Transformation Theory

This work develops Structure Theory, which treats structure not as a secondary phenomenon but as a fundamental ontological principle. The theory introduces three universal laws of structural dynamics that explain when systems remain stable, when they transform, and when new orders emerge. By defining structure as the necessary condition for existence, change, and observation, it provides a unified framework for analyzing transformations across physical, biological, cognitive, and social systems. The theory is formulated in a falsifiable way and distinguished from existing approaches by grounding order itself, rather than processes within order.

The objective of this third edition is the same as in previous two editions: to provide a broad coverage of various mathematical techniques that are widely used for solving and to get analytical solutions to Partial Differential Equations of first and second order, which occur in science and engineering. In fact, while writing this book, I have been guided by a simple teaching philosophy: An ideal textbook should teach the students to solve problems. This book contains hundreds of carefully chosen worked-out examples, which introduce and clarify every new concept. The core material presented in the second edition remains unchanged. I have updated the previous edition by adding new material as suggested by my active colleagues, friends and students. Chapter 1 has been updated by adding new sections on both homogeneous and nonhomogeneous linear PDEs, with constant coefficients, while Chapter 2 has been repeated as such with the only addition that a solution to Helmholtz equation using variables separable method is discussed in detail. In Chapter 3, few models of non-linear PDEs have been introduced. In particular, the exact solution of the IVP for non-linear Burger's equation is obtained using Cole-Hopf function. Chapter 4 has been updated with additional comments and explanations, for better understanding of normal modes of vibrations of a stretched string. Chapters 5-7 remain unchanged. I wish to express my gratitude to various authors, whose works are referred to while writing this book, as listed in the Bibliography. Finally, I would like to thank all my old colleagues, friends and students, whose feedback has helped me to improve over previous two editions. It is also a pleasure to thank the publisher, PHI Learning, for their careful processing of the manuscript both at the editorial and production stages. Any suggestions, remarks and constructive comments for the improvement of text are always welcome. With the remarkable advances made in various branches of science, engineering and technology, today, more than ever before, the study of partial differential equations has become essential. For, to have an in-depth understanding of subjects like fluid dynamics and heat transfer, aerodynamics, elasticity, waves, and electromagnetics, the knowledge of finding solutions to partial differential equations is absolutely necessary. This book on Partial Differential Equations is the outcome of a series of lectures delivered by me, over several years, to the postgraduate students of Applied Mathematics at Anna University, Chennai. It is written mainly to acquaint the reader with various well-known mathematical techniques, namely, the variables separable method, integral transform techniques, and Green's function approach, so as to solve various boundary value problems involving parabolic, elliptic and hyperbolic partial differential equations, which arise in many physical situations. In fact, the Laplace equation, the heat conduction equation and the wave equation have been derived by taking into account certain physical problems. The book has been organized in a logical order and the topics are discussed in a systematic manner. In Chapter 0, partial differential equations of first order are dealt with. In Chapter 1, the classification of second order partial differential equations, and their canonical forms are given. The concept of adjoint operators is introduced and illustrated through examples, and Riemann's method of solving Cauchy's problem described. Chapter 2 deals with elliptic differential equations. Also, basic mathematical tools as well as various properties of harmonic functions are discussed. Further, the Dirichlet and Neumann boundary value problems are solved using variables separable method in cartesian, cylindrical and spherical coordinate systems. Chapter 3 is devoted to a discussion on the solution of boundary value problems describing the parabolic or diffusion equation in various coordinate systems using the variables separable method. Elementary solutions are also given. Besides, the maximum-minimum principle is discussed, and the concept of Dirac delta function is introduced along with a few properties. Chapter 4 provides a detailed study of the wave equation representing the hyperbolic partial differential equation, and gives D'Alembert's solution. In addition, the chapter presents problems like vibrating string, vibration of a circular membrane, and periodic solutions of wave equation, shows the uniqueness of the solutions, and illustrates Duhamel's principle. Chapter 5 introduces the basic concepts in the construction of xi Preface to the First and Second Edition ( , ) 0 . F x y z xy z 1 1 1 2 x C C C § • ¨© ¹ (8) 1 c ξ and 2 c η are families of straight lines parallel to the axes as shown in Fig. 1.1(b). From Eqs. (1.50) and (1.51), we have 2 2 2 2 ( ) *( ) 2 2 2 2 0

The attachment theory of John Bowlby has had an enduring impact on our understanding of child development. But these ideas are a neglected and forgotten discourse in adult education. In this paper concepts such as secure and insecure attachments, internal ...

