1.1. On mathematics and technologies influencing more-than-human music

Although mathematics are representative frameworks constructed by humans, the properties of mathematically generated structures can contain non-human influences and agencies. This can be observed, for instance, in the way mathematics operate as an agential influence in the field of architecture. Daniel Norell argues ‘that the mathematics that facilitates architectural production has agency’ (Norell Reference Norell2021: 255). Norell demonstrates this through a discussion of how the mathematics underpinning physics simulation software serves as a hidden technological agency ‘that silently governs design’ (ibid.: 263). Norell points to Zeynep Çelik Alexander’s argument (Alexander Reference Alexander, Alexander and May2020), that rather than a simplistic ‘cause-and-effect relationship between object and subject … an approach that focuses on “technics”, a term encompassing both object and procedure’ (Norell Reference Norell2021: 260) must be considered to understand the reciprocal relation between architectural practice and the mathematics underpinning it. In this sense, the dialogue between human and non-human actors constitutes the creative process, whereby both are co-reliant on one another.

As abstract and virtual as mathematics can be, scholars have nonetheless recognised its capacity to act as a material agent. For instance, Elizabeth de Freitas and Nathalie Sinclair have taken a new materialist ontological approach to argue for ‘a theory of embodiment that better recognizes the ways in which mathematical concepts function as material agents in mathematical activity’ (De Freitas and Sinclair Reference De Freitas and Sinclair2013: 454). While at the core of mathematics is virtuality, they seek to locate mathematics in the physical world, specifically through discussing the learning of mathematics as an embodied process. In rethinking ‘the borders of the body, the nature of matter, and the ontology of mathematics’ (ibid.: 457), the assemblage of human with non-human components embraces the body as unfinished, recognising emergence in the encounter. This collaborative construction is also well captured in Karen Barad’s notion of ‘intra-action’, a neologism signifying ‘the mutual constitution of entangled agencies’ (Barad Reference Barad2007: 33; emphasis in original). Within music-making processes involving mathematics, the intra-action between maker and mathematic agent can lead towards the constitution of a more-than-human music – one born of the coming together between human and non-human system. While a more-than-human music entirely devoid of human intervention could be a possibility, I argue that the coming together of human with material agency (in this instance, the material agency of mathematics) can lead to a music that goes beyond the anthropocentric.

The question remaining is what could a more-than-human music that is co-created between human and non-human be? I believe the answer is highly contingent on the materials and intra-action in question. The increasing inclusion of artificial intelligence and machine learning into musical creation is an interesting area to look at because the creative contribution by non-humans can be substantial. Whether or not AI contributes genuine creative ideas is up for debate, although a new materialist perspective would affirm them as actant contributors. Musician-scholars Martin Ullrich and Sebastian Trump argue that AI should be respected as genuine creative contributors, and further consideration needs to be made of what they require, their conditions and the consequences of this (Ullrich and Trump Reference Ullrich and Trump2023). Even if AI and machine learning were considered only to be a tool, their datasets and underlying computational processes still have major impact on the artistic process and resulting work. With AI and machine learning potentially having a huge impact by moving music into increasingly more-than-human terrain, how non-humans exert influence on music creation becomes an even more pressing concern.

Technologic tools and design components have been recognised as affecting the development of musical instruments, frequently with James J. Gibson’s theory of affordances (Gibson Reference Gibson1979) cited to explain how objects, tools or design elements afford certain actions above others. Gibson’s theory explains how environments provide possibility for action to humans or animals operating within them, proposing an ecological psychology of perception. It has been applied within music-making to discuss performers’ responsivity to spatial elements, social factors, compositional frameworks, instruments and technologies (Einarsson and Ziemke Reference Einarsson and Ziemke2017). Homing in on musical instrument design, as Atau Tanaka states, ‘Affordance is a concept fundamental to interaction design practice’ (Tanaka Reference Tanaka2010: 89). Any given interface affords certain modes of action over others, thereby directing a performer to use it in those specific ways. Design practice must take into account the affordances offered by differing interfaces. For instance, William Gaver explains how a horizontal door handle inherently affords pushing, whereas a vertical handle affords pulling, and that differing computer interface objects (buttons, scrollbars) afford certain user interactions (clicking, vertical/horizontal tracking) (Gaver Reference Gaver1991). In creating musical instruments, the affordances of component interface elements must be considered in organising gesture-sound mapping and instrument layout. Yet within the application of affordance theory to design practice, the human decision and selection of affordance-carrying interfaces tends to remain paramount. If human decision is instead framed as being guided by non-human factors – the material agency of interfaces, technologies, or otherwise – then design elements could be understood to propose affordances, thereby guiding the overarching making process. Through intra-action between maker and technology, decentring of the Anthropos and emergence of a more-than-human music can result.

