Music theory: An Introduction to Tonal Theory by Peter Westergaard.
Comparing this book to others is almost unfair, because in a sense, this is the only book on its subject matter that has ever been written. Other books purporting to be on the same topic are really on another, wrong(er) topic that is properly regarded as superseded by this one.
However, it’s definitely worth a few words about what the difference is. The approach of “traditional” texts such as Piston’s Harmony is to come up with a historically-based taxonomy (and a rather awkward one, it must be said) of common musical tropes for the student to memorize. There is hardly so much as an attempt at non-fake explanation, and certainly no understanding of concepts like reductionism or explanatory parsimony. The best analogy I know would be trying to learn a language from a phrasebook instead of a grammar; it’s a GLUT approach to musical structure.
(Why is this approach so popular? Because it doesn’t require much abstract thought, and is easy to give students tests on.)
Not all books that follow this traditional line are quite as bad as Piston, but some are even worse. An example of not-quite-so-bad would be Aldwell and Schachter’s Harmony and Voice Leading; an example of even-worse would be Kotska and Payne’s Tonal Harmony, or pretty much anything you can find in a non-university bookstore (that isn’t a reprint of some centuries-old classic like Fux).
I’ve always found traditional music theory to be useless if not actively damaging (seems to train people in bad thought habits for writing/appreciating music). Can you summarize Westergaard’s approach? I know why the typical methods are bad, but I’m interested in what exactly his alternative is.
Can you summarize Westergaard’s approach? I know why the typical methods are bad, but I’m interested in what exactly his alternative is.
In ITT itself, Westergaard offers the following summary (p.375):
we can generate all the notes of any tonal piece from the pitches of its tonic triad by successive application of a small set of operations, and moreover
the successive stages in the generation process show how we understand the notes of that piece in terms of one another
(This, of course, is very similar to the methodology of theoretical linguistics.)
Westergaard basically considers tonal music to be a complex version of species counterpoint—layers upon layers of it. He inherits from Schenker the idea of systematically reversing the process of “elaboration” to reveal the basic structures underlying a piece (or passage) of music, but goes even further than Schenker in completely explaining away “harmony” as a component of musical structure.
Notes are considered to be elements of lines, not “chords”. They operations by which they are generated within lines are highly intuitive. They essentially reduce to two: step motion, and borrowing from other lines.
A key innovation of Westergaard is to unify pitch-operations and rhythmic operations. Every operation on pitch occurs in the context of an operation on rhythm: segmentation, delay, or anticipation of a timespan. This is arguably implicit in Schenker (and even in species counterpoint itself) but Westergaard makes it explicit and systematic. Hence he arrives at his “theory of tonal rhythm” which is the core of the book (chapters 7-9).
The table of contents, at the level of chapters, should give you an idea of how different Westergaard’s book is from other texts:
Part I. Problems and Assumptions
What are we talking about?
Notes
Lines
Part II. A First Approximation: Species Counterpoint
4. Species counterpoint 5. Simple species 6. Combined species
Part III A little closer to the real thing—a theory of tonal rhythm
7. Notes, beats and measures 8. Phrases, sections, and movements 9. Performance
Appendix: Constructing a pitch system for tonal music
EDIT: 1,2,3 under Part II and Part III should be 4,5,6 and 7,8,9 respectively, which is what I typed. I mostly like the comment formatting system here, but that is one hell of a bug.
You can defeat automatic list formatting if your source code looks like this:
4\. Species counterpoint##
5\. Simple species##
6\. Combined species
except with spaces instead of “#” (to prevent the list items from being wrapped into one paragraph). Edit: If the list items have blank lines between them, the trailing spaces are not necessary.
Interesting, thanks. I don’t know if that sounds right or even useful, but it definitely sounds interesting, I’ll be putting it on my “books to check out” list. I get the impression that it’s very reductionist approach, which is a promising sign.
I’ve found traditional music theory to be useless in the practical sense mainly because of their GLUT nature. You essentially have to look at written music to learn the practical part of it, with reference to the GLUT, which you can do without and is essentially useless because all it does is giving names to different stuff in music theory. They aren’t functional names either, like names that helps with the functionality of the implementations of the theories. Some stuff like rules of the octave are rather practical. The scale degrees have so many different names, totally unnecessary. These things, GLUT, basically have made the subject, music theory, more complex than it really needs to be. Do you really need to know that the 5th degree, is called the 5th degree, dominant, sol, etc.
I can’t really think of any other subject that’s as of a joke as music theory.
I have been using Harmony and Voice Leading for a little while. Is An Introduction to Tonal Theory really that much better?
