Ultra-Low Frequency ‘Pulse’ / ‘Hypermeter’ in Dinnerstein’s Bach

I    wanted it to have a feeling of spaciousness, and of ‘breath’ ... I was influenced by Marie-Claire Alain’s [interpretation of Bach’s ‘Goldbergs’] in Paris, who played a Bach recital... an incredible sense of rhythm, almost an organic sense of rhythm ... [to] these large rhythmic structures. But the smaller beats [in Alain’s interpretation] were very free. [The overall effect of Alain’s approach was such that] it had [qualitatively, an astrophysical] feeling—almost like the turning of the Earth [on its axis] ... that there was an inevitability to the way the pulse was going, but within it there was this kind of feeling of ‘give-and-take’ in the rhythm. And I think that [by contrast to Alain,] a lot of the approach to rhythm has [for other pianists] been very ‘motoric’, and influenced by having a very strong feeling of a ‘small’ pulse. And I wanted to experiment with freeing myself of the small pulse and feeling a larger pulse.”
  —  Simone Dinnerstein interviewed by Sarah Dallas of www.moreintelligentlife.com, commenting on rhythmic freedom, breathing, phrasing, and why the silences in Bach, are as important as the notes, 2007.

S imone Dinnerstein catapulted to fame on the merits of her recording of Bach’s Goldberg Variations worldwide in August 2007. Her Berlin recording has also been very well-received, and she has a very busy touring schedule. Dinnerstein is a graduate of The Juilliard School where she was a student of Peter Serkin. Last night she performed in Kansas City in the Friends of Chamber Music’s piano series.

H er interpretation of Bach is highly unusual, as some reviews and the blockquote above attest. Despite the written time signatures, Bach’s ‘French Suite’ No. 5 gives the impression of a compound meter. The groupings into hierarchies of nested two-, four-, six-, eight-, 10-, 12-, 16-, 20-, 24-, and 32-bar structures contribute to this.

A rtists’ interpretations of these are widely varied, and the variations are especially pronounced where the hypermeter or polymeter structures change—for example, become more accentuated or obvious, or become less obvious, or become more or less competitive or conflictual. Some artists affirm or cooperate with the hypermetric and polymetric structures; other artists fight the imposition of a hypermeter by regularizing the rhythms and defeating the agogic and rubato cues that are in the score.

S imone Dinnerstein engages in a style of interpretation that alternately cooperates with and, at times, opposes Bach’s cues for improvisational and rubato freedom. At times, she also liberally expands upon the cues that are in the musical text. Characteristically in her playing she sheds light on longer-range, larger-scale hypermeter structures than might have been perceptible to us before. As a result of her emphases and rhythmic decisions, we hear what we might imagine to be the Earth spinning on its axis, one revolution every 24 hours. Maybe more accurately, we imagine the Earth’s axis precessing, one cycle every 26 thousand years. We hear celestial mechanics operating in timeless, wheels-within-wheels majesty. Music of the spheres...

P  robably the most beautiful written-out rubatos in all keyboard music are those in the Andante of [Bach’s] Italian Concerto, BWV 971. With the aid of rhythmic notation of unparalleled subtlety, Bach wrote down the free rhythms that a contemporary singer or instrumentalist schooled in the Italian style might have improvised in performance... To be sure, even Bach’s notation cannot wholly render the infinite subtleties which characterize great artistry in performance. A sensitive harpsichordist will emphasize the B-flat here by playing it just a shade early in order to be able to sustain it slightly longer. This rhythmic flexibility enables good harpsichordists to overcome the relatively inflexible dynamics of their instrument and still play cantabile.”
  —  Paul Badura-Skoda, p. 66.

M ost studies of agogicity and rubato address small temporal deviations with regard to the beat, as defined by the prevailing time signature as-written—they address the ‘shimmering’ effect on a timescale of tens of milliseconds up to several seconds, usually. But hypermeter also has a pulse—it’s just not an explicitly written one, and not readily apprehended on the timescales of our usual attention-span. There can be agogic and rubato effects that arise on the very long timescales of hypermetric patterns—what I might only half in jest call an astrophysical, nebula-esque shimmering on a timescale of tens of seconds to hundreds of seconds: eons, musically speaking. Here are the longest hypermetric timescales that occur in Dinnerstein’s account of BWV 816.