2.1. Methodology

Although the agency of mathematics and technologic systems can be observed in the work of other artists – pertinent to this paper is the influence of hyperbolic equations in the architectural design and musical composition practice of Iannis Xenakis – to understand how non-human agents can guide the process of creation, this article specifically focuses on the act of making. To achieve this, the methodology used here is an autoethnographic account of an artistic-research project created by the author. By presenting reflections from across the project’s development, insights are provided into how my own reasoning and decision-making were influenced by non-human agency, and how a more-than-human artwork emerged from this entangled making. The following presents how a mathematic relationship, namely that defining a hyperbolic paraboloid plane, as well as how the technologic systems I worked with, had agential capacity in the making of the exhibition piece A Sonic Exploration of Hyperbolic Paraboloids.

2.2. Architectural basis

As part of the exhibition Of Time and Place presented at the Huddersfield Art Gallery from 10 February to 5 April 2023, I worked with the architecture firm Feilden Clegg Bradley Studio to create an audience-interactive artwork exploring the mathematics of hyperbolic paraboloids. The exhibition broadly celebrated the culture of Huddersfield, with a large focus on the architectural significance of Huddersfield’s Queensgate Market (Figure 1). Completed in 1970, Queensgate Market is a significant modernist building due to its unique design. Its roof is constructed out of 21 freestanding asymmetrical hyperbolic paraboloid, or hypar, umbrella-like concrete shell structures (Figure 2) (Marsden and Evans Reference Marsden and Evans2009). Although following in the footsteps of earlier hypar architectural developments, notably by the Spanish/Mexican artist Felix Candela, Huddersfield’s market is a novelty in that symmetrical hypar structures were the norm. The Queensgate Market’s architectural novelty helped give the building a Grade II heritage listed status, and, in addition to the good condition of its concrete, mean that the building will be preserved and integrated into the council’s redevelopment of the area. As such, to celebrate Huddersfield’s heritage, and to help draw attention to the building’s unique geometrical form, I was asked to create a sound work about the building.

Figure 1. Queensgate Market as viewed from exterior. The freestanding hypar shells dramatically cantilever above Fritz Steller’s large ceramic panels ‘Articulation in Movement’.

Figure 2. One of the Queensgate Market’s freestanding concrete shells photographed in 1968 during the market’s construction. Copyright: estate of Kenneth Davis.

The resulting work, A Sonic Exploration of Hyperbolic Paraboloids, sonified the mathematics behind the Queensgate Market’s structure. Hyperbolic paraboloids are planes whereby one dimension is defined by a parabolic curve while the other is defined by a hyperbola, typically expressed with the equation $z = {{{y^2}} \over {{b^2}}} - {{{x^2}} \over {{a^2}}}$ . Visible in the saddle shape, hyperbolic paraboloids can be found in structures such as Le Corbusier and Iannis Xenakis’s Phillips Pavilion and the American chip food, the Pringle. In the case of the Queensgate Market, each roof column contains four hyperbolic paraboloid planes intersected together. The same hyperbolic paraboloid equation can result in a variety of curvature planes, depending on the selection of coefficient variables and the limits set. In addition, the combination of multiple hypars together can result in a wide range of shapes. The smooth curvatures and flowing forms resulting from the hyperbolic paraboloid equations have been featured in many visually striking buildings, such as the Iglesia de Santa María Inmaculada in Longuelo, Bérgamo (1960–63) designed by the Italian Pino Pizzigoni (Deregibus and Pugnale Reference Deregibus and Pugnale2009) and Felix Candela’s L’Oceanogràfic in Valencia (2003). Prior to the thin-shell construction which afforded the use of hypars in architecture, in sculpture the constructivist and modernist art movements used hyperbolic paraboloids; for instance, Antoine Pevsner’s Developable Surface (1938–39) or Barbara Hepworth’s Winged Figure (1963).

I was interested in presenting exhibition visitors with a way for interactively exploring different types of hyperbolic paraboloid shapes, through both visual and sonic means. The most fitting means for achieving this goal became apparent when I encountered the software IanniX, a sequencing software inspired by Iannis Xenakis’s approach to converting mathematic, geometric and architectural ideas into music. IanniX allowed for the creation of hypar forms in three-dimensional space as well as creating temporal sequences based on those shapes.

