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Transfiguring Conventional Music Elements a Mathematically Informed Approach to Composition

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Transfiguring Conventional Music Elements a Mathematically Informed Approach to Composition Estonian Academy of Music and Theatre Giovanni Albini Transfiguring Conventional Music Elements A Mathematically Informed Approach to Composition A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Music) Supervisor: Prof. Kerri Kotta Tallinn 2021 Abstract As a composer, when I deal with the issue of musical legacy I feel the impact of two opposing strands: the one that can be traced back to modernism, thus overcoming tradition, and the other under the influence of post-modernism, that often reduces tradition to a mere distant material. Because neither of them, taken independently, satisfies me, my composition practice responds to the need for a third approach, not renouncing the desire for novelty nor the awe-inspiring aura of the established and intelligible material of musical legacy. In fact, the concern for tradition, the crave for novelty and beauty, and the mathematical means have been the key points of my whole activity as a musician and as a composer. In this context, the general questions that had been the starting point of my artistic research are the following. How can mathematics serve to shape musical structures that grant a neat focus on traditional music elements and yet put them in a different perspective? And which of the several ways I could find are closer to my individual character, aesthetics and aims? To answer these broad questions I took into analysis from a musicological standpoint the process of my former composition practice, recognizing some mathematically informed traits that I could then reduce and formalize in three concepts of a specific structured compositional strategy of combinatorial nature, that I named 1) completeness, 2) exhaustiveness and 3) equality in repetition. These three concepts naturally emerged in my own composition practice. Thereby, such study let me narrow down the aforesaid questions to a more specific one: the very research question of this text. How and why the three concepts of completeness, exhaustiveness and equality in repetition can serve to transfigure conventional musical elements offering useful tools for composers? Since the three concepts originated in the specific harmonic context of major and minor triads, my method to answer the question had been to inductively generalize them and then test the derived strategies in different contexts, giving rise to new scores. Therefore, I took the documentation of the composition process and its outputs into analysis. Finally, I investigated and discussed the wider function and potential of the core of the new strategies trying to understand not only how and why they have been useful in my own composition practice, but rather how and why they could be useful, relevant and effective also from a general standpoint. Table of Contents 1. Introduction ………………………………………………………………………. 1 2. A mathematically informed approach to music composition ………………………. 3 2.1 Some aesthetic remarks ………………………………………………………………. 5 2.2 Mathematics in the composition practice ………………………………………. 8 2.3 Completeness, exhaustiveness and equality in repetition ………………………. 9 3. Graphical strategies dealing with triads ………………………………………………. 13 3.1 Hamiltonian cycles in the chicken-wire torus ………………………………………. 13 3.2 Rearranging the past ………………………………………………………………. 16 3.3 Escaping conventions ………………………………………………………………. 17 4. Combinatorial strategies handling diatonic trichords ………………………………. 23 4.1 A comprehensive strategy ………………………………………………………. 24 4.2 Layering techniques ………………………………………………………………. 27 5. Beyond harmony ……….…………………………………………………………….... 31 5.1 Engaging performance ………………………………………………………………. 31 5.2 Rhythmical elements ….………………………….………………………………... 36 5.3 A computational approach ……….……………………………………………… 40 5.4 Symbolic meanings ………………………………………………………………. 42 6. Conclusions ………………………………………………………………………. 47 Bibliography ………………………………………………………………………. 51 List of doctoral concerts ………………………………………………………………. 55 T>> l?hikokkuv@te ………………………….……………………………………………. 