I’ve always felt that the way they explain concepts is very hand wavy and doesn’t really explain anything and I tend to prefer things to be more mathematical or abstract.
I’ll probably pick this book up on your suggestion.
I have been using Harmony and Voice Leading for a little while. Is An Introduction to Tonal Theory really that much better?
Yes.
Don’t get me wrong, Aldwell and Schachter are about the best you can do while still remaining in the traditional “vocabulary of chords” paradigm. (You can even see how they tried to keep the number of “chords” down to a minimum.) Unfortunately, that paradigm is simply wrong.
Also, Aldwell and Schachter, brilliant musicians though they may be (especially Schachter), lack the deeper intellectual preoccupations that Westergaard possesses in abundance. One should perhaps think of their book as being written for students at Mannes or Julliard, and of Westergaard’s as being written for students at Columbia or Princeton. (There is a certain literal truth to these statements.)
I’ve always felt that the way they explain concepts is very hand wavy and doesn’t really explain anything and I tend to prefer things to be more mathematical or abstract.
“One should perhaps think of their book as being written for students at Mannes or Julliard and of Westergaard’s as being written for students at Columbia or Princeton. (There is a certain literal truth to these statements.)”
As a graduate of Juilliard I am curious about this assertion. Care to elaborate? Not that I personally have ever had much use as a performer for abstract notions about music theory. My experience has been that it gets in the way of actually performing music. Which leads to the question ‘why should this be so’ ? Those of my colleagues who were great adepts at theory were uninspired performers of the music they seemed to understand so well. All head and no heart. But why? I can understand that they are different skill sets, but why should they not be complementary skill sets?
I imagine that on this site, alarm bells may go off as I make an observation from experience, but I do not think that it would be possible to use any sort of methodology or system analysis to determine who is and who is not an inspired performer. Just try figuring out how orchestral auditions are run! Now that is a sloppy business!
Regarding textbooks: have any of you read W.A. Mathieu’s
W.A. Mathieu
Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression (1997) Inner Traditions Intl Ltd. ISBN 0-89281-560-4.
As a graduate of Juilliard I am curious about this assertion. Care to elaborate? Not that I personally have ever had much use as a performer for abstract notions about music theory. My experience has been that it gets in the way of actually performing music. Which leads to the question ‘why should this be so’ ? Those of my colleagues who were great adepts at theory were uninspired performers of the music they seemed to understand so well. All head and no heart. But why? I can understand that they are different skill sets, but why should they not be complementary skill sets?
It’s a complicated question, but the short answer is that what usually passes for “music theory” is the wrong theory. At least, it’s certainly the wrong theory for the purposes of turning people into inspired performers, because as you point out, it doesn’t.
But then, if you’ll forgive my cynicism, that isn’t the purpose of music theory class, any more than the purpose of high-school Spanish class is to teach people Spanish. The purpose of such classes is to provide a test for students that’s easy to grade them on and makes the school look good to outside observers.
(Nor, by the way, do students typically show up at Juilliard for the purpose of turning themselves from uninspired into inspired performers; rather, in order to get there in the first place they already have to be “inspired enough” by the standards of current musical culture, and are there simply for the purposes of networking and career-building.)
But music theory isn’t inherently counterproductive to or useless for becoming a good performer or composer; it’s just that you need a different theory for that. Ultimately, inspired performers are that way because they know certain information that their less-inspired counterparts don’t; to see what this sort of information looks like when written down, see Chapter 9 of Westergaard. (And after reading that chapter, tell me if you still think that knowledge of music theory “gets in the way of actually performing music”.)
Is this text useful for actually learning to write harmony, or does it teach about music theory in a more abstract kind of way?
I’m preparing for an exam for a teaching diploma in a few months’ time, and I need to relearn harmony and counterpoint. (I was okay enough at them a few years ago to get by, but never really mastered them.) Also, I want to learn them for their own sake, it’s just a useful skill to have.
I was planning on getting Lovelock’s textbooks on harmony—they come recommended with the warning that it’s very much harmony-by-the-numbers, but that they teach it systematically. I reckon a healthy skepticism towards his advice would minimize the damage done.
Is this text useful for actually learning to write harmony, or does it teach about music theory in a more abstract kind of way?
It depends on what you mean by “write harmony”. I will say that if “abstract” is a bad word for you, you probably won’t like it. However, that isn’t typically an issue for LW readers.