This paper presents a conceptual and mathematical investigation of twelve-tone equal temperament (12-TET), examining its structure through the lenses of group theory,
 
information theory, and recent developments in mathematical music theory. We analyze the cyclic properties of 12-TET, its geometric representation in relation to π, and propose Information Structure Wave (ISW) Theory as a conceptual model for emergent structural resonance across mathematical and perceptual domains. While ISW Theory remains in a formative state, this work provides formal definitions, reproducible algorithms, and quantitative data to support the non-randomness and optimality claims for 12-TET. We critically assess alternative explanations—including cultural, cognitive, and technological factors—and conclude with an agenda for future mathematical and empirical validation.

This document presents Bhagath's Exponential Triad Algorithm (BETA), a mathematical transformation inspired by the Collatz Conjecture. By applying powers of 3 to odd numbers and dividing even numbers by 2, the algorithm reveals predictable convergence points and fixed cycles in integer sequences.

This document presents Bhagath's Exponential Triad Algorithm (BETA), a mathematical transformation inspired by the Collatz Conjecture. By applying powers of 3 to odd numbers and dividing even numbers by 2, the algorithm reveals predictable convergence points and fixed cycles in integer sequences.

Fork is a framework written in the SuperCollider programming language for algorithmically composing music to chess games in real time in a way that meaningfully conveys the progress of the game in a musically coherent manner. Fork is intended to serve as a dynamic, immersive soundtrack to app-based chess where the players have direct control over the resulting generation, thus turning chess itself into a musical endeavor alongside its normal gameplay. The principles behind Fork straddle the boundary between more traditional note structures and compositional use of uniquely electronic parameters such as timbre and spatialization in musically effective sonification of data.

We introduce the Orbit-Convergence Transformation Space, a mathematical framework for analyzing the behavior of iterative transformations. The framework defines a quadruple consisting of a state space, a transformation function, a distance metric, and a flow function. This paper demonstrates the application of this framework to several domains, including the Collatz conjecture, twin prime distribution, and game-theoretic strategies. We show how the framework provides new insights into the structure and patterns within these mathematical domains through numerical simulations and visualizations. The results suggest that the Orbit-Convergence framework offers a powerful unified approach for studying diverse mathematical problems through the lens of transformation spaces.

This article examines how the Tonnetz, widely considered the emblematic diagram of “Western” music theory, emerged in its modern form from a global network of scholars spanning Japan, Germany, and India. Part one consists of four microhistories of transnational connection. The first two focus on Tanaka Shōhei’s 1890 presentation of the Tonnetz as representative of a “mutually civilizing” discourse between Germany and Japan, and Hugo Riemann’s adaptations of Tanaka’s diagram in re-centering Germany’s musicological status. The second two focus on G. S. Khare’s use of the Tonnetz in 1918 to bolster claims of Indian “priority” in the development of tonality, and the influence of Khare’s associates on Tanaka’s later efforts to bolster a Pan-Asianist narrative that cast Japan as the last bastion of an uncolonized Asian spirit. Part two applies a macrohistorical lens in analyzing how these Tonnetze played a constitutive role in the development of musical modernity, which I interpret as an emergent phenomenon of global processes of integration by building on the work of Sebastian Conrad and Sanjay Subrahmanyam. Each of these Tonnetze is thus a “global Tonnetz,” stemming from the drive of each nation to participate in the construction of musical modernity on a global stage. I conclude by acknowledging tensions between my histories of connection and integration, as the former traces threads of causality while the latter embraces emergence. I nevertheless argue in favor of a framework of historical duality that can hold both approaches simultaneously when dealing with global phenomena.

