and Historical Perspectives . Ed. Bozarth George S. 139– 64. Oxford: Clarendon Press. ———. 2003. Brahms: The Four Symphonies . New Haven, CT: Yale University Press. Gabrielsson Alf. 1988. “Timing in Music Performance and Its Relations to Music Experience.” In Generative Processes in Music: The Psychology of Performance, Improvisation, and Composition . Ed. Sloboda John. 27– 51. Oxford: Oxford University Press. Gotham Mark. 2015. “ Attractor Tempos for Metrical Structures.” Journal of Mathematics and Music  9 ( 1): 23– 44. Google Scholar CrossRef Search ADS   Hasty Christopher. 1997. Meter as Rhythm . New York; Oxford: Oxford University Press. James William. 1890. The Principles of Psychology . Vol. 1 and 2. London: Macmillan. Kalbeck Max. 1913. Johannes Brahms . Berlin: Deutsche Brahms-Gesellschaft. Kirnberger Johann P. 1982. The Art of Strict Musical Composition . Ed. and Trans. Beach David W., Thym Jurgen. New Haven, CT: Yale University Press. Kolisch Rudolf. 1993a. “ Tempo and Character in Beethoven’s Music.” Musical Quarterly  77 ( 1): 90– 131. Google Scholar CrossRef Search ADS   ———. 1993b. “ Tempo and Character in Beethoven’s Music (Continued).” Musical Quarterly  77 ( 2): 268– 342. CrossRef Search ADS   Komma Karl Michael. 1967. “Das ‘Scherzo’ der 2. Symphonie von Johannes Brahms: Eine melodisch-rhythmische Analyse.” In Festschrift für Walter Wiora: zum 30. Dezember 1966 . Ed. Wiora W., Finscher Ludwig, Mahling Christopher. 448– 57. Kassel: Bärenreiter. Leech-Wilkinson Daniel. 2009. “Recordings and Histories of Performance Style.” In Cambridge Companion to Recorded Music . Ed. Cook Nicholas, Clarke Eric, Leech-Wilkinson Daniel, Rink John. 246– 62. Cambridge: Cambridge University Press. Google Scholar CrossRef Search ADS   Lerdahl Fred, Jackendoff Ray. 1983. A Generative Theory of Tonal Music.  Cambridge, MA: MIT Press. Levin Tamara Y. 1993. “ Integral Interpretation: Introductory Notes to Beethoven, Kolisch, and the Question of the Metronome.” Musical Quarterly  77 ( 1): 81– 89. Google Scholar CrossRef Search ADS   London Justin. 2002. “ Some Non-Isomorphisms between Pitch and Time.” Journal of Music Theory  46 ( 1–2): 127– 51. Google Scholar CrossRef Search ADS   ———. 2012. Hearing in Time: Psychological Aspects of Musical Meter . Oxford: Oxford University Press. McClelland Ryan. 2010. Brahms and the Scherzo: Studies in Musical Narrative . Burlington, VT: Ashgate. Murphy Scott. 2009. “ Metric Cubes in Some Music of Brahms.” Journal of Music Theory  53 ( 1): 1– 56. Google Scholar CrossRef Search ADS   Musgrave Michael. 1994. The Music of Brahms . Oxford: Clarendon Press. Parncutt Richard. 1994. “ A Perceptual Model of Pulse Salience and Metrical Accent in Musical Rhythms.” Music Perception: An Interdisciplinary Journal  11 ( 4): 409– 64. Google Scholar CrossRef Search ADS   Pascall Robert. 1976. “ Unknown Gavottes by Brahms.” Music & Letters  57 ( 4): 404– 11. Google Scholar CrossRef Search ADS   ———. 2013. Brahms beyond Mastery: His Sarabande and Gavotte, and Its Compositions . Burlington, VT: Ashgate. Pascall Robert, Struck Michael. 2001. Symphonie Nr. 2, D-Dur, Opus 73.  Neue Ausgabe sämtlicher Werke / Johannes Brahms. Ser. 1, Bd. 2. München: G. Henle. ———. 2004. Symphonie Nr. 2, D-Dur, Opus 73.  Study Edition following the Neue Ausgabe sämtlicher Werke / Johannes Brahms. Ser. 1, Bd. 2. München: G. Henle. Polak Rainer. 2010. “ Rhythmic Feel as Meter: Non-Isochronous Beat Subdivision in Jembe Music from Mali.” Music Theory Online  16 ( 4). Quantz Johann Joachim. 1966. On Playing the Flute . Ed. and Trans. Reilly E. R. London: Faber and Faber. Rink John. 1995. “Playing in Time: Rhythm, Metre and Tempo in Brahms’s Fantasien op. 116.” In The Practice of Performance: Studies in Musical Interpretation . Ed. Rink John. 254– 82. Cambridge: Cambridge University Press. Google Scholar CrossRef Search ADS   Schoenberg Arnold. 1917 [1994]. Coherence, Counterpoint, Instrumentation, Instruction in Form (Zusammenhang, Kontrapunkt, Instrumentation, Formenlehre) . Ed. Neff Severine. Trans. Charlotte M. Cross and Severine Neff. Lincoln: University of Nebraska Press. ———. 1951. Style and Idea . London: Williams and Norgate. Sherman Bernard D. 1997. “ Tempos and Proportions in Brahms: Period Evidence.” Early Music  25 ( 3): 463– 78. Google Scholar CrossRef Search ADS   ———. 2003. “Metronome Marks, Timings, and Other Period Evidence regarding Tempo in Brahms.” In Performing Brahms: Early Evidence of Performance Style . Ed. Musgrave Michael, Sherman Bernard D. 99– 130. Cambridge: Cambridge University Press. Swafford Jan. 1998. Johannes Brahms: A Biography . London: Macmillan. Volk Anja. 2008. “ Persistence and Change: Local and Global Components of Metre Induction Using Inner Metric Analysis.” Journal of Mathematics and Music  2 ( 2): 99– 115. Google Scholar CrossRef Search ADS   Wöllner Clemens, Deconinck F. J. A., Parkinson Jim, Hove Michael J., Keller Peter E. 2012. “ The Perception of Prototypical Motion: Synchronization Is Enhanced with Quantitatively Morphed Gestures of Musical Conductors.” Journal of Experimental Psychology: Human Perception and Performance  38 ( 6): 1390– 1403. Google Scholar CrossRef Search ADS PubMed  © The Author(s) 2018. Published by Oxford University Press on behalf of The Society for Music Theory. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Music Theory Spectrum Oxford University Press

“Attractor Tempi” in Brahms’s Symphony No. 2/III

Music Theory Spectrum , Volume Advance Article (1) – May 2, 2018

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Abstract

Abstract This analysis assesses tempo choice in the third movement of Brahms’s Second Symphony. It is shown that, at key moments of the movement, the average tempi used in a sample of commercial recordings align closely with those suggested by the “attractor” model of cognitively preferable (easiest) tempi (Gotham 2015). Between these stable checkpoints, the movement navigates through a variety of other metrical structures at (formally) the same tempo, each with their own attractor tempi. These necessarily align less well with the tempi used, thus providing a pattern of relative tension/relaxation, which the new model helps to elucidate. 1. INTRODUCTION TO “ATTRACTOR TEMPI” Ever since William James’s pioneering work in the nineteenth century,1 a growing body of evidence in the cognitive sciences has substantiated the intuitive notion that there is a basic, human preference for pulses in a certain tempo range: that we are better able to deal with a range of “moderate” pulse rates than with anything extremely fast or slow. Apart from the extensive music-psychological literature,2 music theorists may be most familiar with these ideas from the notion of metrical “projection,” including observations such as: “Since mensural determinacy is gradually attenuated, evidence of projection or projective potential will, as a rule, become progressively weaker as duration increases.”3 The reader may wish to test the idea of pulse preference by means of a simple experiment. Try tapping along with the second hand of a clock, look away for a few seconds, come back and see how well you kept time. Try the same experiment for an interval of two seconds (without mentally subdividing) and it is likely that you will have fared less well: two seconds is usually found to be a harder duration to project than one. The ramifications that this individual pulse preference might have for tempo preference in (multi-leveled) metrical structures have been alluded to in the literature,4 and my prior work develops a theoretical model of “attractor tempi” for meter-specific tempo preference, systematically deducing those tempi at which different meters may be most easily entrained to and parsed. This modeling is achieved by adopting the view of meter as an interaction between coinciding periodic pulse levels in simple proportions,5 and modeling tempo preference as a trade-off between the competing calls of the various pulse levels involved. Each pulse may “want” to be individually optimized for salience, but the combination of levels and groupings (the meter) is optimized by a tempo, which balances those individual pulses. For instance, where there are just two pulse levels involved, the combined attractor value will split the difference between the individual ones. This can be thought of in terms of a kind of “tempo see-saw” (level and fulcrum): adding “weight” (e.g., metrical levels) to one end of the system moves the “center of mass” (attractor tempo) further in that direction. This manifests in different ways for different metrical structures, but the principle remains: for any meter, the attractor tempo is that which “balances” the combination. Appendix 1 provides a more technical overview of the inner working of the model, including a short summary of the mathematics involved. In short, the model argues that, given any metrical structure, we may be drawn toward the tempo that optimizes salience in that context (the attractor tempo). These attractor tempi are determined by the number of levels represented and their proportional relationships (metrical schemes) in largely simple, categorical ways. To be absolutely clear from the outset, the model does not claim to generate tempi that are “correct,” and which therefore “should” be used. Rather, the attractor tempi represent a kind of notional default to which performers may find they are drawn, and from which composers and performers will very frequently deviate, using deliberately fast or slow tempi to expressive ends. The goal is to engage with a form of relative “tempo dissonance” (divergence from attractor tempi) and its use as a compositional parameter. 