\'Attractor Tempos\' in Brahms 2/III

‘Attractor Tempos’ in Brahms 2/III Mark Gotham [Pre-print. Article to appear in Music Theory Spectrum 40/1, 2018.]

Abstract This analysis assesses tempo choice in the third movement of Brahms’ Second Symphony. It is shown that at key moments of the movement, the average tempos used in a sample of commercial recordings align closely with those suggested by a new ‘attractor’ model of cognitively preferable (easiest) tempos. Between these stable checkpoints, the movement moves among a variety of other metrical structures at (formally) the same tempo, each with their own attractor tempos. These necessarily align less well with the tempos used, thus providing a pattern of relative tensionrelaxation across the whole which the new model helps to elucidate.

1

Introduction to ‘Attractor Tempos’

Ever since William James’s pioneering work in the nineteenth-century (James 1890), 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 (see especially London (2012) for a summary), 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’ (Hasty 1997, p.183). You may wish to test the idea pulse preference for yourself 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 2 seconds (without mentally sub-dividing) and it is likely that you will have fared less well: 2 seconds is usually found to be a harder duration to project than 1. The ramifications that this individual pulse preference might have for tempo preference in (multi-levelled) metrical structures have been alluded to in the literature,1 and Gotham (2015) develops a theoretical model of ‘attractor tempos’ for meter-specific tempo preference, systematically deducing those tempos at which different meters may be most easily grasped and entrained to. This is achieved by adopting the view of meter as an interaction between coinciding periodic pulse levels in simple proportions (Lerdahl & Jackendoff 1983), and modelling 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 (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). This manifests in different ways for different metrical structures, but the principle remains: for any meter, the attractor tempo 1

See London (2012) once again, as well as London (2002) and Parncutt (1994), for instance.

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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 towards the tempo which optimizes salience in that context (‘the attractor tempo’). These attractor tempos 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 tempos which are ‘correct’, and which therefore ‘should’ be used. Rather, the attractor tempos 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 tempos to expressive ends. The goal is to engage with a form of relative ‘tempo dissonance’ (divergence from attractor tempos) and its use as a compositional parameter.

Attractor tempos 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 heuristic (in analysis). This analysis tests that heuristic claim, using Brahms 2/III as a suitably challenging case-study. To do so, it necessarily puts asides questions over the fundamentals of the model. It is assumed for present purposes that the notion of attractor tempos 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 in relation to it. The focus of the article is therefore 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 tempos predicted by the model, the tempos actually used (whether in an individual recording, or in a corpus average), and the metrical structure of the work or passage (including level usage). 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. The model 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 tempos used in a recording may (theoretically) be objectively verified, though summarising 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 interonset interval values for ‘steady-state’ passages deliberately selected to avoid areas of expressive variation in the timing. Further discussion of method follows below at 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 judgement. Reliance on such analytical judgement might be seen as a shortcoming of this analysis (at least from the systematic musicology perspective). These decisions are readily disputable (as with any analysis), though I have attempted to set out the analysis in such a way that readers may have a different view of level usage and 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 in that 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.

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Attractor tempos 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 ‘48’ is metrically much simpler than the Brahms in that there are no (explicitly notated) tempo or metrical changes in the work, and that the piece does not obviously 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 pulse levels of h1, 2, 4, 8i and proportional values of [2,2,2], for which the attractor tempo model suggests a quarter note of 70.8 beats per minute (‘bpm’).2 However, chords changes take place at the whole note (measure) level, which would suggest a metrical structure with an extra level – pulse levels of h1, 2, 4, 8, 16i, and proportions of [2,2,2,2] – for which the attractor tempo is quarter = 100bpm. The model thus suggests two strategies for ‘moderate’ tempo in this work: 70.8bpm for metrical levels up to the half note level (the figural pattern), versus 100bpm 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 balanced 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: h1, 2, 4, 8, 16i is centred by positioning the 4-level (quarter note) in the centre at 100 bpm, while h1, 2, 4, 8i keeps its 2- and 4- levels equidistant from that centre, leading to the slower attractor tempo. For illustration, try singing, playing, or just imagining this piece to see what tempos you find ‘easiest’ or most ‘natural’. First, try emphasising half note ‘beats’, then focus on whole note ‘beats’. The model expects you to find it easier to project whole note beats at a faster tempo. 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 & Zanette (2015) provide a corpus of 48 recordings of this work,3 and the first 100 onsets (sixteenth note, equating to 25 quarter note beats) provide a relatively ‘steady-state’ section in most recordings. The average tempo for this section is 69.3bpm. This in indeed faster than the average tempo for the whole work (67.1bpm), and it does corresponds well to other steady-state passages, such as the second measure alone (69.7bpm). The performers’ average preference of 69.3bpm corresponds closely to the attractor tempo option of 70.8bpm, and thus to a meter with the half note (figure) as its highest level.

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Introduction to Brahms 2/III

This 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 consonance / dissonance across the whole. The third movement of Brahms’ Second Symphony 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.4 2

Once again, see Appendix 1 for a more technical explanation of how these values are reached. I am grateful to Fernando Benadon for sharing this data. 4 See, for instance, Epstein (1995) and Murphy (2009) as discussed in the main text. 3

3

Tempo 1

2

? 1

Section A(i) B(i) (ii) (i) (iii) A(i) (ii) B’(iv rel. ii) (v) (vi) (vii) A(i)

measures 1–32 33–50 51–62 63–100 101–106 107–113 114–125 126–155 156–187 188–189 190–193 194–240

Meter 3/4 (‘6/4’) Binary ‘2/2’ (‘3/1’) Binary 2/4 (‘3/2’) 3/4 (‘6/4’) 3/4 or ‘9/8’ ‘6/8’ (‘18/8’) ‘6/8’ (‘12/8’ 3/8 (‘9/8’) ‘6/4’ or (‘3/2’) 3/4 (‘6/4’)

Pulse levels h1, 2, 6(, 12)i h1, 2, 4, 8(, 16)i h1, 2, 4, 8(, 24)i h1, 2, 4, 8(, 16)i h1, 2, 4(, 12)i h1, 2, 6(, 12)i h1, 2, 4, 12i or h1, 3, 9i h1, 3, 6(, 18)i h1, 3, 6(, 12)i h1, 3(, 9)i h1, 2, 6(or4), 12i h1, 2, 6(, 12)i

Proportion [2,3(,2)] [2,2,2(,2)] [2,2,2(,3)] [2,2,2(,2)] [2,2(,3)] [2,3(,2)] [2,2,3] or [3,3] [3,2(,3)] [3,2(,2)] [3(,3)] [2,3,2] / [2,2,3] [2,3(,2)]