Full version available at UC Press: https://online.ucpress.edu/jams/issue/77/2

In this paper, we attempt to explore mathematical structures of tonal music from the 18th and 19th centuries. We review the known mathematical/musical structures and, noting lacking features of this geometrical construct as a useful model of tonal harmony, we propose generalizations that may be better suited to tonal music. We bring several mathematical devices to bear on the Tonnetz (or Tone Network) in new ways, including Cayley graphs and Coxeter hyperbolic representations. We conclude with three-dimensional geometric models that represent the four-note seventh chords which are ubiquitous in 18th and 19th century music.

Music analysis represents the most useful way of exploration and innovation of musical interpretations. Performers who use music analysis efficiently will find it a valuable method for finding the kind of musical richness they desire in their interpretations. The use of Schenkerian analysis in performance offers a rational basis and an unique way of interpreting music in performance.

This essay argues that the search for a scientific theory of transformation is ill-conceived. Postcommunist transformation is not a scientific project but a political project. It therefore requires a political theory rather than a scientific theory of transformation. The distinction is important because social scientists as political actors have played a siginificant role in the transformation process. Several examples are provided to illustrate the relationship between social science and transformation. In political theories of transformation, social science knowledge is subordinated and instrumental. This does not reduce the significance of social science, but rather reconceptualizes it. The legitimate functions of social science in transformation theory have critical, constructive and applied dimensions.

This extended article introduces Adam Ockelford's 'zygonic' theory of music-structural understanding, which holds that imitation, which can occur in all domains of perceived sound and at all levels, is the ultimate organising force in music. Hence the theory is potentially of value not only in theoretical terms (shown here in relation to the first movement of Mozart's Piano Sonata K. 333), but metatheoretically too, as a tool to interrogate other systems of musical analysis (an example is provided in relation to Allan Forte's 'set-theoretical' method). The zygonic approach also enables the powers of influence at work in group improvisation to be captured, permitting the evolution of musical ideas to be charted as they unfold in time between performers, and a zygonic analysis of a short, improvised song with piano accompaniment is provided by way of illustration. Finally, zygonic theory prospectively offers an epistemological link between the sister (though sometimes apparently incompatible) disciplines of music psychology and music theory-an avenue that is explored briefly in conclusion.

The goal of this research is to maximize chord-based composition possibilities given a relatively small amount of information. A transformational approach, based in group theory, was chosen, focusing on chord intervals as the components of a modified Markov process. The Markov process was modified to balance between average harmony, representing familiarity, and entropy, representing novelty. Uniform triadic transformations are suggested as a further extension of the transformational approach, improving the quality of tonality. The composition algorithms are demonstrated given a short chord progression and also given a larger database of albums by the Beatles. Results demonstrate capabilities and limitations of the algorithms.

Film has influenced the public as an important cultural form since the 20th century, and film music, as an integral part of film, plays an important role in shaping its content. Film music was influenced by late European Romantic music, and the Hollywood film score system was established based on composers of European descent, followed by the fusion of numerous emerging musical styles, such as Jazz, Rock, and EDM, etc. The orchestral music as the main composition of the film score has a sense of aural expectation in the sound expression, which largely comes from the chromaticization of the harmonies. When analyzing Hollywood film music, it is inevitable that chromatic harmony progression cannot be accurately expressed by traditional tonal music analysis. This study analyzes the chromatic harmony progression of chromatic mediant relationship in the film scores of four composers from the perspective of neo-Riemannian theory. And the correlation between the four sets of transformations...

My sincere thanks to Professor Richard Vella, my principal supervisor, whose wide knowledge, insightful suggestions and incisive questioning were a constant source of inspiration and wise counsel throughout the long roads and occasional detours of my researches. Thanks also to Colin Spiers, whose acerbic wit and practical experience provided some helpful reality checks in the early stages of this project. I would also like to thank the faculty, staff and graduate students in the music discipline of the School of Creative Industries at University of Newcastle, who contributed to a constructive and stimulating environment in which to test ideas and expand horizons. I owe a special debt of gratitude to Sean Lowry. His friendship, enthusiasm and unflagging support, both as an expert sounding board and as a creative collaborator, have helped me to crystallise and better articulate my theoretical thinking about music and art, as well as furthering my development as a practising artist. Finally, my deepest thanks go to my family, whose curiosity and encouragement motivated me to persevere whenever it looked as if this thesis might never be completed.

