04265oam 2200481 450 991043757260332120190911112726.03-642-39587-210.1007/978-3-642-39587-1(OCoLC)864999647(MiFhGG)GVRL6YHR(EXLCZ)99371000000003127020131017d2013 uy 0engurun|---uuuuatxtccrThe languages of western tonality /Eytan Agmon1st ed. 2013.Heidelberg [Germany] :Springer,2013.1 online resource (xxvii, 280 pages) illustrations, musicComputational Music Science,1868-0305"ISSN: 1868-0305."3-642-39586-4 Includes bibliographical references and index.Chap. 1  Prototonal Theory: Tapping into Ninth-Century Insights -- Part I  Prototonality -- Chap. 2  Preliminaries -- Chap. 3  Communicating Pitches and Transmitting Notes -- Chap. 4  The Conventional Nomenclatures for Notes and Intervals -- Chap. 5  Communicating the Primary Intervals -- Chap. 6  Receiving Notes -- Chap. 7  Harmonic Systems -- Chap. 8  Prototonality -- Part II  The Languages of Western Tonality -- Chap. 9  Tonal Preliminaries -- Chap. 10  Modal Communication -- Chap. 11  Topics in Dyadic and Triadic Theory -- Chap. 12  Modes, Semikeys, and Keys: A Reality Check -- Chap. 13  A Neo-Riepelian Key-Distance Theory -- Chap. 14  Tonal Communication -- Chap. 15  The Tonal Game -- App. A  Mathematical Preliminaries -- App. B  Z Modules and Their Homomorphisms -- Index.Tonal music, from a historical perspective, is far from homogenous; yet an enduring feature is a background "diatonic" system of exactly seven notes orderable cyclically by fifth. What is the source of the durability of the diatonic system, the octave of which is representable in terms of two particular integers, namely 12 and 7? And how is this durability consistent with the equally remarkable variety of musical styles — or languages — that the history of Western tonal music has taught us exist? This book is an attempt to answer these questions. Using mathematical tools to describe and explain the Western musical system as a highly sophisticated communication system, this theoretical, historical, and cognitive study is unprecedented in scope and depth. The author engages in intense dialogue with 1000 years of music-theoretical thinking, offering answers to some of the most enduring questions concerning Western tonality. The book is divided into two main parts, both governed by the communicative premise. Part I studies proto-tonality, the background system of notes prior to the selection of a privileged note known as "final." After some preliminaries that concern consonance and chromaticism, Part II begins with the notion "mode." A mode is "dyadic" or "triadic," depending on its "nucleus." Further, a "key" is a special type of "semi-key" which is a special type of mode. Different combinations of these categories account for tonal variety. Ninth-century music, for example, is a tonal language of dyadic modes, while seventeenth-century music is a language of triadic semi-keys. While portions of the book are characterized by abstraction and formal rigor, more suitable for expert readers, it will also be of value to anyone intrigued by the tonal phenomenon at large, including music theorists, musicologists, and music-cognition researchers. The content is supported by a general index, a list of definitions, a list of notation used, and two appendices providing the basic mathematical background.Computational music science.MusicologyData processingTonalityMusicWestern influenceMusicologyData processing.Tonality.MusicWestern influence.781.258Agmon Eytanauthttp://id.loc.gov/vocabulary/relators/aut1065560MiFhGGMiFhGGBOOK9910437572603321The Languages of Western Tonality2546635UNINA03698nam 22005415 450 991030009880332120200706114300.03-319-90321-710.1007/978-3-319-90321-7(CKB)4100000004243955(DE-He213)978-3-319-90321-7(MiAaPQ)EBC6311370(PPN)227403681(EXLCZ)99410000000424395520180521d2018 u| 0engurnn|008mamaatxtrdacontentcrdamediacrrdacarrierExploring Mathematics Problem-Solving and Proof /by Daniel Grieser1st ed. 2018.Cham :Springer International Publishing :Imprint: Springer,2018.1 online resource (XXI, 298 p. 77 illus., 15 illus. in color.) Springer Undergraduate Mathematics Series,1615-20853-319-90319-5 Introduction -- 1 First explorations -- 2 Recursion – a fundamental idea -- 3 Mathematical induction -- 4 Graphs -- 5 Counting -- 6 General problem solving strategies -- 7 Logic and proofs -- 8 Elementary number theory -- 9 The pigeonhole principle -- 10 The extremal principle -- 11 The invariance principle -- A A survey of problem-solving strategies -- B Basics on sets and maps -- List of symbols -- Glossary -- Lists of problems, theorems and methods -- Hints for selected exercises -- References.Have you ever faced a mathematical problem and had no idea how to approach it? Or perhaps you had an idea but got stuck halfway through? This book guides you in developing your creativity, as it takes you on a voyage of discovery into mathematics. Readers will not only learn strategies for solving problems and logical reasoning, but they will also learn about the importance of proofs and various proof techniques. Other topics covered include recursion, mathematical induction, graphs, counting, elementary number theory, and the pigeonhole, extremal and invariance principles. Designed to help students make the transition from secondary school to university level, this book provides readers with a refreshing look at mathematics and deep insights into universal principles that are valuable far beyond the scope of this book. Aimed especially at undergraduate and secondary school students as well as teachers, this book will appeal to anyone interested in mathematics. Only basic secondary school mathematics is required, including an understanding of numbers and elementary geometry, but no calculus. Including numerous exercises, with hints provided, this textbook is suitable for self-study and use alongside lecture courses.Springer Undergraduate Mathematics Series,1615-2085MathematicsMathematics—Study and teaching Mathematics, generalhttps://scigraph.springernature.com/ontologies/product-market-codes/M00009Mathematics Educationhttps://scigraph.springernature.com/ontologies/product-market-codes/O25000Popular Science in Mathematicshttps://scigraph.springernature.com/ontologies/product-market-codes/Q26000Mathematics.Mathematics—Study and teaching .Mathematics, general.Mathematics Education.Popular Science in Mathematics.510Grieser Danielauthttp://id.loc.gov/vocabulary/relators/aut65750MiAaPQMiAaPQMiAaPQBOOK9910300098803321Exploring Mathematics2102299UNINA