Tonal prisms: iterated quantization in chromatic tonality and Ravel's ‘Ondine’

Abstract

The mathematics of second-order maximal evenness has far-reaching potential for application in music analysis. One of its assets is its foundation in an inherently continuous conception of pitch, a feature it shares with voice-leading geometries. This paper reformulates second-order maximal evenness as iterated quantization in voice-leading spaces, discusses the implications of viewing diatonic triads as second-order maximally even sets for the understanding of nineteenth-century modulatory schemes, and applies a second-order maximally even derivation of acoustic collections in an in-depth analysis of Ravel's ‘Ondine’. In the interaction between these two very different applications, the paper generalizes the concepts and analytical methods associated with iterated quantization and also pursues a broader argument about the mutual dependence of mathematical music theory and music analysis.

Keywords:

• chromaticism
• harmony
• scales
• Ravel
• voice leading
• geometry
• maximal evenness

Thanks to Dmitri Tymoczko for many conversations that have shaped the work in this paper. Thanks also to Marek Žabka, Julian Hook, Richard Plotkin, and Jack Douthett for comments on an earlier draft, to all the participants of the 2012 John Clough Memorial Conference, and especially to Richard Cohn for bringing together such an impressive group of music theorists.

Notes

1. The J.S. Bach examples in [2] are a notable exception.

2. Rounding is the same as adding a constant of 0.5 before applying the floor function. Note that this implies the round-up ‘tiebreaker rule’. Tymoczko's contribution to this volume makes a case for rounding rather than using the floor function.

3. See also the illustrations in [6, p.101–106,7, Fig. S5].

4. Cohn's solution is to recognize diatonic regions within the Tonnetz and confer a special status to progressions that ‘teleport’ between the boundaries of this region, treating it loosely as a modular subspace. The mathematical foundations of this approach might be examined beginning from the special properties of diatonic subregions, which are far from being arbitrary selection from amongst many possible Tonnetz parallelograms.[15] This may result in some convergence between Cohn's approach and the present one.

5. See, for example, Kopp's [16] detailed discussion of the voice-leading attributes of chromatic mediant relationships.

6. These terms are not restricted to quantizations of ME sets, either. I do not consider quantizations of non-ME chords in this paper, but they do appear frequently in [5].

7. In their contribution to this issue Plotkin and Douthett also derive the P–R cycle using an 8-hole filter. This similarly has little obvious explanatory connection to the Brahms examples and others like it beyond the feature of voice-leading parsimony, because there is no apparent use of octatonic scales in the passages. One caveat is that this 3→8→12 space may have some relevant application to tonal passages whose harmonic logic is governed more by enharmonic reinterpretations of diminished seventh chords than diatonic contexts.

8. Note that anchored progressions on triads generalize hexatonic cycles, including, for example, varieties with diminished triads.

9. This fixed point is equivalent to a fixed beacon position in filtered point-symmetry. Douthett [3] shows how dynamical systems with a fixed beacon can generate hexatonic cycles.

10. Translation Mattias Müller, from the Henle edition.

11. Other analysts, such as Howat [18] and Bhogal [19], characterize this design in sonata-form terms.

12. Bhogal [19] brings out the interesting metrical aspects of this accompaniment as it develops over the course of the piece.

13. Throughout this analysis I use note-names to indicate the transpositions of scalar collections (such as ‘B acoustic’ or ‘C♯ diatonic’) according to musical convention, without intending to imply anything about perceived tonic or root. More neutral nomenclature is available, most notably Hook's [21] method of naming spelled heptachords, which uses circle-of-fifths positions. I adhere to the conventional nomenclature here only to make the analysis easier to follow.

14. On Ravel's use of the octatonic, see [23].