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Aaron Hughes
Aaron Hughes

A Theory of Harmony: How Ernst Levy Revolutionized Music Theory with the Harmonic Series



A Theory of Harmony: A Book Review




If you are interested in music theory, you may have heard of a book called A Theory of Harmony, written by Ernst Levy and published by SUNY Press in 1985. This book is considered to be one of the most original and influential works on harmony in the 20th century, and it has inspired many musicians, composers, and scholars to rethink the foundations of music. But what is this book about, and why is it so important? In this article, I will give you an overview of the main concepts, applications, and criticisms of Levy's theory of harmony, and help you decide whether you should read this book or not.




A Theory of Harmony (SUNY Series in Cultural Perspectives) book pdf



What is the book about?




A Theory of Harmony is a book that presents a new way of understanding harmony and tonality in music. Levy argues that harmony is not a set of rules or conventions that govern musical practice, but a natural phenomenon that arises from the physics of sound and the perception of the human ear. He bases his theory on the harmonic series, which is a sequence of pitches that are related by simple ratios and form the basis of all musical intervals and chords. He also introduces a new tool for analyzing harmony, called the harmonic matrix, which is a grid that shows all the possible combinations of pitches in a given tonal system. He then uses this tool to explain how harmony works in different musical styles and genres, from ancient modal music to modern atonal music.


Who is the author?




Ernst Levy was a Swiss-American composer, pianist, musicologist, and teacher who lived from 1895 to 1981. He studied music in Basel, Paris, and Berlin, and was influenced by composers such as Debussy, Schoenberg, and Bartok. He taught music at various universities in Europe and America, including Harvard, MIT, and Chicago. He composed over 200 works for various instruments and ensembles, including symphonies, concertos, chamber music, piano music, vocal music, and electronic music. He was also a prolific writer who published several books and articles on music theory, history, aesthetics, and philosophy.


Why is the book important?




A Theory of Harmony is important because it challenges some of the common assumptions and dogmas that have dominated music theory for centuries. For example, Levy rejects the idea that harmony is based on triads (three-note chords) or tertian structures (chords built from thirds), which are considered to be the norm in Western classical music. Instead, he shows that harmony can be based on any interval or combination of intervals that are derived from the harmonic series. He also rejects the idea that harmony is hierarchical or functional (that some chords are more stable or important than others), which are considered to be essential for creating musical form and direction. Instead, he shows that harmony can be symmetrical or dualistic (that any chord can be inverted or reversed without changing its identity or quality), which allows for more freedom and creativity in musical expression.


The Main Concepts of the Book




Harmony and tonality




Levy defines harmony as "the simultaneous sounding of two or more tones" (p. 1), and tonality as "the organization of tones with reference to a tonic" (p. 3). He argues that harmony and tonality are not arbitrary or artificial constructs, but natural and universal phenomena that are rooted in the physics of sound and the physiology of the ear. He explains that when two or more tones are sounded together, they produce a complex sound wave that contains not only the original tones, but also other tones that are called overtones or partials. These overtones are the result of the vibration of the sound source, and they form a series of pitches that are related by simple ratios and decrease in amplitude as they increase in frequency. This series is called the harmonic series, and it is the basis of all musical intervals and chords.


The harmonic series




The harmonic series is a sequence of pitches that are generated by dividing a string, a pipe, or any other sound source into smaller segments that vibrate faster and produce higher pitches. The first pitch of the series is called the fundamental, and it is the lowest and loudest pitch that is heard. The second pitch of the series is called the first overtone, and it is obtained by dividing the sound source into two equal parts that vibrate twice as fast and produce a pitch that is an octave higher than the fundamental. The third pitch of the series is called the second overtone, and it is obtained by dividing the sound source into three equal parts that vibrate three times as fast and produce a pitch that is a fifth higher than the first overtone. The fourth pitch of the series is called the third overtone, and it is obtained by dividing the sound source into four equal parts that vibrate four times as fast and produce a pitch that is a fourth higher than the second overtone. And so on.