• Allemande, 4/4, two 12-bar repeats: 104 sec
  
• Courante, 3/4, two 16-bar repeats: 48 sec
  
• Sarabande, 3/4, 16-bar repeat and 24-bar repeat: 130 sec and 196 sec
  
• Gavotte, 2/2, 8-bar repeat and 16-bar repeat: 19 sec and 38 sec
  
• Bourée, 2/2, 10-bar (4:6) repeat and 20-bar (8:12) repeat: 52 sec and 104 sec
  
• Louré, 6/4 (2), two 8-bar repeats: 34 sec
  
• Gigue, 12/16 (2), 24-bar repeat and 32-bar repeat: 86 sec and 116 sec

    [30-sec clip, Simone Dinnerstein, Bach, ‘French Suite No. 5, BWV 816, Saraband’, 1.2MB MP3]


    [30-sec clip, Simone Dinnerstein, Bach, ‘French Suite No. 5, BWV 816, Courante’, 1.2MB MP3]

A    century before Bach, Heinrich Schütz had perceived regular time to be ‘as it were, the soul of music’ [preface to ‘Auferstehungs-historia’, Dresden, 1628] ... It is revealing to compare the ‘slow beat’ mentioned here with the remarks about Bach’s conducting... Between the firm ‘pillars’ of the beats there was a large degree of freedom. With the exception of pieces in perpetuum mobile style, there were all kinds of rubato, breathing pauses, breaks, and thematic dovetailing running counter to the beat.”
  —  Paul Badura-Skoda, pp. 16-17.

I f you were a listener who had never seen the BWV 816 score and you did not know that the dances that are “in 2” are separate movements, or that the dances with meters that are “in 3” are separate movements I suppose you might impute even longer-scale hypermetric patterns in French Suite No. 5—ones that transcend inter-movement boundaries.

G lenn Gould, Andras Schiff, Angela Hewitt, Piotr Anderszewski, Constance Keene, Edward Aldwell, and others—almost devoid of these hypermetrical effects and tremendously different from Simone Dinnerstein in myriad other ways.

T he spontaneity and greater temporal elasticity of, for example, Dinnerstein’s Sarabande—reduce our attention to the longer-scale structure. But the more motoric dances draw our attention to longer-scale hypermetric structure and the rubber-banding that Simone is doing with time and meter.

W e take a look at how this works quantitatively, using the mathematical/statistical formalism of ‘copulas’. Copulas are functions that link univariate (single-variable) marginal values to their statistical variations with multiple other variables—to their joint multivariate distributions. Copulas are useful because they allow us to discover concurrent or joint multivariable correlations with any set of univariate marginal distributions that may happen to exist. Each ‘voice’ in the left-hand part(s) and the right-hand part(s) in ‘French Suite’ No. 5 has its own rhythmic marginal distribution (of inter-beat time intervals, quantized to 4 millisecond time-resolution or better). Figuring out how Simone Dinnerstein makes this thing ‘breathe’ the way she does is possible by computing the copulas, and comparing them to, say, Aldwell’s or Goode’s or Gould’s.



C opulas are a way of quantitatively characterizing the dependence structures of vectors of multiple random variables. Although copulas were first studied more than 50 years ago, they were not extensively applied. Interest in copulas has been renewed recently in biostatistics, reliability engineering, and financial engineering, and other fields. Within the past 10 years in finance, copulas have become a standard analytical technique for multi-asset basket pricing (especially CDOs and other credit derivatives), equities portfolio modeling, risk management, etc.


A nja Volk, in her paper at the 2004 ICMPC meeting, considers all metric pulses arising from equally spaced notes’ onsets (the so called local meters). All external information (bar lines and time signature) was ignored. Only the internal relations of the actual as-composed notes were analyzed. Anja modeled the inner metric structure of music mathematically, and contrasted it with the outer metric structure given by the accent hierarchy of the as-written time signature. The inner metric weight for each note is calculated on the basis of the meters m_s,d,k it participates in and which have a minimum length that is not smaller than a fixed value l. The variable s denotes the starting point, d denotes the period (the duration between consecutive onsets of the local meter) and k the length of the local meter. The contribution of each local meter to the metric weight depends solely on its length k, not on its starting point s nor its period d. She denotes m_s,d,k= m(k) and gain for each note’s onset o the metric weight w_l,p as the weighted sum of the length k of the corresponding local meters:


whereas the minimum length l and the weighting parameter p can be varied. The calculation of the inner metric weight of a musical piece (or a passage, a part, a passage from a part) can be easily understood from figure below.


A n algorithm calculates all local meters (i.e. arithmetic sequences) within the set of onsets under consideration (there are five such local meters in the short example). The metrical weight encodes the incidences of different meters, i.e. the metric weight of an onset is determined by the set of all local meters passing through it.

M    aintenance of the metre depends on the basic rhythmic character of a work. An arioso allows greater rhetorical freedom—more rubato, that is—than, for example, a gavotte or a minuet. The greatest agogic freedom is permissible in freely structured forms such as the stylus phantasiticus of the toccata and the fantasia. Works of this kind had their origins in instrumental improvisation, such as the free invention of preludes.”
  —  Paul Badura-Skoda, p. 18.

H ave a listen to Dinnerstein, remarking on her intentions with regard to Bach’s ‘Goldberg Variations’ (YouTube):



I n my day-job, among other things I perform signal-processing and mathematical modeling of time-series. Sometimes the timeseries are physiologic monitoring variables related to health and survival in critical care units; other timeseries I have modeled include financial trading timeseries and geophysics/seismic timeseries. I think that the spectral analysis software and mathematical tools that are used in those areas have broad applicability to quantifying what is going on in music, including helping to account for what is so unusual about Simone Dinnerstein’s Bach interpretations and rhythmic/metrical decisions.

