The Math Behind the Music (Outlooks) [Hardcover]

Leon Harkleroad (Author)

Book Description

Publication Date: August 14, 2006 | Series: Outlooks

Mathematics has been used for centuries to describe, analyze, and create music. In this book, Leon Harkleroad explores the math related aspects of music from its acoustical bases to compositional techniques to music criticism, touching on - overtones, scales, and tuning systems - the musical dice game attributed to Mozart and Haydn - the several-hundred-year-old style of bell-playing known as ringing the changes - the twelve-tone school of composition that strongly influenced music throughout the 20th century and many other topics involving mathematical ideas from probability theory to Fourier series to group theory. He also relates some cautionary tales of misguided attempts to mix music and mathematics. Both the mathematical and the musical concepts are described in an elementary way, making the book accessible to general readers as well as to mathematicians and musicians of all levels. The book is accompanied by an audio CD of musical examples.

Editorial Reviews

Review

"...a text which relates music and mathematics in a manner accessible to many individuals, and provides a bibliography of further readings on each chapter's topics, so the best use of this book may be as a jumping-off point for further examinations of whichever topic interests and individual reader.The Math Behind the Musicdoes not require advanced understanding of either mathematics or music: in fact most of the text could be easily understood by an interested high school student."
Sarah Boslaugh
MAA Reviews


"Leon Harkleroad has written a concise and accessible survey of some of the interesting mathematics of music, and he has done so in a way that requires a minimum of preparation on the reader's part in either music or mathematics."
Gareth Loy, Gareth, Inc.


"The book can be recommended as a first survey. Those who want to know more can find many suggestions for further study in the extensive bibliography. The book addresses readers who do not have much mathematical knowledge. Everything is explained clearly and at great length, and the theoretical facts are illustrated by many pictures, scores, and the tracks on the accompanying CD."
The Mathematical Intelligencer

"The material is very well presented and well explained, it comes with many (musical and mathematical) examples and illustrations (the book even comes with a CD-ROM from which the reader can listen to the musical examples of the book), and many details side aspects, anecdotes and references are provided."
Christian Krattenhaler, Mathematical Reviews

Book Description

Mathematics has been used for centuries to describe, analyze, and create music. In this book, Leon Harkleroad explores the mathematics related to aspects of music from its acoustical bases to compositional techniques to music criticism. His clear, elementary exposition will be accessible to general readers as well as to mathematicians and musicians of all levels.

Product Details

• Hardcover: 158 pages
• Publisher: Cambridge University Press (August 14, 2006)
• Language: English
• ISBN-10: 0521810957
• ISBN-13: 978-0521810951
• Product Dimensions: 9.2 x 6.2 x 0.7 inches
• Shipping Weight: 15.2 ounces
• Average Customer Review: 4.0 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #2,787,710 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

25 of 27 people found the following review helpful:

4.0 out of 5 stars Interesting and informative, December 12, 2006

By 

Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews
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This review is from: The Math Behind the Music (Outlooks) (Paperback)

If one is familiar with the physics of sound, with its accompanying use of sophisticated mathematics, such as Fourier series, partial differential equations, and so on, it is not surprising to learn that some mathematicians have tried to give music theory a mathematical foundation. If one studies the historical record one will find that their efforts go back for centuries, and musicians have used their results to varying degrees of success. Philosophers of aesthetics have also attempted to find formal or mathematical theories of art, and in a few cases have completely embarrassed themselves in doing this, primarily because of their misunderstanding of the mathematics. Mathematics can be a powerful tool, and it continues its domination in science, technology, business, and industry, but it must be used where it is relevant, and not distorted to make it fit a particular scenario. This is not to say that a successful theory of mathematical aesthetics could not be developed. Indeed, it is belief of this reviewer that such a theory could be developed and would encompass music, art, and dance, and would add much to the appreciation of all these areas.

In general this short book gives a good overview of what is and has been done in the systematization and composition of music using mathematics. The author has given enough to wet the reader's appetite for more in-depth coverage by perusing the many references at the end of the book. The mathematics is kept at a very elementary level, making the book more accessible to a general readership (elementary group theory plays a central role). Therefore readers with a more sophisticated mathematical background may be disappointed. Musicians, both amateur and professional, would definitely gain an appreciation of just how mathematics can encapsulate musical compositions, and how indeed these compositions can be created using various mathematical constructions. The accompanying CD is of course helpful since it music must be heard in order to be fully appreciated. The book could also be of assistance to those who are interested in fully automated musical composition such as now being done in some research programs in artificial intelligence (some of this research is discussed briefly in the book). Of course, a machine that was able to use the same reasoning patterns to do mathematics as to compose music would signal a major advance in machine intelligence.

Although the book is not one on music theory, the author realizes the importance of understanding the elementary physics behind musical sounds, as well as the various tonal systems, such as the Pythagorean `circle of fifths'. He discusses these concepts early in the book, making it more self-contained. This is followed by a somewhat detailed overview of how to use group theory to create musical compositions. Towards the end of the book one finds an interesting application of L-systems to musical compositions. The book ends with the author's views on how various approaches to mathematical music have failed or have been too ad hoc to be useful.

3 of 3 people found the following review helpful:

5.0 out of 5 stars An amazing book that links math and music, which have a great deal in common, September 8, 2008

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
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This review is from: The Math Behind the Music (Outlooks) (Paperback)

It is no coincidence that the three areas of human endeavor where there are child prodigies are mathematics, music and chess. Success in each requires a similar form of mental reasoning, with music and mathematics being the two that are most related.
Harkleroad has written an amazing book, after the base introduction in chapter one, chapter two covers the concept of pitch, in other words the fundamentals of how sounds are different. Chapter three then uses this idea to describe how different sounds can either clash or reinforce each other. In chapter four, you learn how to vary a theme mathematically; it is here where group and subgroup operations are used to alter tunes to make new ones that still sound pleasing. Chapter five covers bell-ringing, where groups and their cosets are used to describe the permutations in the order of bell-ringing. Creating music by using random processes is the topic of chapter six, while it seems odd to think of random processes creating noise having a pleasing structure; some composers have been able to do it. Chapter seven deals with some of the patterns found in music, chapter eight, which is called "Sight Meets Sound", starts with an explanation of "millimetrization." This is the process where the rises and falls of a tune are used to trace out a graph and vice-versa. Composer Heitor Villa-Lobos used photographs of scenes such as mountains and skylines to construct the graph, from which he would compose his music. The ninth and last chapter has the title "How Not to Mix Mathematics and Music." In it, attempts to do things like using numeric sequences such as the Fibonacci numbers to compose music are explored and the reasons why they failed explained. A CD containing the musical pieces referenced in the text is included with the book.
Although I played the saxophone in elementary school and am a regular attendee at the local symphony, I make no claim to being knowledgeable in music. Yet, I was able to read and follow this entire book and truly came away with an appreciation for how mathematics can be used to explain the structure of musical pieces.

6 of 16 people found the following review helpful:

4.0 out of 5 stars Good book, April 28, 2007

By 

K. Gandhi (Orange County, CA USA) - See all my reviews
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This review is from: The Math Behind the Music (Outlooks) (Paperback)

Has a lot of good math theory behind the numbers, and is not that difficult to read, assuming you've had a physics class. Very interesting in some parts, too.