1.1. ATTRACTOR TEMPI IN PRACTICE This model invites scrutiny of various kinds. One might assess how successfully it accounts for metrical listening (with cognitive tests) and musical practice (with corpus studies), and there is also plenty of scope for testing the model’s main claim to have explanatory power as an analytical tool. This article tests that claim, using Brahms’s Symphony No. 2, III, as a suitably challenging case study. To do so, it necessarily puts aside questions concerning the fundamentals of the model. It is assumed, for present purposes, that the notion of attractor tempi does speak to an aspect of our musical experience and that the shape of the theoretical model is sufficiently accurate to make meaningful comments about tempo choice. The focus of the article is thus not on proving or disproving the model, though any use of the model should be well informed of its strengths and shortcomings. This section sets out the variables involved in applying the model, engaging some debatable areas; the discussion is then illustrated by a repertoire application. There are three interacting variables in play here: the attractor tempi predicted by the model, the tempi actually used (whether in an individual recording, or in a corpus average), and the metrical structure of the work or passage (including the possible usage of hypermetrical levels). The model itself is new and is based on tempo preference in general—a large, unruly issue, which abstract, context-independent models can only ever hope to approximate. It attempts a balance between accurately corresponding to the data available without “over-fitting” to any individual experiment, and without unduly complicating what can only hope to be an approximate heuristic. The tempi used in a recording may (theoretically) be objectively verified, though summarizing a work or section with a single value presupposes a meaningful “steady state,” or “main tempo,” which may be more or less evident in a given recording. Extensive discussion of average types is beyond the scope of this study; suffice to say that the approach here has been to take mean inter-onset interval (IOI) values for “steady-state” passages deliberately selected to avoid areas of expressive variation in the timing. Further discussion of method follows below, with the introduction of the new corpus data. The metrical structure may seem to be clear and not variable at all; however, the presence or absence of (particularly hyper-) metrical levels is under-determined by the notation and must be deduced by analytical judgment. Reliance on such analytical judgment might be seen as a shortcoming of this analysis. These decisions are indeed readily disputable (as with any analysis), though I have attempted to set out the analysis in such a way that readers with a different view of level usage can still trace a course through the piece in relation to attractors for the levels they consider to be present. To that effect, the two most likely metrical structures for each passage (one usually involving a single additional level to the other) are presented throughout. More limiting is a shortcoming of the attractors model itself: it deals with levels only in terms of presence or absence. There is not yet a mechanism for finer shades of relative usage in a work or performance. 1.2. ATTRACTOR TEMPI IN BACH’S C-MAJOR PRELUDE (BWV 846) A short repertoire example will help clarify how the model works and the effect that these variables have. The first prelude from Bach’s Das Wohltemperierte Klavier is, metrically, much simpler than Brahms’s Symphony No. 2, III. There are no (explicitly notated) tempo or metrical changes in the work, and the piece does not overtly call for a particularly fast or slow tempo. One might, therefore, expect the average approach to be a reasonable reflection of moderate tempo for its metrical structure. The metrical levels self-evidently in use are those of the sixteenth note (the level of the individual notes), as well as the eighth, quarter, and half notes (the last being the length of the repeated, figural pattern). This gives a metrical structure with pulses of ⟨1, 2, 4, and 8⟩ unit lengths and therefore proportions between those four pulse levels of [2, 2, 2], for which the attractor tempo model suggests a quarter note of 70.8 beats per minute (bpm).6 However, chord changes take place at the whole-note (measure) level, which would suggest a metrical structure with an extra level—pulse levels of ⟨1, 2, 4, 8, 16⟩, and proportions of [2, 2, 2, 2]—for which the attractor tempo is quarter note = 100 bpm. The model thus suggests two strategies for “moderate” tempo in this work: 70.8 bpm for metrical levels up to the half-note level (the figural pattern), versus 100 bpm for levels up to the whole-note (measure) level. This highlights a potentially surprising aspect of the model: that the addition of a single metrical level can give rise to a very different attractor tempo. The model seeks to make the levels in use as salient as possible, and adding a level has a strong effect on the best value for this. Think again of the “tempo see-saw”: if you have one pulse level, it is balanced when positioned at the fulcrum, in the center; if you have two, they balance when equidistant from that center. By adding one metrical level, the balanced (centered) position for the continuing level changes markedly. So it is with the metrical options for the Bach: ⟨1, 2, 4, 8, 16⟩ is centered by positioning the four-level (quarter note) in the center at 100 bpm, while ⟨1, 2, 4, 8⟩ keeps its two- and four-levels equidistant from that center, leading to the slower attractor tempo. For illustration, try singing, playing, or just imagining this piece to see what tempi you find “easiest” or most “natural.” First, try emphasizing half-note beats, then shift focus to whole-note beats. The model expects you to find a faster tempo easier when working with the whole-note level. Assuming that performers do (on average) share the view of the work as being in a moderate tempo, then we can look at their tempo choices to see which of the metrical strategies they adopt. Benadon and Zanette (2015) provide a corpus of forty-eight recordings of this work,7 and the first 100 onsets (sixteenth notes, equating to twenty-five quarter-note beats) provide a relatively “steady-state” section in most recordings. The average tempo for this section is 69.3 bpm. This is indeed faster than the average tempo for the whole work (67.1 bpm), and it does correspond well to other steady-state passages (such as the second measure alone, at 69.7 bpm). The performers’ average preference of 69.3 bpm corresponds closely to the attractor tempo option of 70.8 bpm, and thus to a meter with the half note as its highest level. 2. INTRODUCTION TO BRAHMS’S SYMPHONY NO. 2, III The Bach example has provided a short exposition of how the theory works, though to test the full explanatory power of such a model, one needs a more complex work with evident changes of tempo-metrical arrangement, which the model can describe in terms of relative tempo relaxation/tension (or consonance/dissonance) across the whole. The third movement of Brahms’s Symphony No. 2 provides an ideal case study: its metrical disposition is complex and varied, and—partly for that reason—the choice of tempo has been debated at length.8 Two main tempo choices are to be made by performers of this movement: one at the start, and another at the introduction of compound meters in m. 126. This analysis will argue that the choices made by conductors tend toward the attractor tempi for the metrical structures that begin these sections at the expense of the other metrical structures (with different attractor tempi) that follow. Brahms’s numerous changes of meter generate changing attractor tempi; no single tempo can correspond to all of these. This precludes alignment with attractor tempi throughout, and it ensures a process of change in this parameter that plays out as a pattern of relative tempo relaxation/tension across the whole. As such, prior to the close analytical reading itself, a brief overview of the movement’s form is in order to contextualize these changes in terms of a possible compositional strategy that may have a bearing on performers’ strategies for tempo choice. 2.1. FORM IN THIS MOVEMENT (AND IN SOME OTHER, RELEVANT PIECES) The movement appears to divide neatly into five parts, with the Allegretto grazioso (Quasi Andantino) A sections alternating with the Presto ma non assai B sections (ABABA). Example 1 outlines these broad sections and the subsections given by changes to the metrical structure (the most useful and relevant basis of subdivision for the purposes of this tempo-metrical analysis).9 Allegretto grazioso (Quasi Andantino) and Presto ma non assai are designated by ‘A’ and ‘B’ respectively. Subsections serve to identify metrical change. The unit pulse in each meter is the eighth note. Time signatures in inverted commas (“”) serve as a shorthand for the whole structure under discussion, including hypermetrical levels. In each row, two meters are set out, usually distinguished by the absence or presence of one debated hypermetrical level. That debatable hypermetrical level (and its corresponding attractor) is given in soft brackets—“()”—in each relevant column. This does not mean that the version with the hypermetrical level is less preferable (indeed, more often the opposite is true); the notation is merely used to facilitate the reading of this table. Example 2 illustrates each subsection with representative musical material (predominantly melodic). Excerpts are notated in the original time signatures (Brahms’s own); comments above identify the larger metrical units involved in the debated hypermetrical levels (referred to in Ex. 1 and throughout). EXAMPLE 1. View largeDownload slide Form and meter in the movement EXAMPLE 1. View largeDownload slide Form and meter in the movement EXAMPLE 2. View largeDownload slide Representative melodic material