Autocorrelation and Agogicity: Measuring Goode’s Bach

W    hat one remembers most from Goode’s playing is not its beauty—exceptional as it is—but his way of coming to grips with the composer’s central thought, so that a work tends to make sense beyond one’s previous perception of it... The spontaneous formulating process of the creator [becomes] tangible in the concert hall.”
  —  David Blum, New Yorker, 29-JUN-1992.

To say exactly why we like or dislike a particular artist’s interpretation of a piece of music is difficult. And to compare and contrast two different artists’ interpretations in a way that can be objectively assessed and broadly agreed upon is also hard. The language of analysis and criticism readily devolves into nebulous descriptions or into abstruse technical terms that get in the way of conversation and real understanding. But the superb performance last night by Richard Goode in Kansas City in a Bach-Chopin program that was part of the Friends of Chamber Music’s ‘Master Pianists Series’ prompted me to take a whack at remedying this.

I thoroughly enjoy everything Richard Goode does, always. But in this recital I especially liked his accounts of Bach’s Sinfonias (BWV 791-801) and Preludes and Fugues from Das Wohltemperirte Clavier, Book II (BWV 870, 878, 885, 889). And I very much admire his accounts of Bach’s Partitas... one of Richard’s encore pieces was the Sarabande from the D major Partita No. 4 (BWV 828). How to describe it? His interpretation is not as ‘dry’ as others’ Bach interpretations; it is optimistic and expansive—and it projects a distinctive, quintessentially ‘American’ attitude of curiosity and humor about life.

I confess I always feel that flowery, subjective descriptions are less than satisfying though. When I tell you why I like Richard’s interpretations, I want instead to be able to give you something that you can independently and reliably and objectively assess for yourself. I want to compare what Richard does quantitatively to, say, Glenn Gould’s and others’ interpretations—and calculate how they differ in various passages, and why those differences cause us to perceive and respond to the interpretations in such varied ways. How shall we do that?

One way to do it is to take digital recordings of the performances and perform a number of signal-processing measurements on them; examine the timings within critical phrases or across longer expressive spans; calculate various time-domain metrics from the digitized music; perform statistical tests on the metrics, to see whether and how they differ; and so on. I regularly do that, as a hobby—not just with baroque music but with other things, too, when I’m trying to understand why I am feeling a certain way about a piece. (Somewhat weird hobby, I admit, but it’s just in my engineer’s nature to want to take things apart and figure out what makes them ‘tick’.) Usually I use MATLAB or MAPLE or MATHEMATICA other fancy math software to do this. But it’s actually possible to do a number of interesting and useful analyses with something as simple as Microsoft Excel. Here goes!

Think, for example, about the difference between Goode and Gould when playing Bach. Actually, to make a clean ‘apples-to-apples’ comparison, let’s think about just Bach’s 6 Partitas and, within those, let’s think about just one of the movement forms that has a consistent time-signature—for example, the Allemandes.

As an historical dance usually in 4/4 time, an Allemande can have a tempo ranging from largo almost up to presto. Gould’s tempi are quite fast; Goode’s are generally more leisurely, at least ‘New Yorker leisurely’. But just noting the large-scale differences in tempi is too crude—it doesn’t explain very much about why their interpretations are so different. The ‘pulse’ is a function of time, operating and varying on a multi-dimensional, fractal time scale: rubato—small-scale, intra- and inter-beat and inter-measure variations; and larger-scale accelerando/ritardando decisions.


But the differences arise from more even than that. The durations that particular notes are held are also distinctively different. ‘Agogic’ is the fancy music-theory word that means varying a note’s (or beat’s) emphasis by varying its duration. A dry, flat, highly ‘motoric’ performance like Gould’s doesn’t have much agogic variation. A more romantic performance like Goode’s has many places where lots of agogic emphasis is used.

A    •gog•ic [uh-goj-ik]
(noun) the theory that accents within a musical phrase can be produced by modifying the duration of certain notes rather than by increasing dynamic stress.
(adj) stressed or accented through a variably prolonged duration.”