In “classical” music theory, the minor triad has always been considered subordinate to the prevailing, and more “natural”, major triad (both for major and minor tonalities).
Harmonic dualism is a principle that ascribes equal importance to the major and minor triads approached from dialectical and acoustical point of view, i.e. in terms of undertones and overtones, mediation of contrasting forces and physiological-structural understanding (following Oettingen’s (1866), Hauptmann’s (1853), and Riemann’s (1905 and 1915) studies). However, these studies do not seem to convincingly justify the existence of the minor triad; all attempts to trace the foundation of the minor triads in Oettingen’s series of undertones appears equally weak. Being the minor triad — and tonality — an indisputable piece of evidence, the David Lewin’s A Formal Theory of Generalized Tonal Functions (1982) is the only one able to demonstrate the structural-formal explanation of harmonic dualism in terms of inversion of the same intervals which belong to the major triad.
Considering that the tonal praxis is likely to come from transposition of psalm-tones in the instrumental music of the XVII century (following Barnett’s (1998), and Powers’s (2014) papers), we seek the ontological basis of harmonic dualism in the praxis of temperaments; depending on the higher or lower consonance of the two triads in meantone temperaments (for simplicity 1/3 and 1/4 of syntonic comma (s.c.) meantone temperaments), we can trace back a double dualism, in which the third in question is ontologically prevalent, and the complementary one is its dual. However, temperaments do not allow modulations without a transposition to distant keys. Conversely, Lewin’s model allows generating “dual” triads by mathematical manipulation of their constituents — i.e., inversion (“TDINV” operation) and/or conjugation (“CONJ” operation). By looking at the modal diatonic sets on which psalmodic-tones are based (the natural notes with added B ) and by arranging the pitches of these tones as thirds, one can obtain two palindromes which represent all the triadic possibilities for these modal sets, which confirms the structure theorized by Lewin (which is a palindrome as well).
Our approach contradicts the theories of “classical” harmonic dualism, which focuses only on the single constituents of the triads (regarding the topics cited above). Moreover, such approaches seek to identify how the model could be adapted to the temperament independently, i. e. without including the temperament itself into the model. All these aspects fail to provide a convincing explanation of the harmonic dualism, that wasn’t achieved until Lewin’s formulation. By considering chords as autonomous entities, different from the ordinary superposition of scalar intervals, Lewin provided for the first time a formally convincing explanation based on the equal temperament, i.e. without making comparable consonance and “naturalness” of the major and minor triad a consequence of a specific intonation (following Lewin’s 1982 paper).
My paper aims at highlighting the relevance of a historically informed genealogy of the harmonic dualism, in relation with the musical practices that led to the emergence of tonality through the use of temperaments and their interval structure.

CHAPTER 3. FURTHER NEW RESULTS ON N-DEMENSIONAL LAPLACE AND INVERSE LAPLACE TRANSFORMATIONS 3.1. Introduction 3.2. The Image of Functions with the Argument ' 89 3.2.1. Applications of Theorem 3.2.1 3.2.2. Laplace Transforms of some Elementary and Special Functions with n Variables ^ 98 3.3. The Original of Functions with the Argument ^ 106 3.3.1. Examples Based Upon Theorem 3.3.1 Ill 3.4. The Image of Functions with the Argument 2pi(x 114 3.4.1 Applications of Theorem 3.4.1 119 3.5. The Original of Functions with the Argument 121 3.5.1. Example Based Upon Theorem 3.5.1 125 CHAPTER 4. THE SOLUTION OF INITIAL-BOUNDARY-VALUE PROBLEMS (IBVP'S) BY DOUBLE LAPLACE TRANSFORMATIONS 127 4.1. Introduction 127 4.2. Non-homogenous Linear Partial Differential Equations (PDEs) of the First Order 129 4.2.1. Partial Differential Equations of Type Ux+u.y = f{x,y),Q<x<«o,0<y<oo 129 4.2.2. Partial Differential Equations of Type au^ + buy + eeu = fix,y), 0<a:<<», 0<y<~ 136 4.3. Non-homogenous Second Order Linear Partial Differential V Equations of Hyperbolic Type 4.3.1. Partial Differential Equations of Type Ujcy = f(x,y),),0<x«>o,0<y<oo 4.3.2. The Wave Equation 4.4. Non-homogenous Second Order Partial Differential Equations of Parabolic Type 4.4.1. Partial Differential Equations of Type u"+2u^+Uyy + KU = fix,y\0<x<oo,0<y<oo CHAPTER 5. CONCLUSIONS AND FUTURE DIRECTIONS 154 5.1. Conclusions 154 5.2. Future Directions 155