and metrical contexts EXAMPLE 2. View largeDownload slide Representative melodic material and metrical contexts However, the form is more equivocal than the neat Allegretto–Presto alternation would suggest. The alternation bears the hallmarks of the classical minuet (or scherzo) and trio form that would be standard fare for a symphonic third movement. Accordingly, the form has been described as a scherzo,10 as ABA′CA″,11 and as ABA′B′A″.12 However, according to that expectation, the intervening trios would ordinarily be slower, and so this movement has also been described as a “back to front scherzo and trio,” or a kind of triple trio with two scherzi (sic, rather than the more usual way around).13 Here a parallel is often drawn with two other works by Brahms that exhibit a similar A–B–A–B–A pattern of slow–fast–slow–fast–slow in their central movements: the String Quintet No. 1, Op. 88, and the Violin Sonata No. 2, Op. 100. The tempo-metrical arrangement of the Op. 88 movement especially bears an uncanny resemblance to the present one: both have A sections in 43 and B sections of which one occurrence is in simple time, and the other is in a compound meter.14 According to a model of symphonic form, this could be viewed as an elision of the second, slower movement (accounting for the anomalously slow A section) and the third (represented by the alternating form and the A section’s triple time).15 However, of these three works, it is the symphony that remains in four movements (retaining a separate slow movement), and the chamber works that have three. Moreover, even the basic, five-part formal division discussed above has been accused of underplaying the “interpenetration” of material between the sections.16 The connection between sections is such that some argue the case for variation form.17 Continuous variation is also intimately linked to that archetypally Brahmsian device, the developing variation.18 Importantly for the present purposes, analysts are keen to note the inclusion of tempo and meter as part of this technique. Walter Frisch notes that it is common to see the “mobile bar line . . . linked to both motivic development and formal articulation” in this way.19 3. TEMPO CHOICE 3.1. CONTEXTUAL, HISTORICAL, AND NOTATIONAL CUES The fact that the main variations in this movement are achieved through rhythmic-metrical transformation motivates the present tempo-metrical analysis and has also led some scholars to draw an alternative formal parallel with the Baroque suite.20 This forms part of a wider range of apparent retrospection in this movement, which Frisch describes as “particularly striking after the Adagio [second movement], one of the least historically retrospective movements in all of Brahms.”21 The formal parallel with Op. 88 may be significant in this light as Pascall has shown the A and second B sections are based on pre-existing movements explicitly identified as Sarabande and Gavotte by the composer.22 The same dance forms may well have been invoked for the symphonic movement in this study. In the A section, the secondary melodic idea introduced in m. 11 could be read as a textbook Sarabande rhythm (see the second system of Ex. 2, noting the 2–3–1 rhythmic pattern and the associated second beat emphasis).23 If the Op. 88 B´ section counts as a Gavotte in Brahms’s reckoning, then the first Presto of the symphonic movement could well be, too—both are marked presto, but can readily be heard in a moderate 2 (indeed, the quintet is notated in cut common time).24 Stylistic allusion could well have an effect on the tempo choices made by Brahms and indeed performers of this work. Unfortunately, however, “very little” can be deduced about specific tempi in Baroque music “either from notation or from contemporary evidence” and the evidence that exists is often extremely inconsistent.25 Instead, this analysis concentrates its efforts on the evidence of contemporary performers’ attitudes to tempi in this movement as expressed in their recordings; no more is said about the possibility of Baroque models.26 As for the compound-time Presto section, this would appear to diverge from Baroque models in any case, fitting rather with a class of post-Beethoven scherzi commonly used by Brahms. As Komma has it, “Brahms’s Scherzos follow the Beethovenian type: a fast, often staccato 86.” Accordingly, they operate at a “stormy, agitated” tempo—that is, a fast one.27 As mentioned above, Brahms instructs conductors to make two primary tempo choices for this movement, at mm. 1 and 126. The half-note tempo of the first Presto (m. 33) should be equal to the chosen quarter-note tempo for the opening A section.28 Accordingly, the opening notated tempo is, at least theoretically, in operation until the (approach to) the second Presto (section B′) commencing at m. 126. At that point, a second tempo selection is to be made for the compound meter passage’s beginning. That second tempo holds until the retransition to A (Tempo 1), which arrives at m. 194. Brahms initially indicated that the new dottedquarter note for this second tempo should be equal to the quarter note of the A section (and half note of the first Presto), though he later withdrew this instruction.29 Despite the withdrawal, this scheme has found some support,30 and other proportional schemes have also been suggested.31 Brahms does not provide metronome marks for this movement (nor indeed for most of his music—his “blood” did not “go well” with the “mechanical instrument”),32 though the tempo terms he uses may be instructive, as are the changes he made to them.33 Brahms’s many changes to these tempo indications consist largely of minor tinkering with the Italian terminology; however, one more substantial and systematic change appears to present itself. For each of the three tempi provided in Symphony No. 2, III, Brahms added qualifiers suggesting a slower tempo than the original designation: (Quasi Andantino) was added to the A section’s Allegretto grazioso, while ma non assai was added to the two Presto passages (via ma non troppo in the case of the first). This may assume significance in relation to attractor tempi for salience. Yet at the same time, Brahms also suggested the Andante cantabile con moto of Beethoven’s First Symphony to be a good model for the tempo of this movement.34 Unlike Brahms, Beethoven was an enthusiast for the metronome, which was invented during his mature career. Beethoven gives a metronome mark for most of his works; in the case of the Andante cantabile con moto, eighth note = 120, which is quick for any allegretto or andantino. Unfortunately, even if Brahms viewed these two movements as being related in tempo, we do not know whether he had Beethoven’s tempo or something else altogether in mind.35 3.2. TEMPO AS EXPRESSED IN PERFORMANCES OF THIS WORK Brahms’s oblique comments, therefore, join the possible models of Baroque dances in providing interesting (but ultimately conjectural) ideas for tempo selection in this work. More specific and quantifiable evidence is to be had from existing recordings of this work, and it is here that an analysis of attractor tempi is best put to use. This analysis is based on two data sources: Murphy’s corpus of recordings (see Murphy 2009), and a new data set focusing on tempo changes by section (as detailed in Appendix 2). Murphy’s corpus appears to provide a relatively good representation of public, commercial, and professional recordings of this work: the sample size is relatively large (67), and the corpus includes a relatively broad representation of recording years (1928–2005) and nationality of both conductor and orchestra.36 Further analysis could compare the year of recording with the conductor’s birth year, as it has been suggested that the generation in which the conductor came of age may be a more relevant determinant of their approach than that in which the performance is given.37 The corpus provides a useful sense of the range of tempi employed by performers of this movement, and—most usefully for the present purposes—the notion of “average tempi” for the passages in question that can be assessed in relation to the “attractor tempi” of my previous work (or indeed any other models of reference points such as the suggestions of Baroque treatises).38 This analysis focuses on the average tempi partly for practical reasons of space and simplicity, but primarily in the interests of comparing “average” practice with the attractor tempi “defaults.” I reiterate, once again, that there is no intention to prescribe the “correct” tempi for this (or any other work), but neither is there any pejorative connotation intended by the notion of an “average” performance. Indeed, quantitatively averaged forms are often assessed to be the most attractive. This has been shown directly for musical interpretations, as well as for numerous other aesthetic judgments.39 Fittingly, the movement in question appears to lend itself more readily to the notion of average tempo than many of Brahms’s works; Sherman’s data on historical changes in performance practice appear to indicate that the choice of tempo for this movement has been more consistent and less susceptible to the global fashions in tempi that he identifies for other works.40 Example 3 sets out the data for the two tempi used by the performers represented in Murphy’s corpus. The distributions of tempi are selected by performers is shown in the two histograms: the first, simple-time tempo on the left; the second, compound-time tempo on the right. The distribution for the first tempo choice centers on a mean of 91 bpm, and a median of 90 (for the quarter note in the A sections and half note in the first B section). The range is 78–104, and the standard deviation 5.54. The distribution for the second, B´ section’s tempo has a faster beat-level (dotted-quarter-note beat: mean 