jwilzewske: The Composer Vanishes: Solving the Diffusion Equation Shows that Adès’s ‘Arcadiana’ is Really a Quintet with a Fifth Un-voiced ‘Phantom’ Part

DSM. Entropy-diffusion from composer Thos. Adès, in absentia, to and through string quartet artists performing ‘Arcadiana’, iii

B oth form and content in Adès’s music depend upon the multivalent conjunctions that may be derived from relatively elemental forms of continuity… Analyses of ‘Arcadiana’ illustrate the paradoxical nature of postmodern musical time, in which a narrative linear interpretation may emerge from the higher-level perception of locally-concurrent periodicities and trends… Rhythmic analysis entails identifying projections of duration and pitch, as well as processes of timbre and dynamics… In some cases, these interactions are mutually reinforcing; in others, they may embody reorientation or conflict, not to mention ironic deception.”
— John Roeder, Music Analysis 2006; 25:121-54.

A dès’s music merits special attention because of the consistency with which it employs elemental continuities, and because of the variety of temporalities that the composer derives from them... Continuity [is] defined as an association between two percepts, formed when the second realises a mental projection that was made as part of the first. Generally, continuity manifests itself in sequences of successive pitches, or in series of pitch percepts that are not literally successive... It may also subsist in more abstract projections and realisations of duration...”
— John Roeder, p. 122.

T he Jupiter String Quartet delivered a wonderful account of Thomas Adès’s String Quartet Op. 12, ‘Arcadiana’ (Venezia notturna; Das klinget so herrlich, das klinget so schön; Auf dem Wasser zu singen; Et... (tango mortale); L’embarquenment; O, Albion; Lethe) [1994] in Kansas City on Friday night in the Friends of Chamber Music 2008-2009 series—this in a program that also included Mendelssohn’s String Quartet in A minor, Op. 13, and Beethoven’s String Quartet in E minor, Op. 59, No. 2.

Nelson Lee and Meg Freivogel, violins
Liz Freivogel, viola
Daniel McDonough, cello

I n 2008, the Jupiter String Quartet received the prestigious Avery Fisher Career Grant, and in 2007 the Quartet won the Cleveland Quartet Award from Chamber Music America. This year is the final year in the Jupiters’ three-year residency with Lincoln Center’s ‘Chamber Music Society Two’. From 2004 to 2006, the Quartet was enrolled in the Professional String Quartet Training Program at New England Conservatory, earning Master of Music degrees in Chamber Music.

A dès wrote his first and, to date, only string quartet in 1994, under commission from Endellion Quartet in the U.K. The work is highly regarded by both performers and by theorists, and has been the subject of several important published analyses. The first several bars of the third movement (‘Auf dem Wasser zu singen’ [To be sung on the water]) are especially amenable to a kind of analysis that I think can shed light on the theme of the whole work—‘vanishing’ and mortality. That is what this post is about...

F ranz Schubert’s ‘Auf dem Wasser zu singen’ D. 774 (1823) is a setting of Friedrich Leopold Graf zu Stolberg-Stolberg’s poem, written in 1782—which Adès took as an inspiration/calibration for this third movement.

M itten im Schimmer der spiegelnden Wellen
Gleitet, wie Schwäne, der wankende Kahn:
Ach, auf der Freude sanftschimmernden Wellen
Gleitet die Seele dahin wie der Kahn;
Denn von dem Himmel herab auf die Wellen
Tanzet das Abendrot rund um den Kahn.

Über den Wipfeln des westlichen Haines
Winket uns freundlich der rötliche Schein;
Unter den Zweigen des östlichen Haines
Säuselt der Kalmus im rötlichen Schein;
Freude des Himmels und Ruhe des Haines
Atmet die Seel im errötenden Schein.

Ach, es entschwindet mit tauigem Flügel
Mir auf den wiegenden Wellen die Zeit;
Morgen entschwinde mit schimmerndem Flügel—
Wieder wie gestern und heute die Zeit—
Bis ich auf höherem strahlendem Flügel
Selber entschwinde der wechselnden Zeit.

[Amid the glimmer of sparkling waves,
The bobbing boat glides like a swan:
Ah, the soul glides onward like the boat,
On shimmering, gleaming waves of joy;
For the sunset glow, shining down from heaven
Upon the waves, dances all ‘round the boat.

The rosy light winks at us
From above the treetops of the western wood;
Beneath the branches of the eastern wood,
The reeds whisper in the light—
In the reddening glow, the soul breathes
The joy of heaven, the peace of the grove.

Ah, alas, Time itself vanishes on dewy wings and I with It,
Into the weighty cradle of the waves.
Tomorrow, Time will fly away on glistening wings—
As it did today and yesterday and the day before—
Until I myself fly away from Time’s inconstancy
On yet loftier, more radiant wings.] ”
— Friedrich Leopold Graf zu Stolberg-Stolberg, ‘Auf dem Wasser zu singen’ [To be sung on the Water], 1782.

H ow does this piece work? What makes ‘Arcadiana’ ‘tick’? I look first at my rhythm metrics... These are not Fibonacci series... Ah, this quartet feels like a diffusion problem! Analytical solutions to heat- and mass-transfer problems reduce to solving partial differential equations (PDEs)—in this case, the so-called ‘heat equation’, treating the medium as a homogeneous solid under appropriate initial and boundary conditions. Hypothetically, the IC and BC can include convective and radiative interactions with the environment, but in the present example of Adès’s ‘Arcadiana’ I ignore convection and radiation possibilities.