57 1. Introduction “The present slips and vanishes like sand between the fingers, acquiring material weight only in its recollection.” Andrei Arsenyevich Tarkovsky “Beauty. My steps in music have all been focused on it, looking for it, struggling for it.” (Albini 2021: 249) Accordingly, I believe that art is not simply the pursuit of beauty, rather the pursuit of the celebration of beauty, and “I consider every technique I deepen meaningful if it does nurture my own aesthetic purposes, the ‘beautiful’ music I wish to write.” (Albini 2021: 249) Thereby, headed by my nature and instinct somehow, I have always been drawn by two specific concepts, two sides of the same ideal of beauty1 I am pursuing. On the one side, there is the ecstatic charm of mathematically informed2 poetics: music and mathematics have been constantly bounded in my composition practice and their relationship became my own uncharted music territory to explore: beautiful means. On the other side, I have always been fascinated by the monumental, timeless fascination of very familiar conventional music elements of Western culture, such as for instance triads and diatonic frameworks: a beauty to be celebrated. I personally feel that both sides, if taken alone, tend to lead to a sort of emptiness and triviality. A straight mathematical approach to composition risks to result in what I believe is a pure and superficial formalism, conceptualism or structuralism. In fact, in my personal view, the prioritisation of formal details, structures or external concepts make music shrinking to a sole translator of other – eventual – forms of beauty: an empty musical vessel. At the same time, the use of very exploited musical objects makes it easy to fall into clichés, getting to stereotypes of beauty. Thus, the preliminary question: may they together enhance mutually their so fragile beauties? Ultimately, mathematics may help revealing undiscovered paths and solutions in the use of traditional musical materials, bringing with it the beautiful seal of all its ontological purity. But at the same time the musical materials themselves, being so rooted in Western culture and minds, no matter how much alienated, could keep some of their expressive contents, of their history, of their memories of beauty. 1 My concept of beauty, that is much related to the categories of wonder and discovery, is tackled in Paragraph 2.2. 2 In the context of this work, the term ‘mathematically informed’ does not mean any formal way of composing, but simply suggests the general availment of mathematics in my own compositional strategies, practice, and imagination. 1 This leads to the general question of my own artistic research: how can mathematics serve to shape musical structures that grant a neat focus on conventional music elements and yet transfigure them? Methodologically, I have started looking for an answer to this broad question analyzing my scores and habits in approaching the composition process, generalizing from them some mathematically informed strategies that could be then developed and tested in a new – more aware – artistic practice, researching in an endless virtuous cycle: from practice to theory, from theory to practice. This is a vast subject: during my research I noted that several answers could be given and that in the past I had both deliberately and unawarely developed many different strategies to shape musical structures granting a focus on conventional music elements and yet transfiguring them so to get to something new3, that could sound unique, unfamiliar and, above all, mine. In fact this thesis is also an opportunity for recollecting ideas that I have developed in almost two decades of artistic endeavors and theoretical research. For this reason, I will refer also to many of my previous papers and scores. Moreover, some strategies born when I was dealing with conventional harmonic elements seemed to have been particularly recurrent in my composition practice. Therefore, I have decided to focus on them, outlining their framework and formalizing their properties, so that I was then able to test them in other – even distant – contexts, beyond the harmonic one in which they emerged. This led to new possibilities that I highlighted and discussed, explaining how and why they could possibly be useful, relevant and effective tools also for other composers. Thus, I start Chapter 2 by introducing the natural regulations I could trace from my ordinary mathematically informed process of music composition and I discuss it from an aesthetic point of view. Against this background I define three features of mathematical nature that describe the core of the aforesaid recurrent strategies; I named them 1) completeness, 2) exhaustiveness and 3) equality in repetition (Albini 2018b).4 In Chapter 3, I deepen the specific harmonic structure dealing with consonant triads from which I could identify and theorize the three features, I introduce some extended strategies for it and their applications in some sections of my compositions. In Chapter 4, I discuss a new strategy involving the three features that applies them to diatonic trichords and their test in the composition of new scores. In Chapter 5, I address the extension of the three features beyond harmony and tackling further methods of applications and their symbolic implications. Case studies of scores and techniques I composed and elaborated are again offered. Chapter 6 is dedicated to a discussion about the new knowledge that my research has created, also in term of future
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