The present paper describes structure and functioning of J-Analyzer, a computational tool for assistedanalysis. It integrates a research project intended to investigate the complete song collection by Brazilian composer Antônio Carlos Jobim, focusing on the aspect of harmonic transformation. The program is used to determine the nature of transformational relations between any chordal pair of chords present in a song, as well as the structure of the chords themselves.

The present paper describes structure and functioning of J-Analyzer, a computational tool for assistedanalysis. It integrates a research project intended to investigate the complete song collection by Brazilian composer Antônio Carlos Jobim, focusing on the aspect of harmonic transformation. The program is used to determine the nature of transformational relations between any chordal pair of chords present in a song, as well as the structure of the chords themselves.

The present paper describes structure and functioning of J-Analyzer, a computational tool for assistedanalysis. It integrates a research project intended to investigate the complete song collection by Brazilian composer Antônio Carlos Jobim, focusing on the aspect of harmonic transformation. The program is used to determine the nature of transformational relations between any chordal pair of chords present in a song, as well as the structure of the chords themselves.

Music analysis represents the most useful way of exploration and innovation of musical interpretations. Performers who use music analysis efficiently will find it a valuable method for finding the kind of musical richness they desire in their interpretations. The use of Schenkerian analysis in performance offers a rational basis and an unique way of interpreting music in performance.

The Planet-4D model introduced by Baroin 2011b is a richer model of pitch-class relationships than the standard cyclic model of the clock-face diagram: instead of the distribution of twelve points on a circle in a 2D plane, the Planet-4D model places the twelve pcs on a 4D hypersphere. Beyond the usual T/I group of symmetries, which is still featured in this new musical space, the additional dimensions yield other isometries (most notably now including the M5/M7 operations), which appear here in animated visualizations of musical pieces. The present article elucidates how these isometries are in fact well-known symmetries that do not preserve shape in most previous models with fewer dimensions. The hyper-spherical environment grants each symbol an equivalent physical position, and therefore allows for more symmetries than any 3D model. This article also shows that the model provides a visually intuitive geometric setting of pcs, in accordance with our perception of 3D Euclidean spac...

In 1987, David Epstein wrote Music complexity is in general a stumbling block to analysis, for the analytical method often seeks a unitary, if not unified, view of things. Singularity is in some ways an intrinsic analytical necessity, for the act of analysis is by prerequisite selective (1987, 6). These are courageous words because they introduce a doubt within a certain well-established way of thinking. Schenker, for example, in the first pages of his Free Composition argues, on the contrary, that music is simple, even if that simplicity is not visible on the surface. "Every surface he adds seen for itself alone, is of necessity confusing and always complex" (1979: xxiii). In this sentence we read a typical testimony of a certain worldview of the late nineteenth century. "Complexity" and "confusion" are put on the same level, that is: complexity is confusion. Moreover, the depth and surface levels are clearly distinguished: the first is the privileged territory of the scientific approach, the place where an "order" and a unitary sense of coherence can be reconstructed against the apparent disorder of the surface. This theme is well known to the philosophical reflection of the 20th century music theory. In a famous chapter of his Archaeology of Knowledge, Michel Foucault wrote: "The history of ideas usually credits the discourse that it analyses with coherence. If it happens to notice an irregularity in the use of words, several incompatible propositions, a set of meanings that do not adjust to one another, concepts that cannot be systematized together, then it regards it as its duty to find, at a deeper level, a principle of cohesion that organizes the discourse and restores to it its hidden unity." (1972: 149

Por ocasião do III Congresso da TeMA realizado conjuntamente com o IV Congresso Internacional de Música e Matemática, organizados pelo Programa de Pós-Graduação em Música da UFRJ, foram realizadas duas entrevistas com convidados palestrantes do evento. A primeira foi conduzida pelo prof. Liduíno Pitombeira e o entrevistado é o prof. Patrick McCreless da Yale University, EUA. A segunda entrevista, conduzida pelo prof. Carlos Almada, foi realizada com o professor Edgardo José Rodríguez da Universidad Nacional de La Plata, Argentina.