S o-called ‘adaptive filters’ offer advantages over Wiener filters for time-varying processes. They are used for mathematical ‘deconvolution’ of seismic data which exhibit non-stationary, time-varying behavior, much like Dinnerstein’s ‘waves’ of slowly altering meter. Different algorithms for adaptive filtering exist. The least-mean-squares (LMS) algorithm, because of its simplicity, has been widely applied to data from different fields that fall outside geophysics. And that is what I have used, on my measurements of Dinnerstein’s Berlin recording—I bought her CD, ripped ‘French Suite No. 5’ tracks’ MP3s from it, and performed Fourier transform and Wavelet transform calculations on those digital audio files, annotating each bar against my pdfs of the score.

T he ‘stacking velocities’ I used to model the time-varying alterations of meter do not need to be known very accurately because—just as in deep-reflection seismography—the residual deviations (‘move-outs’) are small on a beat-to-beat basis and have only a minor influence on the results of the adaptive time-slicing.


W hat I discover is what amounts to a very low frequency (VLF = 0.01 to 0.1 Hz) ambient ‘seismic’ hypermetric architecture of Bach’s French Suite No. 5, something I think is heretofore unreported. Maybe it’s just Simone; maybe it’s really there; we will have to measure other elite performers’ interpretations and see if we can find (less pronounced) evidence of it. That such a thing should seem novel is particularly likely in recent years, as the trend has been toward highly ‘motoric’ species of virtuosity, especially with Bach interpretations.

B ut here in Dinnerstein we have ‘ocean waves’ that “couple” into ‘seismic waves’ through (what appears to be, from my measurements and analysis thus far—) a quadratic nonlinearity of the metrical surface boundary condition. The quadratic nonlinearity may be entirely under the artists’ conscious control; or maybe it is not, or at least not entirely. The nonlinearity couples pairs of slowly propagating metrical ocean-like waves of similar frequency to a high metrical phase-velocity component that has approximately double the frequency. These “approximately 2f” relationships are easy to discern in the FFT spectra of French Suite No. 5, with its hierarchical organization of phrases and repeats.

B esides seismology, gravitational wave physics also offers some useful tools and analytical methods for investigating metrical and hypermetrical phenomena in music. Waves familiar from other areas of physics such as water waves, sound waves, and electromagnetic waves are able to carry energy, momentum, and angular momentum. By carrying these away from a source, waves are able to dissipate energy and ‘rob’ that source of its energy and linear or angular momentum. Gravitational waves do the same thing in space. For example, a binary star-pair loses angular momentum as the two orbiting stars spiral towards each other: the angular momentum of the mutually-orbiting pair is radiated away by gravitational waves.

H ere’s what I think: a prevailing musical pulse and meter manifests a particular musical metrical ‘mass’ that exerts a quasi-gravitational force on the space-time continuum through which it passes and which we occupy. If the performer suddenly alters the pulse, it is as though a gravitational wave is created, which propagates forward and may make space-time seem to either dilate or contract, compared to the previous regime. If the performer imparts a hypermetric periodicity, that can be detected or sensed, via neurophysiological monitoring and analytics not too dissimilar from methods to detect gravitational waves.

T he transmission of hypermetrical ‘waves’ is not only through the ‘physical’ medium (sound through the air) but also through ‘cognitive’/’neurophysiological’ media. The latter media have an ‘impedance’ and transfer-function, with rate-dependent,time-dependent, remanent properties—almost like thixotropic or rheopectic non-Newtonian liquids and gels, or like non-Newtonian star-stuff in space. So the transmission of hypermeter waves is imperfect—the waves suffer delays and phase-lags; depending on its spectral properties, the waves experience more or less attenuation and dissipation as they propagates; they undergo refraction at ‘apertures’ (phrase boundaries; movement boundaries) and interferometric interaction with other hypermeters that exist concurrently in the same space-time.

W hen the musical polyphony is ‘shallow’, the hypermeter waves undergo wavefront-steepening and the wave amplitude increases like tsunami waves; and so on. This would seem to me to be a rich space for cross-disciplinary study in the future. To my knowledge, cross-pollination between music theory and physics has not explored these possibilities to-date. Maybe you will be intrigued enough to pick up this line of analysis and give it a ‘go’.

I   t can be seen in accompanied recitatives that tempo and metre must be frequently changed in order to rouse and still the rapidly alternating affections. The metric signature is in many such cases more a convention of notation than a binding factor in performance.”
  — C.P.E. Bach, ‘Versuch über die wahre Art, das Clavier zu spielen’, Leipzig, 1753, p. 153 (trans, W. Mitchell, ‘Essay on the True Art of Playing Keyboard’, 1949).

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