Here is what Westergaard says in the preface (in the “To the teacher” section):
This book was developed for a first-year two-semester college-level course in tonal theory. (I cover either Chapters 1-5 in the first semester and 6-9 in the second, or, if the students are up to it, Chapters 1-6 in the first semester and 7-9 in the second.) You could, however, also use Part II (Chapters 4-6) separately for a one-semester course in tonally oriented species counterpoint for students who have already had at least one year of traditional harmony. You could also use Part III (Chapters 7-9) separately to introduce more advanced students to the problems of tonal rhythm. While the degree of abstraction may seem higher than that of many music theory textbooks, I have not found it too high for college freshmen. On the contrary, college freshmen are conditioned by their other courses to expect this kind of argument. [N.B.: Westergaard taught at highly elite universities. -k.] The exceptions are those students who can handle relationships between sounds so well intuitively that they resent the labor of having to think through the implications of those relationships.
The best way to know if you’ll like the book would be to take a look at it and see. Failing that, my advice would be as follows: if you want to actually learn how music works, this is the book to read. If you merely want to pass some kind of exam without actually learning how music works in the process, you probably don’t need it.
By abstract, I meant like Schenker (I then saw that you compare Schenker and Westergaard’s approaches elsewhere in the thread). Schenker was pretty adamant that his method was for analysis only, and not a compositional tool. So I was wondering if the book gave an overview of how Westergaard thinks music works, or if it does this and also teaches how to do harmony exercises, perform species counterpoint, and the like.
To break it down into my goals: I have a general goal of learning how music actually works (I’ve got a reasonably good grasp as it is; kinda important to me professionally), hence the interest in music cognition. However, as a specific goal I need to pass this exam!
It certainly looks interesting; it seems a little too expensive for me to get right now, but if I can get a cheap copy or a loan, I’ll look into it.
So I was wondering if the book gave an overview of how Westergaard thinks music works, or if it does this and also teaches how to do harmony exercises, perform species counterpoint, and the like.
Oh, the book certainly contains exercises, and is definitely intended as a practical textbook as opposed to a theoretical treatise (in fact, I actually wish a more comprehensive treatise on Westergaardian theory existed; the book is pretty much the only source). It’s true that Westergaard’s theory itself is descended from Schenker’s, but his expository style is quite different! Part II of the book is basically a species counterpoint course on its own.
What the book doesn’t contain is “harmony” exercises in the traditional sense. (In fact, I think the passage I quoted above might be the only time the word “harmony” occurs in the book!) However, this is not an omission, any more than the failure of chemistry texts to discuss phlogiston is. “Harmony” does not exist in Westergaard’s theory; instead, its explanatory role is filled by other, better concepts (mainly the “borrowing” operation introduced in Section 7.7 -- of which the species rule B3 of Chapter 4 is a “toy” version).
So in place of harmony exercises, it has Westergaardian exercises, which are strictly superior.
It certainly looks interesting; it seems a little too expensive for me to get right now, but if I can get a cheap copy or a loan, I’ll look into it.
If you have access to a university library, there’s a good chance you can find a copy there; at the very least, you should be able to get one through interlibrary loan.
Right. Well to pass this exam, seeing as I’ll be required to perform harmony exercises, I will possibly keep the other approach in mind.
My college library doesn’t have a copy according to the online database; besides I’m actually finished my degree so I can’t borrow stuff from there from next month on anyway. I’ll try convince someone to get it out for me from another college.
Music theory: An Introduction to Tonal Theory by Peter Westergaard.
Comparing this book to others is almost unfair, because in a sense, this is the only book on its subject matter that has ever been written. Other books purporting to be on the same topic are really on another, wrong(er) topic that is properly regarded as superseded by this one.
However, it’s definitely worth a few words about what the difference is. The approach of “traditional” texts such as Piston’s Harmony is to come up with a historically-based taxonomy (and a rather awkward one, it must be said) of common musical tropes for the student to memorize. There is hardly so much as an attempt at non-fake explanation, and certainly no understanding of concepts like reductionism or explanatory parsimony. The best analogy I know would be trying to learn a language from a phrasebook instead of a grammar; it’s a GLUT approach to musical structure.
(Why is this approach so popular? Because it doesn’t require much abstract thought, and is easy to give students tests on.)
Not all books that follow this traditional line are quite as bad as Piston, but some are even worse. An example of not-quite-so-bad would be Aldwell and Schachter’s Harmony and Voice Leading; an example of even-worse would be Kotska and Payne’s Tonal Harmony, or pretty much anything you can find in a non-university bookstore (that isn’t a reprint of some centuries-old classic like Fux).
I’ve always found traditional music theory to be useless if not actively damaging (seems to train people in bad thought habits for writing/appreciating music). Can you summarize Westergaard’s approach? I know why the typical methods are bad, but I’m interested in what exactly his alternative is.
