On Musical Self-Similarity Acta Semiotica Fennica XXXIX Approaches to Musical Semiotics Series Acta Semiotica Fennica Editor: Eero Tarasti Associate Editors: Paul Forsell Richard Littlefield Editorial Board: Honorary Member: Thomas A. Sebeok † Pertti Ahonen Henri Broms Jacques Fontanille André Helbo Altti Kuusamo Ilkka Niiniluoto Pekka Pesonen Hannu Riikonen Kari Salosaari Vilmos Voigt Gabriel Pareyon On Musical Self-Similarity Intersemiosis as Synecdoche and Analogy Acta Semiotica Fennica XXXIX Approaches to Musical Semiotics 13 International Semiotics Institute at Imatra Semiotic Society of Finland 2011 This book is a publication of The International Semiotics Institute • Imatra (ISI) http://www.isisemiotics.fi [email protected] tel. +358 20 617 6639 tel. +358 20 617 6700 fax +358 20 617 6696 Cover design by Gabriel Pareyon Cover picture by Matías Vidal Layout by Paul Forsell and the author Copyright 2011 by Gabriel Pareyon All rights reserved Printed by Yliopistopaino, Helsinki (April, 2011) ISSN 1235-479X Acta Semiotica Fennica XXXIX ISSN 1458-492 Approaches to Musical Semiotics 13 ISBN 978-952-5431-32-2 (paperback) ISBN 978-952-10-6970-3 (PDF) Abstract Self-similarity, a concept taken from mathematics, is gradually becoming a keyword in musicology. Although a polysemic term, self-similarity often refers to the multi-scalar feature repetition in a set of relationships, and it is commonly valued as an indication for musical ‘coherence’ and ‘consistency’. This investigation provides a theory of musical meaning formation in the context of intersemiosis, that is, the translation of meaning from one cognitive domain to another cognitive domain (e.g. from mathematics to music, or to speech or graphic forms). From this perspective, the degree of coherence of a musical system relies on a synecdochic intersemiosis: a system of related signs within other comparable and correlated systems. This research analyzes the modalities of such correlations, exploring their general and particular traits, and their operational bounds. Looking forward in this direction, the notion of analogy is used as a rich concept through its two definitions quoted by the Classical literature: proportion and paradigm, enormously valuable in establishing measurement, likeness and affinity criteria. Using quantitative–qualitative methods, evidence is presented to justify a parallel study of different modalities of musical self-similarity. For this purpose, original arguments by Benoît B. Mandelbrot are revised, alongside a systematic critique of the literature on the subject. Furthermore, connecting Charles S. Peirce’s synechism with Mandelbrot’s fractality is one of the main developments of the present study. This study provides elements for explaining Bolognesi’s (1983) conjecture, that states that the most primitive, intuitive and basic musical device is self-reference, extending its functions and operations to self-similar surfaces. In this sense, this research suggests that, with various modalities of self-similarity, synecdochic intersemiosis acts as ‘system of systems’ in coordination with greater or lesser development of structural consistency, and with a greater or lesser contextual dependence. Keywords analogy, autosimilarity, Gestalt, intersemiosis, intersemiotic continuum, invariance, musical coherence, proportion, self-dissimilarity, self-reference, self-similarity, similarity, synecdoche, translatability. v Contents page Abstract v Acknowledgements x Part I Theoretical – Methodological Frame and Basic Definitions 1. Introduction 1 1.1. About this study 1 1.1.1. General background 3 1.1.2. Recent investigation on the subject 8 1.1.3. The ideal reader: musicologist or composer? 11 1.1.4. Empirical evidence and concept testing 13 1.2. Chief lines of exploration 14 1.2.1. The Post-Structuralist view 14 1.2.2. Mental Spaces in Lakoff and Fauconnier 16 1.2.3. The concept of cognitive domain 18 1.2.4. Other points of reference 21 1.3. Methodology 24 1.3.1. Methodological summary 24 1.3.2. Special uses and codes 26 1.3.3. Mathematical language 28 1.3.4. Narrow use of the term fractal 29 1.4. Content organization 33 vii 2. Common notions 35 2.1. Relation 35 2.2. Repetition 37 2.3. Symmetry 40 2.4. Functional similarity 49 2.5. Statistical similarity 52 3. Special notions 55 3.1. The two modes of analogy 55 3.2. Synecdoche 59 3.3. Self-similarity 63 3.4. Invariance 78 3.5. Gestalt 85 3.6. Self-reference 89 3.7. Recursion 94 3.8. Intersemiosis 100 3.8.1. Intersemiotic translation 102 3.8.2. Synecdochic intersemiosis 110 3.8.3. Applying Peircean semiotics to the IC theory 111 3.9. Philosophical implications 115 3.9.1. Perturbation of the creative centrism 115 3.9.2. A self-comprehending map 117 3.9.3. Stochastic distribution 118 3.9.4. On the concept of chaos 120 3.9.5. Power laws 122 3.9.6. Intersemiosis and power laws 124 4. Intersemiotic variety in musical self-similarity 127 4.1. In search of sound’s elementary particle 129 4.2. Mechanical self-similarity 138 4.3. Biological self-similarity 150 4.4. Structural