The script editor function of IanniX allowed for the automatic creation of multiple sets of lines defining hyperbolic paraboloid planes. Each line can be assigned a cursor, which serves to convert the length of the line to duration. These cursors can be set to move from one side of the line to its ending point once, or they can be set to oscillate from start to end to start in a continuous loop. The intention of the artwork was to create continuous sound, so the choice of looping cursors was made.

2.3. Overview of completed artwork

Before discussing the human and technologic entangled development process, an outline of the completed artwork will be given. This will help to clarify aspects of the following discussion.

The work, A Sonic Exploration of Hyperbolic Paraboloids (Figures 3 and 4), consisted of a custom interface that allowed exhibition visitors to manipulate the IanniX environment, and consequently affect a Max patch generating the sonic material. Visitors could manipulate three elements on the interface (Figure 5). First, with three LED illuminable buttons, they could select between three different hyperbolic paraboloid versions (to be discussed in the following sections). Second, using an arcade-style control stick, they could rotate the three-dimensional perspective on the currently selected hypar version. Third, using two LED illuminable triangle-shaped buttons, they could change the direction (forwards and backwards, looping when time equalled zero) and speed at which the IanniX cursors moved (with four speed levels). These components were affixed to a laser cut housing which contained an Arduino Uno. The Arduino was connected by USB cable to a Macbook hidden beneath a plinth. The Macbook in turn was connected via HDMI to a monitor displaying in full screen the IanniX visuals. In the background, Max was generating audio based on the selected hypar version and the IanniX cursor movements. These sounds could be heard by a gallery visitor in headphones which they could pick up from the side of the plinth.

Figure 3. The audience-interactive exhibition piece ‘A Sonic Exploration of Hyperbolic Paraboloids’.

Figure 4. A member of the public engaging with the installation. Photograph by Laura Mateescu.

Figure 5. Custom-built interface for visitors to explore the hypars and their sounds.

While controls received by the Arduino would control IanniX, the signal flow was such that the Arduino sent serial messages to Max first. Button selections of different hypar versions would trigger a patch change, as each hypar form corresponded with a different sonification. Messages from the viewpoint rotation joystick and backwards/forwards buttons were reformatted within constraints then piped to IanniX via User Datagram Protocol (UDP). In turn, IanniX’s cursors movements and triggers were sent via UDP to Max to control each hypar versions’ respective sonification patch.

3.1. Initial trial

During my initial trial sonifying a hyperbolic paraboloid plane in IanniX, an immediate conflict arose between my musical imaginary and the mathematics behind the hyperbolic paraboloid. In its most basic form, a hyperbolic paraboloid can contain lines that are all of equal length. However, even if the curvature shape appears more complex than a flat plane, any line defined from one edge to its opposing edge will remain the same length. For the purposes of sequencing musical events, the linearity of this system provides an uninteresting output, as the equal length lines produce no differentiation in their behaviour.

An initial attempt to sequence musical events was produced in IanniX that exemplifies this conflict. Figures 6 and 7 show two views of the same hyperbolic paraboloid plane that I generated using the script editor function of IanniX. Despite the curvature apparent in Figure 7, the lines along each axis are equal in length, as is evident in Figure 6. The orange cursors, which travel along a line’s length, resultantly traverse the plane at an equal rate. As each cursor outputs its distance travelled scale from zero to one, whereby zero is the starting point and one is the line’s end-point, even though each cursor traversed differing x–y–z coordinates, no variation occurred between cursors for this version of the hyperbolic paraboloid equation.

Figure 6. A flat view of an initial attempt at generating a hyperbolic paraboloid in IanniX. Each line on the respective axis is equal in length.

Figure 7. A rotated view of the same initial attempt at generating a hyperbolic paraboloid plane in IanniX. Although the saddle curvature is apparent, the lines defining the shape on two axes are equal in length.

In this attempt to solve the task at hand, Hutchins’s recognition that tools simultaneously are representational media and ‘provide constraints on the organization of action’ (Hutchins Reference Hutchins1995: 154) becomes more apparent. While a different software may have output the exact coordinates of each cursor as it moved in three-dimensional space, resulting in each cursor containing inbuilt data variance due to their different location in x–y–z space, the software’s design produced zero-to-one output per cursor based on their respective line’s length. Had the software produced an x–y–z coordinate output system, this would have only created an illusion that variance existed in this instance of the hypar, as all cursors would still have been moving in parallel motion. The specific coming together of the hyperbolic paraboloid’s formula with its implementation in the IanniX environment, entangled with my search for data-output variance, resulted in a need to experiment with other hypar implementations. In this sense, the technologies of mathematic formula and computer software were actants; they altered my understanding of the task, requiring that I reorganise my actions. Although these two technologies contain within them many layers of human thought, being human-constructed abstractions, in this artmaking process their non-human agency became apparent in their refusal to provide the human-desired variance I was searching for.

