Showing posts with label porous media. Show all posts
Showing posts with label porous media. Show all posts

Friday, October 21, 2011

Tokyo Quartet: Meta-Messengers, Physics of Flow in (Musical) Porous Media

Tokyo Quartet
M   usic has values which are above the ‘ordinary’ realm—unchangeable and not subject to mortal instabilities. They are not ‘human-made’ as such, but are domiciled in more esoteric realms of our musical nature. We must turn to the immaterial, spiritual aspects of music in order to find them... The veiled secrets of Art dwell in a region of visionary irrationality... the composer [and, for that matter, the performers-] can never enter this region but can only be elected as its messenger(s).”
  —  Paul Hindemith, A Composer’s World: Horizons and Limitations, pp. 2, 221.
J uxtaposing the ‘familiar’ with the ‘unfamiliar’ is tremendously helpful as a ‘programming pattern’, and this program performed by Tokyo Quartet in the Friends of Chamber Music series was a case in point. Juxtapositions like this reveal new facets of the familiar works and provide context for apprehending the facets of the unfamiliar ones. Juxtapositions produce dynamic contrasts that boost our minds’ “signal-to-noise” ratio, and we are able grasp ideas and feelings that we otherwise might totally miss.
  • Haydn – Quartet in G Major, Op. 77, No. 1
  • Hindemith – Quartet in C Major, Op. 22, No. 4 (formerly No. 3)
  • Schumann – Quartet in A Major, Op. 41, No. 3
I ’d never heard the Hindemith before. The whole program was a treat, but the Hindemith was especially cool. The Tokyos’ lyricism is a fine match for this Hinde-mythic cosmic, rapt vision. The slow Sehr Langsam and Ruhige Viertel movements have a persistent impulse keeping them going… an impulsion that is (‘stets fließend’!) almost feminine in its other-regarding flow.

T he fast passages of the Schnell Achtell, Massig schnelle Viertel, and Rondo movements are robust and motoric, as they should be. Seasoned and mature, without any compulsion to use any work as a technical warhorse. (The whole piece is extremely challenging; the viola and cello parts of Op. 22 present monumental, virtuosic demands, say, but the Tokyo Quartet’s playing never feels self-referential. It is about going to the ‘heart’ of the music... communicating its true message(s)... which is precisely as things should be!) Phenomenal to hear the deep, ‘heart-felt-but-deliberated’ results of musicians’ living for 40+ years with all of these pieces!

I listen... listen to Op. 22 and try to place it, understand it, fit it with other things I think I understand... Yet, it defies fitting. This can’t be ‘counterpoint’—Can it?—if the parts/voices are not dependably running ‘counter’. It is like Hindemith had invented a kind of parallelization/fluidization not unlike what happens in cloud computing these days—with multiple, parallel Hadoop 'map-reduce' jobs.

M utants of a theme ‘melt’ and percolate as subordinate voices, which in turn are witnessed by the other voices, moving in different pores or channels. The ‘stets fließend’ mutants accompany the theme itself, and we get a complex texture or ‘network’ of intertwining thematic lines.

T he network of relations among the various motives in this Hindemith quartet make me look at Fiore’s and Satyendra’s 2005 article (link below). The more direct the relations, the more intuitive they are, and the more of our attention they command. The more direct, intuitive ones then seem to “take precedence” over the more indirect or prosaic ones. Their flow seems to accelerate; each seems to streamline in its channel or crevice as they pass other slower-moving expressions that other members of the quartet are playing at that moment. In that regard, Dembske’s 1995 article mentions Rahn’s work exploring musical “hydraulics” and ‘paths of least resistance’… After the Tokyo Quartet performance, I guess I need to look at Rahn now, too…

A stretch with recurrent, penetrating C-sharps played by the viola (‘mit dämpfer’) in the Ruhige Viertel movement of Op. 22: acknowledging no relationship to the other ‘parts’/‘molecules’/‘reducers’ they have just passed or displaced in the flow… Fragments rejoining each other after each has percolated through different crevices and tubules and pores before converging again… And at passages with fermatas and subsequent resumptions of flow, the dynamics of the musical parts “feel like” moving-boundary integro-differential equations—equations characterizing the flow of a mixture of long-chain macromolecules, a suspension of ‘beads’ undergoing melting as it flows: phase-change equations (Stefan Problem) for flow through a 3-D (or even higher-dimensional) porous-media network. To me, it feels like there is even ‘porosity’ and inter-movement ‘trans-phase mass transport’ between the first and second movements, and between the fourth and fifth. Control-volumes of stets fließend musical ‘beads’ percolate between these movements. It is hard to imagine Hindemith writing such elegant complexity as this at the tender age of 26 years...

P hysics and computational methods for modeling flows in porous media would, I believe, be highly relevant and productive, for better understanding this wonderful music—this graphic Hindemith Vision—quantitatively (please see links below). It remains only for some enterprising young, conservatory-based music theory student to do it! Some of the computational fluid dynamics (CFD) software packages that would be relevant to such a project are in the links below. Also several links for CFD papers that have been delivered at IRCAM conferences in recent years, for your interest. [It’s regrettable that the science and finite-element and Galerkin and other applied mathematics to do this were only developed after Hindemith’s lifetime, because I am confident that he would have enjoyed what they may reveal about the sonic imagery he created—this string quartet and other of his works.]

A ll in all, a tremendously exciting performance, vivid, thrilling!
Bruhn book



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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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