Gradus ad Parnassum

In my previous post, I wrote briefly about a historically important pedagogical work in music theory entitled Gradus ad Parnassum, or in English, The Steps of Parnassus, (Parnassus being the mythical home of the Muses). This work first appeared in 1725 and has never fallen out of use, being used by composers and teachers ranging from Mozart to Brahms to Richard Strauss. The author was Joseph Fux, a major composer of the late Baroque period. Although his fame as a composer has not survived as well as some of his contemporaries, J.S Bach and Handel being obvious examples, he was highly admired in his own time and occupied during his career the most prestigious musical post in the world: Music Director of the Viennese Court. I would like to speak in more detail about the Gradus because I feel there is much that we can learn from Fux’s approach that can be applied to the teaching of mathematics and physics. It may be surprising to some that a parallel could be drawn between mathematics and music pedagogy, but the actual art of composition is more systematic that some would think. It is part art and part science, much like mathematics itself.

In any math problem of a particular level of sophistication, it is the general aim of the student to arrive at the correct answer.  In other words, there is some unknown value or function (in the case of a differential equation) and the object of the game is to derive an expression for the unknown. This is most systematic level of mathematics. For example, given a Hookean spring with particular characteristics attached to a mass with certain initial conditions, a differential equation can be written from which the motion of the spring can be obtained. Beyond obtaining that function, there is little that can be drawn from that problem. One step above in sophistication would be to study the problem more generally, studying the system while directly specifying as little as possible. This would enter the region of mathematical proof (as in, to prove that the period of a Hookean spring does not depend on the amplitude of the oscillation). It has been my experience that this kind of mathematics has something in it akin to art: we take certain facts and try to arrange them, or follow their implications in such a way that a whole new fundamental truth is revealed, not merely the specific truth or a particular system with particular values. In music it is somewhat similar, to arrange sounds and pitches in a new and meaningful way. The difference is that in mathematics, the aim is consistency, which is objective; in music, the aim is aesthetic appeal, which is partially subjective. Music theory is the system of rules and conventions that attempt to formalize ‘correctness’ in composition. Unlike in mathematics, these rules are not unbreakable but rather “for the guidance of the wise and the obedience of fools”. By this I mean that good composers break these rules when they desire a particular effect, while bad composers break them because they lack the skill to follow them.  Turning now to the Gradus ad Parnassum, Fux outlines his pedagogical philosophy in his introduction:

There have certainly been many authors famous for their teaching and competence, who have left and abundance of works on the theory of music; but on the practice of writing music they have said very little, and this little is not easily understood. Generally, they have been content to give a few examples, and never have they felt the need of inventing a simple method by which the novice can progress gradually, ascending step by step to attain mastery in this art. I shall not be deterred by the most ardent haters of school, nor by the corruptness of the times … Seeking a solution to this problem, I began, therefore, many years ago, to work out a method similar to that by which children learn first letters, then syllables, then combinations of syllables, and finally how to read and write. And it has not been in vain. When I used this method in teaching I observed that the pupils made amazing progress within a short time. So I thought I might render a service to the art if I published it for the benefit of young students, and shared with the musical world the experience of nearly thirty years, during which I served three emperors (in which I may in all modesty take pride).

The work itself is written as an ongoing dialogue between a pupil Josephus and the teacher Aloysius (representing the Renaissance master Giovanni Pierluigi da Palestrina). Aloysius begins with a review of consonances and dissonances (which he assumes the student to already know) and then proceeds to introduce the three kinds of motion: direct, contrary, and oblique. He concludes the introduction with the four fundamental rules of counterpoint:

1.     From one perfect consonance to another perfect consonance, one must proceed in contrary or oblique motion.

2.     From a perfect consonance to an imperfect consonance, one may proceed in any of the three motions.

3.     From an imperfect consonance to a perfect consonance, one must proceed in contrary or oblique motion.

4.     From one imperfect consonance to another imperfect consonance one may proceed in any of the three motions.

These rules form basically all the information that the reader (living vicariously through Josephus) needs for the first lesson. Aloysius presents the student with a melody (or cantus firmus) and has him harmonize it in simple note-on-note counterpoint. This is the musical equivalent to a simple word problem. The fictional student presents his answer and explains to the teacher how he arrived at it. However, the student’s answer is not perfect and Josephus corrects it. Some of the mistakes are from breaking one of the Four Rules. Some mistakes are subtler, such as not setting the counterpoint in the same mode as the cantus firmus.