The harmonic series can be represented by a diagram like this:


Fundamental 1st Overtone 2nd Overtone 3rd Overtone 4th Overtone 5th Overtone 6th Overtone ... --- --- --- --- --- --- --- --- C C' G' C'' E'' G'' Bb'' ... 1/1 2/1 3/2 4/3 5/4 6/5 7/6 ... The numbers below each pitch indicate the ratio between the frequency of that pitch and the frequency of the fundamental. For example, the ratio between C' and C is 2/1, which means that C' has twice the frequency of C. The ratio between G' and C is 3/2, which means that G' has three times the frequency of C divided by two. And so on.


The harmonic series contains all the possible intervals and chords that can be formed by combining pitches that are related by simple ratios. For example, the interval between C and G' is a perfect fifth, which has a ratio of 3/2. The interval between C and E'' is a major third, which has a ratio of 5/4. The chord formed by C, G', and E'' is a major triad, which has a ratio of 4/5/6. And so on.


The harmonic matrix




The harmonic matrix is a tool that Levy invented to analyze harmony in music. It is a grid that shows all the possible combinations of pitches in a given tonal system. For example, here is a harmonic matrix for the tonal system of C major:


/ C' D' E' F' G' A' B' --- --- --- --- --- --- --- --- C''/C' = 2/1 (octave) C'' (1) D'' (9) E'' (5) F'' (15) G'' (3) A'' (27) B'' (45) G'/C' = 3/2 (fifth) G' (2) A' (18) B' (10) C'' (6) D'' (54) E'' (30) F#'' (90) E'/C' = 5/4 (third) E' (4) F#'(20) G#'(12) A'(28) B'(16) C#''(36) D#''(52) B'/C' = 15/8 (seventh) B'(8) overtones of the fundamental pitch, and the columns represent different pitches within an octave. The numbers in each cell indicate the ratio between the pitch and the fundamental pitch, and the numbers in parentheses indicate the order of the harmonic in the harmonic series. For example, the cell at the intersection of the row C''/C' and the column G' shows the pitch G'', which has a ratio of 3/2 with C'', and is the third harmonic of C'. The cell at the intersection of the row E'/C' and the column B' shows the pitch B', which has a ratio of 16/15 with E', and is the 16th harmonic of C'. And so on.


The harmonic matrix can be used to identify and classify different types of chords and intervals based on their ratios and harmonics. For example, a chord that consists of pitches that are adjacent in a row is called a harmonic chord, because it is formed by harmonics of the same fundamental pitch. A chord that consists of pitches that are adjacent in a column is called a subharmonic chord, because it is formed by subharmonics (or inverted harmonics) of the same fundamental pitch. A chord that consists of pitches that are not adjacent in either a row or a column is called a mixed chord, because it is formed by a mixture of harmonics and subharmonics of different fundamental pitches. And so on.


The harmonic dualism




The harmonic dualism is a concept that Levy developed to explain how harmony can be symmetrical or reversible in music. He argues that any chord or interval can be inverted or reversed without changing its identity or quality, as long as it preserves its ratio and harmonics. For example, the interval C-G', which has a ratio of 3/2, can be inverted to G'-C'', which has a ratio of 2/3, and still be considered a perfect fifth. The interval C-E', which has a ratio of 5/4, can be reversed to E'-C', which has a ratio of 4/5, and still be considered a major third. And so on.