M any analytical solutions of the heat equation refer just to the simple one-dimensional situation, with a ‘heat capacity’ c, a ‘diffusion coefficient’ k, and a ‘loss term’, φ, to account for dissipation of energy as diffusion proceeds.

DSM. Diffusion equation with dissipation term φ

T he Endellion Quartet recording of ‘Arcadiana’ is the only one currently available. But, obviously, it is not possible to separate the four parts in the stereo channels. So I transcribed each part into Finale®, played each track individually, ripped myself MP3s from those, converted to WAV files, and began examining entropy as a function of time using FAWAV® and MATLAB®’s signal-processing toolbox.

DSM. Wavelet-transform entropy analysis of viola part, mm. 1-4, ‘Arcadiana’, iii

T he results surprise even me. Using Crank-Nicholson finite-difference math to solve the 1-dimensional diffusion equation for propagation of entropy S(t), I find that the entropy pulse in the viola part diffuses to the cello, which appears to be sitting a “virtual 1,600 msec” away. Then the same entropy pulse appears, diffusing to the vn2 part with a smaller peak amplitude and more spread-out, consistent with a non-zero ‘dissipation’ within the expressive ‘medium’ constituted by the music. The vn2 appears to be located, according to the Crank-Nicholson solution of the diffusion equation, at a distance of about 2,100 msec from the cello. Further downstream, vn1 experiences the diffusing entropy pulse, at a virtual distance 1,200 msec away from vn2.

Example 11a from John Roeder, ‘Co-operating continuities in the music of Thos. Adès’, Music Anal 2006; 25:121-54

A dès often combines such elemental durational and pitch continuities within a single stream so that their realised projections are made to reinforce one another ... a series of regularly increasing and descending pitch-intervals unfolds [in ‘Arcadiana’, iii] within a sequence of regularly decreasing durations.”
— John Roeder, p. 126.

T he effects of harmonic confluence may lead to a blurring of streams, and thus to a clear sense of temporal interaction, not merely co-operation. Example 11a renotates and simplifies the first eight bars of the third movement of ‘Arcadiana’ in order to foreground the two streams of activity which characterise this passage. In the pizzicato stream, the instruments alternate transpositions of the accelerating expanding-interval motive... The registral and transformational termination of the motive’s accelerating free-fall imparts a frustrated potential energy to its final interval that finds only tentative release in the contrasting tremolos of hexatonically stacked perfect fifths...”
— John Roeder, p. 141.

T hese results by themselves were fascinating to me. But then I got to thinking. Who is the ‘source’? The patterns account for the palpable, organically-sensible relative locations of four performers experiencing an entropy wave. But where is the entity who emitted this entropy wave? Can we use the PDE solution to impute where that ‘phantom’ source is ‘located’ with respect to the 1-D positions of the other players? (Click on the jpeg below to download the Excel spreadsheet with the C-N results in it.)

DSM, Crank-Nicholson solution for entropy-diffusion in mm.1-4 of Thos. Adès’s ‘Arcadiana’, iii

A nd the answer is, ‘Yes’! If we take the ‘source’ of the entropy wave to be the composer in-absentia, then the diffusion equation says that that invisible, inaudible phantom is located about 3,200 msec upstream of the viola (jpeg at top of this CMT post). This piece is a quasi-quintet where the fifth player is absent!

A ctually, encouraged by the preliminary findings above today I’ve begun to do more calculations from other sections of this movement. And it appears that not only do the four string parts move around with respect to each other, as evidenced by the diffusion of entropy pulses propagating between and amongst them, so too does the phantom of Adès move, hovering at times very nearby one part and at other times watching the quartet at what is apparently a great distance.

John Roeder, Mus Analysis 2006; 25:121-54, Example 1c, Arcadiana, iii, mm. 1-2

John Roeder, Mus Analysis 2006; 25:121-54, Table 1

J ohn Roeder’s wonderful 2006 article in the journal Music Analysis—on coordinated continuities in Adès—is a great contribution to our understanding, illuminating a number of the features and mechanics of what Adès is doing. But I think what we have here (in entropy-diffusion evidence) is far deeper and far more symbolic (of Adès’s theme of mortality and vanishing) than Roeder and others have thus far recognized. What Adès has constructed here is nothing less than a statistical physics ‘diffusion chamber’ in which the four living performers together create an acoustic entropy field that strongly implies the existence/presence of the music’s creator. We are each familiar with the diffusion of a solute like sugar as it dissolves in a cup of tea; we are each familiar with the diffusion of heat by conduction through the cup from the hot inside to the surface that we hold in our hand. And the diffusive/dissipative effects that Adès has crafted here also strike the performer/listener as intuitively, empirically, palpably familiar. The fact that we can quantitatively model them and analyze them with pretty fancy math tools does not in any way diminish the fact that the underlying cognitive physics that Adès is using is utterly elemental, utterly organic.

A rcadiana’ is very difficult to play, yes. But when played well, the quartet evokes a physical reality that belies the technical difficulty. The very memory and spirit of the person who is absent are reanimated by the performers. We sense Adès’s presence through the implied [silent] fifth voice that is the source of the entropy waves that percolate throughout the other four parts. I find this deeply beautiful, as I hope you do as well. Thank you, Thomas! Thank you, Jupiters!

W e are particularly fond of the ‘Arcadiana’ quartet. We wish Thomas would get busy and write more quartets.”
— Daniel McDonough, Jupiter String Quartet, 20-FEB-2009, remark over supper.

jwilzewske

Sunday, February 22, 2009

The Composer Vanishes: Solving the Diffusion Equation Shows that Adès’s ‘Arcadiana’ is Really a Quintet with a Fifth Un-voiced ‘Phantom’ Part

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