3.2. Three hyperbolic paraboloid forms for sonification variation

In reconsidering how to sonify the hyperbolic paraboloid form such that the lengths of lines defining the plane would differ, three versions were made. These three solutions were used in the artwork. Visitors could select different hypar forms via the specially designed interface.

3.2.1. Version 1: A saddle-shaped hypar

The first version (Figure 8; Video Example 1) defined the hyperbolic paraboloid plane by effectively rotating the grid of the initial attempt by 45 degrees. In defining lines diagonal to the x–y axis, the hypar equation of $z = a*{x^2} - b*{y^2}$ (where a and b are coefficients) results in lines with varying length. The saddle shape of the hypar remains apparent, while also affording the variation in line length I desired to create an interesting impact on the sound.

Figure 8. Version 1, a saddle-shaped hypar created in IanniX for the interactive exhibition piece.

The result of this calculation produces a series of different length lines that, when the cursors are set to move along them, produce a wave-like pattern. Figure 9 presents this, showing an x–y viewpoint after 20 seconds have elapsed in IanniX. Although this perspective makes all the lines appear to be of equal length, the z dimension resulting from the hyperbolic paraboloid function produces a curvature. With the cursors set to loop once they arrive at their line’s end-point, this wave becomes more complex over time.

Figure 9. An x–y view of the hypar used in version 1 after 20 seconds has passed in IanniX. A wave-like shape is produced because this version of the hypar’s formula created lines of differing length.

To better define the plane visually, a second series of lines was made with the x-value per line decreasing instead of increasing, resulting in a perpendicular array. As such, the shape that was used for the artwork is symmetrical.

The process of sonification involved experimentally applying the cursors’ movements to different sound-controlling parameters in Max. Throughout this process, the audio software entered into the entanglement, contributing its own agency. The groundwork of Max as an object-oriented programming environment, and one operating with low-level signal processing (its building blocks being fundamental signal types such as sine or sawtooth waves), had direct impact on the musical results. As the installation’s computer generated all sound, the basis for synthesis was these fundamental building blocks – this patch used as its core signal generator a sawtooth multiplied by a sinewave, then passed through a resonant bandpass filter and Max’s inbuilt ‘soft clipping’ overdrive object. The agency of Max as a digital tool also became a contributor, in that the digital allows for rapid and identical cloning of the same patch. To reflect the 30 independent cursor movements in IanniX,Footnote ³ I created a synthesizer bank and assigned it with a chord of 30 equal tempered pitches. Data from the cursors’ movement controlled a number of parameters affecting the synthesizer, including amplitude, filter Q factor, overdrive and modulation. Mainly apparent was the cursors’ impact on amplitude, which would translate the wave patterns caused by the hyperbolic paraboloid into gradually morphing planes of sound. While the initial wave of sound is most evident – whereby the chord fades from silence with a frequency curve corresponding to Figure 9 – as the cursors’ wave-like movements grow too complex for a pattern to be visually discernible, sonic planes can still be heard.

3.2.2. Version 2: A simple hypar plane for triggering samples

The second version (Figure 10; Video Example 2) created a basic 11×11 grid on the x–y plane, whereby the z dimension is calculated based on the hyperbolic paraboloid equation. As is a property of hypars, in this orientation the lines dictating the plane are all straight. However, the lines with the greatest z-value are longer than the others, affording the variance in cursor movement I was looking for. Similar to version 1, a wave-like pattern emerges over time.

Figure 10. Second hyperbolic paraboloid shape. Circles trigger sound files when a cursor encounters them.

For this version, I decided to place triggers along the grid’s diagonal. In IanniX, triggers send single messages when a cursor encounters them (Scordato 2017: 19). As this arrangement is also symmetrical, triggers are executed simultaneously when each pair of cursors (those corresponding along the x-axis and y-axis) passes it. This redundancy was permitted so that the hypar’s planar nature remained visual.

