We here present the Arabic text, with an English translation, of certain pages dedicated by al-Khayyām to the mathematical theory of music. Our edition is based on a manuscript extant in a library in Manisa, and corrects the mistakes found in another transcription. Lastly, we compare the theory of al-Khayyām with other Arabic theories of Music, and with those coming from other traditions. Résumé Nous présentons ici le texte arabe, avec traduction anglaise, de certaines pages (...) consacrées par al-Khayyām à la théorie mathématique de la musique. Notre édition se fonde sur un manuscrit conservé dans une bibliothèque de Manisa et corrige les erreurs d'une autre transcription. Nous comparons enfin la théorie d'al-Khayyām avec d'autres théories arabes de la musique, et avec certaines théories issues d'autres traditions. (shrink)

The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der Töne. It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication of Geometrie der Töne in 1990. The conceptual basis has been vastly generalized to topos-theoretic foundations, including a corresponding thoroughly geometric musical logic. The theoretical models and results now include topologies for rhythm, melody, and harmony, as well as a (...) classification theory of musical objects that comprises the topos-theoretic concept framework. Classification also implies techniques of algebraic moduli theory. The classical models of modulation and counterpoint have been extended to exotic scales and counterpoint interval dichotomies. The probably most exciting new field of research deals with musical performance and its implementation on advanced object-oriented software environments. This subject not only uses extensively the existing mathematical music theory, it also opens the language to differential equations and tools of differential geometry, such as Lie derivatives. Mathematical performance theory is the key to inverse performance theory, an advanced new research field which deals with the calculation of varieties of parameters which give rise to a determined performance. This field uses techniques of algebraic geometry and statistics, approaches which have already produced significant results in the understanding of highest-ranked human performances. The book's formal language and models are currently being used by leading researchers in Europe and Northern America and have become a foundation of music software design. This is also testified by the book's nineteen collaborators and the included CD-ROM containing software and music examples. (shrink)

One is a science, the other an art; one useful, the other seemingly decorative, but mathematics and music share common origins in cult and mystery and have been linked throughout history. Emblems of Mind is Edward Rothstein’s classic exploration of their profound similarities, a journey into their “inner life.” Along the way, Rothstein explains how mathematics makes sense of space, how music tells a story, how theories are constructed, how melody is shaped. He invokes the poetry of Wordsworth, (...) the anthropology of Le;vi-Strauss, the imagery of Plato, and the philosophy of Kant. Math and music, Rothstein shows, apply comparable methods as they create their abstractions, display similar concerns with ratio and proportion, and depend on metaphors and analogies to create their meanings. Ultimately, Rothstein argues, they reveal the ways in which we come to understand the world. They are images of the mind at work and play; indeed, they are emblems of Mind itself. Jacques Barzun called this book “splendid.” Martin Gardner said it was “beautifully written, marvelous and entertaining.” It will provoke all serious readers to think in new ways about the grand patterns in art and life. “Lovely, wistful. . . . Rothstein is a wonderful guide to the architecture of musical space, its tensions and relations, its resonances and proportions. . . . His account of what is going on in the music is unfailingly felicitous.”— New Yorker “Provocative and exciting. . . . Rothstein writes this book as a foreign correspondent, sending dispatches from a remote and mysterious locale as a guide for the intellectually adventurous. The remarkable fact about his work is not that it is profound, as much of the writing is, but that it is so accessible.”— Christian Science Monitor. (shrink)

The Consolations of Philosophy by Boethius, whose English translators include King Alfred, Geoffrey Chaucer, and Queen Elizabeth I, ranks among the most remarkable books to be written by a prisoner awaiting the execution of a tyrannical death sentence. Its interpretation is bound up with his other writings on mathematics and music, on Aristotelian and propositional logic, and on central themes of Christian dogma. -/- Chadwick begins by tracing the career of Boethius, a Roman rising to high office under the (...) Gothic King Theoderic the Great, and suggests that his death may be seen as a cruel by-product of Byzantine ambitions to restore Roman imperial rule after its elimination in the West in AD 476. Subsequent chapters examine in detail his educational programme in the liberal arts designed to avert a threatened collapse of culture and his ambition to translate into Latin everything he could find on Plato and Aristotle. -/- Boethius has been called `last of the Romans, first of the scholastics'. This book is the first major study in English of a writer who was of critical importance in the history of thought. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, there (...) was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

