On the Geometric Realisation of Equal Tempered Music (original) (raw)
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The Mathematics Behind the Music Volume 2
The relation to music and Mathematics is quite a thick topic to discuss because, in a very general sense, music is mathematics. It is a difference and consistency of changes in melody and rhythm that generates sound in a way that we enjoy and appreciate it. Many different applications have been used to describe these changes from logarithmic fluctuation in sound wave frequency to the difference and scale relations of transposed music. This topic interests me in particular because I have always loved composing music and the differences in melodies via Music theory. Being both a mathematician and a musician I've never been terribly impressed with, and consequently interested in, attempts to combine the two. It is perhaps even more surprising that music, with all its passion and emotion, is also based upon mathematical relationships. Such musical notions as octaves, chords, scales, and keys can all be demystified and understood logically using simple mathematics. Mathematical models can be found from theoretical analysis to actual composition or sound production. Increasingly in the last few decades, musical scholarship has incorporated modern mathematical content. One example is the application of methods from Algebraic Combinatorics, or Topology and Graph Theory, to the classification of different musical objects. However, these applications of mathematics in the understanding of music have also led to interesting open problems in mathematics itself. The reach and depth of the contributions on mathematical music theory presented in this volume is significant. Each contribution is in a section within these subjects: (i) Algebraic and Combinatorial Approaches; (ii) Geometric, Topological, and Graph-Theoretical Approaches; and (iii) Distance and Similarity Measures in Music.
Tuning of Musical Notes through Mathematics
2012
Mathematics, an enigma in numbers and calculations, often accompanied by feelings of rejection and disinterest while Music, a flow with emotions, feelings and life. Motivation for investigating the connections between these two apparent opposites’ poles is attempted in this paper. A correlation between Mathematics and Music is shown.. Music theorists sometimes use mathematics to understand music. Mathematics is "the basis of sound" and sounds itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical". In today’s technology, without mathematics it is difficult to imagine anything feasible. In this paper we have discussed the relation between music and mathematics. How piano keys are interrelated with mathematics, frequencies are correlated and discussed. With the aid of mathematical tools, regression, geometric progression, tuning frequency can be calculated and further re...
Quantifying Harmony: The Mathematical Essence of Music
This research delves into the intricate correlation between music and mathematics, examining how mathematical principles play a fundamental role in shaping various aspects of musical theory and composition. Starting from Pythagoras' early insights on harmonic intervals to the complex patterns found in the Fibonacci sequence and fractal geometry, this study uncovers the deep connections between numerical concepts and musical notes. By extensively exploring historical perspectives, theoretical frameworks, and practical applications, we gain valuable insight into how mathematical ideas influence the melodies, harmonies, and rhythms that elevate our auditory experiences.
Music and Math: Foundations of Music
2011
Mathematicians throughout history primarily designed the western musical scale. This thesis tells the story of the mathematicians that contributed to this process of developing the scale and how mathematical concepts were applied. The story spans from Pythagoras around to Rene Descartes and examines the development of the musical scale most commonly used today.
ICONEA PUBLICATIONS ISBN 978-1-716-47825-3, 2020
Relative and absolute pitches Dating the texts Systemic duplicity Span and system The nature of intervals Simultaneous intervals and polarity How can dichords fit in with text U.7/80 More philology Allocation of relative pitches Enneatonic pitch-set construction from U.3011 Tridecachord or heptachord The nature of the intervals of the tridecachord Provisional analysis of the intervals terms in CBS 10996 What was the purpose of text CBS 10996 Memory tuning Conclusions Text CBS 1766 First conclusions Heptatonic construction Interlude
The Mathematics of Music and Art
The Mathematics of Music and Art, 2023
This book explores the relationships between music, the sciences, and mathematics, both ancient and modern, with a focus on the big picture for a general audience as opposed to delving into very technical details. The language of music is deciphered through the language of mathematics. Readers are shown how apparently unrelated areas of knowledge complement each other and in fact propel each other’s advancement. The presentation as well as the collection of topics covered throughout is unique and serves to encourage exploration and also, very concretely, illustrates the cross- and multidisciplinary nature of knowledge. Inspired by an introductory, multidisciplinary course, the author explores the relationships between the arts, sciences, and mathematics in the realm of music. The book has no prerequisites; rather it aims to give a broad overview and achieve the integration of the three presented themes. Mathematical tools are introduced and used to explain various aspects of music theory, and the author illustrates how, without mathematics, music could not have been developed.