So, gee, you can take an MP3 and use a cheap digital audio editor software app; you drag the cursor over the beginning of a note and then rubber-band-drag the cursor to the end of the note; you measure how many milliseconds that time-interval is. You can do that for all of the notes in 4 or 6 or 8 bars of a passage that contains what, for you, is emblematic of the interpretive difference you’re interested in. Do that for one artist’s recording, and then do the same thing for the same bars in another artist’s recording. You put the information into a spreadsheet or database. You then quantitatively calculate the ‘agocity’ of the two artists’ performances/interpretations.

W hich is what I did, after being stimulated by Richard’s wonderful performance last night. I just had to figure out why it is that I like his Bach so much while I simultaneously like various other artists’ Bach interpretations very much, too. And I wanted to be able to say objectively how their characteristic personal rhythmic decisions differ in crucial passages. I wanted to put together a little ‘tool’ that I could share with you, which would enable all of us to conveniently figure these things out in the future—not just for Bach but for anything. That way we won’t be frustrated by nebulous words that don’t really say very much or that have hopelessly imprecise meanings.

Autocorrelation (of beat-to-beat time-intervals; or of note-to-note durations) is a statistical function that measures repeating patterns, such as the presence of a periodic signal which is (partly) obscured by variability.

It’s used frequently in signal processing for analyzing series of values, so-called time-domain signals or time-series. Mathematically, it represents the quantitative similarity between observations as a function of the time separation between them. The time-domain is divided up into units—in this case, my measurements were in milliseconds for note/beat durations over passages over various lengths (up to 6 bars at a time in the portions of the Partitas that interested me). Then I ‘normalized’ those values to the average milliseconds per beat across the number of measures in the passages I compared, and I set the normalized tempo for Goode to 100% and the normalized tempo for Gould to 100%. Next, to take into account that Gould plays almost everything substantially faster than Goode, I calculated the ‘within-performer-within-phrase’ autocorrelations from the percentage deviations from those by-performer normalized 100%-meter values. The autocorrelation can be expressed as an equation involving the covariance and variance of these time-series from time i to time t as a function of the beats in between (lag k):


The results of this little experiment are in this Excel spreadsheet. You can click on the screenshot below to open the file or download a copy of it. If you like, you can use the spreadsheet to make your own measurements of selected bars of pairs of performances you’d like to compare.


To me, the significance is that I now understand much better what it is about Goode’s Bach that pleases me so much: it is this particular autocorrelation pattern—a characteristic species of agogicity that the autocorrelogram graphically shows. There are long-range, high, positive autocorrelations at various inter-beat distances that line up more or less on the first-beat-of-measure boundaries [and also the phrase and hyper-meter boundaries], and those autocorrelation peaks taper off quite slowly the further apart any two notes are (the more ‘lag’ units there are between them in the sequence of notes or beats). By contrast, Gould (and other ‘drier’ Bach interpreters) has a high autocorrelation for adjacent notes (at Lag = 1), which damps out pretty quickly the further apart any two notes are.

W hat the autocorrelation analysis tells us is that Goode’s playing manifests a high degree of ‘associativity’ of time intervals over longer time-scales—and this is equally true for intervals that he compresses to be shorter than the nominal meter, and for intervals that he agogically prolongs and stresses. The peaks in the autocorrelogram amount to a sort of ‘polymeter’ that is distinctively Goode’s.

By contrast, Gould’s playing manifests a ‘memory-lessness’—nearly devoid of associativity of rhythms and durations on long time-scales for non-adjacent notes. Both Goode and Gould perform these motoric Bach dance suite movements with motoric rhythmicity. But their motor-like rhythms are remarkably and measurably different—the differences in our cognitive and emotional perceptions when we hear these disparate interpretations arise from their deliberately different, highly personal and unmistakeably different autocorrelations. Here we have quantitatively different ‘agogicity’ as a measure of interpretive style.