Attractor ♩ = 85.2 (90.4)

= 70.8 (100)

= 70.8 (73.5)

= 70.8 (100)

= 50 (54.2) ♩ = 85.2 (90.4) tactus = 54.2 or 100 ♩‰ = 118 (142) ♩‰ = 118 (185) ♩‰ = 57.8 (100) ♩ = 90.4 (108) ♩ = 85.2 (90.4)

Note 2-measure level strong. 4-measure until 49 Sixteenths sub-metrical Rel. b.33. 4-measure until c. 91 3-measure iterations of motive

3/8 form of b.51 Compare b.63 Compare b.101 Hemiola implication

Table 1: Form and meter in the movement. Allegretto grazioso (Quasi Andantino) and Presto ma non assai are designated by ‘A’ and ‘B’ respectively. Sub-sections 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 whole structure under discussion, including hypermetrical levels above (longer than) the notated meter. 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 hypemetrical level is less preferable (indeed, more often the opposite is true); the notation is merely used to facilitate the reading of this table. 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 b.126. This analysis will argue that the choices made by conductors tend towards the attractor tempos for the metrical structures which begin these sections, at the expense of the other metrical structures (with different attractor tempos) which follow. Brahms’ numerous changes of meter generate changing attractor tempos; no single tempo can correspond to all of these. This precludes alignment with attractor tempos throughout, and it ensures a process of change in this parameter that plays out as a pattern of relative tempo consonance / dissonance (or 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.

Form in this movement (and in some other, relevant pieces) The movement appears to divide neatly into five parts, with Allegretto grazioso (Quasi Andantino) – A, and Presto ma non assai – B, sections alternating (ABABA). Table 1 outlines these broad sections and the sub-sections given by changes to the metrical structure (the most useful and relevant basis of subdivision for the purposes of this tempo-metrical analysis).5 Figure 1 illustrates each sub-section with representative musical material (predominantly melodic). 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 (Abdy Williams 1909, p.226), as ABA’CA” (Komma 1967, p.449), and as ABA’B’A” (McClelland 2010, p.267).6 However, according to that expectation, the intervening ‘trios’ would 5

Please note that the table 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) p.18ff. for another specifically metrical analysis of this movement, especially his Figure 9 for comparison with this table. 6 Others identify the ‘intermezzo’ as an alternative kind of norm with which Brahms ‘replaced the symphonic scherzo’, (Swafford 1998, p. 440; my emphasis). For Pascall (2013), ‘Stylized dance-types are brought into an overall rondo form’ (p.39).

4

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A section (i): bar 1 ff. 3/4 x 2 bar hypermetre. j j œfi œfi

#3 & 4œ

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A cont.: bar 11 ff. No metrical change. 'Sarabande' rhythm.

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# 2 . . . >. & 4œ œ œ œ

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B section (i): bar 33 ff. 2/4 x 2 x 2 (binary metre up to 4-bar level, at least initially).

p leggiero

# 2 >Ϫ & 4

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j œ.

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B(ii): bar 51 ff. 2/4 x 2 x 3? (= '3/1').

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B(iii): bar 101: 2/4 x 3 (= '3/2').

pp

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A(ii): bar 114. Triplets versus duplets.

∑

B # 43 œ

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Strings (+ 8va, 8vb) 3 p

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B(iv): the 'pivotal moment' from b.126. 3/8 x 2 x 3? (= '18/8'). Compare this excerpt (b.132-7) with B(ii).

p

#3 & 8 bœ. nœ.

#œ. J

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B(v): bar 156 ff. 3/8 x 2 x 2.

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B(vi): bar 188-9. Explicit 9/8 metre (compare motif of iii)

dim.

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. . . . . œ œ Œ™ 43 Œ #œ #œœ #œœ # Œœ Œ . . . . .œ . .j œ n œ # œ. œ. #œ. #œ. 43 #œœ ‰ Œ nœ Œ

. . nœ #œ œ . . œ Œ #œ

. œ Œ œ #œ œ . . .

B(vii): bar 190ff. 3/4 with hemiola (rel. iii and vi)

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Figure 1: Representative melodic material and metrical contexts. Excerpts are notated in the original time signatures (Brahms’ own); comments above identify the larger metrical units involved in the debated hypermetrical levels (referred to in Table 1 and throughout).

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ordinarily be slower, and so this movement has also been described as a ‘back to front scherzo and trio’(Brinkmann 1995, p.164): a kind of triple trio with two scherzos (sic, rather than the more usual way around).7 Here a parallel is often drawn with two other works by Brahms which exhibit a similar A-B-A-B-A pattern of slow-fast-slow-fast-slow in their central movements: the op.88 quintet, and the op.100 violin sonata. The tempo-metrical arrangement of the op.88 movement especially bears an uncanny resemblance to the present one: both have A sections in 3/4 and B sections of which one occurrence is in simple time and the other is in a compound meter.8 According to a model of symphonic form, this could be viewed as an elision of the second, slow movement (accounting for the anomalously slow A section) and the third (represented by the alternating form and the A section’s triple time).9 However, of these three works, it is the symphony which remains in four movements (retaining a separate slow movement), and the chamber works which have three. What is more, even the basic, five-part formal division discussed has been accused of underplaying the ‘interpenetration’ of material between the sections (Frisch 2003, p.80). The connection between sections is such that some argue the case for variation form.10 Continuous variation is also intimately linked to that archetypally Brahmsian device, the ‘developing variation’.11 Importantly for the present purposes, analysts are keen to note the inclusion of tempo and meter as part of this technique. Frisch notes that it is common to see the ‘mobile bar line . . . linked to both motivic development and formal articulation’ in this way (Frisch 1990, p.155).