This thesis describes a system of neural networks that performs harmonic analysis on musical compositions. The system is split into four groups of Networks: Key Network, Root Network A (used to detect secondary functions), Root Network B, and Quality Network. The system is trained on 20 J.S. Bach chorales and tested on another 18 Bach chorales. Accuracies of over 90% were obtained for all four groups of Networks. Comparison with a modified version of Krumhansl's key-finding algorithm indicates Key Network outperforms it by more than 16% in overall accuracy. This work has potential for extension into music of other composers or genres. Also, this work can be used by a computer accompanist to determine the key of an improvising soloist. I would like to thank my brothers and sisters in Paul Fellowship for praying for me, my cousin Alan Wong for his continual encouragement to go on, my parents and grandmother for encouraging me to pursue graduate school, and Brenda Chan for keeping me sane in the days and nights when work is much but progress is few. Finally, I give thanks to my Lord Jesus Christ for leading me through this earthly travail. Without Him, nothing is possible.

The problematic in tonality and tonal composition are not tied to harmony or counterpoint, but are the result of their combination. In the more elaborate compositions, harmony and counterpoint have very well defined functions, thus resulting a set of clear basic principles in composition development. One of these principles consists in the idea that, harmonic progressions are the structure of the composition and the contrapuntal progressions and chords are the ones that enrich and create the flow of it.

The present paper describes structure and functioning of J-Analyzer, a computational tool for assistedanalysis. It integrates a research project intended to investigate the complete song collection by Brazilian composer Antônio Carlos Jobim, focusing on the aspect of harmonic transformation. The program is used to determine the nature of transformational relations between any chordal pair of chords present in a song, as well as the structure of the chords themselves.

Contextual inversion, introduced as an analytical tool by David Lewin, is a concept of wide reach and value in music theory and analysis, at the root of neo-Riemannian theory as well as serial theory, and useful for a range of analytical applications. A shortcoming of contextual inversion as it is currently understood, however, is, as implied by the name, that the transformation has to be defined anew for each application. This is potentially a virtue, requiring the analyst to invest the transformational system with meaning in order to construct it in the first place. However, there are certainly instances where new transformational systems are continually redefined for essentially the same purposes. This paper explores some of the most common theoretical bases for contextual inversion groups and considers possible definitions of inversion operators that can apply across set class types, effectively decontextualizing contextual inversions.Accepted manuscrip

Music theorists have proposed two very different geometric models of musical objects, one based on voice leading and the other based on the Fourier transform. On the surface these models are completely different, but they converge in special cases, including many geometries that are of particular analytical interest.

Richard Cohn's eagerly anticipated monograph on nineteenth-century chromaticism crystallizes Cohn's decades of influential work developing an analytical framework for chromatic harmony. It also fulfills the need of presenting a self-contained, accessible introduction to Cohn's theory, one that will be of great value to readers less receptive to the mathematical orientation of many of the articles that mark essential milestones in the development of Cohn's theories. But Audacious Euphony also goes further, marking a stride forward in the unification of Cohn's theories and their extension into new analytical concepts and techniques that promise to be widely influential in future work on nineteenth-century harmonic practices. (1) [2] The first half of Audacious Euphony follows the general contours of the historical development of Cohn's thinking about chromaticism. He begins, in Chapter 1, by advancing the non-integrationist argument, that nineteenth-century chromaticism is not simply an extension of Classical harmonic practice, which has been a persistent theme of his work beginning with Cohn 1996. In particular, Cohn argues that chromatic harmony is based on the voice-leading logic of the consonant triad in chromatic space, and is therefore independent from Classical harmony, in which triadic relationships are mediated by diatonic scales. Chapter 2 introduces hexatonic regions, the topic of Cohn 1996, while chapters 3 and 4 develop the idea of Weitzmann regions as companions to hexatonic regions, the subject of Cohn 2000. The ultimate goal of these arguments is the unified model of voice-leading between consonant and augmented triads, Douthett and Steinbach's (1998) "Dancing Cubes" network. Cohn similarly united hexatonic cycles and Weitzmann regions in Cohn 2000, but in Audacious Euphony, the idea has noticeably matured. Most importantly, Cohn follows Tymoczko (2009) in recognizing Cube Dance as a faithful model of voice-leading distance, unlike, e.g., the Tonnetz (84-85). From this other significant properties emerge, such as the cyclic arrangement of sum classes, which Cohn dubs "voice-leading zones" (102-6). [3] In parallel with the construction of the Cube-Dance model of triadic relationships, Cohn also develops the technology of neo-Riemannian transformations, in particular the Tonnetz, the subject of his highly influential "Neo-Riemannian Operations,

