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Music: A Mathematical Offering
Since the time of the Ancient Greeks, much has been written about the relation between mathematics and music: from harmony and number theory, to musical patterns and group theory. Benson provides a wealth of information here to enable the teacher, the student, or the interested amateur to understand, at varying levels of technicality, the real interplay between these two ancient disciplines. The story is long as well as broad and involves physics, biology, psycho acoustics, the history of science, and digital technology as well as, of course, mathematics and music. Starting with the structure of the human ear and its relationship with Fourier analysis, the story proceeds via the mathematics of musical instruments to the ideas of consonance and dissonance, and then to scales and temperaments. This is a must-have book if you want to know about the music of the spheres or digital music and many things in between.
426 pages, Hardcover
First published January 1, 1989
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Displaying 1 - 7 of 7 reviews
May 12, 2022
This is a good 'music for mathematicians' book. It covers how human ears work; the physics of instruments; the details of scales, harmonics, equal temperaments etc; and digital synthesis.
These are all covered in much more mathematical detail than I have found elsewhere, and goes further than you might find in a physics course on waves. For example: the difference between waveforms of strings plucked vs stroked with a bow.
I skimmed a lot of the heavy mathematical sections since I had already studied them. I can imagine these sections putting off a reader who isn't already familiar with Fourier analysis, waves, DSP, group theory etc.
I found that the section on harmonics and tonal theory to be very useful in understanding why scales and music are as they are, although it was tough to understand what is going on since I only had access to an equal-temperament keyboard. There is a big section in the middle of this which is a load of examples of just-intonation chords that I can't play.
I wish that this section spent more time slowly going through the ramifications of this for equal-temperament, which is what 95% of western songs and instruments use. A good supplement for this is https://mugglinworks.com/chordmaps/.
There is then a whole chapter about other alternative scales and temperaments. I guess these are cool to think about in the abstract but they don't sound good unless you spend your life listening to them, so I didn't find this interesting.
There are a few points where he gets a bit too swept up in the mathematics of things without there being a clear musical application. Particularly in the final chapter where he gives an intro to group theory up to orbit-stabiliser, but the payoff is understanding the motivation behind modern twelve-tone music which sounds awful.
Also maybe I am just a curmudgeon but I found the inclusion of comics and not-terribly-relevant images to be a tad twee.
So overall I would recommend this book if you are musically minded and have studied physics or maths. It won't make you a much better musician but it is fun and answers a lot of questions about how music works. This book would be fantastic as an interactive website.
These are all covered in much more mathematical detail than I have found elsewhere, and goes further than you might find in a physics course on waves. For example: the difference between waveforms of strings plucked vs stroked with a bow.
I skimmed a lot of the heavy mathematical sections since I had already studied them. I can imagine these sections putting off a reader who isn't already familiar with Fourier analysis, waves, DSP, group theory etc.
I found that the section on harmonics and tonal theory to be very useful in understanding why scales and music are as they are, although it was tough to understand what is going on since I only had access to an equal-temperament keyboard. There is a big section in the middle of this which is a load of examples of just-intonation chords that I can't play.
I wish that this section spent more time slowly going through the ramifications of this for equal-temperament, which is what 95% of western songs and instruments use. A good supplement for this is https://mugglinworks.com/chordmaps/.
There is then a whole chapter about other alternative scales and temperaments. I guess these are cool to think about in the abstract but they don't sound good unless you spend your life listening to them, so I didn't find this interesting.
There are a few points where he gets a bit too swept up in the mathematics of things without there being a clear musical application. Particularly in the final chapter where he gives an intro to group theory up to orbit-stabiliser, but the payoff is understanding the motivation behind modern twelve-tone music which sounds awful.
Also maybe I am just a curmudgeon but I found the inclusion of comics and not-terribly-relevant images to be a tad twee.
So overall I would recommend this book if you are musically minded and have studied physics or maths. It won't make you a much better musician but it is fun and answers a lot of questions about how music works. This book would be fantastic as an interactive website.
May 6, 2013
The cover art of this book is the striking visual of the normal modes of a vibrating drum head made visible by sand coalescing into patterns on a kettledrum head. Known as Chladni patterns after their 18th Century discoverer, these same images caught my imagination as an undergraduate on the cover of the Third Edition of Boundary Value Problems (David L. Powers). From that time until know, I have always felt modeling music is a particularly enlightening way to understand ODEs, Fourier analysis, the wave equation, and … music.
This book does not necessarily require greater mathematical knowledge than that imparted by something like the Powers textbook and no real close understanding of music is required, either. But, with a fundamental knowledge of the wave equation and solving ODEs along with at least a passing interest in music, this text will raise the avid reader’s knowledge and appreciation of subjects both mathematical and musical.
The material in this book sprung from the author’s fascination with the early purchase of a second-hand synthesizer which developed into an undergraduate mathematics course taught over the period of a few years. It is then natural for this book to start as it does from very basic concepts of defining sound, sound mechanics in the human ear, trigonometry basics, damped harmonic motion, and resonance. Fourier theory is introduced and built up to Bessel functions, the Hilbert transform and related topics. These preliminaries are complete in the first two chapters and are there if required, or a prepared reader can dive right into the encyclopedic third chapter, “A Mathematician’s Guide to the Orchestra”.
The third chapter is the first of seven that overview a musical topic in a methodical, organized approach. Each such chapter breaks its subject up into classifications make this work both a guided tour and a reference work on mathematics and music. Chapter 3 starts from the wave equation for strings to mathematically modeling and determining the vibration modes for stings, wind instruments, drums, horns, xylophones and more. Each section on a class of instruments includes suggestions for further reading. An additional chapter considers consonance and dissonance, including a fascinating section on musical paradoxes where consideration is made of ways to trick the ear.