119, median 118, range 106–136, and standard deviation 7.02). The histogram bars represent spans of two MM values, and a normal distribution of best fit is included in both cases. EXAMPLE 3. View largeDownload slide Histograms showing the distribution of tempi used by performers in Murphy’s corpus: the first tempo selection for the A and simple time B sections (left); the second tempo selection for the compound meter Presto, section B’ (right) EXAMPLE 3. View largeDownload slide Histograms showing the distribution of tempi used by performers in Murphy’s corpus: the first tempo selection for the A and simple time B sections (left); the second tempo selection for the compound meter Presto, section B’ (right) There may only be two distinct tempi in this movement, but there are many changes of meter (cf. Exx. 1 and 2), each with its own attractor. As such, a secondary analysis is needed to assess the extent to which conductors’ tempi change from section to section (despite their being notated at a constant tempo). This requires collecting new data and an opportunity to look at recordings not included in Murphy’s study and to extend the total number addressed. Accordingly, the data set newly collected for this study makes use of six recordings chosen from outside of Murphy’s collection. Of these, three are “modern” (since 2000) and three are “historical.” The results are set out in Example 4 on which there is one data point for each recording (or attractor) and section. Diagonal lines are used to connect those points and lead the eye; they do not imply smooth transitions between these tempi (nor, for that matter, is the x-axis proportional to duration). For the sake of comparison, the tempi are given by tactus beats corresponding to the quarter note (A section), half note (B section), and dotted-quarter note (B’ sections). Thin gridlines are also included for reference: two horizontal lines for the average tempi in Murphy’s corpus, and five vertical lines for the start of each major section (A, B, A, B’, and A). Full details of the recordings are provided in Appendix 2: Examples 7 and 8 contain the tempo data and recording information respectively. EXAMPLE 4. View largeDownload slide Tempo change by section for the six newly studied recordings and the “attractor” tempi for the two most plausible metrical structures EXAMPLE 4. View largeDownload slide Tempo change by section for the six newly studied recordings and the “attractor” tempi for the two most plausible metrical structures As with Murphy’s corpus, the method has been for the author to tap along to the excerpt with basic software that converts tapping into an average tempo count. Several steps have been taken to ensure accuracy and objectivity. I outline the basic method here and provide a summary as part of the data in Example 7. To ensure consistency, three trials were conducted for each section in each recording. The representative ranges used were selected to give the largest continuous part of the section (to provide the most representative averages), while still avoiding any areas with clear rallentandi, or pauses (notated or otherwise).41 Obviously, important exclusions include the first beat of a new start as the onset cannot be reliably predicted. For objectivity, I looked away from the BPM counter during the trial (to avoid attempting to emulate prior results) and conducted a cross-check of my method both with Murphy’s (using a sample of his recordings) and with an automatic beat counter: the “tempo and beat tracker” plug-in of the Sonic Visualiser software package.42 Sections were once again set at a consistent length before I proceeded to run the software, manually check the results for “octave” and phase errors (still relatively common in automatic beat extraction), export the annotation, and take the average. In short, the new results match exactly with the semiobjective measurement of Sonic Visualiser. They also match Murphy’s results with the singular exception of Sanderling’s second (B´ section) tempo choice. The summarized data is provided in Appendix 2 (Ex. 9), with the anomalous value identified by an asterisk. The resulting tempi are discussed in the course of the text in relation to possible attractors. In the meantime, Example 4’s illustration enables a broad summary. As shown by the roughly horizontal lines across the first seven sections (an A–B–A unit), the recordings do broadly preserve the notated equivalence across the A and first Presto (B) section. The tactus of the second, compound time B´ section—from B´(iv)—is faster in all recordings, and again the tempo equivalence within this section is broadly preserved in most recordings. Knappertsbusch’s recording very clearly does not preserve consistent within-section tempi. This recording is the most variable in tempo from section to section and often follows the vicissitudes of the attractor tempi more closely than do the others, as will be discussed below. 4. ANALYSIS: METER AND TEMPO, SECTION BY SECTION The primary changes of metrical structure in this movement appear in the summarizing Examples 1 and 2, above. This analysis focuses on the three-way relationship between those metrical structures, the attractor tempi suggested for them, and the tempi used in the recordings studied (with a primary focus on Murphy’s averages). In total, the analysis traces a process of (non-)alignment that stands to comment on the performers’ view of this movement’s structure. It outlines each of the metrical forms encountered and discusses what the attractor tempo model has to say about these selections. Professional performers (such as those represented in the corpus) are almost certain to select tempi on the basis of a consideration of the whole movement, or even multi-movement work in question. Most trivially, this involves choosing a speed at which the fastest passages are feasible, but it applies as well to preferred tempi for phrasing and other factors. This analysis engages some elements of the interpretative form of that process, including whether and which sections may be optimized for salience and, relatedly, the presence or absence of hypermetrical levels as part of the meter. Level usage is debated throughout in reference to perceptual limits on the upper length of what can be considered “metrical” (cf. discussion in Gotham 2015 after London 2012) and to all relevant musical parameters including grouping boundaries delimited by changes in motif, phrase, harmony, and orchestration. 4.1. THE A SECTIONS As Example 1 shows, the minimum number of metrical levels that are unequivocally represented by the notation of the A section are those of the eighth note, quarter note, and dotted-half note ( 43 measure): a metrical structure in the ratio ⟨1, 2, 6⟩.43 The “attractor tempo” for this three-level metrical scheme is given at quarter note = 85.2 bpm, and is shown on the lower attractors line on Example 4. This is relatively close to the average of the conductor corpus (mean 90, median 91 bpm) already, though it is not the whole picture—we must consider the possible inclusion of one or more hypermetrical level. There is a strong case for the presence of a two-measure level here. Binary pairing would be the normal assumption for classical phrasing of this kind, and the relevant pulse length for this two-measure level is under 4 seconds, thus falling within the acceptable limits for pulse projection. Finally and most importantly, the two-measure level is strongly borne out by the musical design throughout the prototypical first A section, and almost consistently in the reprises thereof. Unsurprisingly, Brahms makes artful changes to this hypermetrical structure later in the movement,44 but these alternations are sufficiently few and occur sufficiently late that they must be viewed as deviations from a well-established two-measure norm. Thus the two-measure level is strongly represented. By contrast, additional hypermetrical levels are equivocal both analytically and perceptually in terms of projecting very slow pulses: the possibility of a consistent four-measure level is quickly ruled out (especially at mm. 11 and 23), and the prospects for projecting a pulse of ca. 8 seconds are remote. Murphy’s analysis accords with this view; he includes the two-measure level but none higher.45 Adding the two-measure hypermetrical level into the consideration gives a metrical structure of ⟨1, 2, 6, 12⟩ and an attractor tempo of quarter note = 90.4 bpm, which aligns exactly with the average of the conductor corpus (mean 90, median 91). This is shown on Example 4 by the alignment of the upper attractors line (Attractors 2) with the lower of the two thin, solid gridlines (the corpus average), and is highlighted by the first large arrow. The spread of individual recordings in Example 4 gives a sense of the range in the new set. For this opening A section, three of the recordings align very closely, one is slower (Knappertsbusch), and two are faster (Gardiner, Jurowski). Averages for this group are discussed below. 