Flora Levin explores how and why music was so important to the ancient Greeks. She examines the distinctions that they drew between the theory of music as an art ruled by number and the theory wherein number is held to be ruled by the art of music. These perspectives generated more expansive theories, particularly the idea that the cosmos is a mirror-image of music's structural elements and, conversely, that music by virtue of its (...) cosmic elements - time, motion, and the continuum - is itself a mirror-image of the cosmos. These opposing perspectives gave rise to two opposing schools of thought, the Pythagorean and the Aristoxenian. Levin argues that the clash between these two schools could never be reconciled. Her book shows how the Greeks' appreciation of the profundity of music's interconnections with philosophy, mathematics, and logic led to groundbreaking intellectual achievements that no civilisation has ever matched. (shrink)

The property theory of musical works says that each musical work is a property that is instantiated by its occurrences, that is, the work's performances and playings. The property theory provides ontological explanations very similar to those given by its popular cousin, the type/token theory of musical works, but it is both simpler and stronger. However, type/token theorists often dismiss the property theory. In this essay, I formulate a version of the property theory that identifies (...) each type (thus, each musical work) with a unique property. I then scrutinize the arguments offered for thinking that types, including musical works, are distinct from properties. I respond that no such argument is forceful and conclude that the property theory of musical works is superior to the type/token theory. (shrink)

Degrees of belief; Dempster's rule of combination; Simple and separable support functions; The weights of evidence; Compatible frames of discernment; Support functions; The discernment of evidence; Quasi support functions; Consonance; Statistical evidence; The dual nature of probable reasoning.

Perfect Harmony and Melting Strains assembles interdisciplinary essays investigating concepts of harmony during a transitional period, in which the Pythagorean notion of a harmoniously ordered cosmos competed with and was transformed by new theories about sound - and new ways of conceptualizing the world. From the perspectives of philosophy, literary scholarship, and musicology, the contributions consider music's ambivalent position between mathematical abstraction and sensibility, between the metaphysics of harmony and the physics of sound. Essays examine the late medieval (...) and early modern history of ideas concerning the nature of music and cosmic harmony, and trace their transformations in early modern musico-literary discourses. Within this framework, essays further offer original readings of important philosophical, literary, and musicological works. This interdisciplinary volume brings into focus the transformation of a predominant Renaissance worldview and of music's scientific, theological, literary, as well as cultural conceptions and functions in the early modern period, and will be of interest to scholars of the classics, philosophy, musicology, as well as literary and cultural studies. (shrink)

A theory of musical narrative. An introduction to narrative analysis : Chopin's prelude in G major, op. 28, no. 3 ; Perspectives and critiques ; A theory of musical narrative : conceptual considerations ; A theory of musical narrative : analytical considerations ; Narrative and topic -- Archetypal narratives and phases. Romance narratives and Micznik's degrees of narrativity ; Tragic narratives : an extended analysis of Schubert, piano sonata in B flat major, D. 960, first movement ; (...) Ironic narratives : subtypes and phases ; Comic narratives and discursive strategies ; Summary and conclusion. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This paper examines the causal basis of our ability to attribute emotions to music, developing and synthesizing the existing arousal, resemblance and persona theories of musical expressivity to do so. The principal claim is that music hijacks the simulation mechanism of the brain, a mechanism which has evolved to detect one's own and other people's emotions.

The aim of this paper is to describe and analyze the epistemological justification of a proposal initially made by the biomathematician Robert Rosen in 1958. In this theoretical proposal, Rosen suggests using the mathematical concept of “category” and the correlative concept of “natural equivalence” in mathematical modeling applied to living beings. Our questions are the following: According to Rosen, to what extent does the mathematical notion of category give access to more “natural” formalisms in the modeling of (...) living beings? Is the so -called “naturalness” of some kinds of equivalences (which the mathematical notion of category makes it possible to generalize and to put at the forefront) analogous to the naturalness of living systems? Rosen appears to answer “yes” and to ground this transfer of the concept of “natural equivalence” in biology on such an analogy. But this hypothesis, although fertile, remains debatable. Finally, this paper makes a brief account of the later evolution of Rosen’s arguments about this topic. In particular, it sheds light on the new role played by the notion of “category” in his more recent objections to the computational models that have pervaded almost every domain of biology since the 1990s. (shrink)

This work aims to demonstrate that a universal model of man can be constructed using the precise mathematical formulation of a few philosophical theses about the nature of human beings. The author uses it to find theoretical solutions for several unexplained phenomena in psychophysics.