Two chapters take on the idea of scales from the basic notion of the Pythagorean scale and the cycle of fifths to such alternative scales as those of Harry Partch and Wendy Carlos. This has at its heart an examination of and application of continued fractions. An additional two chapters consider digital music starting from the basics of digital signals and dithering and includes a granular overview of the MP3 and WAV file formats. Fourier transforms are brought in for analyzing synthesis of sounds and practical application is made through examples in the CSOUND programming language.
A final fascinating chapter is “Symmetry in Music”. Through modular arithmetic and some group theory (Cayley, Burnside, etc.) basic concepts of symmetry in music are made clear. Like the other chapter and related chapter groups, “Symmetry in Music” can stand alone both on the musical topic and as an example of an application of group theory.
This work is fully indexed and includes several pages of bibliography and reference works. There is also an “octave” of eight appendices that cover Bessel functions, scales, MIDI charts, a table of intervals referenced back into the text, music theory, and a lengthy list of recommended recordings. (Many sections of the work have context-specific recommended recordings.)
Cambridge University Press allows the author to keep a free online version of this text at http://www.maths.abdn.ac.uk/~bensondj....
This book does not necessarily require greater mathematical knowledge than that imparted by something like the Powers textbook and no real close understanding of music is required, either. But, with a fundamental knowledge of the wave equation and solving ODEs along with at least a passing interest in music, this text will raise the avid reader’s knowledge and appreciation of subjects both mathematical and musical.
The material in this book sprung from the author’s fascination with the early purchase of a second-hand synthesizer which developed into an undergraduate mathematics course taught over the period of a few years. It is then natural for this book to start as it does from very basic concepts of defining sound, sound mechanics in the human ear, trigonometry basics, damped harmonic motion, and resonance. Fourier theory is introduced and built up to Bessel functions, the Hilbert transform and related topics. These preliminaries are complete in the first two chapters and are there if required, or a prepared reader can dive right into the encyclopedic third chapter, “A Mathematician’s Guide to the Orchestra”.
The third chapter is the first of seven that overview a musical topic in a methodical, organized approach. Each such chapter breaks its subject up into classifications make this work both a guided tour and a reference work on mathematics and music. Chapter 3 starts from the wave equation for strings to mathematically modeling and determining the vibration modes for stings, wind instruments, drums, horns, xylophones and more. Each section on a class of instruments includes suggestions for further reading. An additional chapter considers consonance and dissonance, including a fascinating section on musical paradoxes where consideration is made of ways to trick the ear.
Two chapters take on the idea of scales from the basic notion of the Pythagorean scale and the cycle of fifths to such alternative scales as those of Harry Partch and Wendy Carlos. This has at its heart an examination of and application of continued fractions. An additional two chapters consider digital music starting from the basics of digital signals and dithering and includes a granular overview of the MP3 and WAV file formats. Fourier transforms are brought in for analyzing synthesis of sounds and practical application is made through examples in the CSOUND programming language.
A final fascinating chapter is “Symmetry in Music”. Through modular arithmetic and some group theory (Cayley, Burnside, etc.) basic concepts of symmetry in music are made clear. Like the other chapter and related chapter groups, “Symmetry in Music” can stand alone both on the musical topic and as an example of an application of group theory.
This work is fully indexed and includes several pages of bibliography and reference works. There is also an “octave” of eight appendices that cover Bessel functions, scales, MIDI charts, a table of intervals referenced back into the text, music theory, and a lengthy list of recommended recordings. (Many sections of the work have context-specific recommended recordings.)
Cambridge University Press allows the author to keep a free online version of this text at http://www.maths.abdn.ac.uk/~bensondj....
May 26, 2018
As a professional physicist with a maths degree who spends pretty much all of his spare time making and thinking about music, this book was right up my alley. To read it in its entirety requires substantial mathematical knowledge, more than the average person - but the author is adept at directing the uninterested reader in avoiding the more technical segments, and the book is intended to be of use to both those that can and cannot follow the mathematical expositions in detail.
I filled a pad of paper with solutions to the technical exercises and variations thereof as I went through it. But I would heartily recommend it to someone with a much less detail-oriented interest in the subject; there's absolutely no need to be so wilfully pedantic as I was in reading through it. Music makes much more sense when maths is allowed to attach to it, and there's useful info here even for those for whom maths is no pleasure.
I filled a pad of paper with solutions to the technical exercises and variations thereof as I went through it. But I would heartily recommend it to someone with a much less detail-oriented interest in the subject; there's absolutely no need to be so wilfully pedantic as I was in reading through it. Music makes much more sense when maths is allowed to attach to it, and there's useful info here even for those for whom maths is no pleasure.
May 13, 2011
If you are a math or physics major and you also enjoy playing instruments, this is the book for you. It is full of fun applications of physics and mathematics to modern music theory.
February 28, 2021
Really nice, especially for the parts covering the scales and temperaments.
It does not cover, however, the more strictly music theory part, which is summarized nicely (but I know nothing about it, so...) in an appendix.
It does not cover, however, the more strictly music theory part, which is summarized nicely (but I know nothing about it, so...) in an appendix.
January 8, 2009
I used this book to dive into my current obsession with the drum. I wrote a paper that used Partial Differential Equations to describe the sound of a square drum. The next step was watching the film "Touch the sound".
April 16, 2009
I read through a bit of this and found I needed a Computer Science degree with Advanced Linear Equations to understand the rest of the book. The information that i DID get, was amazing. Maybe once I finish a PhD in Science, I can go back and read the rest.
Displaying 1 - 7 of 7 reviews