Iannis Xenakis and Sieve Theory An Analysis of the Late Music (1984-1993) A Dissertation Presented in Fulfilment of the Requirements for the Degree of Doctor of Philosophy Goldsmiths College, University of London by Dimitrios Exarchos Volume 1 of 2: Text Dissertation Supervisor: Dr Craig Ayrey November 2007 DECLARATION I declare that the work presented in this thesis is my own. Dimitrios Exarchos 2 Abstract Iannis Xenakis and Sieve Theory An Analysis of the Late Music (1984-1993) Dimitrios Exarchos 2007 This thesis is divided in three parts, the first two of which are theoretical and the third analytical. Part I is an investigation of Iannis Xenakis’s general theory of composition, the theory of outside-time musical structures. This theory appears in many of Xenakis’s writings, sometimes quite idiosyncratically. The aim of this part is to reveal the function of the non-temporal in Xenakis’s musical structures, by means of a historical approach through his writings. This exploration serves to unveil certain aspects discussed more thoroughly through a deconstructive approach. The deconstructive is demonstrated in the classification of musical structures and aims partly at showing the nature of Time in Xenakis’s theory. Part II is preoccupied with Xenakis’s Sieve Theory. In the earlier writings on Sieve Theory he presented a slightly different approach than in the later, where he also provided an analytical algorithm that he developed gradually from the mid 1980s until 1990. The rationale of this algorithm and the pitch-sieves of 1980-1993 guides Part III, which is preoccupied with a methodology of sieve analysis, its application, and an exploration of the employment of sieves in some of Xenakis’s compositions of the 1980s. When possible, the analysis takes in consideration the pre-compositional sketches, available at the Archives Xenakis, Bibliothèque Nationale de France. The sketches reveal aspects of the application of Sieve Theory, not included in Xenakis’s theoretical writings. 3 As with the application of other theories, Xenakis progressed to less formalised processes. However, this does not mean that Sieve Theory ceased to inform the process of scale-construction. As the conclusion of this dissertation indicates, he employed Sieve Theory in order to achieve structures that conform to his general aesthetic principles, that relate to various degrees of symmetry and periodicity. 4 Contents Acknowledgements..........................................................................................8 Abbreviations..................................................................................................9 Introduction ..................................................................................................10 PART I 1 Outside-Time Structures ..............................................................................20 1.1 Literature ......................................................................................................20 1.2 Overview........................................................................................................21 1.3 Symbolic Logic (‘Symbolic Music’ – 1963) ..................................................23 1.4 Two Natures (‘La voie de la recherche et de la question’ – 1965)...............28 1.5 Tomographies Over Time (‘Towards a Metamusic’ – 1965).......................31 1.6 ‘Towards a Philosophy of Music’ (1966)......................................................33 1.7 Ontological/Dialectical (‘Une note’ – 1968)..................................................34 1.8 Arts/Sciences: Alloys (1976)...........................................................................36 2 Outside-Time Structures as Writing............................................................38 2.1 The Third Term ............................................................................................38 2.2 The Critique of Serialism..............................................................................40 2.2.1 General Harmony .........................................................................................40 2.2.2 Magma...........................................................................................................42 2.3 The Temporal as Outside-Time....................................................................44 2.4 Outside-Time as Supplement........................................................................46 2.5 Symmetry ......................................................................................................47 2.5.1 Series..............................................................................................................47 2.5.2 Non-Retrogradable Rhythms .......................................................................50 2.6 Spacing ..........................................................................................................51 PART II 3 Sieve Theory..................................................................................................54 3.1 The Sieve of Eratosthenes.............................................................................54 3.2 Logical Operations........................................................................................57 3.2.1 Union .............................................................................................................57 3.2.2 Intersection....................................................................................................58 3.2.3 Complementation..........................................................................................59 3.3 Transcription.................................................................................................60 3.3.1 Transcription of unions ................................................................................61 3.3.2 Transcription of intersections.......................................................................62 3.4 Types of Sieves ..............................................................................................63 3.4.1 Symmetry ......................................................................................................64 5 3.4.1.1 Symmetric Sieves ..........................................................................................65 3.4.1.1.1 Symmetric Sieves with Even Number of Intervals ......................................66 3.4.1.1.2 Symmetric Sieves with Odd Number of Intervals........................................68 3.4.1.2 Asymmetric Sieves ........................................................................................70 3.4.2 Periodicity .....................................................................................................71 3.4.2.1 Prime Sieves ..................................................................................................71 3.4.2.2 Composite Asymmetric Sieves......................................................................72 3.5 Metabolae of Sieves (Transformations)........................................................74 3.5.1 Residues.........................................................................................................74 3.5.1.1 Inversion........................................................................................................74 3.5.1.2 Cyclic Transposition .....................................................................................76 3.5.1.3 Variables........................................................................................................79 3.5.2 Moduli ...........................................................................................................80 3.5.3 Unit ................................................................................................................81 4 Sieve Theory and Primes ..............................................................................82 4.1 Canonical Form.............................................................................................82 4.2 Limitations ....................................................................................................86 4.3 Types of Formulae ........................................................................................86 4.3.1 Decomposed Formula ...................................................................................86 4.3.2 Simplified Formula .......................................................................................88 4.4 Program: Generation of Points ....................................................................91 4.5 Sieve Theory and Sieves................................................................................94 4.6 Symmetries/Periodicities...............................................................................95 PART III 5 Methodology................................................................................................101 5.1 Inner Periodicities and Formulae Redundancy.........................................101 5.2 Construction of the Inner-Periodic Simplified Formula ...........................104 5.3 Analytical Algorithm: Early Stage.............................................................107 5.4 The Condition of Inner Periodicity ............................................................110 5.5 Inner-Periodic Analysis ..............................................................................112 5.6 Interlocking Periodicities............................................................................116 5.7 Analytical Algorithm: Final