4.2. B(i) SECTIONS: MM. 33–49; 63–90 The B section begins (mm. 33–49) and is reprised (mm. 63–90) with simple binary meters that are strongly articulated with pulse levels from the eighth note right up to the four-measure unit. However, whatever the number of levels in operation here, there is no sense of an alignment between the average tempi (90–91 bpm). For an even number of binary metrical levels (in this case ⟨1, 2, 4, 8⟩ without the four-measure level) the attractor for the half-note level is 70.8 bpm, while for an odd number (⟨1, 2, 4, 8, 16⟩ with that four-measure level) it is 100 bpm.46 On average, then, a tempo has been selected, which is close to an attractor for the A section(s), and far from those of the first Presto (B) section, at least at the start.47 This establishes a tension between the average tempo used in this Presto section and the attractor for its metrical structure. This tension will play out in a most interesting way as the metrical groupings change, specifically as levels of three-groupings are introduced and migrate from the remotest to the most salient levels. In this situation, where the tempo is supposed to remain constant but no longer aligns with any possible attractor, any tempo changes that do take place in performance may assume significance. In the new sample shown on Example 4, four of the six recordings opt for a clearly slower tempo in this B section. If either of the possible attractors is having an effect here, it is possible that the slower, 70.8 bpm (Attractor 1) may be drawing conductors toward a slower tempo. Particularly notable is the strict alignment of the slowest recording (Knappertsbusch) with the Attractors 1 line across the first four sections (A[i], B[i], B[ii], and B[i]). 4.3. FRAGMENTATION AND TRANSITION. B(ii) AND B(iii); MM. 49FF. AND 91FF The ends of those B(i) sections (mm. 49–51; 91–100) fragment the four-measure level into a two-measure form as part of a transition into new metrical structures. In the first instance, this leads to a regrouping of the two-measure fragment in 3 seconds at m. 51 (Rehearsal A) to give a highest (possible) level of six measures, with the hypermetrical structure 13.48 This is designated B(ii) and set out on the fourth system of Example 2. That highest level (grouping two-measure pulses into sets of 3 rather than 2) marks the first deviation from simple binary grouping in the B section. At the average tempo, this six-measure level equates to a plausible period of approximately 4 seconds, though it is musically equivocal. The formal division is clear—and analytically important, in view of what follows—but those six-measure units are not necessarily a strong candidate for metrical pulse projection. The reader may wish to listen to a sample of different recordings of this passage and consider whether (or how often) they (a) do in fact project a six-measure pulse, even at its second iteration (mm. 57–62); (b) take the change of texture after four measures as indicative of a new start which is cut short (a division of 4 + 2 rather than 6);49 or (c) do not bother with metrically projecting any grouping at a higher level than the two-measure pulse. Assuming option (a), the metrical structure is ⟨1, 2, 4, 8, 24⟩,50 optimized by half note = 73.5 bpm; option (b) presents the nearest attractor tempo to the average tempo used (half note = 100), but it is thwarted as a metrical strategy by the six-measure units; option (c) rejects periods above the two-measure level, resulting in the binary structure ⟨1, 2, 4, 8⟩ much as at the start of the Presto (half note = 70.8). Both viable attractors (73.5, 70.8) are far from the average used (90–91), while the attractor at 100, which is closer to the average is invalidated by the six-measure units. Whatever the listening strategy, either the tempo or the irregular phrasing perpetuates the dissonance here. The second, equivalent two-measure fragmentation (m. 91) leads to another new grouping. As discussed above, m. 63 returns to the binary metrical grouping with which the Presto began. The parallel is strengthened by the reprised material at Rehearsal B (m. 79). Once again, the four-measure level is initially clear, but is then subtly fragmented to two-measure exchanges of a one-measure idea (mm. 93–100). The single measures are then regrouped in 3 s as a retransition to the A section: mm. 101–3, 104–6. This last transitional passage presents a new structure of 42 measures with three-measure hypermeter, which is set out on the fifth system of Example 2 described by the time signature 23, and designated B(iii). According to Brahms’s instructions, this 23 tempo matches exactly the 43 of the A section that follows, and—sure enough—there is also a thematic retransition in the wind and cello parts: indeed, m. 101 can easily be heard as a stronger formal division than 107, contrary to the visual appearance of the notation. As Epstein observes, “Where Brahms changes tempi through a true alteration of beat duration, he invariably sets up some rhythmic example prior to the change, which serves as the reference for the changed pulse itself. The change is thus composed into the music, so to speak.”51 This resonates with the technique of developing variation (briefly mentioned above) and, indeed, to supposedly “later” compositional developments such as metric modulation. The three-measure pulse here is just over 3 seconds at the average speed: an eminently feasible duration for projection, though a much weaker one than the alternative (two-measure) grouping (as at the previous incarnation of this material at mm. 41ff.). The six-measure grouping must be excluded. It is both at the limits of pulse length, and without a second iteration there is little chance to engage it as a meter. Here, we have a new attractor tempo to consider for the grouping (of ⟨1, 2, 4, 12⟩, excluding 24) that equates to a half note = 54.2 (quarter note = 108). This is even further from the average (90–91) than were the binary groupings with which the B section began (attractor half note = 70.8 or 100 depending on the level usage). Thus the tension briefly increases before returning to the neatly attractor-aligned A section at m. 107. 4.4. REPRISE OF A(i) AND OBSERVATIONS FROM THE NEW DATA SET The newly studied recordings suggest that the tempo fluctuates less within the B section, than from A to B (though once again, Knappertbusch’s tempo dives obligingly in the direction of the extremely slow attractors for the B[iii] section just discussed). Returning to the A(i) section at the reprise (m. 107), the new data set suggests a markedly slower tempo for many of the recordings at the reprise (average 87.6—slightly below the attractor of 90.4) than was the case at the opening (average 94.3—slightly above it). While some conductors (Knappertbusch, Jurowski) bring the tempo back up for the A reprise (following a slower B section), even these come “back” to tempi slower than those with which they began. In this A section, reprise is also an area of conspicuously slower tempo in most recordings (not notated). As such, the new data set introduces an additional A(ii) section for mm. 114ff. (Reh. C, and shown on the sixth system of Ex. 2), whose average is 81.7 bpm. As can be seen in Examples 4 and 7, this A(ii) section is slower than any of the tempo-equivalent sections (within A and B) in all but one of the recordings (Knappertbusch being the exception once again). This is a fascinating (and complicated) area in terms of a possible motivation from attractor tempi. The most conspicuous changes to the metrical structure here appear at the level below the tactus, with the addition of triplets (strings, leading to what could be notated as 89), and then dotted rhythms (winds, superimposed on ongoing triplets in the strings). The psychological literature suggests that “there is no such thing as [a perceptual mechanism for] polymeter,”52 but rather that listeners tend to hear one of the subdivision classes as a rhythm in the context of the other as a meter. Accordingly, the two attractor tempi put forward by the model for this moment correspond respectively to a meter of 89 with three levels (tactus = 100, attractors 2, the higher line on Ex. 4) and a binary meter with four levels (tactus = 54.2, attractors 1, lower line). Of course, some conductors may have felt it beneficial to adopt a slower tempo for other reasons. For instance, a slower tempo would allow the simultaneous triplets and dotted rhythms to be more clearly distinguished; at anything other than a very slow tempo, they tend to collapse into a single metrical category. 4.5. B´(IV): MM. 126FF.—THE PIVOTAL MOMENT After the A section reprise, the Presto returns at m. 126, but in 83—that is, with three-unit grouping at the level of the dotted-quarter-note beat. This section is labeled as an altogether new C section in some accounts, as discussed above. It is also the moment at which a new tempo is to be chosen, the mean and median tempo for the conductor corpus being 119 and 118 respectively. In Example 4, this tempo (118.5) is given as the upper (other) horizontal gridline, and the extent to which the new recordings fit with this can be seen at a glance (their average comes in at the slightly lower 117, thanks largely to Knappertsbusch’s leisurely interpretation of “presto”). As well as being a significant moment in the traditional formal sense, it is also pivotal in the ongoing process of metrical change in the movement. So far, the single level of ternary grouping (hereafter “three-level”) has migrated from a 13 grouping to a 23, and, via the A section’s 43, now arrives at 83. Simultaneously, this moment begins the same process for a second level of ternary grouping as those 83 measures are grouped in six-measure units exactly as at m. 51.53 The equivalence between the material is made especially clear at m. 132 (refer again to Ex. 2, comparing the seventh and fourth systems—B´[iv] and B[ii]). This generates an “ 818” (to match the 13 above), which will undergo the same process as the three-level migrates to neighbor the beat level as part of a (notated) 89 at m. 188. The proportional scheme for this whole process of three-level migration in the B section can be seen in the “Proportion” column of Example 1 and is clarified diagrammatically in Example 5. In Example 5, numbers are given at the bottom, while 2 seconds and 3 seconds represent proportions between consecutive levels, and arrows indicate the migration described. Levels are aligned according to their exact or nearest durational equivalent, with the various note values, and notated “measure”-levels included for ease of reference. EXAMPLE 5. View largeDownload