Abstract
Motivated by an investigation of the historical roots of set theory in analysis, this paper proposes a generalisation of existing spectral synthesis methods, complemented by the idea of an experimental algorithmic composition. The background is the following argument: already since 19th century sound research, the idea of a frequency spectrum has been constitutive for the ontology of sound. Despite many alternatives, the cosine function thus still serves as a preferred basis of analysis and synthesis. This possibility has shaped what is taken as the most immediate and self-evident attributes of sound, be it in the form of sense-data and their temporal synthesis or the aesthetic compositional possibilities of algorithmic sound synthesis. Against this background, our article considers the early phase of the foundational crisis in mathematics (Krise der Anschauung), where the concept of continuity began to lose its self-evidence. This permits us to reread the historical link between the Fourier decomposition of an arbitrary function and Cantor’s early work on set theory as a possibility to open up the limiting dichotomy between time and frequency attributes. With reference to Alain Badiou’s ontological understanding of the praxis of axiomatics and genericity, we propose to take the search for a specific sonic situation as an experimental search for its conditions or inner logic, here in the form of a decompositional basis function without postulated properties. In particular, this search cannot be reduced to the task of finding the right parameters of a given formal frame. Instead, the formalisation process itself becomes a necessary part of its dialectics that unfolds at the interstices between conceptual and perceptual, synthetic and analytic moments, a praxis that we call musique axiomatique. Generalising the simple schema of additive synthesis, we contribute an algorithmic method for experimentally opening up the question of what an attribute of sound might be, in a way that hopefully is inspiring to mathematicians, composers, and philosophers alike.
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Notes
- 1.
- 2.
The “crisis” of intuition was called the “Krise der Anschauung” in the German discourse [17].
- 3.
They continue with acoustic examples: “Besides applying it to our radar image processing interests, we also found the transform provided a very good analysis of actual sampled sounds, such as bird chirps and police sirens, which have a chirplike nonstationarity, as well as Doppler sounds from people entering a room, and from swimmers amid sea clutter” [11].
- 4.
In the general case, these partials need not be linearly independent, and the coefficient need not be unique for a given resulting function. It is convenient, however, if we know a coefficient that cancels the contribution of the respective partial (typically zero). This means that depending on the combinator G, we need different scaling functions for each partial. With an explicit generalised scaling function, and a neutral element e with regard to G (usually, the neutral element, i.e. zero for addition and one for multiplication), we can write: \(f(t) = \underset{\scriptscriptstyle {i = 1}}{\overset{\scriptscriptstyle {n}}{G}} \; c_i g_i(t, 1) + (1 - c_i) e\)
- 5.
In all ‘conventional’ series, the combinator is just the iterated addition. \(G = g_1(t, c_1) + g_2(t, c_2) + \dots g_n(t, c_n)\), or conveniently \(\sum _{i = 1}^{n}{g_i(t, c_n)}\), where usually \(g_i(t, c_i) = c_i g_i(t)\). In the general form, however, a combinator is thought of as any interpretation of ‘\(+\)’, thus any form of ‘one more’.
- 6.
- 7.
This general schema does not lead to any method to calculate the coefficients for a given case and neither does it guarantee that it is orthogonal, unique, and linearly independent. But as we shall see more clearly in the next section, these properties need not be secured in advance where no type can be given anyhow.
- 8.
We are aware that the term generic may lead to misunderstandings, in particular due to the existing terminology in topology. We use the term to mark a distance from the idea of ‘generalisation’, following Alain Badiou’s and Paul Cohen’s concept of a generic set, as briefly explained in the last part of section 2. We have to leave open to what degree the precise ramifications of this concept remain adequate to its origin.
- 9.
Note that in the SuperCollider signal semantics, the time parameter t is usually factored out: UGens are essentially arrows, similar to the description given by Hughes [8].
- 10.
‘One more step’ here simply means ‘one more f unction applied’. Note that this is a case where the order in which the partials are combined influences the outcome (the operation of function composition is in general noncommutative). Furthermore, the coefficient scaling function is a little more complicated: a coefficient of zero must result in the identity function \(f(x) = x\), when applied to a partial g.
- 11.
The concatenative language Steno is embedded in SuperCollider. See https://github.com/telephon/Steno. For examples of generic additive synthesis, see: https://github.com/musikinformatik/Generic-Additive-Synthesis
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Acknowledgements
In the process of experimenting with generic additive synthesis in a multichannel laboratory environment, the inspiring contributions by Hans W. Koch and Florian Zeeh were essential. We would also like to thank Guerino Mazzola for his ideas on frequency modulation in the present context. This paper would have lacked much of what we like about it without the continuing exchanges with Gabriel Catren, Maarten Bullynck, Renate Wieser, Tzuchien Tho and Alberto de Campo. The clarity of James McCartney’s programming language design choices made it easy to develop these ideas. Last but not least, Frank Pasemann and Till Bovermann have given extremely valuable comments on the terminology and formalisation used – it goes without saying that we take full responsibility for remaining errors.
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Rohrhuber, J., Lach Lau, J.S. (2017). Generic Additive Synthesis. Hints from the Early Foundational Crisis in Mathematics for Experiments in Sound Ontology. In: Pareyon, G., Pina-Romero, S., Agustín-Aquino, O., Lluis-Puebla, E. (eds) The Musical-Mathematical Mind. Computational Music Science. Springer, Cham. https://doi.org/10.1007/978-3-319-47337-6_27
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