Musicae Scientiae, 2016
There has been a long-standing discussion about the connections between music and mathematics. Music involves patterns and structures that may be described using mathematical language. Some compositions are even constructed around certain mathematical ideas, and composers have used various kinds of symmetries and transformations to shape their works. Many concepts from music theory can be expressed using some basic mathematical formalisms. And of course, in the study of musical sounds, the relation between music and mathematics becomes quite obvious. In his book From Music to Mathematics, Gareth Roberts explores some of the connections between these two disciplines. Organized into eight chapters, the book covers a range of topics starting with well-known relationships such as the Pythagorean theory of musical scales and simple ratios, harmonic consonance and overtones series, as well as musical symmetries and group theory. More curious connections exist in scenarios such as change ringing, twelve-tone music, or mathematically inspired modern music. The book is written with care, and in it Roberts reveals his passion for both music and mathematics. Each chapter starts with a certain musical aspect which leads to a mathematical problem. This problem is then formalized and treated in more detail. Reading through the book is like listening to a pleasant medley, in which one encounters some favorite tunes as well as new, surprising perspectives. Beyond simply pointing out interesting connections between music and mathematics, Roberts notes that one main goal of this book is to use music in order to illuminate important mathematical concepts. By doing so, the author tries to overcome the first hurdle many students are confronted with when studying abstract mathematics. Although this book is not meant to replace a proper textbook on algebra, number theory, combinatorics, trigonometry, or differential calculus, it does not refrain from using proper mathematical notation, all the while giving students a glimpse into the mathematical realm and its beauty. Rather than giving elaborate introductions in the different mathematical fields, Gareth Roberts covers different topics in an anecdotal and elementary form, which makes the book accessible for a wide readership including undergraduate and even advanced high school students. The following paragraphs will address the individual chapters of the book. Music is typically organized into temporal units or pulses, referred to as beats. Repeating sequences of stressed and unstressed beats and sub-beats, in turn, form higher temporal patterns, which are related to what is called the rhythm of music. In Chapter 1, the book relates the musical notions of beat and rhythm to the fundamental mathematical concept of counting. First it discusses the role of note durations, which are specified in terms of rational numbers multiplied by the underlying beat duration. Looking at the basic note types (whole, half, quarter, eighth, etc.) leads the writer to the mathematical concept of geometric series, which is then discussed in greater detail. Furthermore, 672822M SX0010.1177/1029864916672822Musicae ScientiaeBook review book-review2016 Book review
JOHN COLLINS - MUSIC AND MATHEMATICS 2003
The connection and between mathematics and music has been recognised from times immemorial. The same ancient Greek root word that means 'flowing' is found in both rhythm and arithmetic, the subjects of arithmetic, geometry, music and astronomy were taught within the same combined quadrivium course in medieval European universities, early astronomers like Kepler tried to relate the movements of the heavenly bodies to musico-mathematical formulas and the philosopher Liebniz believed the musical sensibility was founded on a hidden mathematical element of consciousness. As will be discussed, more recent attempts to correlate music and mathematics include the acoustic laws of the physicist Herman Helmholtz 1 , the deep musical structural templates of Heinrich Schenker and the set theories of serial composers such as Arnold Schoenberg. But first let us turn to antiquity.
Intrinsic Harmonic Spaces: A Solution to the Ancient Problem of Perfect Tuning
2021
In this article we solve this ancient problem of perfect tuning in all keys and present a system were all harmonies are conserved at once. It will become clear, when we expose our solution, why this solution could not be found in the way in which earlier on musicians and scientist have been approaching the problem. We follow indeed a different approach. We first construct a mathematical representation of the complete harmony by means of a vector space, where the different tones are represented in complete harmonic way for all keys at once. One of the essential differences with earlier systems is that tones will no longer be ordered within an octave, and we find the octave-like ordering back as a projection of our system. But it is exactly by this projection procedure that the possibility to create a harmonic system for all keys at once is lost. So we see why the old way of ordering tones within an octave could not lead to a solution of the problem. We indicate in which way a real mu...