3

Tempo choice

3.1

Contextual / notational matters

Possible historical models for tempo selection The fact that the main variations in this movement are achieved through rhythmic-metrical transformation gives motivation to the present tempo-metrical analysis, and it has also led some scholars to draw an alternative formal parallel with the baroque suite.12 This forms part of a wider range of apparent retrospection in this movement which Frisch notes to be ‘particularly striking after the Adagio [second movement], one of the least historically retrospective movements in all of Brahms’ (Frisch 2003, p.80). The formal parallel with the op.88 quintet may be significant in this light as Pascall has shown the A and second B sections of that work to be based on pre-existing movements explicitly identified as ‘sarabande’ and ‘gavotte’ by the composer.13 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 7

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 which the attractor tempos help to address. 8 In the quintet: A (3/4), B (6/8), A (3/4), B’ (cut-common time), A (3/4); while in the symphony: A (3/4), B (2/4), A (3/4), B’ (3/8), A (3/4). By contrast, the op.100 sonata movement alternates 2/4 andantes with 3/4 vivaces: andante – vivace – andante – vivace di pi` u – andante – vivace (short). 9 Pascall observes Brahms’ ‘interest in radical middle-movement forms’ (Pascall 2013, p.39). 10 The movement is ‘thoroughly infused with variation technique’ (McClelland 2010, 267). Compare, for instance, b.1–2 with 33–34 and 176–7. The same motive is reprised in the three different metrical contexts (3/4 – 2/4 – 3/8). Indeed, this motive is an inversion of that which opens the symphony and even dominates the finale of preceding symphony no. 1. 11 See Schoenberg’s origination of the term in his iconic ‘Brahms the Progressive’ (Schoenberg 1951, pp.398–441) and also Frisch (1990) among more recent scholarship. 12 See comments from Frisch (2003) p.80, Komma (1967) p.448-9 (quoting Willi Lahl), and Pascall (2013) p.39, for instance. 13 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’ re-workings of these neo-baroque pieces into not only the op.88 quintet, but also the second string sextet and the clarinet quintet.

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in b.11 could be read as a textbook sarabande rhythm (see the second system of Figure 1, noting the 2-3-1 rhythmic pattern and the associated second beat emphasis).14 And if the op.88 B’ section counts as a gavotte in Brahms’ 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).15 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 tempos in baroque music ‘either from notation or from contemporary evidence’ (Donington 1973, p.243) and the evidence which exists is often extremely inconsistent. Instead, this analysis will concentrate its efforts on the evidence of contemporary performers’ attitudes to tempos in this movement as expressed in their recordings; no more is said about the possibility of baroque models. For an introduction to possible tempo choices in Baroque dances, see Donington (1963)’s translation of Johann Joachim Quantz’s seminal treatise (or Quantz (trans. / ed. E. R. Reilly 1966) for a modern translated edition of the original commentary); and Epstein (1995)’s summary of treatises by L’Affilard, Lachapelle, Onzembray, and Choquel. 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’ Scherzos follow the Beethovenian type: a fast, often staccato 6/8’; accordingly, they operate at a ‘stormy, agitated’ tempo – that is, a fast one.16 Tempo according to Brahms’ notations and comments As mentioned above, Brahms instructs conductors to make two primary tempo choices for this movement, at measures 1 and 126. The half note tempo of the first presto (b.33) should be equal to the chosen quarter note tempo for the opening A section.17 Accordingly, the opening notated tempo is at least theoretically in operation until the (approach to) the second presto (section B’) commencing at b.126. At that point a second tempo selection is to be made for the compoundtime passage beginning there. That second tempo holds until the re-transition to the A section (Tempo 1) which arrives at b.194. Brahms initially indicated that the new dotted quarter 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.18 Despite the withdrawal, this scheme has found support from some,19 and other proportional schemes have also been suggested.20 Brahms does not provide metronome marks for this movement (nor indeed for most of his 14 That said, the section has also been likened to the minuet and indeed the L¨ andler. See Brinkmann (1995) p.160, for instance. For Max Kalbeck (an important contemporary and biographer of Brahms himself), the A section is a minuet-like L¨ andler [‘menuettartiger L¨ andler]’, and the two B sections are best described as a ‘Galopp’, and ‘tinglingly fast waltz’ [‘prickelnder Geschwind-Walzer’] respectively (Kalbeck 1913, III/1, p.171). 15 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 2/4 avoids commitment to any hypermetrcial grouping. Incidentally, Kirnberger (a contemporary baroque musician and theorist) complains about music notate in incorrect metrical displacements. For a modern, translated edition see Kirnberger (trans. / ed. D. Beach and J. Thym 1982). 16 ‘Brahms’ Scherzi folgen dem Beethoven’schen Typus, stehen in schneller, oft staccartierter 6/8’ (Komma 1967, p.448); st¨ urmisch bewegte (ibid.). 17 Murphy’s corpus (discussed below) assumes that conductors honour 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. 18 See Pascall & Struck (2004) or Murphy (2009), p.23 for a discussion and list of sources. 19 See Epstein (1995), for instance. 20 Murphy (2009) proposes a 3:4 tempo ratio, asserting that this is followed by the 31 of the 67 recordings (46%) in his corpus (discussed below) to an accuracy within the bounds of a 5% ‘just noticeable distance’ (JND). See Figure 8 on 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.

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music – his ‘blood’ did not ‘go well’ with the ‘mechanical instrument’),21 though the tempo terms he uses may be instructive, as are the changes he made to them.22 Brahms’ 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 tempos, 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 tempos 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.23 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 this case, eighth note = 120 which is very much on the fast side of any allegretto or andantino. Unfortunately, even if Brahms viewed these two movements as being related in tempo, we don’t know whether he had Beethoven’s 120 in mind for them, or something else altogether.24

3.2

Tempo as expressed in performances of this work

Brahms’ 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 tempos is put to best use. This analysis is based on two data sources: Murphy (2009)’s corpus of recordings, and a new data set focussing on tempo changes by section. Details of both are fully set out in Appendix 2. Murphy’s Corpus Murphy’s corpus (see Table 3) appears to provide a relatively good representation of public, commercial, professional recordings of this work: the sample size is relatively large (67), and the corpus includes a relatively broad representation of recording year (1928–2005, mean 1974) and nationality of both conductor and orchestra.25 This study makes no further comment on the representativeness of the sample. 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.26 21

Sherman observes that Brahms himself ‘made it clear that he did not believe that there is one ideal tempo for a work’ (Sherman 1997, quoted on p.469) and – furthermore – that ‘[i]f Brahms ever said a kind word about the metronome, we have no record of it’ (Sherman 2003, p.99). Brahms’ disinclination is associated with the incompatibility of performance with the cold, mechanical objectivism of the metronome. The clearest extant expression of Brahms’ 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’ (Avins 1997, p.559). 22 See Pascall & Struck (2001)’s edition of the Symphony in the Neue Ausgabe S¨ amtlicher Werke for full details of designations and sources (p.270–4 for this movement). 23 See Pascall & Struck (2004), p.viii. Presumably, Brahms was comparing his own quarter note tempo (in 3/4) to Beethoven’s eighth note (in 3/8). 24 What is more, Beethoven’s notated tempos have been the subject of heated contention since he first committed them to paper. For a considered defence of Beethoven’s markings, see the Kolisch’s famous essay (re-translateed and re-printed as (Kolisch 1993a,b)) along with Levin (1993)’s commentary. This also includes a rumination of the role of the metronome in a more general sense. 25 Nationalities, within the major European and American centres 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 which this corpus covers. 26 See Leech-Wilkinson (2009) 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

