Higher-Order Geometry: Murray Perahia and Chopin’s Polonaises

M   usically, the most important consequence of the geometry is the following: since a nearly-even N-note chord occupies the same cross-section of the [metric] space as its transposition by 12/N semitones, such chords can be linked by particularly efficient voice-leading. Thus, nearly-even two-note chords are very close to their six-semitone transpositions; nearly-even three-note chords are very close to their 4- and 8-semitone transpositions; nearly-even 4-note chords are very close to their 3-, 6-, and 9-semitone transpositions and so on.”
  — Dmitri Tymoczko, Geometry of Music, p. 97.

M y enjoyment of last night’s excellent concert by Murray Perahia was enhanced by paying special attention to the intervals and root-motions of the chords—especially in the Chopin pieces on the program.

A nd, while Dmitri Tymoczko’s ‘Geometry of Music’ has in the year since the book was published reaped some objections and a certain notoriety (David Headlam and Rob Schneiderman (AMS), among other commenters), I do feel that considering the quantitative and statistical properties of a musical work can help some of us to understand the work better, be moved more deeply by it, and, possibly, learn how we ourselves might compose music more successfully, or measure and figure out how something we have composed compares, for better or worse, to other beautiful and effective compositions.


F rom the score of the Polonaise Op. 26 No. 1 in C-sharp minor that Perahia performed last night, I transcribed a number of the intervals contained in the 138-bar, 414-beat-long piece. Like Tymoczko and others, I used a root-motion log-frequency representation of intervals in my array, so that distances of notes N-semitones apart are uniform in any octave. I then utilized the ‘phom’ package in the R system to examine the distribution of those intervals in topological S³ space. This relies on theory set forth by Gunnar Carlsson and coworkers at Stanford, on persistent homology. I had occasion to collaborate with Gunnar about 5 years ago on analysis of critical-care ICU timeseries data. Persistent homology computes topological invariants of filtered sequences of simplicial complexes, constructed from my transcribed Chopin Polonaise interval dataset. The relevance of this to music theoretic analysis is that conventional topological invariants are unsuitable for characterizing the geometry of music since they are simultaneously too sensitive, and too weak. They are too “weak” because of their homotopy invariance, such that the underlying harmonic/intervallic structure can be stretched in any way (as long as it is not ripped) without changing its homology groups.

L ots of additional empirical work would be needed, to accurately and adequately characterize just this one Chopin Polonaise, so beautiful and evocative as performed by Perahia. I leave that to the card-carrying music theorists. But, unlike Professor Headlam and others, I do think that such effort would be rewarded. As an amateur musician, I have no academic career in music theory to advance, no tenure review to worry about, no political axes to grind in my analysis and writing—posting this tiny, speculative blogpost as I am here, and pointing interested readers to a new open-source R package that might aid others’ endeavors.

I n summary, it is simply a deeply pleasurable thing for me, to roll up my own sleeves and bring novel empirical tools and concepts to bear and see what I can learn. And, having corresponded in years past with Dmitri Tymoczko several times by email, I truly believe that his aims and aspirations are as wholesome as this as well, not so cynical or bombastic as Schneiderman and others allege. For some of us, loving something magical like this Chopin music [or, alternatively, loving someone] means, among other things, persistently and deeply wanting and striving to comprehend it [or them], by numerical quantitative means as well as by subjective, qualitative means, that’s all.

B y the way, Gunnar Carlsson’s papers are very readable, if you care to visit the links to them below...

• Murray Perahia website
  
• Harriman-Jewell Series
  
• Chopin F. Polonaises, Op. 26, score at IMSLP.org
  
• Baez J. Mathematics of music at Univ Chicago. The n-Category Cafe blog, 31-MAY-2009.
  
• Callender C, Quinn I, Tymoczko D. Generalized voice-leading spaces. Science 2008;320:346-8.
  
• Carlsson G. Topology and data. Bull Am Math Soc 2009;46:255–308.
  
• Tymoczko reply to Headlam, 2012.
  
• Milne A. Metrics for pitch collections. ICMPC 2011.
  
• Rothstein W. ‘Transformation of Cadential Formulae in the Music of Corelli and his Successors.’ In Essays from the Third International Schenker Symposium, ed. A. Cadwallader, 245–78. Georg Olms Verlag, 2006.
  
• Sanguinetti G. Chopin’s Polonaise Op. 53. Analitica 2000;1:-.
  
• Schneiderman R. Notices of AMS 2011;58:1408-9.
  
• Tausz A. Persistent Homology, R package ‘phom’ 02-NOV-2011.
  
• Tymoczko D. Three conceptions of musical distance. Manuscript, 2009.
  
• Tymoczko D. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. Oxford Univ, 2011.
  
• Woo A. What does the space induced by this metric on R/Z look like? MathOverflow, 11-MAY-2011.
  
• Zomorodian A, Carlsson G. Computing persistent homology. Discrete Comput. Geom. 2005;33:249–74.