Rhythm, Tunings, Number Theory, and Abstract Algebra

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

Sets and Numbers. We assume familiarity with the basic notions of set theory, such as the concepts of element of a set, subset of a set, union and intersection of sets, and function from one set to another. We assume familiarity with the descriptors one-to-one and onto for a function.

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.

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...

The Mathematical Intelligencer, 2019

Your article is protected by copyright and all rights are held exclusively by Springer Science+Business Media, LLC, part of Springer Nature. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to selfarchive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com".

The purpose of this paper is to show through particular examples how group theory is used in music. The examples are chosen from the theoretical work and from the compositions of Olivier Messiaen (1908-1992), one of the most influential twentieth century composers and pedagogues. Messiaen consciously used mathematical concepts derived from symmetry and groups, in his teaching and in his compositions. Before dwelling on this, I will give a quick overview of the relation between mathematics and music. This will put the discussion on symmetry and group theory in music in a broader context and it will provide the reader of this handbook some background and some motivation for the subject. The relation between mathematics and music, during more than two millennia, was lively, widespread, and extremely enriching for both domains. This paper will appear in the Handbook of Group actions, vol. II (ed. L. Ji, A. Papadopoulos and S.-T. Yau), Higher Eucation Press and International Press.

Notices of the American Mathematical Society, 2023

Mircea Pitici, ed., _The Best Writing on Mathematics 2010_ (Princeton University Press), 2011

"The dialectic between soul and science, body and number, freedom and discipline, self and non-self – dare I say it? That’s culture in a nutshell. It is this very dialogue - this sustained interaction between ourselves and the world around us - that I wish to make audible through music." A version of this article originally appeared in The Guardian: https://www.theguardian.com/music/2009/oct/15/fibonacci-golden-ratio

International Journal of Research, 2014

Music is the pleasure the human soul experiences from counting without being aware that it is counting.-Gottfried Wilhelm von Leibniz, a German mathematician who co-discovered calculus.

2006

The intimate association between mathematics and music can be traced to the Greek culture. It is well-represented in the prevailing Western musical culture of the 18th and 19th centuries, where the traditional cycle of fifths provides a mathematical model for classical harmony that originated with the well-tempered, and later the equal-tempered, keyboard. Equal-temperament gives equivalent status to all twelve tonal centres in the chromatic scale, leading to a high degree of symmetry and an underlying group structure. This connection seems to endorse the Pythagorean concept of music as exemplifying an ideal mathematical harmony. This paper examines the relationship between abstract mathematics and music more critically, challenging the idealized view of music as rooted in pure mathematical relations and instead highlighting the significance of music as an association between form and meaning that is negotiated and pragmatic in nature. In passing, it illustrates how the complex and s...