Hierarchy and position usage in ‘mixed’ metrical structures

Abstract

Musical metre is commonly formalised in hierarchical terms, and defined or represented on the basis of regular, interacting pulse streams. These hierarchies are often merely asserted a priori, though systematic studies by Palmer and Krumhansl and Prince and Schmuckler suggest that such hierarchies are also strongly manifest in the relative usage of metrical positions: a strong metrical position is also a frequently used one. If position usage provides an insight into metrical structure, then this may provide a way of engaging with a wider range of metrical structures, including ‘mixed’ metres (5/8, 7/8 ) which are excluded by many systems of well-formedness invoked in the canonical studies of ‘simple’ metres (2/4, 3/4 ). This article assesses whether position usage in mixed metres is similarly indicative of the metrical structures asserted for them by music theory. The complete set of Bartók’s solo piano works in 7/8 (223) provides the primary case study repertoire. This is complemented by a look at the more ambiguous 8/8 (323), as part of considering how position usage might relate to distinctions between syncopation and mixed metre. An introductory discussion of sample size also provides new data and observations for larger corpora of common practice music by Bach. Brief analytical comments keep the study grounded in ‘the music’ throughout.

Keywords:

• metre
• rhythm
• Bartók
• corpus analysis
• music information retrieval

Notes

Faculty of Music, University of Cambridge, 11 West Road, Cambridge CB3 9DP, UK.

1 Cf. London (2012)’s ‘many metres hypothesis’ and the emerging discipline of micro-timing analysis which encompasses an array of different musics—Iyer (2002), Moelants (2006), Kvifte (2007), Benadon (2009), and Polak & London (2014).

2 See London, Polak, and Jacoby (2016) for a discussion of this issue in the context of rhythmic-metrical corpus analysis.

3 Cf. Lerdahl & Jackendoff (1983)’s ‘metrical well-formedness rules’, relaxations of which to admit mixed metres are discussed on pp. 97–98. From a wider repertorial perspective, London (2016) notes that ‘2:3 ratios for the duration of successive beats are characteristic of metres with complex beat patterns’.

4 See the discussion of representative corpora in London (2013), for instance.

5 Another more statistical study could assess the most suitable methods for quantifying this relative variance.

6 The full quote reads ‘Very much in the style of Gershwin. Gershwin’s tonality, rhythm, and color. American song feeling. Moderate tempo but vital, crisp, and accented’. Quoted in Suchoff (1983, p. 138).

7 Rice also assesses how ‘Bulgarian’ these pieces really are stylistically on the basis of the metre, tempo, possible quotation, and other idiomatic factors such as the use of a motoric quaver pulse (which is particularly relevant here).

8 Other accounts exclude some of those listed here. Rice (2000) excludes no. 151 for the reasons discussed above, and Moelants (2006)’s list doesn’t mention 115, or 148.

9 Series 1 no. 3 is in fact notated in alternating 4/8–3/8 bars (30 bars in total, equating to 15 bars of 7/8). It also exists in two versions: an alternative is given in the Appendix which is identical but for a thicker texture with octave doublings of the melodic lines.

10 Other classical repertoire with sections explicitly in 8/8 (323) include the first movements of Ravel’s Piano Trio in A minor (which is identified in terms of the Basque ‘Zortziko’ dance), and Einojuhani Rautavaara’s Piano Sonata No. 2, ‘The Fire Sermon’ (1970).

11 Cf. Gotham (2015b) for comments on rotational stability and Prokofiev’s Piano Sonata No. 7, movement 3 (‘Precipitato’) for a fast and consistent 7/8 (232) for piano solo.