universalism: the linguistic model 159 4.5. Stylistic endomorphisms 170 4.6. Transcultural self-similarity 177 4.7. Tension between grammar and style 186 4.8. Dynamics between idiolect, ecolect and grammar 192 viii Part II Self-similarity in musical information and proportion: From Simple Synecdoche to Complex Intersemiosis 5. Self-similarity as information 205 5.1. On the concept of musical information 207 5.2. Structure and randomness 220 5.3. Music in noise 238 5.4. Noise in music 256 5.5. Determinism and indeterminism in cooperation 298 6. Self-similarity as proportion 319 6.1. Dot and plot: basic analogies 325 6.2. Transition: from simplicity to complexity 336 6.3. Golden mean 372 6.4. Tessellations and brocades 403 6.5. Self-replacement strings 432 6.6. Asymmetry and antiproportion 455 Conclusions 472 Bibliography 485 (e) Electronic sources 522 Index of names 525 Index of subjects 534 ix Acknowledgements This study was supported by many people whose generosity I would like to gratefully acknowledge. Firstly, I would like to thank my mentor, Professor Dr. Eero Tarasti, for his moral support and intellectual encouragement, and for his help looking for valuable information at the right moments during the development of this investigation. Professor Tarasti, despite his many responsibilities in the Department of Musicology at our university, and heading numerous academic projects, always found time to provide wise counsel and other advice. I also thank him for introducing me to external teachers and consultants who contributed to the improvement of my dissertation. This study would not have been possible without the guidance of Dr. Alfonso Padilla, who patiently and helpfully stood by my side revising my text during the sessions in our Seminar of Musicology in Spanish, at the University of Helsinki. His valuable advice and human qualities, together with his methodological help, will always be appreciated. Besides having created a space for musicological discussion at the University of Helsinki, Doctor Padilla has been a sharp critic and a tireless mentor during my doctoral studies, from the first to the last day of this project. Many thanks, dear Alfonso. I am particularly grateful to Professor Marc Leman (Department of Musicology, University of Ghent) and Professor Mark Reybrouck (Department of Musicology, Catholic University of Leuven), for the revision they generously carried out to the final draft of this dissertation. In spite of having manifold occupations as teachers and researchers, both scholars devoted many hours to make detailed criticism and constructive suggestions that contributed to the improvement of this study. I owe my deepest gratitude to my Master’s supervisor, Professor Clarence Barlow, who had a positive influence in helping me gain a clearer research focus. Professor Barlow oriented my interests about music, language, and mathematics, so they could have confluence at the beginning of this study. His helpful feedback and guidance, his encyclopaedic knowledge, and his generous friendship left a deep mark on my own musical ideals. x I am heartily thankful to Professor Raymond Monelle, for having facilitated my access to the John Rylands Library (Manchester University); and to the specialized libraries at the Royal Northern College of Music and the University of Edinburgh. Professor Monelle also provided me with moral support. In spite of his many academic tasks and suffering from health problems, he offered his assistance in the hardest moments during the development of this study. His work is testament to a lifelong devotion to the love of music. I gratefully acknowledge my colleagues in our Seminar of Musicology in Spanish for their solidarity, patience and careful reading of my drafts. Their criticisms and suggestions were taken into account, and are reflected in a significant improvement to this study. In particular, I would like mention Marianela Calleja, Silvia Herrera Ortega, Rafael Junchaya, Mercedes Krapovickas, Grisell MacDonel, Álvaro Menanteau, Sergio Natali, Camilo Pajuelo and Clara Petrozzi-Stubin; their friendship and spirit of cooperation contributed to making the development of this work enjoyable and exciting. Many thanks are owed to Irma Vierimaa, from our Department of Musicology, for her constant and patient advice solving practical problems regarding my research, and to Jaakko I. Tuohiniemi, our school’s librarian, for his very efficient help. Special thanks go to Dario Martinelli for his accurate recommendations. His innovative work and ideas for the renewal of musicology motivated my own enthusiasm for many subjects in this investigation. I also want to thank to Paul Forsell for his help in giving this work its book form; and to my friend Boris Calvo, for his kind support during my studies in Finland. Regarding the English revision of this text, I want to thank the help provided by Peter Field (Language Centre, University of Manchester) and Dr. Christopher Hull (Dept. of Spanish, Portuguese and Latin American Studies, University of Nottingham).