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This book offers an in-depth analysis of musical variation through a systematic approach, heavily influenced by the principles of Grundgestalt and developed variations, both created by the Austrian composer Arnold Schoenberg (1874-1951). The author introduces a new transformational-derivative model and the theory that supports it, specifically crafted for the examination of tonal music. The idea for this book emerged during a sabbatical at Columbia University, while the content is the product of extensive research conducted at the Federal University of Rio de Janeiro, resulting in the development of the Model of Derivative Analysis. This model places emphasis on the connections between musical entities rather than viewing them as separate entities. As a case study, the Intermezzo in A Major Op.118/2 by Brahms is selected for analysis. The author's goal is to provide a formal and structured approach while maintaining the text's readability and appeal for both musicians and mathematicians in the field of music theory. The book concludes with the author's recommendations for further research.

Music—Mathematics. --- Mathematics. --- Mathematics in Music. --- Applications of Mathematics. --- Math --- Science

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Sounds can be harmonic, number sequences too - a coincidence? This book deals with a musical theory of proportions, i.e. the ancient doctrine of proportions as the oldest and most important common anchorage of the two cultural sciences mathematics and music. The musical theory of tones, intervals, tetrachords, sounds and scales is in fact the exact musical image of the laws of arithmetic and its symmetries in the set of rules of playing with numbers, their proportions and their medievals. Alone the miracle of the so-called Harmonia perfecta maxima 6 - 8 - 9 - 12, whose proportions determine the fifth as well as the fourth, form the octave and have the brazen whole tone in their center, shaped the musical edifice of Pythagorean music for thousands of years. This elementary chain of proportions 6 : 8 : 9 : 12 is, moreover, completely symmetrical and built up from the arithmetic as well as from the harmonic medieta of the octave numbers 6 and 12. This book develops the theory of proportions as a mathematical science and always contrasts it with the musical motivation by means of numerous examples. The main idea is the derivation of a theory of symmetry from the Harmonia perfecta maxima to the Harmonia perfecta infinita abstracta, a process of unlimited tone generations by Babylonian mean iterations. From this, both the classical-antique diatonic is simultaneously extracted and the path "from the monochord to the organ" is re-examined. Finally, the work contains a mathematically guided introduction to the ancient tetrachordics as well as to the church tonal scales and concludes with an excursion into the sound worlds of the organ. Here the "foot-number rule of the organ" leads us by means of examples into the world of the tonal dispositions of this instrument and shows the omnipresence of the ancient theory of proportions. This book is suitable for anyone with an interest in mathematics and music. The Author Prof. Dr. Karlheinz Schüffler is a mathematician and musician. As a mathematician he teaches at the University of Krefeld as well as at the University of Düsseldorf. As a musician he practices church music with a focus on organ, Gregorian chant and choir conducting. This book is a translation of an original German edition. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation.

Music --- Mathematics in Music. --- Mathematics. --- Music theory --- Musical intervals and scales. --- Ratio and proportion.

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This book explores the profound and ancient relationship between music and astronomy. Throughout history, Music has occupied a significant place among the disciplines of the Quadrivium, which also include Geometry, Arithmetic, and Astronomy. The captivating bond between these two realms has not only inspired eminent scientists like Kepler, Newton, and Einstein, but has also captured the imagination of NASA and astronauts in modern times. The author delves into various aspects of the intersection between music and astronomy, encompassing everything from ancient cosmological beliefs to groundbreaking discoveries such as the cosmic background radiation and gravitational waves. This enthralling theme has not only stimulated renowned artists like David Bowie and Elton John, but has also served as a muse for movies like Close Encounters of the Third Kind. Within the book, readers will find an extensive photo gallery and a specially curated soundtrack that enhances the reading experience. It caters to a broad audience, appealing to those with a general interest in both music and astronomy, as well as to specialized individuals in either field of study.

Music. --- Music --- Astronomy. --- Anthropology and the arts. --- Mathematics in Music. --- Philosophy of Music. --- Astronomy, Cosmology and Space Sciences. --- Anthropology of the Arts. --- Mathematics. --- Philosophy and aesthetics.

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Galileo Galilei said he was “reading the book of nature” as he observed pendulums swinging, but he might also simply have tried to draw the numbers themselves as they fall into networks of permutations or form loops that synchronize at different speeds, or attach themselves to balls passing in and out of the hands of good jugglers. Numbers are, after all, a part of nature. As such, looking at and thinking about them is a way of understanding our relationship to nature. But when we do so in a technical, professional way, we tend to overlook their basic attributes, the things we can understand by simply “looking at numbers.” Tom Johnson is a composer who uses logic and mathematical models, such as combinatorics of numbers, in his music. The patterns he finds while “looking at numbers” can also be explored in drawings. This book focuses on such drawings, their beauty and their mathematical meaning. The accompanying comments were written in collaboration with the mathematician Franck Jedrzejewski.

Arithmetic. --- Mathematics. --- Math --- Graph theory. --- Graph Theory. --- Mathematics, general. --- Mathematics in Music. --- Mathematics --- Set theory --- Calculators --- Numbers, Real --- Science --- Graph theory --- Graphs, Theory of --- Theory of graphs --- Combinatorial analysis --- Topology --- Extremal problems

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This book features original papers from 27th International Symposium on Frontiers of Research in Speech and Music (FRSM 2023), jointly organized by Sardar Vallabhbhai National Institute of Technology, Surat, India, and Sir C.V. Raman Centre for Physics and Music, Jadavpur University, Kolkata, India, during 4–5 August 2023. The book is organized into four main sections, considering both technological advancement and interdisciplinary nature of speech, music, language and their applications. The first section includes chapters related to computational, modelling and cognitive aspects of the speech signal. The second part contains chapters covering the foundations of both vocal and instrumental music processing with the signal, computational and cognitive aspects. The third section relates to the variety of research being done in the peripheral areas of languages and linguistics with special focus on regional languages of India. A lot of research is being performed within the speech and music information retrieval domain which is potentially interesting for most users of computers and the Internet. Therefore, the fourth and final section is dedicated to the chapters related to multidisciplinary applications of speech and music signal processing.

Computational intelligence. --- Signal processing. --- Image processing --- Computer vision. --- Music --- Computational Intelligence. --- Signal, Speech and Image Processing. --- Computer Imaging, Vision, Pattern Recognition and Graphics. --- Mathematics in Music. --- Philosophy of Music. --- Digital techniques. --- Mathematics. --- Philosophy and aesthetics.