8

The corpus provides a useful sense of the range of tempos employed by performers of this movement, and – most usefully for the present purposes – the notion of ‘average tempos’ for the passages in question which can be assessed in relation to the ‘attractor tempos’ of Gotham (2015) (or indeed any other models of reference points such as the suggestions of baroque treatises).27 This analysis focusses on the average tempos partly for practical reasons of space and simplicity, but primarily in the interests of comparing ‘average’ practice with the ‘attractor’ tempos’ ‘defaults’. I reiterate once again that there is no intention to prescribe the ‘correct’ tempos 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 judgements.28 Fittingly, the movement in question appears to lend itself more readily to the notion of average tempo than many of Brahms’ 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 tempos that he identifies for other works.29 Figure 2 sets out the data for the two tempos used by the performers represented in Murphy’s corpus. The distribution of tempos selected by performers are 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 centres on a mean of 91bpm, and a median of 90 (for the quarter note in the A sections, and half note in the first B section). The range is from 78 to 104, and the standard deviation 5.54 (as shown on Table 3). 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.) A new data set There may only be two distinct tempos in this movement, but there are many changes of meter (c.f. Table 1 and Figure 1), each with its own attractor. As such, a secondary analysis is needed to assess the extent to which conductors’ tempos change from section to section (despite their being notated at a constant tempo). This requires the collecting of new data, and so an opportunity to look at recordings not included in Murphy’s study and thus extend the total number that we have collectively 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 Figure 3, and full details of the recordings are provided in the Appendices: Tables 4 and 5 provide the tempo data and recording information respectively. As with Murphy’s corpus, the method has been for the author to tap along to the extract 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 for the rest of their lives’ (p.250), and ‘that being so, it is more important to know a performer’s birth year than the date of the recording’ (fn.14). This would move the average back several decades. Assuming very roughly that conductors come of age at 20, and that the mean age at the time of recording is 55 (half way between 40 and 70), that would generate a new ‘conductor coming of age’ mean at approximately 1939. 27 One brief disclaimer must be passed from Murphy’s analysis (p.25, fn.21) to the present one. Deviations in tempo are included in the construction of those tempos used. Given the numerous possibilities for local ritardandos, this may make some of the tempos given slower than accurate. This therefore equates to the first of Gabrielsson (1988)’s 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. 28 ‘Empirical evidence confirm[s] that morphed (quantitatively averaged) human faces, musical interpretations, and human voices are preferred over most individual ones’ (W¨ ollner et al. 2012, p.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. 29 This material can be found in Sherman (1997) (see especially the table on p.467) or in Sherman (2003).

9

Figure 2: Histograms showing the distribution of tempos used by performers in Murphy’s corpus. Left, the first tempo selection for the A and simple time B sections (mean 91, median 90); right, the second tempo selection for the compound meter presto, section B’ (mean 119, median 118). The measures represent spans of two MM values, and a normal distribution of best fit is included in both cases. part of the data in Table 4. 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 accurate averages), while still avoiding any areas with clear rallentandos, or pauses (notated and otherwise).30 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 objective, automatic beat counter: the ‘tempo and beat tracker’ plug-in of the Sonic Visualiser software package (Cannam et al. 2010).31 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 semi-objective 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 as Table 6, with the anomalous value identified by an asterisk. The resulting tempos are discussed in the course of the text in relation to possible attractors. In the meantime, Figure 3’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. One recording which very clearly does not preserve consistent within-section tempos is Knappertsbusch’s. This recording is the most variable in tempo from section to section, and often follows the vicissitudes of the attractor tempos more closely than do the others, as will be discussed in the text.

4

Analysis: meter and tempo, section by section

The primary changes of metrical structure in this movement have been shown in the summarising Table 1 and Figure 1, above. This analysis focusses on the three-way relationship between those 30 31

The shortest sections (with the fewest taps) often gave the most consistent averages. I am grateful to Scott Murphy for providing the exact extracts he used.

10

Figure 3: Tempo change by section in the six newly studied recordings and in the ‘attractor’ tempos for the two most plausible metrical structures. 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 to imply smooth transitions between these tempos (and nor, for that matter, is the xaxis proportional to duration). For the sake of comparison, the tempos 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 tempos in Murphy’s corpus, and five vertical lines for the start of each major section (A, B, A, B’, and A). metrical structures, the attractor tempos suggested for them, and the tempos used in the recordings studied, with a primary focus on Murphy’s averages. Altogether, the analysis traces a process of (non-)alignment that stands to make a significant statement about the performers’ view of this movement’s structure. The analysis outlines each of the metrical forms encountered and comments on 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 tempos 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 obviously applies as well to preferred tempos 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’ (c.f. 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 Table 1 shows, the minimum number of metrical levels that are unequivocally represented by even the notation of the A section are those of the eighth note, quarter note and dotted half note (3/4 measure): a metrical structure in the ratio h1, 2, 6i.32 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 32

Here, I exclude the fleeting melodic triplet eighth notes from a metrical consideration, though its increased use at figure C, b.114 may be considered grounds for investigating a 9/8 meter there.

11

Figure 3. 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; the pulse length is under 4 seconds, so falls within the acceptable limits for pulse projection; and (most importantly) it 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,33 but these alternations are sufficiently few and occur sufficiently late that they must be viewed as deviations from a well established two-measure norm. So the two-measure level is strongly represented. By contrast, additional hypermetrical levels are equivocal both analytically and also perceptually in terms of projecting very slow pulses: the possibility of a consistent four-measure level is quickly ruled out (especially at b.11 and b.23) and the prospects for projecting a pulse of c.8 seconds are remote. Murphy’s analysis accords with this view: he includes the two-measure level but none higher.34 Adding the two-measure hypermetrical level into the consideration gives a metrical structure of h1, 2, 6, 12i and an attractor tempo of quarter note = 90.4bpm which aligns exactly with the average of the conductor corpus (mean 90, median 91). This is shown on Figure 3 by the alignment of the upper attractors line (Attractors 2) with the lower of the two thin, solid gridlines (the corpus average) and highlighted by the first large arrow. The spread of individual recordings in Figure 3 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: bb.33–49; 63–90.