When I read this example, I thought at the time that it seemed somewhat sloppy, throwing information at the student after the fact. However, speaking for myself, the phrase “set the counterpoint in the same mode as the cantus firmus” was easier for to understand from the mistaken example than from the rule itself. The student’s work then becomes an instructional example, with the teacher demonstrating what he could do to avoid the error. This is the general pattern of the book. Very small amounts of information are introduced at a time, which allows for a steady but never overwhelming increase in complexity. The work progresses while remaining in two-part harmony, through the first, second, and third species of counterpoint, ligatures, and florid counterpoint. Having exhausted this type of composition, Fux then goes back to the beginning and does it all over again in three-part and four-part harmony.

When I read this book, barely more than 100 pages, I wonder what specifically it is that made it so effective. What struck me first was how old fashioned the compositional style being taught was. The Baroque period was coming to a close at the time of writing. Renaissance polyphony, which had been essentially replaced with Baroque counterpoint, was quickly giving way to the Rococo style. What Fux was intending with this book was to solidify the old way of doing things, not because he felt that people ought to write that way, but because he saw that the young composers were ignorant of the historical foundations of their art and were becoming reckless and unbounded by any standards of taste. He says this explicitly in his introduction where he states that that his purposes “do not tend – nor do I credit myself with the strength – to stem the course of a torrent rushing precipitously beyond its bounds. I do not believe that I can call back composers from the unrestrained insanity of their writing to normal standards.” Music pedagogy then, was either too theoretical or heavy with information, or it was too undisciplined, prompting a great deal of activity yet resulting in an overall decline of quality and technical mastery.

I wonder sometimes if these two errors are still present to some extent in mathematics (and physics) pedagogy. I’m always struck when I tutor high school students what tomes their textbooks are, and how difficult it is to find what I’m looking for in them. This would be an example of too much information, particularly when the underlying concept is very simple. It frustrates me enormously whenever I work with a first-year university student, and find that after two years of high school physics they still can’t do any kind of projectile motion problem without flipping back and forth through the textbook, or define for me Newton’s 2^nd Law or the Work-Energy Theorem (and this happens a lot). The textbook almost functions for them as a book of spells instead of an instructional tool. The other extreme is when creativity and discovery is emphasized at the expense of formal instruction, in the expectation that students through exploration will spontaneously discover the principles they need. (Re, “We Don’t Want No Education” by Theodore Dalrymple). I expect that in some circumstances this may work very well, but I wonder at the same time if it can allow serious misconceptions slip past the teacher. Also, I wonder if making a subject appear more subjective that it really is can inadvertently encourage in students a relativist epistemology. Whatever the virtues of the present age, I don’t believe due respect to Principle of Non-Contradiction is one of them.

I believe that part of the brilliance of Fux’s text is the skill with which he takes the middle road. In modern parlance, it is very ‘student-centered’ in that the student in the dialogue does the bulk of the work, and information is introduced very gradually. There is no deluge of information but rather, knowledge of music theory is grown naturally in the student through the use of exercises. At the same time, Fux does not beat about the bush with respect to the student’s ignorance of the subject. He presents his rules and methods as law and not as a matter of debate. He sometimes explains his rationale to the student and sometimes refuses to, if he feels the explanation is premature or would confuse the student more. At one humorous point in the dialogue, Aloysius advises Josephus not to get over-confident and to remember his place:

I want to remind you again and again to make every effort to overcome the great difficulties of the study you have undertaken; and neither to become discouraged by hard work; not to allow yourself to be deterred from unflagging industry by flattery of such skill as you have already achieved. If you will work thus you will be delighted to see the way in which light gradually illuminates had been obscure and how in some manner the curtain of darkness seems to be drawn away.

To conclude, part of me wonders what a math or physics instruction book would look like written in this manner, and if it would even be possible. Being a dialogue, it is not particularly useful as a reference, but I think this format makes it especially effective as an instruction tool. It is possible to read a textbook and come away thinking that you’ve learned something: perhaps you have, and perhaps you haven’t. However, it would be just about impossible to read Gradus ad Parnassum and convince yourself that you’ve learned something if you haven’t also done the exercises. Yet history shows us that someone who diligently works his way through the entire book could gain an enormous amount of skill and understanding. As such, Fux’s work is not so much a textbook in the modern sense but an actual pedagogical program. I won’t speculate on how effectively this method could be transplanted from music theory to mathematics, but as someone who has had to learn both, the natural similarities between these two subjects make me believe that anyone who wants to learn how to present mathematics in an intuitive guise and a sensible order may glean many useful ideas from this work.