The trouble with rhizomes...

In the sense proposed by Deleuze and Guatarri, some of the earliest examples of rhizomatic works in Western Art music, include, Karlheinz Stockhausen Klavierstück XI [1956], John Cage Concert for Piano [1958], Pierre Boulez Third Piano Sonata [1963-], Mauricio Kagel Prima Vista [1962-3] and Earle Brown Event Synergy II [1967]. (It is perhaps notable that it was not necessary to negotiate functional harmony in a rhizomatic context in any of these works). These are works allowing for a multiplicity of re-orderings of different but determinate pathways to be explored, not simply exhibiting the non-linear “asignifying ruptures” found in Stravinsky’s Symphonies of Wind Instruments [1920] [19] or Charles Ives Holidays Symphony [1913] [D. Thurmaier. Time and Compositional Process in Charles Ives’s Holidays Symphony. PhD Indiana University, 2006, 26-82.
], or “lines of flight” [J. Gilbert.  Deleuze and Music. Edinburgh U.P., 2004] as found in jazz improvisation [J. Barham. Rhizomes and plateaus: Rethinking
jazz historiography and the jazz-‘classical’ relationship. Jazz Res. J., 3, 2, 2010, 171-202. ].
Boulez, for example, describes the Third Piano Sonata as a Labyrinth, in which,

The itinerary is left to the interpreter’s initiative, he must direct himself through a tight network of routes. This form, which is both fixed and mobile, is situated, because of this ambiguity, in the centre of the work for which it serves as a pivot, as a centre of gravity [P. Boulez. Sonate, que me veux-tu?. Perspectives of New 
Music, 1, 2, 1963, 32–44: 41].

A number of problems beset the first generation of paper-based rhizomatic scores:

• since the audience always experiences the works in a linear fashion, sequentially in time, their indeterminacy is unverifiable: the audience cannot compare the pathways chosen to those that were not. In this sense, the rhizomatic qualities are evident to the performer(s) alone;
• performances are arguably undermined by the fact that any particular instantiation may have been potentially been less satisfying than another;
• there can be no overarching cartographical representation of its the structural potentials for the audience inhibiting communication of the structural/performative principles in play;
• the length and complexity of alternate “pathways” are limited to passages accommodated by the printed score (Klavierstück XI uses a very large sheet of paper (53 x 94 cm), and a balsa wood frame to stand it upright);
• the quality of “immanent choice” that is one of the affordances of a rhizomatic structure, can only be executed by a single performer unless conductor(s) are used, (Event Synergy II), otherwise the route through the structure must be pre-determined (Concert for Piano, Third Piano Sonata, Prima Vista);
• indicating and limiting the number of potential connections between pathways is extremely difficult;
• Western music notation is almost exclusively read horizontally from left to right, meaning that pathways remain on a single plane and cannot easily be joined together 2-dimensionally;
• although pathways may consist of varied musical materials and therefore result in diverse musical outcomes, the sounds themselves remain situated in the instruments that make them, excluding the possibility of communication of the rhizomatic structure through the form-bearing [S. McAdams. Psychological Constraints on Form-Bearing Dimensions in Music. Contemporary Music Review, 4, 1, 1989, 181-198.] parameter of spatialisation.

As Žižek noted in relation to the cinematic qualities of novels immediately prior to the emergence of film, the burst of rhizomatic musical works in the 1960s seems “to point towards a new technology that will be able to serve as a more ‘natural’ and appropriate “objective correlative” [S. Žižek. The Art of the Ridiculous Sublime. Seattle Washington: University of Washington Press, 2000, 39.]. The appropriate technology was graphical computing, but its emergence was still more than 30 years away and compositional concerns moved on to other diverse issues including Spectralism and Minimalism. Despite the permeation of rhizomatic concepts in New Media [R. W. Sweeny. Dysfunction and Decentralization in New Media Art and Education. Fishponds: Bristol, 2015, 79-108] and literature [G. P. Landow. Hypertext 3.0: Critical Theory and New Media in an Era of Globalization. Johns Hopkins U.P., 2006.] from the 1990s onward, musical notation proved stubbornly resistant to adaptation to the screen.