This article introduces a type of harmonic geometry, Fourier phase space, and uses it to advance the understanding of Schubert’s tonal language and comment upon current topics in Schubert analysis. The space derives from the discrete Fourier transform on pitch-class sets developed by David Lewin and Ian Quinn but uses primarily the phases of Fourier components, unlike Lewin and Quinn, who focus more on the magnitudes. The space defined by phases of the third and fifth components closely resembles the Tonnetz and has a similar common-tone basis to its topology but is continuous and takes a wider domain of harmonic objects. A number of musical examples show how expanding the domain enables us to extend and refine some the conclusions of neo-Riemannian theory about Schubert’s harmony. Through analysis of the Trio and Adagio from Schubert’s String Quintet and other works using the geometry, the article develops a number of concepts for the analysis of chromatic harmony, including a geom...

The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces, and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the twelve pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. The present article examines how these duplications can relate to important musical concepts such as key or pitch-height, and details a method of removing such redundancies and the resulting changes to the homology the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth-chords, scalar tetrachords, scales, etc., as well as Zeitnetze on a common types of cyclic rhythms or timelines. Their different topologies -whether orientable, bounded, manifold, etc. -reveal some of the topological character of musical concepts.

This article introduces two vectors intended to formalize some triadic transformations, considering specially the Chromatic Transformational System by David KOPP (2002). The numeric content of vector K describes concisely the processes associated to a given operation that must be applied for transforming a referential perfect triad onto a derived one. Vector G informs the spatial position of an operation considering its geometric projection on a referential two-dimensional plan (a Tonnetz). A practical application concerning analysis by computational means is presented in the last section of the study.

Common principles of voice-leading can be represented using an orbifold (ie a quotient space) in which each point is a chord, and the smoothness of voice-leadings corresponds to the closeness of their chords in the space (Tymoczko 2005). The set of ...

problem of unknown quantities outside the sample would be eliminated. Perhaps the study could involve only those elements which have finite qualities. It will be interesting to read from the author and others about further refinements in his method.

The main theme of the present paper is to establish a formula for calculating the Inverse Laplace transformation of functions of the form pl(s'/2)/p,,(~~/~) F[(P~(s'/~))~]. This result is used to obtain the Inverse Laplace transforms of generalized hypergeometric and Lommel functions of 7~ variables.

He has published a number of articles on the music of Babbitt, Carter, Schoenberg, and Webern, as well as other composers; and his book, An Introduction to the Music of Milton Babbitt, is published by the Princeton University Press. He is also active as a composer.

Hanninen's book presents and exemplifies a 'theory of analysis', not a theory of musical structure per se-a premise that will be addressed in this overview and at the end of the entire review. A Theory of Musical Analysis consists of four parts: an introduction; a presentation of the theory of analysis; six analyses, each of which focuses on a specific work (by Beethoven, Debussy, Nancarrow, Riley, Feldman and Morris); and a final part that addresses Hanninen's theory in relation to other recent music theoretical trends and some implications for its further development. Hanninen's theory sets out 'criteria and mechanisms for object formation and the interrelations of objects of analytical interest' (p. 3)-or, in other words, it provides guidelines for determining musical segments and their associations. Hanninen argues that the theory is not a 'methodology' that lays out which musical features determine a segment but, rather, it is a 'multidimensional conceptual space' that 'supports' music analytical thought. Thus both 'objects' and 'interrelations' are analytical entities determined by an analyst. Hanninen's theory not only recognizes but further encourages analytical agency, 'supporting' an analyst's individual interpretation (p. 4). While Hanninen's theory emphasizes analytical agency, the 'criteria and mechanisms' establish some constraints on analytical observation-these constraints forming the core of her theory of analysis. 2 She defines two dimensions for the space of music analytical thought.