Auf Grund der Vorstellung, dass die Erregungsleitung auf einer Wiedererregung der benachbarten Gewebebezirke durch lokale bioelektrische Ströme beruht, wurde vorher eine mathematische Theorie der Fortpflanzung der Erregung im Nerv entwickelt, welche einige Tatsachen befriedigend darstellt. In der vorliegenden Arbeit wird die Theorie auf den Fall angewandt, dass die Erregung von einem Gewebe auf ein anderes übertragen wird, wobei die beiden Gewebe verschiedene elektrische Eigenschaften haben. Es zeigt sich, dass dabei gewisse Bedingungen für die Möglichkeit der Übertragung der Erregung erfüllt sein (...) müssen, welche an den vonL. Lapique geforderten Isochronismus erinnern.Dans deux mémoires précédents nous avons développé une théorie mathématique de la propagation de l'excitation nerveuse, basée sur l'hypothèse, que cette propagation est due à une réexcitation par les courants bioélectriques. Dans le mémoire présent nous étudions le cas de deux tissus adjacents, différents dans leurs constantes électriques, du point de vue du passage de l'excitation d'un tissu sur l'autre. Il se trouve, que pour que ce passage put avoir lieu, certaines relations doivent être satisfaites, des relations qui rappellent l'isochronisme postulé parL. Lapique. (shrink)

In this paper we present an informal review of our recent work whose goal is to develop a mathematical theory of the physical phenomenon of emission and absorption of radiation by systems of nonrelativistic matter such as atoms and molecules.

Several prominent philosophers of art have worried about whether Kant has a coherent theory of music on account of two perceived tensions in his view. First, there appears to be a conflict between his formalist and expressive commitments. Second (and even worse), Kant defends seemingly contradictory claims about music being beautiful and merely agreeable, that is, not beautiful. Against these critics, I show that Kant has a consistent view of music that reconciles these tensions. I argue (...) that, for Kant, music can be experienced as either agreeable or beautiful depending on the attitude we take toward it. Although it is tempting to think he argues that we experience music as agreeable when we attend to its expressive qualities and as beautiful when we attend to its formal properties, I demonstrate that he actually claims that we are able to judge music as beautiful only if we are sensitive to the expression of emotion through musical form. With this revised understanding of Kant's theory of music in place, I conclude by sketching a Kantian solution to a central problem in the philosophy of music: given that music is not sentient, how can it express emotion? (shrink)

Evidence Theory is a branch of mathematics that concerns combination of empirical evidence in an individual’s mind in order to construct a coherent picture of reality. Designed to deal with unexpected empirical evidence suggesting new possibilities, evidence theory is compatible with Shackle’s idea of decision-making as a creative act. This essay investigates this connection in detail, pointing to the usefulness of evidence theory to formalise and extend Shackle’s decision theory. In order to ease a proper framing (...) of the issues involved, evidence theory is compared with sub-additive probability theory and Ewens’s infinite alleles model. Furthermore, the original version of evidence theory is presented along with its most recent developments. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and (...) updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

DESPITE CONSIDERABLE DIFFERENCES OF IDEOLOGY, objective, and style, these theorists join in giving voice to what is perhaps the most deeply rooted conviction in modern aesthetics: that aesthetic experience is, in some essential respect, ineffable. In apprehending a work of art we come to know something we cannot put into words.

In this paper, I present the beginnings of a resemblance theory of representation. I start by surveying the contemporary philosophical debate surrounding musical representation and reveal that its main interlocutors share a conception of artistic representation as a mode of meaningful communication. I then show how conceiving of artistic representation in this way severely limits music’s possibilities as a medium for representation. Next, I propose an alternative conception of representation that, despite its widespread acceptance outside of the philosophy (...) of art, has been largely denigrated within it: namely, that representation fundamentally depends on structural resemblance. Finally, I demonstrate how conceiving of artistic representation as grounded upon structural resemblances, rather than as a mode of meaningful communication, provides us with a more accurate picture of both (a) how music represents and (b) how we perceive and appreciate musical representations. (shrink)