The harmonic dualism can be illustrated by using the harmonic matrix. For example, here is how to invert or reverse an interval or chord using the matrix:


/ C' D' E' F' G' A' B' --- --- --- --- --- --- --- --- C''/C' = 2/1 (octave) C'' (1) D'' (9) E'' (5) F'' (15) G'' (3) A'' (27) B'' (45) G'/C' = 3/2 (fifth) G' (2) A' (18) B' (10) C'' (6) D'' (54) E'' (30) F#'' (90) E'/C' = 5/4 (third) E' (4) F#'(20) G#'(12) A'(28) B'(16) C#''(36) D#''(52) B'/C' = 15/8 (seventh) B'(8) To invert an interval or chord, simply move it to the opposite side of the diagonal line that runs from C'' to B'. For example, to invert the interval C-G', move it from the cell at row C''/C' and column G', to the cell at row G'/C' and column C''. The inverted interval is G'-C'', which has the same ratio and harmonics as C-G', but in reverse order. To invert the chord C-E'-G', move it from the cells at row C''/C' and columns E' and G', to the cells at row E'/C' and columns C' and G'. The inverted chord is E'-G'-C'', which has the same ratio and harmonics as C-E'-G', but in reverse order.


the horizontal line that runs from C'' to C'. For example, to reverse the interval C-E', move it from the cell at row C''/C' and column E', to the cell at row E'/C' and column C'. The reversed interval is E'-C', which has the same ratio and harmonics as C-E', but in reverse order. To reverse the chord C-E'-G', move it from the cells at row C''/C' and columns E' and G', to the cells at row G'/C' and columns C' and E'. The reversed chord is G'-C'-E', which has the same ratio and harmonics as C-E'-G', but in reverse order.


The harmonic dualism allows for more flexibility and variety in harmonic progression and modulation in music. For example, a chord can be inverted or reversed to create a different chord that has the same function or quality, but a different sound or color. A chord can also be inverted or reversed to create a different chord that has a different function or quality, but a similar sound or color. And so on.


The Applications of the Book




Music analysis




One of the main applications of Levy's theory of harmony is music analysis. Levy uses his harmonic matrix to analyze various musical examples from different periods and styles, and shows how his theory can explain and illuminate their harmonic structure and logic. For example, he analyzes Bach's chorales, Mozart's sonatas, Beethoven's symphonies, Chopin's nocturnes, Debussy's preludes, Schoenberg's serial music, and more. He demonstrates how his theory can account for both the conventional and the unconventional aspects of harmony in these works, such as modulations, chromaticism, enharmonicism, parallelism, polytonality, atonality, etc. He also shows how his theory can reveal the hidden symmetries and dualities that underlie the harmonic organization and expression of these works.


Music composition




Another application of Levy's theory of harmony is music composition. Levy uses his harmonic matrix to generate new musical ideas and materials for his own compositions, and encourages other composers to do the same. He shows how his theory can provide a rich and diverse source of harmonic possibilities for creating original and innovative musical works. For example, he shows how his theory can help composers to explore different tonal systems and modes, such as the Pythagorean system, the just intonation system, the equal temperament system, the harmonic series system, etc. He also shows how his theory can help composers to experiment with different types of chords and intervals, such as harmonic chords, subharmonic chords, mixed chords, complex chords, microtonal chords, etc. He also shows how his theory can help composers to manipulate and transform chords and intervals using inversion, reversal, transposition, permutation, combination, etc.


Music education




A third application of Levy's theory of harmony is music education. Levy uses his harmonic matrix to teach harmony and tonality to his students, and suggests that other teachers should do the same. He argues that his theory can offer a more natural and intuitive way of learning harmony and tonality than the traditional methods that are based on rules and conventions. He claims that his theory can help students to develop a better ear for harmony and tonality by exposing them to the physics and perception of sound. He also claims that his theory can help students to develop a better understanding of harmony and tonality by showing them the logic and coherence of sound. He also claims that his theory can help students to develop a better appreciation of harmony and tonality by introducing them to the diversity and beauty of sound.