Wonder and awe are often considered to be synonyms. But though these emotions are similar, a more nuanced look at their musical representations reveals that they are not entirely the same. This thesis examines the difference between musical wonder and awe in music from The Lion, The Witch and the Wardrobe, a 2005 film about siblings who find a magical world in a wardrobe. After a review of the literature on musical wonder, the music from two scenes is analyzed. The wardrobe scene, in which the youngest child discovers the land of Narnia, depicts wonder visually. Certain musical characteristics, both harmonic transformations and non-harmonic factors, correspond with this visual representation. The other scene analyzed is the battle scene, in which the children fight against the army of the evil White Witch. Visually, this scene is more complex and represents a variety of emotions, including power, fear, and awe. Again, both harmonic and non-harmonic factors contribute to a musical cu...

Die Propaganda des faschistischen Italiens setzte die Antike prominent in Szene. Diese Instrumentalisierung der Antike in moderner Propaganda hatte sehr konkrete Rückwirkungen auf das Bild, das man sich von der Antike machte. Teilweise direkt unter dem Eindruck faschistischer Propaganda begannen damals Altertumswissenschaftler, moderne Propaganda in der Antike zu erkennen: Antiken Texten und Monumenten wurde nun eine ganz andere Wirkungskraft und -absicht zugeschrieben als noch in den Jahrzehnten zuvor. Die zeitbedingten Anachronismen sind offenkundig, doch die faschistische Inszenierung der Antike offenbarte eine Wirkungsmöglichkeit antiker Monumente, die man so vorher nicht gesehen hatte und die nicht per se ‚falsch‘ ist. Dass dabei die römische Kaiserzeit im Vordergrund stand, ist bezeichnend: Die auffallende Omnipräsenz von Bildern, der Fokus der Repräsentation auf städtische Räume und die besonders ausgeprägte Zuspitzung dieser Repräsentation auf die Person des Herrschers ließen die römische Kaiserzeit innerhalb der Vormoderne für moderne Anachronismen besonders anschlussfähig erscheinen. Die Auseinandersetzung mit der modernen Übertragung des Propagandabegriffs auf die Antike zielt daher nicht bloß auf eine Offenlegung von Anachronismen, sondern kann als heuristisches Instrument den Blick für die Besonderheit der Antike innerhalb der Vormoderne schärfen.

To date, calculating the frequencies of musical notes requires one to know the frequency of some reference note. In this study, first-order ordinary differential equations are used to arrive at a mathematical model to determine tonal frequencies using their respective note indices. In the next part of the study, an analysis that is based on the fundamental musical frequencies is conducted to theoretically and neurobiologically explain the consonance and dissonance caused by the different musical notes in the chromatic scale which is based on the fact that systematic patterns of sound invoke pleasure. The reason behind the richness of harmony and the sonic interference and degree of consonance in musical chords are discussed. Since a human mind analyses everything relatively, anything other than the most consonant notes sounds dissonant. In conclusion, the study explains clearly why musical notes and in toto, music sounds the way it does.

Privatization was one of the key mechanisms in the transformation
from planned to market economies in the former Eastern Bloc
following the collapse of communist regimes. Although this radical
change in ownership structures is most often understood as
belonging to the sphere of the economy, it also profoundly affected
society and shared values. As historians are increasingly turning to
the post-1989 period in Central and Eastern Europe, this introduction
and special issue argue that economic and political history
alone are not sufficient to investigate the process of privatization;
approaches from social and cultural history are also necessary.
Transformation, and privatization in particular, was the result of
complex interactions between the economic policies of nationstates,
the actions of transnational organizations and private corporations,
the development of global capitalism, but also of local
traditions, cultural stereotypes and representations, and the transformation
of institutions other than political and economic ones. By
taking into account this complex nexus of factors, we argue, historical
research can bring a new quality to the existing social science
work on postsocialist privatization and economic transformation
more generally.