I    fixed the action in some of the instruments I play on — and the piano I use for all recordings is now so fixed — so that it is a shallower and more responsive action than the standard. It ... is rather like an automobile without power steering: you are in control and not it; it doesn’t drive you, you drive it. This is the secret of doing Bach on the piano at all. You must have that immediacy of response, that control over fine definitions of things.”
  —  Glenn Gould.

Saying that Gould’s Allemande playing is relatively ‘memory-less’ or anti-agogic is not to say that his playing was ‘mechanical’ or automatic or that his vision of these Allemandes was ‘microscopic’ and eschewed a macro-scale rhythmic architecture. By no means! Goode has and Gould had—each of them—comprehensive conceptions of each of these Allemandes—of each of the Partitas’ and Suites’ movements. But the metric organization and fractal scales that they each use to build their interpretations’ rhythmic structure are dramatically different—in ways that, as you see above, can be measured and compared numerically and displayed graphically. By autocorrelation analysis, we find out exactly how Goode paints with a ‘broader’ somewhat impressionistic brush (Rembrandt?) and Gould with a magnifying glass and a ‘finer’ one (Vermeer?). Both are highly detailed and narrative in their delivery of the Partitas and Suites; both are illusionistic.

So here we have one good, objective way to measure and say how they do what they do, and how it works. Leigh Smith and other music theoreticians have examined things in similar ways in years past [see links below], not for Bach and not with quite the same purpose as what we’re doing here. They have used wavelet transforms and other frequency-domain methods, not just autocorrelation and time-domain methods. Smith in particular discusses the difference between measuring the actual timing (from waveform or other performance data) versus capturing the musician’s intention (from MIDI keystrokes datastream or other empirical data that precede the music being made audible). In my own experimentation to prepare this CMT blog post, I have (obviously) only the features in the waveforms digitized in the (audible) MP3 files to work with and have spent only a few hours working on it. This post and its Excel attachment are meant only as a quick demonstration that these things work and that they are useful to help us understand the music better. Enjoy!


    [50-sec clip, Richard Goode, Bach, Partita No. 1, BWV 825, Allemande, 1.2MB MP3]

    [40-sec clip, Glenn Gould, Bach, Partita No. 1, BWV 825, Allemande, 1.2MB MP3]


    [50-sec clip, Richard Goode, Bach, Partita No. 2, BWV 826, Allemande, 1.2MB MP3]

    [45-sec clip, Glenn Gould, Bach, Partita No. 2, BWV 826, Allemande, 1.2MB MP3]


    [50-sec clip, Richard Goode, Bach, Partita No. 3, BWV 827, Allemande, 1.2MB MP3]

    [45-sec clip, Glenn Gould, Bach, Partita No. 3, BWV 827, Allemande, 1.2MB MP3]


    [50-sec clip, Richard Goode, Bach, Partita No. 4, BWV 828, Allemande, 1.2MB MP3]

    [45-sec clip, Glenn Gould, Bach, Partita No. 4, BWV 828, Allemande, 1.2MB MP3]


    [50-sec clip, Richard Goode, Bach, Partita No. 5, BWV 829, Allemande, 1.2MB MP3]

    [35-sec clip, Glenn Gould, Bach, Partita No. 5, BWV 829, Allemande, 1.2MB MP3]


    [50-sec clip, Richard Goode, Bach, Partita No. 6, BWV 830, Allemande, 1.2MB MP3]

    [40-sec clip, Glenn Gould, Bach, Partita No. 6, BWV 830, Allemande, 1.2MB MP3]

In Goode’s Bach I hear a sort of Walt Whitman-esque, America-flavored ‘agogic freedom’—Goode’s rhythm and emphases and playing style represent to me a sort of ‘democracy trope’. I hear in him an enthusiasm for discovery; an ever-renewing openness for life’s ambiguities and ‘matters-arising’.