The B section begins (bb.33–49) and is reprised (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 tempos used (90–91bpm). For an even number of binary metrical levels (in this case h1, 2, 4, 8i without the four-measure level) the attractor for the half note level is 70.8bpm, while for an odd number (h1, 2, 4, 8, 16i with that four-measure level) it is 100bpm.35 On average, then, a tempo has been selected which is close to the close to an attractor for the A section(s), and far from those of the first presto (B) section, at least at the start.36 This establishes a tension between the average tempo used in this presto section, and the attractor for its metrical structure that 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, the tempo changes that take place contrary to the score instruction of equivalence 33

These are prepared by potential ambiguities in the initial material. The original b.4 is used as a new phrase beginning at figure C, preceded by a 4+3 measure phrase structure from the start of the A section reprise (bb.107– 113). Similarly, the b.8–9 register transfer serves just as well as b.4 for such a trick, and it too is used to ensure that b.207 (in the second reprise) is unequivocally phrase-commencing. That connection is enhanced in readings which emphasize the b.207 theme (which first occurs at b.11 – see the second system of Figure 1) as a parallel of that which commences the sections at b.51, and 132. See, for instance, McClelland (2010) p.268 including fn.3; and Musgrave (1994) p.217 musical example. 34 See the summarising Figure 9 on p.27. 35 These were the meters and attractors discussed in relation to the Bach example. 36 That said, these ‘attractor’ values of 70.8 ad 100 broadly delimit the extremes of the tempos 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.

12

may assume significance. In the new sample (Figure 3), 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 towards 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); b.49ff. and 91ff.

The ends of those B(i) sections (b.49–51; 91–100) fragment the four-measure level into a twomeasure form as part of a transition into new metrical structures. In the first instance, this leads to a re-grouping of the two-measure fragment in 3s at b.51 (rehearsal mark A) to give a highest (possible) level of six measures, with the hypermetrical structure ‘3/1’.37 This is designated B(ii) and set out on the fourth system of Figure 1. 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 6-measure level equates to a quite 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 6-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 6-measure ‘pulse’, even at its second iteration (measure 57–62); b) take the change of texture after 4 measures as indicative of a new start which is cut short (a division of 4 + 2 rather than 6);38 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 h1, 2, 4, 8, 24i,39 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 h1, 2, 4, 8i 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 (b.91) leads to another new grouping. As discussed, b.63 returns to the binary metrical grouping with which the presto began. The parallel is strengthened by the reprised material at figure B (b.79). Once again, the four-measure level is initially clear, but is then subtly fragmented to two-measure exchanges of a one-measure idea (b.93– 100). The single measures are then re-grouped in 3s as a re-transition to the A section: bb.101– 3, 104–6. This last transitional passage presents a new structure of 2/4 measures x 3 measure hypermeter, which is set out on the fifth system of Figure 1, described by the time signature ‘3/2’, and designated B(iii). According to Brahms’ instructions, this ‘3/2’ tempo matches exactly the 3/4 of the A section which follows, and – sure enough – there is also a thematic re-transition in the wind and ’cello parts: indeed, b.101 can easily be heard as a stronger formal division than 107, contrary to the visual appearance of the notation. As Epstein1990 observes, Where Brahms changes tempos through a true alteration of beat duration, he invariably sets up some rhythmic figure 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. (Epstein 1990, p.206–10) 37 As in Table 1, inverted commas are used here to indicate a time signature which would express the full metrical structure for the note values used but which is not actually notated in the score. 38 This is the view of Abdy Williams (1909), for instance. 39 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.

13

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 (2 measure) grouping (as at the previous incarnation of this material at measure 41ff.). The 6-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 h1, 2, 4, 12i, excluding 24) which 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 b.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 (b.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 94.27˙ – 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 tempos 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 measure 114ff. (reh. C, and shown on the sixth system of Figure 1), whose average is 81.72˙ bpm. As can be seen in Figure 3 and Table 4, 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 tempos. 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 9/8), 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’ (London 2012, p.50),40 but rather that listeners tend to hear one of the sub-division classes as a rhythm in the context of the other as a meter. Accordingly, the two attractor tempos put forward by the model for this moment correspond respectively to a meter of 9/8 with three levels (tactus = 100, attractors 2, the higher line on Figure 3) 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): b.126ff. – the pivotal moment

After the A section reprise, the presto returns at b.126 but in 3/8 – that is, with three-unit grouping at the level of the dotted quarter note beat. This section is labelled 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. On Figure 3, this tempo (118.5) is given as the upper (other) horizontal gridline, and the extent to which the 40

See Polak (2010) footnotes 9 and 10 for further references rejecting polymeter, particularly in the context of Jembe music.

14

new recordings fit with this can be seen at a glance (their average comes in at the slightly lower ˙ thanks largely to Knappertsbusch’s leisurely interpretation of ‘presto’). 117.1, 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 ‘3-level’) has migrated from a ‘3/1’ grouping to a ‘3/2’, and (via the A section’s 3/4) now arrives at 3/8. Simultaneously, this moment begins the same process for a second level of ternary grouping as those 3/8 measures are grouped in 6-measure units exactly as at b.51.41 The equivalence between the material is made especially clear at b.132 (refer again to Figure 1, comparing the seventh and fourth systems – B’(iv) and B(ii)). This generates an ‘18/8’ (to match the ‘3/1’ above) which will undergo the same process as the 3-level migrates to neighbor the beat level as part of a (notated) 9/8 at b.188. The proportional scheme for this whole process of 3-level migration in the B section can be seen in the ‘Proportion’ column of Table 1 and is clarified diagrammatically in Figure 4. It may also be significant that this pivot occurs at the mid-point of the movement (b.126 of 240 / c.2’40” of c.5’20”). This would appear to be centre 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. (Agawu 1999, p.136) 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 3-levels occupying both extremes of the structure.

3 

2

M easure

(

2 M easure

‰

3

/ (= 3)

M easure

♩

Bar no.

2 ♩ 2  51

2 ♩ 2  101

Time signature

2/4

2/4

3

‰ 2

M easure (

M easure (

3 ♩‰

3 

/3

(107, A)

126

188

3/4

3/8

9/8

(= 2) 



Figure 4: Migration of 3-level grouping across the two scherzi. Bar numbers are given at the bottom, while 2s and 3s 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. Crucially then, what is the attractor tempo for this pivotal moment? The shorter measures make the 6-measure level more feasible as a metrical pulse for projection than was the equivalent in the first presto (b.51), though similarly unattractive from the perspective of musical phrasing. Including the six-measure level yields a structure of h1, 3, 6, 18i (‘18/8’) which is optimized by a dotted quarter note (measure) pulse of 142 bpm – faster than all of the performances studied and so 41 This process could be conceived as a intra-structural-level form of the latent-emergent-manifest principle that Rink invokes in the context of a related Brahms analysis (Rink 1995, p.273)

15

clearly not operating as an ‘attractor’ for them (see the upper ‘attractors 2’ line of Figure 3). The other viable metrical option (corresponding to ‘option c’ at b.51) involves neglecting levels above that of the two-measure period. This yields a structure of h1, 3, 6i (‘6/8’) 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 on Figure 1. 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 tempos 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 tempos 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: bb.156–end. B(v) and (vi); A.