The Criticisms of the Book




Academic reception




and wrote in the preface of his book: "I am fully aware that this book will not be read by many people. It is too technical for the layman and too unorthodox for the professional" (p. ix). Some of the reasons why Levy's book has been largely ignored or rejected by the academic music community are: - His book is written in a dense and complex style that makes it difficult to read and understand. - His book is based on a single principle (the harmonic series) that he applies to all aspects of harmony and tonality, which makes it seem simplistic and dogmatic. - His book challenges some of the established and accepted theories and practices of harmony and tonality, which makes it seem radical and heretical. - His book does not provide enough empirical evidence or historical context to support his claims and arguments, which makes it seem speculative and subjective. Alternative approaches




Another criticism of Levy's book is that it is not the only or the best way to approach harmony and tonality in music. There are many other theories and methods that have been developed and proposed by other musicians, composers, and scholars that offer different perspectives and insights on harmony and tonality. Some of these alternative approaches are: - The functional theory of harmony, which explains harmony in terms of its role and purpose in creating musical form and direction. - The Schenkerian theory of harmony, which explains harmony in terms of its reduction and derivation from a fundamental structure or background. - The neo-Riemannian theory of harmony, which explains harmony in terms of its transformation and relation to other harmonic entities or regions. - The set theory of harmony, which explains harmony in terms of its classification and organization using mathematical concepts and operations. Limitations and challenges




A third criticism of Levy's book is that it has some limitations and challenges that prevent it from being fully applicable and useful for music analysis, composition, and education. Some of these limitations and challenges are: - His book is focused on harmony and tonality, but does not address other musical elements or parameters, such as melody, rhythm, timbre, texture, etc. - His book is based on the harmonic series, but does not account for other factors that affect the perception and production of sound, such as temperament, tuning, intonation, psychoacoustics, etc. - His book is derived from Western classical music, but does not consider other musical cultures or traditions that have different concepts and practices of harmony and tonality. - His book is intended for advanced musicians, composers, and scholars, but does not provide enough guidance or examples for beginners or intermediate learners.


The Conclusion of the Book Review




In conclusion,A Theory of Harmony is a book that presents a new theory of harmony and tonality based on the harmonic series. It introduces a new tool for analyzing harmony, called the harmonic matrix, and a new concept for understanding harmony, called the harmonic dualism. It applies its theory to various musical examples from different periods and styles, and shows how its theory can be used for music analysis, composition, and education. It also challenges some of the common assumptions and dogmas that have dominated music theory for centuries.


but also dense and complex, unorthodox and radical, speculative and subjective. It is a book that is not widely read or accepted by the academic music community, but has inspired many musicians, composers, and scholars to rethink the foundations of music. It is a book that has some limitations and challenges that prevent it from being fully applicable and useful for music analysis, composition, and education, but also offers a rich and diverse source of harmonic possibilities for creating original and innovative musical works.


If you are interested in music theory, you may want to read this book or not. It depends on your level of musical knowledge and skill, your curiosity and openness to new ideas, your preference and taste for different musical styles and genres, and your purpose and goal for learning harmony and tonality. Whatever your decision, I hope this book review has given you an overview of what this book is about, and why it is important.


FAQs




Here are some frequently asked questions about A Theory of Harmony:


Q: Where can I find the book?




A: You can find the book online or in print from various sources. Here are some links to buy or download the book:


  • Amazon



  • SUNY Press



  • Internet Archive



Q: How can I learn more about the book?




A: You can learn more about the book by reading some articles or reviews that discuss or critique the book. Here are some links to some articles or reviews:


  • A Theory of Harmony by Ernst Levy (review) by David Clampitt



  • A Theory of Harmony by Ernst Levy (review) by John Clough



  • A Theory of Harmony by Ernst Levy (review) by David Lewin



  • A Theory of Harmony by Ernst Levy (review) by Robert Morris



  • A Theory of Harmony by Ernst Levy (review) by James Tenney



Q: How can I apply the book to my own music?




A: You can apply the book to your own music by using the harmonic matrix to analyze or compose music in different tonal systems and modes, using different types of chords and intervals, and using inversion, reversal, transposition, permutation, combination, etc. to manipulate and transform chords and intervals. You can also use the harmonic dualism to create symmetrical or reversible harmonic progressions and modulati


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