In ITT itself, Westergaard offers the following summary (p.375):
(This, of course, is very similar to the methodology of theoretical linguistics.)
Westergaard basically considers tonal music to be a complex version of species counterpoint—layers upon layers of it. He inherits from Schenker the idea of systematically reversing the process of “elaboration” to reveal the basic structures underlying a piece (or passage) of music, but goes even further than Schenker in completely explaining away “harmony” as a component of musical structure.
Notes are considered to be elements of lines, not “chords”. They operations by which they are generated within lines are highly intuitive. They essentially reduce to two: step motion, and borrowing from other lines.
A key innovation of Westergaard is to unify pitch-operations and rhythmic operations. Every operation on pitch occurs in the context of an operation on rhythm: segmentation, delay, or anticipation of a timespan. This is arguably implicit in Schenker (and even in species counterpoint itself) but Westergaard makes it explicit and systematic. Hence he arrives at his “theory of tonal rhythm” which is the core of the book (chapters 7-9).
The table of contents, at the level of chapters, should give you an idea of how different Westergaard’s book is from other texts:
Part I. Problems and Assumptions
What are we talking about?
Notes
Lines
Part II. A First Approximation: Species Counterpoint
4. Species counterpoint
5. Simple species
6. Combined species
Part III A little closer to the real thing—a theory of tonal rhythm
7. Notes, beats and measures
8. Phrases, sections, and movements
9. Performance
Appendix: Constructing a pitch system for tonal music
EDIT: 1,2,3 under Part II and Part III should be 4,5,6 and 7,8,9 respectively, which is what I typed. I mostly like the comment formatting system here, but that is one hell of a bug.
EDIT2:: fixed.
Thanks for the summary. I may get this book.
You can defeat automatic list formatting if your source code looks like this:
except with spaces instead of “
#” (to prevent the list items from being wrapped into one paragraph). Edit: If the list items have blank lines between them, the trailing spaces are not necessary.(The creator of the Markdown format says “At some point in the future, Markdown may support starting ordered lists at an arbitrary number.”)
Thanks, fixed.
Interesting, thanks. I don’t know if that sounds right or even useful, but it definitely sounds interesting, I’ll be putting it on my “books to check out” list. I get the impression that it’s very reductionist approach, which is a promising sign.
I’ve found traditional music theory to be useless in the practical sense mainly because of their GLUT nature. You essentially have to look at written music to learn the practical part of it, with reference to the GLUT, which you can do without and is essentially useless because all it does is giving names to different stuff in music theory. They aren’t functional names either, like names that helps with the functionality of the implementations of the theories. Some stuff like rules of the octave are rather practical. The scale degrees have so many different names, totally unnecessary. These things, GLUT, basically have made the subject, music theory, more complex than it really needs to be. Do you really need to know that the 5th degree, is called the 5th degree, dominant, sol, etc.
I can’t really think of any other subject that’s as of a joke as music theory.
I have been using Harmony and Voice Leading for a little while. Is An Introduction to Tonal Theory really that much better?
I’ve always felt that the way they explain concepts is very hand wavy and doesn’t really explain anything and I tend to prefer things to be more mathematical or abstract.
I’ll probably pick this book up on your suggestion.
Yes.
Don’t get me wrong, Aldwell and Schachter are about the best you can do while still remaining in the traditional “vocabulary of chords” paradigm. (You can even see how they tried to keep the number of “chords” down to a minimum.) Unfortunately, that paradigm is simply wrong.
Also, Aldwell and Schachter, brilliant musicians though they may be (especially Schachter), lack the deeper intellectual preoccupations that Westergaard possesses in abundance. One should perhaps think of their book as being written for students at Mannes or Julliard, and of Westergaard’s as being written for students at Columbia or Princeton. (There is a certain literal truth to these statements.)
You’ll love ITT.
“One should perhaps think of their book as being written for students at Mannes or Julliard and of Westergaard’s as being written for students at Columbia or Princeton. (There is a certain literal truth to these statements.)”
As a graduate of Juilliard I am curious about this assertion. Care to elaborate? Not that I personally have ever had much use as a performer for abstract notions about music theory. My experience has been that it gets in the way of actually performing music. Which leads to the question ‘why should this be so’ ? Those of my colleagues who were great adepts at theory were uninspired performers of the music they seemed to understand so well. All head and no heart. But why? I can understand that they are different skill sets, but why should they not be complementary skill sets?
I imagine that on this site, alarm bells may go off as I make an observation from experience, but I do not think that it would be possible to use any sort of methodology or system analysis to determine who is and who is not an inspired performer. Just try figuring out how orchestral auditions are run! Now that is a sloppy business!