The rest of the scherzo plays out the process of a ‘migrating 3-level’ described above (c.f. Figure 4). During the completion of that process, the new metrical structures are associated with attractor tempos that diverge from the tempo used. At b.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 b.156 with b.132 and b.51 (as shown on Figure 1), this continuity at the two-measure level (but no higher) provides another possible motivation for the h1, 3, 6i (without 12) representation of bb.126–155. This would unite the best part of the section (the two-measure level operates from b.126 right through to b.188), while an emphasis on the change of higher-level structure would divide the section in two at b.156. However, the four-measure level at b.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 (‘12/8’) metrical level is almost inevitable, forcing a divergence of metrical structures from attractor tempos for the passage. The attractor without the four-measure level h1, 3, 6i continues to be 118bpm, while the equivalent including it h1, 3, 6, 12i would be the rather extreme 185bpm. This is even further than the attractor for h1, 3, 6, 18i (a possibility for bb.126–155). 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 honour the notated tempo consistency across this (B’) section, though once again there is a noticeable tendency to slow down (see the averages of Table 4, 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.42 This moment (B’(vi), b.188) is where the three-level migration reaches fulfilment, with the two three-unit levels finally adjacent as shown in the final column of Figure 4. 42

The exceptions are Stokowski who remains broadly constant, and Szell who moves slightly faster.

16

Finally, at b.190 occurs a metrical re-transition even more brilliant than that at b.101. The final system of Figure 1, shows how the motivic content in the upper strings continues across the divide from 9/8 to 3/4. This is invariably performed such that the compound beat of b.189 = the duple beat of 190 (though nowhere is that stated).43 Nevertheless, the motivic parallel is suggestive of a kind of hemiola structure on the 3/4 vs 6/8 level. This has been hinted at in the preceding music (see the entry of voices in bb.180–7) and is also coupled with a much clearer hemiola on the 3/2 vs 6/4 level as lower strings and wind parts have onsets on alternate quarter note beats.44 There is therefore a bewildering range of possible metrical structures to represent, though 3/2 and 6/4 are the most probable along with sub-structures omitting those highest levels (2/4 and 3/4 respectively). The attractor tempos for these meters are all far wide of the average (3/2 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 b.194. It should be noted that the start of this final A section (b.194) is melodic reprise only and not a tonal one, being as it is in the thoroughly remote key of F sharp major. The tonal reprise (of the same material) only comes later, at b.219 (rehearsal mark 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 tempos’ suggested for them by the model advanced in Gotham (2015), and the tempos used in two samples of commercial recordings. The primary observations concern how the average tempos used by performers in a large, pre-existing 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 tempos 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 which follows are necessarily divergent from their attractors. The metrical structure of that 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 b.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 which follow. Therefore, of all the many metrical structures in the presto sections, b.126 is the only one whose attractor tempo aligns with the average tempo of the conductor corpus. What is more, the specific alignments which 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 43

According to Epstein this is ‘implicit’ (Epstein 1990, see the diagram on p.208), and no more or less problematic than the (highly dubious) beat equivalence of b.126 which he marks in the same way. 44 The bass part of this hemiola is also included in Figure 1.

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a representation that includes a single hypermetrical level. In the A section (3/4), this marks the highest (almost) consistently tenable level (generating a h1, 3, 6, 12i total scheme), while in the compound-time scherzo, higher levels are possible, but the focus is apparently on just three levels, h1, 3, 6i. Motivic parallels with earlier forms of this material (at b.51) provide a possible explanation for this, as does a possible attraction to continuity between successive metrical structures, to unite large formal sections. Brahms’ 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, whatever the two ‘primary’ tempos selected. This study assess 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 tempos selected optimize the metrical structure at section beginnings (b.1, b.126) leads to the possibility of the cynical interpretation that the performers optimize tempos at the moment of selection for that moment only, without regard for what is coming in later section. This hardly seems likely to be the case for the commercial recordings studied involving 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 both A’s leisurely, grazioso feel, and the more unsettled quality of B (which is fast and changes meter more frequently). The optimisation of the h1, 3, 6i 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 in the main text). Apart from interpreting other performers’ choices, we might use this heuristic as a part of the basis for planning a performance of such repertoire. To do so, one must address a fundamental question: is it best to employ a tempo which compromises between the various meters involved, or one which 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 which is strongly emphasized in this text: ‘attractor’ tempos are not about dictating ‘correct tempos’ for performers, but rather observing and engaging with the presence of ‘default’ or ‘easiest’ tempos which 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 tempos model could be used to assess other performance matters, such as why certain tempos 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’ (Rink 1995, p.25). In terms of future developments, it would be useful to include in the ‘attractor tempos’ 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.

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References Abdy Williams, C. F. (1909), The rhythm of modern music, London. Agawu, K. (1999), Formal perspectives on the symphonies, in M. Musgrave, ed., ‘Cambridge companion to Brahms’, Cambridge University Press, Cambridge, chapter 6, pp. 133–155. Avins, S., ed. (1997), Johannes Brahms: Life and Letters, Oxford University Press. Benadon, F. & Zanette, D. (2015), ‘A corpus analysis of rubato in Bach’s C Major Prelude, WTC I’, Music Performance Research 7. Brinkmann, R. (1995), Late Idyll: the Second Symphony of Johannes Brahms, Harvard University Press, Cambridge, Mass. ; London. Cannam, C., Landone, C. & Sandler, M. (2010), Sonic visualiser: An open source application for viewing, analysing, and annotating music audio files, in ‘Proceedings of the ACM Multimedia 2010 International Conference’, Firenze, Italy, pp. 1467–1468. Donington, R. (1963), The Interpretation of early music, Faber and Faber, London. Donington, R. (1973), A Performers guide to Baroque music, Faber, London. Epstein, D. (1990), Brahms and the mechanisms of motion: the composition of performance, in Bozarth, ed., ‘Brahms studies : analytical and historical perspectives : papers delivered at the International Brahms Conference, Washington, DC, 5-8 May 1983’, Clarendon Press, Oxford, pp. 191–226. Epstein, D. (1995), Shaping time: music, the brain, and performance, Schirmer, New York. Frisch, W. (1990), The shifting bar line: Metrical displacement in Brahms, in Bozarth, ed., ‘Brahms studies : analytical and historical perspectives : papers delivered at the International Brahms Conference, Washington, DC, 5-8 May 1983’, Clarendon Press, Oxford. Frisch, W. (2003), Brahms: the four symphonies, Yale University Press, New Haven, Conn. ; London. Gabrielsson, A. (1988), Timing in music performance and its relations to music experience, in J. Sloboda, ed., ‘Generative processes in music; the psychology of performance, improvisation, and composition’, Clarendon, Oxford, pp. 27–51. Gotham, M. (2015), ‘Attractor tempos for metrical structures’, Journal of Mathematics and Music 9(1), 23–44. Hasty, C. (1997), Meter as Rhythm, Oxford University Press, New York ; Oxford. James, W. (1890), The Principles of Psychology, Vol. 2, MacMillan, London. Kalbeck, M. (1913), Johannes Brahms, Deutsche Brahms-Gesellschaft, Berlin. Kirnberger, J. P. (trans. / ed. D. Beach and J. Thym 1982), The art of strict musical composition, Music theory translation series; 4, Yale University Press, New Haven. Kolisch, R. (1993a), ‘Tempo and character in Beethoven’s music’, The Musical Quarterly 77(1), 90– 131. URL: http://www.jstor.org/stable/742431