The experience of notes as higher or lower than one another, and of movement within passages of music, underpins many other musical experiences. Several theories of such an experience have been defended, claiming that concepts of space and movement variously play some sort of metaphorical role in our experience, can be eliminated from musical discourse, or apply literally to the music. I argue that all such theories should be rejected in favour of the view that our experience of (...) musical space and movement is imaginative, in much the same way our experience of fiction is widely believed to be. (shrink)

In our previous writings we have outlined a mathematical theory of biological phenomena. In our systematic construction of “mathematical biology,” similar in its aims to mathematical physics, we have started with the fundamental unit,—the living cell. After having established a physico-mathematical theory of the fundamental properties of the cell, we have studied the interaction of several cells. This led us into two different fields. On the one hand we studied such interactions of cells, which (...) determine the form of cellular aggregates, constituting multicellular organisms. On the other hand we studied different types of functional interactions, which determine the reactions of the aggregate as a whole to different environmental changes. While the first field leads us eventually to the theory of organic form, the second brings us to the physico-mathematical theory of behavior. (shrink)

The processes taking place in modern science tend to integrate various scientific disciplines in studying any scientific phenomenon. This explains the search of methodology based on the techniques of different sciences involved in formation of the subject of research, which in this article is represented by the digital music art. Interpreted as a branch of music art with digital specificity, digital music art incorporates electronic music presented by specific, algorithmic, and electronic music. A signature characteristic (...) of digital music art consists in its interdisciplinarity, which suggests the combined effect of separate parts of the system and leads to self-organization, which corresponds with the scientific tasks of synergetics. This substantiates the relevance for elaboration of interdisciplinary methodology for studying digital music art in the context of research mechanisms developed within synergetics. The scientific lies in the analysis of digital art through the prism of the qualities inherent to synergetics. The disciplines united by digital art (mathematics, set theory, probability theory, information theory, combinatorics, game theory, cybernetics, computer science, acoustics, sociology, communication, and biology) are interdisciplinary, which means that in ensemble with other disciplines they form a complex interdisciplinary knowledge of structure of digital music art. At the same time, the disciplines that comprise scientific framework of digital music art are attributed to different sciences: humanities, natural sciences, mathematics, social sciences, and technical sciences. In this context, the research of digital music art is simultaneously of interdisciplinary and multiscientific nature. The author established a peculiar impact of interdisciplinarity upon digital art, which generates the qualities inherent to synergetics – nonlinearity, disequilibrium, openness, chaotic nature, entropy, indeterminacy, and dissipation. (shrink)

Known today mostly as an author of Romantic short stories and fairy tales for children, Prince Vladimir Odoyevskiy was a distinguished thinker of his time, philosopher and bibliophile. The scope of his interests includes also history of magic arts and alchemy, German Romanticism, Church music. An attempt to understand the peculiarity of eight specific modes used in chants of Russian Orthodox Church led him to his own musical theory based upon well-known writings by Zarlino, Leibniz, Euler, Prony. He (...) realized his theoretical ideas in an enharmonical piano with nineteen notes for every octave. His call for a different musical scale remained largely ignored in the nineteenth century, until the topic was raised anew by twentieth-century composers and musicians. (shrink)

This dissertation defends the place of representation in music. Music’s status as a representational art has been hotly debated since the War of the Romantics, which pitted the Weimar progressives (Liszt, Wagner, &co.) against the Leipzig conservatives (the Schumanns, Brahms, &co.) in an intellectual struggle for what each side took to be the very future of music as an art. I side with the progressives, and argue that music can be and often is a representational medium. (...) Correspondence (or resemblance) theories of representation, such as the one I offer, have been much maligned in philosophy since the 1960s. Most theories assimilate representation under “meaning,” which has usually been thought to belong primarily to language. As a result, representational content has been taken to be purely conventional in the way that sentential meaning is. People want to know what music “means,” and these theories interpret this as “what does it refer to?” or “what propositions does it express?” I argue that propositional communication is only one (small) part of the issue. Once we overcome the bias of conceiving of musical works as essentially linguistic items, speech acts (performed) or sentence tokens (written), we can begin to take music on its own terms to discover how it represents—one way in which it “means.” The first step is to “naturalize” music’s representational content. Influenced by recent discussions in the philosophies of mind and science, I argue in Chapter 1 that composers represent extra-musical objects, events, and states of affairs through their works by exploiting antecedent relations (such as similarities in pitch, timbre, and structure) in order to secure reference to them. In Chapter 2, I survey and respond to the main challenges that those skeptical of music’s representational possibilities would raise against my theory of musical representation. In Chapter 3, I explore a number of ways through which music has been claimed to represent in order to show how my theory accounts for these diverse phenomena better than its conventionalist rivals, both in terms of the metaphysics and the epistemology. Chapter 4 extends this discussion by offering an account of how we perceive, understand, appreciate, and interpret sophisticated musical representations. I conclude by teasing out some of my theory’s implications and suggesting areas for further research. (shrink)