slide Migration of three-level grouping across the two scherzi EXAMPLE 5. View largeDownload slide Migration of three-level grouping across the two scherzi It may also be significant that this pivot occurs at the midpoint of the movement (m. 126 of 240; ca. 2’40” of ca. 5’20”). This would appear to be center of something significant; as such, the moment could perhaps be read in terms of Agawu’s observation that, “Deep into each of Brahms’s sonata-form movements (and sometimes others as well) is a significant turning point, a moment of reversal that announces closure.”54 Perhaps “reversal” is not quite indicative of the process here, but there is certainly a strong sense of “pivot” and of an extreme boundary point in the metrical expansion, with the migrating three-level occupying both extremes of the structure. Crucially then, what is the attractor tempo for this pivotal moment? The shorter measures make the six-measure level more feasible as a metrical pulse for projection than was the equivalent in the first Presto (m. 51), though similarly unattractive from the perspective of musical phrasing. Including the six-measure level yields a structure of ⟨1, 3, 6, 18⟩ (“ 818”), which is optimized by a measure-level dotted-quarter-note pulse of 142 bpm—faster than all of the performances studied and so clearly not operating as an “attractor” for them (see the upper “Attractors 2” line of Ex. 4). The other viable metrical option (corresponding to “option c” at m. 51) involves neglecting levels above that of the two-measure period. This yields a structure of ⟨1, 3, 6⟩ ( 86) for which the attractor tempo is dotted-quarter note = 118, aligning precisely with the averages of the conductor corpus and the new data set (as shown by the second arrow in Ex. 4). This is the first and only time in the many and varied meters of the B sections that there is a possible structural representation that even comes close to the tempo used. Against that unpromising background, the structural representation in question is analytically strong, and the tempo alignment is as exact as was the alignment between the first tempo choice and the attractor for the A section’s meter. In summary, it would appear that: The initial tempo aligned well with the attractor tempo for a version of the A with two-measure level (only) leading to a divergence from the attractors tempi for the various meters in the tempo-equivalent first Presto (B). The new tempo at the second Presto (B´) also aligns with the attractor for its meter with two-measure level only, leaving higher levels to the domain of form. Thus, the attractor tempo model suggests a possible motivation for the value of the average tempi used, as well as a view of the metrical level usage, beyond that which is self-evident from the notation, which that tempo suggests. This study does not comment on proportional tempo schemes, but the result is compatible with Murphy’s proposition of a 3:4 ratio. 4.6. AFTER THE PIVOT: MM. 156–END. B(v) and (vi); A The rest of the scherzo plays out the process of a “migrating three-level” described above (cf. Ex. 5). During the completion of that process, the new metrical structures are associated with attractor tempi that diverge from the tempo used. At m. 156, the two-measure level continues to operate, but the higher level moves from three-grouping to two-grouping. In addition to the motivic parallel connecting m. 156 with m. 132 and m. 51 (as shown in Ex. 2), this continuity at the two-measure level (but no higher) provides another possible motivation for the ⟨1, 3, 6⟩ (without 12) representation of mm. 126–55. This would unite the best part of the section (the two-measure level operates from mm. 126–88), while an emphasis on the change of higher-level structure would divide the section in two at m. 156. However, the four-measure level at m. 156 is an eminently salient pulse and is strongly articulated in the music, especially by changes in harmony and orchestration. Such is the strength of the musical articulation that incorporating this four-measure ( 812) metrical level is almost inevitable, forcing a divergence of metrical structures from attractor tempi for the passage. The attractor without the four-measure level ⟨1, 3, 6⟩ continues to be 118 bpm, while the equivalent including it ⟨1, 3, 6, 12⟩ would be the rather extreme 185 bpm. This is even further than the attractor for ⟨1, 3, 6, 18⟩ (a possibility for mm. 126–55). It seems that in this case, a form of tempo-metrical tension is (somewhat paradoxically) the path of least resistance here. As previously, the newly studied recordings do broadly honor the notated tempo consistency across this B´ section, though once again there is a noticeable tendency to slow down (see the averages of Example 7, and the recordings by Jansons and Jurowski in particular). This is consistent with the very slow attractors for B´(vi), at which point four of the recordings adopt a slow tempo.55 This moment (B´[vi], m. 188) is where the three-level migration reaches fulfillment, with the two three-unit levels finally adjacent, as shown in the final column of Example 5. Finally, a metrical retransition even more brilliant than that at m. 101 occurs at m. 190. The final system of Example 2 shows how the motivic content in the upper strings continues across the divide from 89 to 43. This is invariably performed such that the compound beat of m. 189 = the duple beat of 190 (though nowhere is that stated).56 Nevertheless, the motivic parallel is suggestive of a kind of hemiola structure on the 43 versus 86 level. This has been hinted at in the preceding music (see the entry of voices in mm. 180–87) and is also coupled with a much clearer hemiola on the 23 versus 46 level as lower strings and wind parts have onsets on alternate quarter- note beats.57 There is therefore a bewildering range of possible metrical structures to represent, though 23 and 46 are the most probable along with sub-structures omitting those highest levels ( 42 and 43, respectively). The attractor tempi for these meters are all far wide of the average ( 23 coming closest at quarter note = 108), and the rallentando obviously complicates matters further. In short, this section provides a final bout of divergence from the attractors before the final tempo-aligned return to the A section at m. 194. It should be noted that the start of this final A section (m. 194) is a melodic reprise only and not a tonal one, being as it is in the remote key of F♯ major. The tonal reprise of the same material only comes later, at m. 219 (Rehearsal F). One might suppose that tempo would be used to comment on this non-tonic reprise, but the new data set suggests otherwise, with recordings tending to remain constant between the two (excepting Szell and Knappertsbusch). Perhaps tempo is being used to enhance the sense of “false” reprise and the power of the subsequent tonal movement. 5. SUMMARY AND CONCLUSIONS This analysis has examined a three-way interaction between metrical structures (including a close focus on level usage), the set of “attractor tempi” suggested for them by the model advanced in Gotham (2015), and the tempi used in two samples of commercial recordings. The primary observations concern how the average tempi used by performers in a large, preexisting corpus align with the attractors associated with particular representations of important metrical structures in the movement. This story is contextualized and qualified with a new corpus of recordings, with tempo data for each internal section. Formally, just two tempi are to be selected for this movement: one for the Allegretto sections (A) along with the Presto passage, which is proportionally related to them (B); and another for the compound-time Presto section, which occurs later (B´). The average tempo is such that the opening A section aligns with its attractors, and so the various meters in the proportionally related Presto that follows are necessarily divergent from their attractors. The metrical structure of the first Presto changes several times, systematically introducing levels of ternary grouping at the extremes of hypermeter and then bringing them into focus at the most salient levels. The introduction of the second Presto and second tempo at m. 126 marks the midpoint of this long-range metrical process, as well as the middle of the movement. At this pivotal point, there occurs a metrical structure, which completes this process for the first three-unit level and begins this process for the second. Here, the corpus average takes the opportunity of a new tempo to realign this new passage with the attractor associated with its metrical structure, once again leading to divergence from attractors in the metrical structures that follow. Therefore, of all the many metrical structures in the Presto sections, m. 126 is the only one whose attractor tempo aligns with the average tempo of the conductor corpus. The specific alignments that emerge here may stand to illuminate the extent of metrical-level usage by performers of those passages. In both cases, the tempo aligns best with a representation that includes a single hypermetrical level. In the A section ( 43), this marks the highest (almost) consistently tenable level (generating a ⟨1, 3, 6, 12⟩ total scheme), while in the compound-time scherzo, higher levels are possible, but the focus is apparently on just three levels: ⟨1, 3, 6⟩. Motivic parallels with earlier forms of this material (at m. 51) provide a possible explanation for this, as does a possible attraction to continuity between successive metrical structures, to unite large formal sections. Brahms’s variation of the metrical structure ensures that the tempo cannot be aligned with attractors throughout: the meters (and thus attractors) change, while the tempo is expected to stay constant, and so there emerges a pattern of