Monday, May 26, 2008

Tempi and Saliency: Psychophysiology, or Why Deceptive Cadences Are Not Always Deceptive

Sloboda book
W  e do not doubt that attention, recurrent computations, and complexity are important aspects to understand consciousness. However, we propose that these aspects are often trivially necessary rather than sufficient. For example, often it is assumed that consciousness emerges not before several hundreds milliseconds after stimulus onset. Hence, given the short time constants of membranes of neurons, recurrent connections are obviously necessary to store and process the stimulus before consciousness is reached. Complexity is for sure of primary importance for consciousness because networks with the same number of neurons can create trivial as well as complex behavior depending on their connectivity. Therefore, the important question is which kind of connectivity or which exact degree of complexity, determined with which mathematical norm, is sufficient for consciousness ...”
  —  Michael Herzog (Laboratory of Psychophysics, Lausanne), Michael Esfeld (Dept of Philosophy, Univ Lausanne), and Wulfram Gerstner (Computer Sciences and Brain-Mind Institut, Ecole Polytechnique Federale de Lausanne), Neural Networks 2007.
This is a quick post-script on the topic of the previous CMT post, about psychophysiology of modality and tempi.

It turns out that Tibor Bosse and collaborators at the Vrije University in Amsterdam have recently been developing mathematical models of attention, mostly for military applications. Some of these models (and the differential equations underlying them) resemble hydrodynamic models used in cardiovascular physiology (‘windkessel’ models; in German, windkessel means elastic reservoir) and in functional magnetic resonance imaging (fMRI).

3-element windkessel electrical model
The effect of stimulus rate and its interaction with stimulus type on brain activity during the visual or acoustic stimulus has been investigated using fMRI. Different brain regions show differential responses as a function of time and as a function of the ‘tempo’ or stimulus rate. These results differentiate functionally specific responses in rate-dependent and rate-independent areas.

4-element windkessel electrical model, with inductance-inertance element
The best approach to the dynamics, though, is probably not a ‘lumped-parameter’ analytical one, like simple windkessel models. Instead, nonlinear wave-front propagation models (Stefan moving-boundary differential-equations) or stochastic network representations may be more realistic as mathematical formalisms. An intermediate approach, such as mathematical approaches used to solve equations for liquid flow through porous media (like aquifers), may be useful, to represent the mesh-like connections and multi-path propagation of signals related to our brains’ perception of musical suspense and resolution (cadences). Have a look at Herzog’s and colleagues’ paper in the recent special issue of the journal Neural Networks and at Tibor Bosse’s papers for a glimpse of how feasible such a quantitative cognitive modeling project is these days.

The results of fMRI (and, presumably, of mathematical modeling of the rate-dependent effects) may enable us to better understand the inter-relationships between tempi and our apprehending meaning in music. For example, so-called ‘deceptive cadences’ to vi (or VI in the minor key) and similar progressions are normally treated in a time-independent way, as a prolongation of the dominant. But in some instances it is better understood as prolonging the tonic, through I-vi. Understanding it in that way, though, depends not only on the context but also on the tempo. In general, the second alternative only if the vi is a middleground harmony or key region and the tempo is relatively slow. Otherwise, we choose the first way as more compelling. The underlying idea of the expansion of the V is that tension in the dominant is not resolved by the deceptive move to vi, but actually heightened; it is only when the true tonic arrives that the tension is removed. But if the tempo is sufficiently slow, the mind anticipates everything. The mind calculates and re-calculates and gauges and re-gauges all the possibilities. The deceptive cadence becomes instead a meta-commentary on irony itself.

Haydn Sonata Deceptive Cadence Example
In Five Graphic Music Analyses, Schenker showed that the Haydn Sonata analysis includes an extended prolongation of vi within the V that is the underlying harmony of the entire development (see the bass of the top system for the V; see mm. 81-111 of the other levels for the vi, which changes to VI (as V/ii) near the end of the passage).

By definition, a deceptive cadence is any phrase the ends in a way that is different from the anticipated outcome. Deceptive cadences can be disruptive, creating ‘Wow !’ moments in music. Usually, deceptive cadences require a melodic line and an accompaniment. A cadence is deceptive only if the melody ends up on the proper note to end the phrase, but the accompaniment contains unexpected harmony underneath. For example, if your song is in C-Major, you might expect the phrase to end up on a C Major chord, with the melody playing a ‘C’. If, though, the melody ends up on a ‘C’, but the harmony is an A-Minor chord, then this is an unexpected harmony and therefore a deceptive cadence.

Deceptive cadences are an example of how composers blur one phrase into the next. In a deceptive cadence, the melody is finishing off the phrase, while the harmony/accompaniment has already moved ahead, to the beginning of the next phrase. But if the tempo is slow enough, the synapses have had time to caucus with each other and anticipate everything. The ‘Wow!’ is of a totally different type. And the blurrings are no longer blurrings at all: they become meta-commentaries on what transitions are about, and why.

Cadwallader-Gagne book; Mozart (K. 457) I, mm. 46-52
In Mozart’s piano sonata (K. 457, I), there is the tonic of mm. 36-41 leading to II6 in m. 46, suggesting the imminent arrival of an authentic cadence. But instead we get a deceptive cadence in mm. 47-48 that delays resolution to the mediant key. The delaying effect resets our expectations and inhibits our reaching conclusions so quickly or relying on them so much once we’ve reached them. Cadwallader and Gagne’s book is excellent in its extensive coverage of effects like these. They don’t talk about Glenn Gould’s peculiar treatments of Mozart, though.

In Glenn Gould’s 1981 recording of Bach’s Goldberg Variations he takes a much slower tempo for the aria/sarabande that bookends the variations, or Variation 7, for example. In the recordings of Haydn’s six late piano sonatas (Hob. XVI: 42, 48-52) he plays the slow
movements very slowly, and with more interpretive inflection. His interpretive eccentricities came not from perversity but from a deep fascination: a love that demands a lingering largo; an experimenter's desire to find out nature’s secrets by way of stop-motion photography; an introspection that was selfless, timeless, otherworldly. To Gould, the slow, measured tempi were tools to reveal hidden profundities—in much the same way as they are to Simone Dinnerstein, for example.

In summary, my idea of ‘funnels’ or hydrodynamic windkessel mathematical models for representing the psychophysiology of generation of musical meaning is, evidently, not so rash or novel after all. True, nobody has yet applied these techniques with fMRI and musical stimuli, in the way that Bosse and others have done with visual stimuli for naval and aviation psychophysiology. But the maths and experimental methods are today up to such a task. Cheers!

Dayan-Abbott book

Parasuraman book


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