One thing to expect from a theory of absolute music is that it explains what makes it so significant to us. Kant rightly observes that the essence of absolute music is our affective response to it. Yet none of the standard 18 th century theories, arousal theory and aesthetic rationalism, can explain both the universality of a judgment of taste and its subjective emotional content. The paper argues that Kant’s own aesthetic theory of aesthetic ideas (...) is on the right path for explaining the emotive significance of music, but that Kant does not further pursue this route because his interest in a “theory of music” is only peripheral to his systematic interest as a transcendental philosopher. For both philosophical and historical reasons, Kant does not have a “theory of music.” Nevertheless, if we free Kant’s conception of aesthetic idea and its restriction to Geist from the background of its 18th century humanism, then Kant’s aesthetics of free play becomes a good theory to account for the broad variety of emotions in absolute music as a fine art proper. (shrink)

This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...) is discussed. Finally, it is suggested that one reason there is a common structure between Aristotle's account of the continuum and that found in Cantor's definition of the real number continuum is that our intuitions about the continuum have their source in the experience of the real spatiotemporal world. A plea is made to consider Aristotle's abstractionist philosophy of mathematics anew. (shrink)

At the beginning of his career in the 1920s, Adorno sketched a plan to write a major work on the theory of musical reproduction, a task he returned to time and again throughout his career but never completed. The choice of the word reproduction as opposed to interpretation indicates a primary supposition: that there is a clearly defined musical text whose precision exceeds what is visible on the page, and that the performer has the responsibility to reproduce it as (...) accurately as possible, beyond simply playing what is written. This task, according to Adorno, requires a detailed understanding of all musical parameters in their historical context, and his reflections upon this task lead to a fundamental study of the nature of notation and musical sense. In the various notes and texts brought together in Towards a Theory of Musical Reproduction, one finds Adorno constantly circling around an irresolvable paradox: interpretation can only fail the work, yet only through it can musics true essence be captured. While he at times seems more definite in his pronouncement of a musical scores absolute value just as a book is read silently, not aloud his discourse repeatedly displays his inability to cling to that belief. It is this quality of uncertainty in his reflections that truly indicates the scope of the discourse and its continuing relevance to musical thought and practice today. (shrink)

Kennedy shows that Plato gave his dialogues a similar musical structure, dividing each dialogue into twelve parts and inserting symbols at each twelfth to mark a musical note. These passages are either harmonious or dissonant and traverse the ups and downs of a known musical scale. Many of Plato's early followers insisted that Plato used symbols to conceal his own views within the dialogues, but modern scholars have denied this. Kennedy, an expert in Pythagorean mathematics and music theory, (...) is able to show that Plato's dialogues contain a system of symbols that are undetectable by those without knowledge of obsolete Greek mathematics. The book begins with a concise and accessible introduction to Plato's symbolic schemes and the role of allegory in ancient times. The author then annotates the musical symbols in two of Plato's most popular dialogues, the Symposium and Euthyphro, and shows that Plato used the musical scale as an outline for structuring his narratives. (shrink)

Savage et al. and Mehr et al. provide well-substantiated arguments that the evolution of musicality was shaped by adaptive functions of social bonding and credible signalling. However, they are too quick to dismiss byproduct explanations of music evolution, and to present their theories as complete unitary accounts of the phenomenon.

The view that music can have a positive or negative effect on a person’s character has been defended throughout the history of philosophy. This paper traces some of the history of the ethos theory and identifies a version of the theory that could be true. This version of the theory can be traced to Plato and Aristotle and was given a clear statement by Herbert Spencer in the nineteenth century. The paper then examines some of the (...) empirical literature on how music can affect dispositions to behave and moral judgement. None of this evidence provides much support for the ethos theory. The paper then proposes a programme of research that has the potential to confirm the ethos theory. (shrink)