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Kolisch, R. (1993b), ‘Tempo and character in Beethoven’s music (continued)’, The Musical Quarterly 77(2), 268–342. URL: http://www.jstor.org/stable/742558 Komma, K. M. (1967), Das “Scherzo” der 2. Symphonie von Johannes Brahms: Eine melodischrhythmische Analyse, in W. Wiora, L. Finscher & C. Mahling, eds, ‘Festschrift f¨ ur Walter Wiora: zum 30. Dezember 1966’, B¨ arenreiter, pp. 448–457. Leech-Wilkinson, D. (2009), Recordings and histories of performance style, in N. Cook, E. Clarke, D. Leech-Wilkinson & J. Rink, eds, ‘Cambridge companion to recorded music’, Cambridge University Press, Cambridge, pp. 246–262. Lerdahl, F. & Jackendoff, R. (1983), A Generative Theory of Tonal Music, The MIT Press, Cambridge, MA. Levin, T. Y. (1993), ‘Integral interpretation: Introductory notes to Beethoven, Kolisch and the question of the metronome’, The Musical Quarterly 77(1), 81–89. URL: http://www.jstor.org/stable/742430 London, J. (2002), ‘Some non-isomorphisms between pitch and time’, Journal of Music Theory 46(1/2), 127–151. URL: http://www.jstor.org/stable/4147679 London, J. (2012), Hearing in time: psychological aspects of musical meter, 2 edn, Oxford University Press, Oxford. McClelland, R. (2010), Brahms and the Scherzo: Studies in Musical Narrative, Ashgate. Murphy, S. (2009), ‘Metric cubes in some music of Brahms’, Journal of Music Theory 53(1), 1–56. URL: http://www.jstor.org/stable/40606877 Musgrave, M. (1994), The music of Brahms, Clarendon Press, Oxford. Parncutt, R. (1994), ‘A perceptual model of pulse salience and metrical accent in musical rhythms’, Music Perception: An Interdisciplinary Journal 11(4), 409–464. URL: http://www.jstor.org/stable/40285633 Pascall, R. (1976), ‘Unknown gavottes by Brahms’, Music and Letters 57(4), 404–411. URL: http://www.jstor.org/stable/734281 Pascall, R. (2013), Brahms beyond mastery : his Sarabande and Gavotte, and its recompositions, Ashgate, Burlington, VT. Pascall, R. & Struck, M. (2001), Symphonie Nr. 2, D-Dur, Opus 73, Neue Ausgabe s¨amtlicher Werke / Johannes Brahms, Ser. 1, Bd. 2., G. Henle, M¨ unchen. Pascall, R. & Struck, M. (2004), Symphonie Nr. 2, D-Dur, Opus 73, Study Edition following the Neue Ausgabe s¨ amtlicher Werke / Johannes Brahms, Ser. 1, Bd. 2., 2001, G. Henle, M¨ unchen. Polak, R. (2010), ‘Rhythmic feel as meter: Non-isochronous beat subdivision in jembe music from Mali’, Music Theory Online 16(4). Quantz, J. J. (trans. / ed. E. R. Reilly 1966)), On playing the flute, Faber, London. Rink, J. (1995), Playing in time: Rhythm, metre and tempo in Brahms’s Fantasien op. 116, in J. Rink, ed., ‘The Practice of Performance: Studies in Musical Interpretation’, Cambridge: Cambridge University Press, pp. 254–82. 20

Schoenberg, A. (1951), Style and idea, Williams and Norgate, London. Sherman, B. D. (1997), ‘Tempos and proportions in Brahms: Period evidence’, Early Music 25(3), 463–478. Sherman, B. D. (2003), Metronome marks, timings, and other period evidence regarding tempo in Brahms, in M. Musgrave & B. D. Sherman, eds, ‘Performing Brahms: Early Evidence of Performance Style’, Cambridge University Press, pp. 99–130. Swafford, J. (1998), Johannes Brahms : a biography, Macmillan, London. Volk, A. (2008), ‘Persistence and change: Local and global components of metre induction using inner metric analysis’, Journal of Mathematics and Music 2(2), 99–115. URL: http://www.tandfonline.com/doi/abs/10.1080/17459730802312399 W¨ollner, C., Deconinck, F. J. A., Parkinson, J., Hove, M. J. & Keller, P. 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.

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Appendix 1: Attractor Tempos This appendix summarizes the mathematical structure of the Attractor Tempos 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 an heuristic. The model is based on cognitive literature which suggests that: there is a preference for pulses around 100 beats per minute (which equates to an interonset interval (IOI) of 0.6 seconds); pulses shorter than 0.1 seconds cease to be metrically useful; and 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:
2 ! log(x /0.6) S = exp − . 0.18 This gives us a metric for the salience of an individual pulse. Attractor tempos are based on combining these values to give a salience value for the whole metre (‘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
2 ! X log(nx /0.6) M= exp − . 0.18 n=1, p, q, r, ...

Each metrical structure has an ‘attractor tempo’ given by the x-value which maximises its metrical salience (M ). Table 2 sets out these values for some of the simplest meters, with the attractor value given both as an inter onset interval in seconds (x), and as a tempo in beats per minute (‘bpm’ – 60/nx) for a central level (one near 100 bpm) that might act as the tactus. Pulse levels h1i h1, 2i h1, 2, 4i h1, 2, 4, 8i h1, 3i h1, 2, 6i h1, 3, 6i

Proportions (n/a) [2] [2,2] [2,2,2] [3] [2,3] [3,2]

Attractor (x) 0.6 0.424 0.3 0.212 0.346 0.352 0.170

Categorical Attractor? Yes: x = 0.6√ Yes: x = 0.6/ 2 Yes: 2x = 0.6√ Yes: 2x = 0.6/√ 2 Yes: x = 0.6/ 3 No No

BPM x = 100 2x = 70.8 2x = 100 4x = 70.8 3x = 57.8 2x = 85.2 3x = 118

Table 2: Some common meters and their attractor values. The IOI of the fastest level is given by x. All values are given to 3 significant figures and also expressed exactly in relation to the peak pulse salience of nx = 0.6 where possible.