The sonification of the hyperbolic paraboloid movements were here rather simple. When a pair of cursors encountered a trigger, a sample of a gamelan percussion instrument would be triggered. The decision to use gamelan percussion samples was due to the instruments’ fast attacks and long resonance. The samples’ quick onsets articulated the mathematics’ polyrhythmic qualities, while their resonance helped blend the independent lines. The samples were arranged from low pitch to high, with the lowest pitch being at the zero crossing. Similar to the sonification of version one, while the initial series of triggers dictates a simple ascending scale, the wave patterns emerging from the cursors’ movements soon thereafter results in a complex polyrhythmic environment. The resulting patterns appear highly chaotic, although (at least in my perception) there seems to be an underlying organisation evident in the resulting music. This resulting perceptual experience likely is because there is a mathematical structure dictating the music, but yet at this point more-than-human forces feel more at play. While dictated by a logical and structural mathematic relation, the resulting music is mesmerising. It appears beyond anything a human-made organisational endeavour would entail. This intangibly visceral, self-generating behaviour was what I felt so enthralled with as I experimentally exploring the hyperbolic paraboloid formulas, and precisely what I wanted to convey to the visitors as they explored the artwork.

3.2.3. Version 3: Mushroom-shaped hypars

The third version (Figure 11; Video Example 3) resembled most closely the hyperbolic paraboloids used in the Queensgate Market architecture. The building’s free-standing concrete shells have been likened to mushrooms, as the central column supporting the cantilevering edges shares this structure with many mushrooms, such as amanitas or portobello mushrooms. The formation of this outwards spreading shape is made by combining four hyperbolic paraboloid planes together. By duplicating and transforming the plane used in the second version, this mushroom shape can be formed. As such, for this version I wanted to present a sonification that closer reflected the architecture which this project stemmed from. While it would have been possible to create asymmetrical hypars – the closest reflection of the market’s roof – symmetrical hypars were the logical first version to attempt because they are simpler to produce. The sonification that I arrived at from a symmetrical version was interesting in itself, so I never developed an asymmetrical model.

Figure 11. Version 3, containing two mushroom forms, each made from four hyperbolic paraboloid planes.

In effect, the IanniX script I created applied the same approach that was taken in version 2, whereby linear lines defined the plane. However, rather than each line being defined by two points, the lines were created from three points: a start-point, a mid-point, and an end-point. The mid-point defined the curvature leading down from the mushroom’s flat edges to the central ‘dip’. This duplicates the cursor motion of version 2 in the structure itself. If all parameters were equal, a cursor travelling along its line in version 2 would loop at its end-point when the same cursor in version 3 would reach its mid-point. While it would be possible to recreate the same sonification as version 2 by reflecting the triggers, I was more interested in capturing the visually striking nature behind this larger three-dimensional object.

It should be noted that two mushroom shapes of sizes 5×5 and 11×11 would be created at random locations each time a user selected this option. This was only done to provide visual variety and had no effect on the sonification, as the sizes remained the same.

The way that the mushroom shapes were translated into sound was through the cursors effecting a bank of notch filters applied to two field recordings. Here again the agency of Max came into effect: Max’s open sandbox-like interface led me to default to using the basic filter object that I frequently use, the biquad filter. The digital duplicability inherent to the software also meant I could identically replicate the filter patch I made, allowing for quick transferal of data from the multiple IanniX cursors onto independent filters. The field recordings were made in the Queensgate Market after normal trading hours, and were attempts to capture the background hum of the electrical systems running the building.Footnote ⁴ These continuous, noisy droning sounds are quite static, and as such were ideal for shaping with the hypar’s planar movements. Each cursor was applied to a notch filter from low to high, whereby the cursor’s position would control the filtration’s gain. When a cursor was at either the start-point or end-point, the filter would be all-pass, whereas when at the mid-point, the notch was at its strongest. As such, because cursors travelling closer to the central line move slower than those at the extremities, the wave-like motion observable in the other versions appears at first as a mid-frequency bandpass, and then accentuation in the high and low frequencies. As the cursors loop, the wave pattern becomes more complex. The resulting sonification is that of planes of sound gradually forming then dissipating, a continuous and mesmerising soundscape akin to shifting sands.

3.3. Mapping as consequence of software and hardware intra-action

In reflecting on how the hardware interface elements were mapped to affect audiovisual elements, it was recognised that the components exerted non-human agency on my choice-making process, in turn causing intra-action between components and the emergence of an assemblage. The hardware that was chosen were a joystick and buttons,Footnote ⁵ with the basic idea of creating a design akin to an arcade controller. The choice of mapping the joystick to control the hypar models’ perspective and the decision to have two buttons control the forwards and backwards movement of IanniX’s time were a consequence of the hardware components and software intra-acting.