Early in 2015 OSC communication was implemented for the Decibel Scoreplayer [C. Hope, A. Wyatt, & L. Vickery. The Decibel Scoreplayer: Enriched Scores for the iPad. Proc. of the 2015 ICM Conference, Denton TX, 2015] allowing for works combining the scoreplayer with synchronized audio processing and spatialisation on a networked computer. This re-opened the possibility for exploring rhizomatic scores with synchronized audio processing and for controlling the movements of the instruments in structurally significant ways

To address these issues a computer-coordinated solution to the problems of rhizomatic presentation of musical compositions with live performers was developed allowing for the creation of precise, unique but variable, multiple versions of rhizomatic works, in which the both the audience and the performers share in the exploratory immanent choice available in this approach, and spatialisation and digital processing are aligned directly to the emerging formal structure.
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Although each of the works Ubahn c. 1985: the Rosenberg Variations [2012], The Last Years [2012], Sacrificial Zones [2014], detritus [2015] and trash vortex [2015]  utilizes a rhizomatic score comprising a network of connections, the formal structures that emerge in a performance derive from the manner in which the performers traverse the score.

Ubahn is the most programmatic of the works and its structure mirrors this in a game-like manner. The individual parts are free to move around the map unless they reach the 'Alexanderplatz' node at which point they are switched to the East German Ubahn system (the black lines in the top right hand corner of Fig. to the left). Upon transfer to the “Eastern Block” their screen is replaced with a “Graffiti Score” (comprising elements of the East German national anthem overlaid with drawings and images by Jon Rose). The performers play the graffiti score as a piece of indeterminate graphic notation. When all players have reached East 

Berlin the graffiti score begins to peel away revealing a five part harmonization of the East German national anthem in traditional notation, which they perform to end the work. Ubahn then, is a concatenative structure [A. Coenen. Stockhausen's Paradigm: A Survey of His Theories. 
Perspectives of New Music, 32, 2, 1994, 200-225.] comprising a freely rhizomatic first section, an indeterminate graphic notation section and a final traditionally notated section. The rhizomatic section has an idiosyncratic form in the sense that the pathway materials are quiet and combine together as a background layer, while the nodal points contain soloistic material. This arrangement highlights the nodal points and provides a contrasting, indeterminate texture.

detritus explores the “territorializing” [G. Deleuze, & F. Guattari. A Thousand Plateaus: Capitalism 
and schizophrenia. Minneapolis, University 
of Minnesota Press, 1987: 317. ] idea of the refrain, an element of rhizomatic structure also discussed by Deleuze and Guattari. They state that the refrain “organizes a limited space around a center in order to keep “the forces of chaos” outside as much as possible” [ibid]. In detritus the score always commences from the same point, a distinctive passage lasting about 10 seconds. At the conclusion of this passage the pathways trifurcate and continue to progressively proliferate. The structure emerges as a consequence of the repetition of this process for different periods of time (between 19 and 145 seconds), allowing a variety of pathways to be charted.
In this way the consequence of the rhizomatic score structure can be emphasized through the exposition of diverse outcomes originating from the same starting point. The use of a Refrain acts against what Deleuze and Guattari would call the ‘deterritorializing’ effect of indeterminate movement through the rhizomatic score structure.
Each performer in detritus has separate parts. The parts are horizontally (temporally) coordinated in the fashion typical of traditional music. This means that when the performers move together their parts are audibly more synchronized than when they are independent of one another. The semantic graphical score was assembled using rhythms and pitch contours from fragments (detritus) of a traditionally notated ensemble piece, cities sunk in endless slumber [2012] for violin, clarinet and piano.

trash vortex takes something of an inverse approach: each part eventually converges upon successive nodes in the rhizomatic score. As the pathways taken from one node to the next vary in duration, each player pauses once a node is reached, “hovering” there until all players have joined them. Tracking the trajectories of each player allows for electronic processing to reinforce the stasis of successive players through emphasis on spectral manipulation of their sound. This structure might be termed a ‘Convergent Nodal’ form, and is a unique implication of rhizomatic structure.