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Appendix 2: tempo data This appendix provides full data for both Murphy (2009)’s corpus, and the additional recordings used in this study. It also provides commercial release information for the recordings used in the newly collected data, as well as methodological matters concerning the collection of that new data and a formal test of compatibility among the data collecting methods (further discussed in the main text).

Murphy’s Conductor Corpus Orchestra BBC Suisse Romande Seito Kinen Staatskapelle Dresden Royal Phil National Bordeaux Aquitaine Philharmonia of London Los Angeles Phil New York Phil Vienna London Phil London San Francisco San Francisco Academy of St. Martin in the Fields Philadelphia Cleveland Israel Phil Amsterdam Concertgebouw Berlin Vienna Berlin Danish Radio Stuttgart Radio London Classical Players Chicago ? London Scottish Chamber

Conductor Bˇelohl´avek Ansermet Ozawa Sanderling Judd Lombard Klemperer Guilini Masur Monteux Alsop Haitink Monteux Monteux Marriner Muti Ashkenazy Mehta Beinum Karajan Kert´esz B¨ohm Horenstein Celibidache Norrington Barenboim Del Mar J¨arvi Mackerras

(cont. overleaf)

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Year 1998 1963 1991 1972 1994 1990 1958 1981 1992 1963 2005 2003 1951 1945 1998 1988 1990 1992 1955 1986 1965 1950 1972 1975 1993 1993 1993 1988 1997

Tempo 1 104 93 98 90 94 93 89 89 95 97 94 93 96 101 99 93 93 95 94 90 98 85 92 93 103 89 88 90 100

Tempo 2 119 106 114 106 113 112 107 107 115 117 115 113 117 124 121 115 115 118 118 113 123 107 116 117 131 113 112 115 128

Berlin Philadelphia Utah Royal Concertgebouw Cleveland Royal Phil Berlin Boston London Phil Boston Mexico City Phil Pittsburgh Berlin Milwaukee New York Vienna Stuttgart Radio Royal Concertgebouw Brusells BRT Houston Vienna Amsterdam Concertgebouw New York Vienna Berlin Boston Chicago NBC Berlin Royal Philharmonic New York London Philharmonic Bamberg Vienna Philadelphia London Philharmonic New York Berlin Mean Median Range Standard Deviation

Abbado Ormandy Abravanel Chailly Szell Pesek Karajan Munch Sawallisch Leinsdorf Lozano Steinberg Harnoncourt Delfs Walter B¨ ohm Schuricht Kondrashin Rahbari Eschenbach Bernstein Mengelberg Rodzinski Kubelik Furtwngler Haitink Solti Toscanini Kempe Beecham Bernstein Weingartner Swarovsky Furtw¨ angler Stokowski Beecham Damrosch Jochum

1988 1966 1977 1989 1967 1992 1978 1956 1999 1965 1994 1963 1997 2005 1956 1976 1954 1975 1990 1993 1994 1940 1950 1959 1952 1991 1979 1952 1957 1959 1962 1940 1970 1945 1929 1946 1928 1951 1974 1975 1928–2005 20.2

89 87 90 93 89 91 94 99 92 85 89 103 87 92 91 87 80 85 89 90 89 93 94 92 88 83 87 89 88 83 91 89 81 86 78 84 81 84 91 90 78–104 5.54

Table 3: Murphy’s Corpus.

Newly collected data for this study (overleaf )

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114 111 115 120 115 119 123 130 121 112 118 136 115 122 122 116 107 113 120 121 122 127 128 126 121 114 121 124 125 118 132 129 119 128 116 128 124 135 119 118 106–136 7.02

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Averages

Historical Recordings

Recent Recordings

Section Method

B(i) all n 36 95 96.5 92.3˙ 71.3˙ 86.3˙ 97.83˙ 89.8˙

B(ii) all y 25 90 96.3˙ 89 71.16˙ 82.6˙ 93.16˙ 87.05˙

B(i) all y 77 92.16˙ 95.16˙ 89.5 72.3˙ 84 95.5 88.1˙

B(iii) all y 13 92 94.5 89 66.6˙ 82.6˙ 95.83˙ 86.7˙

A(i) 107-13 y 22 90.3˙ 98 88 79.6˙ 81 89 87.6˙

(ii) 114-20 y 22 82.3˙ 89 81.3˙ 79.6˙ 73.6˙ 84.3˙ 81.72˙

B’(iv) all n 30 124 125 120.6˙ 99.3˙ 110 123.6˙ 117.1˙

B’(v) all y 33 121.6˙ 127.6˙ 116 100.6˙ 107 124 116.16˙

B’(vi) all y 7 117.3˙ 125.3˙ 113 91.6˙ 107 127.3˙ 113.61˙

Table 4: Tempo data (and methodological information) for the recordings added in this study.

Measures of section First beat? Number of taps Gardiner Jurowski Jansons Knappertsbusch Stokowski Szell

A(i) b.1-20 n 60 100 104.3˙ 90.3˙ 87.3˙ 91.3˙ 92.3˙ 94.27˙

86 84 89 90.05˙

A(i) -216 n 69 93 98 90.3˙

15 92 97.6˙ 91 77 86 84.6˙ 88.05˙

(iii) 219-223

Conductor Gardiner Jurowski Jansons Knappertsbusch Stokowski Szell

Orchestra Orch. Revolutionnaire et Romantique Philharmonic Concertgebouw Berlin Philharmonic National Philharmonic Orchestra Cologne

Year 2007 2010 2004 1944 1977 1958

Label / Code SDG-703 LPO-0043 RCO Live RCD 05002 Urania SP-4207 Columbia CACD0531 Guild GHCD 2404

Note Live Live Released 2002

Table 5: Information about the recordings used in the newly studied recordings.

Scott Murphy Sonic Visualiser Present author

Sanderling A 90 88.2 88

B(iv) 106* 119.6 119.6˙

Stokowski A 78 77.6 77

B(iv) 116 115.3 116

Steinberg A 103 102.3 101

B(iv) 136 133.7 134

Pesek A 91 91 91

B(iv) 119 117.1 117.6˙

Table 6: Summarized data for the methodological cross-check of Murphy and the present author’s tempo judgements, using the objective analysis of Sonic Visualiser as the benchmark. The anomalous entry discussed in the main text is marked with an asterisk (*).

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