changing tension-relaxation for the tempo-metrical alignment independent of the two “primary” tempi selected. This study assesses the extent to which six recordings do indeed remain constant across these passages and suggests possible attractor-based reasons for why they might diverge as they do in specific sections. In so far as performers do observe the formal tempo continuities in this work, we can interpret their strategy of tempo choice in a number of ways. The fact that the average tempi selected optimize the metrical structure at section beginnings (m. 1, m. 126) leads to the possibility of the cynical interpretation that the performers optimize tempi at the moment of selection for that moment only, without regard for what is coming in later sections. This hardly seems likely to be the case for the commercial recordings studied, which involve professionals at the highest level. A more convincing alternative interpretation is that the A section is optimized at the expense of the first Presto (B) in order to enhance the leisurely, grazioso feel of the A sections, and the more unsettled quality of B (which is fast and changes meter more frequently). The optimization of the ⟨1, 3, 6⟩ structure for the second scherzo may be attractive in that that meter could be representative of a large part of the section (rejecting most hypermetrical possibilities, as discussed). Apart from interpreting other performers’ choices, we might use this heuristic as a part of the basis for planning a performance. To do so, one must address a fundamental question: is it best to employ a tempo that compromises between the various meters involved, or one that prioritizes some meters (as stable forms) and differentiates them from the other (less stable) meters? I would suggest that the answer depends entirely on the piece, but that differentiation is the better strategy in this dynamic, discursive work. This brings us back to an important comment strongly emphasized in the text: “attractor” tempi are not about dictating “correct tempi” for performers, but are, rather, about observing and engaging with the presence of “default” or “easiest” tempi that exist for classes of metrical structures. In this case, the analysis has observed a correlation between those attractors and the averages used. In future work, the attractor tempi model could be used to assess other performance matters, such as why certain tempi are easier to maintain in a given passage than others. These lines of enquiry are important to pursue as temporal matters (of rhythm, meter and tempo) affect the performer “more than any other compositional parameter.”58 In terms of future developments, it would be useful to include in the “attractor tempi” model a quantification of the relative usage of metrical levels, rather than simply asserting that they are present or absent. Volk (2008) provides a context-specific model of metrical weightings through quantification of each metrical position’s usage. Volk’s model is important and encouraging, though it is currently limited to note onsets. Developments to include other considerations would be welcome, especially with respect to parameters such as harmony and orchestration, both of which would appear to be significant articulators of metrical structure in this piece. A connected performance analysis incorporating all of these parameters systematically would take this line of enquiry to the next level. Footnotes 1 See, for instance, James (1890), The Principles of Psychology, Vol. 2. 2 See especially London (2012) for a summary. 3 Hasty (1997, 183). 4 See London (2012) once again, as well as London (2002) and Parncutt (1994), for instance. 5 Lerdahl and Jackendoff (1983). 6 See Appendix 1 for a more technical explanation of how these values are reached from the “Attractor Tempos” model (Gotham 2015). The bracketing convention of angle brackets for sets of pulse lengths, and square brackets for proportions is also adopted from Gotham (2015) and used here throughout. 7 I am grateful to Fernando Benadon for sharing this data. 8 See, for instance, Epstein (1995) and Murphy (2009) as discussed in the main text. 9 Please note that Ex.1 is a point of reference for much of the discussion which follows and, thus, includes a great deal of other detail that will become relevant in due course. See Murphy (2009, 18ff.) for another specifically metrical analysis of this movement, especially his Fig. 9 for comparison with Ex. 1. 10 Williams (1909, 226). 11 Komma (1967, 449). 12 McClelland (2010, 267). Others identify the “intermezzo” as an alternative kind of norm with which Brahms “replaced the symphonic scherzo” (Swafford 1998, 440). For Pascall, “Stylized dance-types are brought into an overall rondo form” (2013, 39). 13 Brinkmann 1995, 164. Note that the change of tempo in this context would ordinarily be given by a lower metronome value for a shared tactus level, whereas Brahms achieves the same result by the arrangement of metrical levels. This speaks to a difficulty we have in discussing even such basic concepts as “fast” and “slow”—a difficulty that the attractor tempi help to address. 14 In the quintet: A ( 43), B ( 86), A ( 43), B′ (cut-common time), A ( 43); while in the symphony: A ( 43), B ( 42), A ( 43), B′ ( 83), A ( 43). By contrast, the Op. 100 Sonata movement alternates 42 Andante with 43 Vivace: Andante–Vivace–Andante–Vivace di più–Andante–Vivace (short). 15 Pascall observes Brahms’s “interest in radical middle-movement forms” (2013, 39). 16 Frisch (2003, 80). 17 The movement is “thoroughly infused with variation technique” (McClelland [2010, 267]). Compare, for instance, mm. 1–2 with 33–34 and 176–77. The same motive is reprised in the three different metrical contexts ( 43– 42– 83). Indeed, this motive is an inversion of that which opens the symphony and even dominates the Finale of preceding Symphony No. 1. 18 Schoenberg’s origination of the term is often associated with his iconic “Brahms the Progressive” (1951, 398–441), but see Severine Neff’s 1994 translation of Schoenberg’s 1917 Coherence, Counterpoint, Instrumentation, and Instruction in Form for a much earlier appearance of this term (Schoenberg 1917 [1994], 39–43]. 19 Frisch (1990, 155). 20 See comments from Frisch (2003, 80), Komma (1967, 448−49; quoting Willi Lahl), and Pascall (2013, 39), for instance. 21 Frisch (2003, 80). 22 See Pascall (1976) for a short article assessing the sources, and including an edition of both pieces, but see especially Pascall (2013) for a more complete exposition of Brahms’s re-workings of these neo-Baroque pieces into not only the Op. 88 Quintet, but also the Second String Sextet and the Clarinet Quintet. 23 That said, the section has also been likened to the minuet and indeed the Ländler. See Brinkmann (1995, 160), for instance. For Max Kalbeck (an important contemporary and biographer of Brahms himself), the A section is a minuet-like Ländler (“menuettartiger Ländler”), and the two B sections are best described as a “Galopp,” and “tinglingly fast waltz” (“prickelnder Geschwind-Walzer”), respectively (1913, 171). 24 Incidentally, the notational custom of beginning Gavottes on the half measure is apparently no impediment here: in the quintet (where the model is known to have been a Gavotte), the custom is ignored; in the symphony, the use of 42 avoids commitment to any hypermetrical grouping. Incidentally, Kirnberger (a contemporary Baroque musician and theorist) complains about music notated in incorrect metrical displacements. For a modern, translated edition, see Kirnberger (1982). 25 Donington (1973, 243). 26 For an introduction to possible tempo choices in Baroque dances, see Donington’s (1963) translation of Johann Joachim Quantz’s seminal treatise, or the E. R. Reilly 1966 translation for a modern edition of the original commentary; and the summary of treatises by L’Affilard, Lachapelle, Onzembray, and Choquel in Epstein (1995). 27 “Brahms’s Scherzi folgen dem Beethoven’schen Typus, stehen in schneller, oft staccartierter 86” (Komma 1967, 448); stürmisch bewegte (ibid.). 28 Murphy’s corpus (discussed below) assumes that conductors honor these equivalence relationships. (I am grateful to the author for private correspondence on this matter.) The new data set introduced here assesses whether and to what extent that is the case. 29 See Pascall and Struck (2004) or Murphy (2009, 23) for a discussion and list of sources. 30 See Epstein (1995), for instance. 31 Murphy (2009) proposes a 3:4 tempo ratio, asserting that this is followed by 3167 recordings (46 percent) in his corpus (discussed below) to an accuracy within the bounds of a 5 percent “just noticeable distance” (JND). See his Fig. 8, p. 24). That said, the JND is not designed for assessments of this kind, but rather experimental contexts in which stimuli are presented side by side. 32 Sherman observes that Brahms himself “made it clear that he did not believe that there is one ideal tempo for a work” (1997, quoted on 469) and—furthermore—that “if Brahms ever said a kind word about the metronome, we have no record of it” (2003, 99). Brahms’s disinclination is associated with the incompatibility of performance with the cold, mechanical objectivism of the metronome. The clearest extant expression of Brahms’s opinion comes in a letter to George Henschel, dated February 1880: “I myself have never believed that my blood and a mechanical instrument go well together” (quoted in Avins [1997, 559]). 33 See Pascall and Struck (2001) edition of the Symphony in the Neue Ausgabe sämtlicher Werke for full details of designations and sources (270–74 for this movement). 34 See Pascall and Struck (2004, viii). Presumably, Brahms was comparing his own quarter-note tempo (in 43) to Beethoven’s eighth note (in 83). 