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Boulez’s Creative Analysis: An Arcane Compositional Strategy in the Light of Mathematical Music Theory

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Patterns of Intuition

Abstract

We investigate part I of the famous composition Structures pour deux pianos by Pierre Boulez with regard to their mathematical construction principles and interpret the analytical results in order to obtain computational schemes for generalized compositions following Boulez’s approach and also in the lines of Boulez’s principle of creative analysis. These generalized schemes are then implemented in rubettes of the software Rubato and yield corresponding compositions. Our analysis confirms the visionary force of Boulez’s innovation in that his matrix methods for part I turn out to be in complete congruence with the category-theoretical situation created by generally addressed points in the spirit of the Yoneda lemma and then systematically used by Alexander Grothendieck.

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Notes

  1. 1.

    For a more philosophical discussion of this approach, we refer to [13, Chap. 7].

  2. 2.

    It should however be noticed that such a creative analysis had been applied in the case of Beethoven’s op. 106 [10] before we knew about Boulez’s idea. The present approach is somewhat more dramatic since we shall now apply Boulez’s idea to two his own works, namely Structures [3, 4].

  3. 3.

    And yes, he is the singer Joan Baez’s cousin, music matters\(\ldots \)

  4. 4.

    We shall come back to this point in the course of our own analytical work using modern mathematics instead of plain combinatorics.

  5. 5.

    Ligeti names it \(R\), but we change the symbol since \(R\) is reserved for retrograde in our notation.

  6. 6.

    “\(\ldots \) schliesslich die Tabellen fetischartig als Mass für Dauernqualitäten angewandt \(\ldots \)”.

  7. 7.

    Title of a Cecil Taylor LP.

  8. 8.

    An affine morphism \(f:M\rightarrow N\) between modules \(M,N\) over a commutative ring \(R\) is by definition the composition \(f = T^t\cdot g\) of a \(R\)-linear homomorphism \(g:M\rightarrow N\) and a translation \(T^t:N\rightarrow N:n\mapsto t+n\). Affine morphisms are well known in music theory, see [12].

  9. 9.

    We represent elements \(x\in \mathbb {Z}_{12}\) by natural numbers \(0\le x\le 11\).

  10. 10.

    All denotators in this discussion will be zero-addressed.

  11. 11.

    In a more recent development, the mouse-driven input is being replaced by a direct gestural input using the hand recognition software Leap Motion.

  12. 12.

    http://www.encyclospace.org/special/restructures.mp3, accessed 4 Jul 2014.

References

  1. Baez C (2006) Quantum quandaries: a category-theoretical perspective. In: French S et al (eds) Foundations of quantum gravity. Oxford University Press, Oxford

    Google Scholar 

  2. Boissiere A (2002) Geste, interprétation, invention selon Pierre Boulez. In: Revue DEMéter, Lille-3 University

    Google Scholar 

  3. Boulez P (1953) Structures, premier livre. UE, London

    Google Scholar 

  4. Boulez P (1967) Structures, deuxième livre. UE, London

    Google Scholar 

  5. Boulez P (1989) Jalons (dix ans d’enseignement au Collège de France). Bourgeois, Paris

    Google Scholar 

  6. Deyoung L (1978) Pitch order and duration order in Boulez’ Structure Ia. Perspect New Music 16(2):27–34

    Article  Google Scholar 

  7. Ligeti G (1958) Pierre Boulez: Entscheidung und Automatik in der Structure Ia. Die Reihe IV, UE, Wien

    Google Scholar 

  8. Ligeti G, Neuweiler G (2007) Motorische Intelligenz. Wagenbach, Berlin

    Google Scholar 

  9. MacLane S, Moerdijk I (1994) Sheaves in geometry and logic. Springer, New York

    Google Scholar 

  10. Mazzola G (1985) Gruppen und Kategorien in der Musik. Heldermann, Berlin

    MATH  Google Scholar 

  11. Mazzola G et al (1996) RUBATO on the internet. http://www.rubato.org. Visited on 24 Aug 2014

  12. Mazzola G et al (2002) The topos of music—geometric logic of concepts, theory, and performance. Birkhäuser, Basel

    Google Scholar 

  13. Mazzola G (2007) La vérité du beau dans la musique. Delatour, Paris

    Google Scholar 

  14. Mazzola G et al (2011) Musical creativity. Springer, Heidelberg

    Book  Google Scholar 

  15. Milmeister G (2009) The Rubato Composer software. Springer, Heidelberg

    MATH  Google Scholar 

  16. Simondon G (1989) Du mode d’existence des objets techniques. Aubier, Paris

    Google Scholar 

  17. Thalmann F (2007) Musical composition with grid diagrams of transformations. MA thesis, University of Bern, Bern

    Google Scholar 

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Authors and Affiliations

  1. School of Music, University of Minnesota, Minneapolis, USA

    Guerino Mazzola

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  1. Guerino Mazzola
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Correspondence to Guerino Mazzola .

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Editors and Affiliations

  1. University of Music and Performing, Institute of Electronic Music and Acoustics, Graz, Austria

    Gerhard Nierhaus

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Mazzola, G. (2015). Boulez’s Creative Analysis: An Arcane Compositional Strategy in the Light of Mathematical Music Theory. In: Nierhaus, G. (eds) Patterns of Intuition. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9561-6_18

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  • DOI: https://doi.org/10.1007/978-94-017-9561-6_18

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  • Online ISBN: 978-94-017-9561-6

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