Regarding textbooks: have any of you read W.A. Mathieu’s
W.A. Mathieu Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression (1997) Inner Traditions Intl Ltd. ISBN 0-89281-560-4.
It’s a complicated question, but the short answer is that what usually passes for “music theory” is the wrong theory. At least, it’s certainly the wrong theory for the purposes of turning people into inspired performers, because as you point out, it doesn’t.
But then, if you’ll forgive my cynicism, that isn’t the purpose of music theory class, any more than the purpose of high-school Spanish class is to teach people Spanish. The purpose of such classes is to provide a test for students that’s easy to grade them on and makes the school look good to outside observers.
(Nor, by the way, do students typically show up at Juilliard for the purpose of turning themselves from uninspired into inspired performers; rather, in order to get there in the first place they already have to be “inspired enough” by the standards of current musical culture, and are there simply for the purposes of networking and career-building.)
But music theory isn’t inherently counterproductive to or useless for becoming a good performer or composer; it’s just that you need a different theory for that. Ultimately, inspired performers are that way because they know certain information that their less-inspired counterparts don’t; to see what this sort of information looks like when written down, see Chapter 9 of Westergaard. (And after reading that chapter, tell me if you still think that knowledge of music theory “gets in the way of actually performing music”.)
For those interested in a scientific perspective, David Huron’s Voice Leading: The Science Behind a Musical Art is really unparalleled.
Is this text useful for actually learning to write harmony, or does it teach about music theory in a more abstract kind of way?
I’m preparing for an exam for a teaching diploma in a few months’ time, and I need to relearn harmony and counterpoint. (I was okay enough at them a few years ago to get by, but never really mastered them.) Also, I want to learn them for their own sake, it’s just a useful skill to have.
I was planning on getting Lovelock’s textbooks on harmony—they come recommended with the warning that it’s very much harmony-by-the-numbers, but that they teach it systematically. I reckon a healthy skepticism towards his advice would minimize the damage done.
It depends on what you mean by “write harmony”. I will say that if “abstract” is a bad word for you, you probably won’t like it. However, that isn’t typically an issue for LW readers.
Here is what Westergaard says in the preface (in the “To the teacher” section):
The best way to know if you’ll like the book would be to take a look at it and see. Failing that, my advice would be as follows: if you want to actually learn how music works, this is the book to read. If you merely want to pass some kind of exam without actually learning how music works in the process, you probably don’t need it.
(Added: I see that you’re interested in reading about music cognition. In that case, you will definitely be interested in Westergaard.)
By abstract, I meant like Schenker (I then saw that you compare Schenker and Westergaard’s approaches elsewhere in the thread). Schenker was pretty adamant that his method was for analysis only, and not a compositional tool. So I was wondering if the book gave an overview of how Westergaard thinks music works, or if it does this and also teaches how to do harmony exercises, perform species counterpoint, and the like.
To break it down into my goals: I have a general goal of learning how music actually works (I’ve got a reasonably good grasp as it is; kinda important to me professionally), hence the interest in music cognition. However, as a specific goal I need to pass this exam!
It certainly looks interesting; it seems a little too expensive for me to get right now, but if I can get a cheap copy or a loan, I’ll look into it.
Cheers for the advice!
Oh, the book certainly contains exercises, and is definitely intended as a practical textbook as opposed to a theoretical treatise (in fact, I actually wish a more comprehensive treatise on Westergaardian theory existed; the book is pretty much the only source). It’s true that Westergaard’s theory itself is descended from Schenker’s, but his expository style is quite different! Part II of the book is basically a species counterpoint course on its own.
What the book doesn’t contain is “harmony” exercises in the traditional sense. (In fact, I think the passage I quoted above might be the only time the word “harmony” occurs in the book!) However, this is not an omission, any more than the failure of chemistry texts to discuss phlogiston is. “Harmony” does not exist in Westergaard’s theory; instead, its explanatory role is filled by other, better concepts (mainly the “borrowing” operation introduced in Section 7.7 -- of which the species rule B3 of Chapter 4 is a “toy” version).
So in place of harmony exercises, it has Westergaardian exercises, which are strictly superior.
If you have access to a university library, there’s a good chance you can find a copy there; at the very least, you should be able to get one through interlibrary loan.
Right. Well to pass this exam, seeing as I’ll be required to perform harmony exercises, I will possibly keep the other approach in mind.
My college library doesn’t have a copy according to the online database; besides I’m actually finished my degree so I can’t borrow stuff from there from next month on anyway. I’ll try convince someone to get it out for me from another college.