35 What is more, Beethoven’s notated tempi have been the subject of heated contention since he first committed them to paper. For a considered defense of Beethoven’s markings, see Kolisch’s famous essay (retranslated and reprinted as Kolisch 1993a, b) along with the commentary in Levin (1993). This also includes a rumination of the role of the metronome in a more general sense. 36 Nationalities, within the major European and American centers for Classical music, that is. There is little representation of artists from outside of Europe, and (among conductors at least) even the few exceptions such as Ozawa tend to have been trained and made their name in the West. That said, this may be broadly reflective of the production of public, commercial, professional recordings of this work, at least during the temporal span that this corpus covers. 37 See Leech-Wilkinson, in which the author suggests that “most recorded musicians for whom we have a lifetime’s output seem to have developed a personal style early in their career and to have stuck with it fairly closely for the rest of their lives” (2009, 250), and “that being so, it is more important to know a performer’s birth year than the date of the recording” (250n14). This would move the average back several decades. Assuming very roughly that conductors come of age at twenty, and that the mean age at the time of recording is fifty-five (half way between forty and seventy), would generate a new “conductor coming of age” mean at approximately 1939. 38 Gotham (2015). One brief disclaimer must be passed from Murphy’s analysis (25n21) to the present one. Deviations in tempo are included in the construction of those tempi used. Given the numerous possibilities for local ritardandi, this may make some of the tempi given slower than is accurate. This therefore equates to the first of Gabrielsson’s (1988) types: the “abstract mean tempo.” The second type—“main tempo”—would have been preferable. This should be borne in mind, but does not affect the substance of the claims made in this analysis. 39 “Empirical evidence confirm[s] that morphed (quantitatively averaged) human faces, musical interpretations, and human voices are preferred over most individual ones” (Wöllner et al. 2012, 1390). The paper from which this quote is taken includes a useful summary of relevant literature, but is primarily concerned with demonstrating a new, practical application: that prototypical conductor patterns are more easily followed than any individual’s. 40 This material can be found in Sherman (1997, see especially the table on p. 467; 2003). 41 The shortest sections (with the fewest taps) often gave the most consistent averages. 42 Cannam et al. (2010). I am grateful to Scott Murphy for providing the exact extracts he used. 43 Here, I exclude the fleeting melodic triplet eighth notes from a metrical consideration, though its increased use at Rehearsal C, m. 114 may be considered grounds for investigating a 98 meter there. 44 These are prepared by potential ambiguities in the initial material. The original m. 4 is used as a new phrase beginning at Rehearsal C, preceded by a 4 + 3 measure phrase structure from the start of the A section reprise (mm. 107–13). Similarly, the mm. 8–9 register transfer serves just as well as m. 4 for such a trick, and it, too, is used to ensure that m. 207 (in the second reprise) is unequivocally phrase-commencing. That connection is enhanced in readings that emphasize the m. 207 theme (which first occurs at m. 11—see the second system of Ex. 2) as a parallel of that which commences the sections at m. 51, and 132. See, for instance, McClelland (2010, 268 including 268n3); and Musgrave (1994, 217). 45 See Murphy (2009), summarizing Fig. 9 on p. 27. 46 These were the meters and attractors discussed in relation to the Bach example. 47 That said, these “attractor” values of 70.8 ad 100 broadly delimit the extremes of the tempi used in the corpus (78–104). It is, therefore, possible that those “extreme” conductors are aligning with attractors for this B section rather than the A section, thus creating an alternative pattern of tension-relaxation for this movement in those individual cases. 48 As in Ex. 1, inverted commas are used here to indicate a time signature that would express the full metrical structure for the note values used but that is not actually notated in the score. 49 This is the view of Williams (1909), for instance. 50 Eighth notes continue to form the faster metrical period involved. Sixteenth notes are only weakly used, and (at the average tempo) are shorter than 0.1 seconds, ruling them out as a metrically useful pulse level. 51 Epstein (1990, 206–10). 52 London (2012, 50). See Polak (2010, nn9–10) for further references rejecting polymeter, particularly in the context of Jembe music. 53 This process could be conceived as an intra-structural-level form of the latent-emergent-manifest principle that Rink invokes in the context of a related Brahms analysis (1995, 273). 54 Agawu (1999, 136). 55 The exceptions are Stokowski who remains broadly constant, and Szell who moves slightly faster. 56 For Epstein, this is “implicit” (1990, see the diagram on p. 208), and no more or less problematic than the (highly dubious) beat equivalence of m. 126, which he marks in the same way. 57 The bass part of this hemiola is also included in Ex. 2. 58 Rink (1995, 25). APPENDIX 1: ATTRACTOR TEMPI This appendix summarizes the mathematical structure of the Attractor Tempi model. Please refer to Gotham (2015) for full details, illustrative diagrams, and a discussion of how the model attempts a balance between accurately corresponding to the data available without “over-fitting” to any individual experiment, and without unduly complicating what is necessarily designed as a heuristic. The model is based on cognitive literature that suggests that: (a) there is a preference for pulses around 100 beats per minute (which equates to an inter-onset interval [IOI] of 0.6 seconds); (b) pulses shorter than 0.1 seconds cease to be metrically useful; and (c) the upper limit for what can be parsed as metrical unit is approximately 6 seconds. To model this salience (S), we need a smoothly continuous curve that accounts for all x-values, peaks at 0.6 seconds, and returns negligibly low values for very fast (x < 0.1) and very slow (x > 6.0) pulses. This is achieved with the following logarithmic Gaussian function:   S=exp⁡-(log⁡(x/0.6))20.18. This gives us a metric for the salience of an individual pulse. Attractor tempi are based on combining these values to give a salience value for the whole meter (“metrical salience,” M), which is operationally defined by the combined saliences of all the pulse levels present in the metrical structure. If we represent the fastest pulse as x, and all higher levels by the relevant multipliers (px, qx, rx…), then this salience value is given by   M=∑n=1,p,q,r,…exp⁡-(log⁡(nx/0.6))20.18. Each metrical structure has an “attractor tempo” given by the x-value that maximizes its metrical salience (M). Example 6 sets out these values for some of the simplest meters, with the attractor value given both as an IOI in seconds for the fastest level, x (in the third column), and as a tempo in beats per minute (“bpm”—60/nx, the last column) for a central level near 100 bpm that might act as the tactus. EXAMPLE 6. View largeDownload slide Some common meters and their attractor values EXAMPLE 6. View largeDownload slide Some common meters and their attractor values APPENDIX 2: TEMPO DATA This appendix provides data for the additional recordings used in this study (such as the commercial release information) as well as methodological matters concerning the collection of that new data and a formal test of compatibility among the data collecting methods (as discussed in the main text). EXAMPLE 7. View largeDownload slide Tempo data (and methodological information) for the recordings added in this study EXAMPLE 7. View largeDownload slide Tempo data (and methodological information) for the recordings added in this study EXAMPLE 8. View largeDownload slide Information about the newly studied recordings EXAMPLE 8. View largeDownload slide Information about the newly studied recordings EXAMPLE 9. View largeDownload slide Summarized data for the methodological cross-check of Murphy and Gotham’s tempo judgments against the results from Sonic Visualiser EXAMPLE 9. View largeDownload slide Summarized data for the methodological cross-check of Murphy and Gotham’s tempo judgments against the results from Sonic Visualiser NEWLY COLLECTED DATA FOR THIS STUDY Works Cited Abdy Williams C. F. 1909. The Rhythm of Modern Music . London: Macmillan and Co. Agawu Kofi. 1999. “Formal Perspectives on the Symphonies.” In Cambridge Companion to Brahms . Ed. Musgrave Michael. 133– 55. Cambridge: Cambridge University Press. Google Scholar CrossRef Search ADS   Avins Styra. 1997. Johannes Brahms: Life and Letters . Oxford: Oxford University Press. Benadon Fernando, Zanette Damian. 2015. “ A Corpus Analysis of Rubato in Bach’s C Major Prelude, WTC I.” Music Performance Research  7 ( 1): 1– 26. Brinkmann Reinhold. 1995. Late Idyll: The Second Symphony of Johannes Brahms . Cambridge, MA: Harvard University Press. 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A., Parkinson Jim, Hove Michael J., Keller Peter E. 2012. “ The Perception of Prototypical Motion: Synchronization Is Enhanced with Quantitatively Morphed Gestures of Musical Conductors.” Journal of Experimental Psychology: Human Perception and Performance  38 ( 6): 1390– 1403. Google Scholar CrossRef Search ADS PubMed  © The Author(s) 2018. Published by Oxford University Press on behalf of The Society for Music Theory. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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Music Theory SpectrumOxford University Press

Published: May 2, 2018

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