Goode’s affect or vision involves more rhetorical freedom than Gould’s concept of Bach, I think, in part because of the sentinel differences in their autocorrelation patterns and ‘agogic polymeter’. Oh, there is of course a difference between rubato (freedom ‘between’ the beats) and agogics (freedom ‘of’ the beats—in where the beats fall). Agogics are often applied at harmonic-pivot points. That’s just where they naturally and logically tend to go.

But Goode’s autocorrelated ‘agogic polymeter’ (or whatever you’d prefer to call it) I think ‘subverts’ or calls into question what we perceive the meter to be—by injecting variable, conflicting metrical structure and by modestly shifting the phase of the metrical accents. This is why it has the strong, friendly effect on me that it does. The initial events of each of these Allemandes provide evidence sufficient for us to establish contexts and referential schematics—mental ‘models’ for what we imagine will follow. The acoustic cues fit into the schematic or framework around which our subsequent anticipation is formed. Deferring or subverting the established meter, as Goode does, forces us listeners to attend to the present moment and continually revise and re-form our expectations. By contrast, with Gould, most of what we anticipate promptly materializes. Gould’s epistemological and perceptual effect and the impact on our cognition and emotions are quite different from Goode’s.

Cognitively, we subconsciously keep track of and group the conflicts and uncertainties/ambiguities that are caused by Goode’s mounting ‘phenomenal agogic accents’. Maintaining these multiple concurrent and conflicting suppositions in our heads as we listen to Goode play—asks that we put a considerable and peculiar sort of effort into it. It is a different sort of attention, I would say, than the attention that Gould’s playing asks of us. Goode’s agogically autocorrelated playing requires our listening strategy to allocate attention and memory toward establishing groupings and processing long timescale expectations. Goode engages us in a distinctive and different sort of conversation/meditation. And the autocorrelation analysis just helps to reveal a quantitative pattern that explains how it’s different.


Compared to Goode’s interpretations, Gould’s Bach has—I go out on a limb here—an ‘anti-expressionist’ aesthetic. From Gould (and others) we get unrelenting introspection, plus transcendent but psychologically ‘dissociative’ experiences, almost PTSD-like experiences.

To me, the ideal affect with Bach also involves a certain amount of rhetorical surprise as well as angst. Agogic accents—a ‘sung’ or ‘danced’ rhythm that either personalizes or depersonalizes—orients or disorients—can make the expression seem as though it were uttered by a particular and known ‘self’ (e.g., Goode; or Bach) or, alternatively, by an ambivalent or profoundly unknowable entity—(e.g, Gould; or God/TheDivine/TheCosmos).

So these Bach suites—and these Allemandes—are no mere dances. Goode ‘dances’ mostly ‘out’; Gould—and others—dance them in a mostly ‘inward’ way. The depth and the originality in the ways that Goode and other artists ‘dance’ them, while they are surely very diverse, never cease to delight and inspire and propel us to take hold of life and figure things out.

The two-hour program last night also included a wonderful collection of Chopin Mazurkas (Op. 6 No. 3 in E major; Op. 7 No. 2 in A minor; Op. 7 No. 4 in A-flat major; Op. 56 No. 2 in C major; Op. 56 No. 3 in C minor), Nocturnes (Op. 15. No. 2 in F-sharp major; Op. 27. No. 2 in D-flat major), and Waltzes (Op. 34 No. 3 in F major; Op 64. No. 2 in C-sharp minor; Op. 64 No. 3 in A-flat major)—24 pieces in all. Both the Op. 39 No. 3 Scherzo in C-sharp minor just before the interval and the Op. 61 Polonaise-Fantasie at the end of the program brought the appreciative audience to its feet. Bravo!

M    aking a break from the first and second notes of a melody is to be recommended where the tempo changes from slow to fast. In passing from the Andante to the Allegro in the Sinfonia of Partita No. 2, for example, it is advisable to retain the slow tempo on the final G major chord and not to embark on the faster tempo until the two semiquavers which follow... A relative degree of independence of the beat also applies in the case of recitative.”
  —  WWPBSD? [What Would Paul Badura-Skoda Do?], p. 18.

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