Brief Note: Axiomatic Approach to Theory

Let me be clear.  Music theory is an insular academic discipline, but things are getting better as the topics get more rigorously codified and generally accepted in academia.  But no one knows better than us music theorists the extent of the problem that persists in the traditional paradigm.  Imprecise language causes disagreements among theorists over the most rudimentary topics. At a more advanced level, the lack of an axiomatic approach to analysis leaves students and professionals to the judgement of their own opinions.

And opinions are dangerous.  Dangerous because they are not the result of rigorous proof, which can only be achieved through a set of axiomatic principles from which all propositions can be ultimately forged.  This is what music theorists should think about in today’s day and age.

I have.  For a year and a half.  And I am only a little closer than I was when I began.  I cannot divulge the details of the project.  But for those of you who have read my notes before, there can make no mistake.  The approach is probably – somehow – related to contour.

4 thoughts on “Brief Note: Axiomatic Approach to Theory

  1. Having read your comment on axiomatic theory I have lots of questions running around in my brain, which I will list as follows:
    1. What exactly do you think is the purpose of music theory? Which problems is it supposed to solve and what sort of questions to answer?
    2. By “axiomatic” I assume you mean based on axioms, the way mathematicians work. The problem will be to formulate axioms which everybody–at least a strong majority–will agree with (otherwise what is the point?) and which will yet be precise enough to allow reasoning to take off from them and solve the above mentioned problems. I don’t expect a complete solution to this problem, but what is your idea on where to hunt for axioms?
    3. You hint that “contour” has something to do with your solution. I am surprised to see that you have chosen a term originating in the field of visual, not aural perception–a contour can be created by taking a lamp and having it throw the shadow of some object on a sheet of paper, then taking a pencil and following the outline of the shadow. Why choose the term and how would the implied metaphor work?
    4. Music theory as currently practiced is multidisciplinary. There is music theory in the narrow sense: harmony, counterpoint, all the stuff a composer needs to have a deep knowledge of. There is the underlying physics (physics is NOT axiomatic by the way, it is empiric). There is music history, there is music sociology, the list is much longer. What sort of music theory do you think can be “axiomated”?
    A final thought: Personally I have more joy with practicing music than with theorizing. I use my knowledge of theory–limited as it is–to help me find a way to better understand, better play and more deeply enjoy a composition I am working on (I am an amateur violinist) or listening to. Music to me has the deepest connection with our existence as humans–animals with culture–of all the arts. It resonates with our bodies (literally in the inner ear and also literally in the case of musicians, especially singers); rhythm is the way almost all our physiological processes work, from the firing of nerve cells to the heartbeat, digestion all the way to hormonal and generational cycles, fast and slow, from presto to largo if you will. Music comes from deep inside our bodies; I think this is the reason it makes us feel a community when we play or sing–because deep inside we are all alike. At the same time it requires brains as well. Classical music in particular also connects us with its tradition–in current practice almost 400 years of it and growing. To be human to me is to make music–and thinking about it is decidedly secondary.

  2. Thank you for the comments, and I will address your points from my perspective.

    1. Music theory has a lot of denotations and connotations that are derived from the way it is taught and understood. The most important thing to understand about the discipline is that music theory tends to change at a slower rate than music. While many people believe that music theory defines a set of rules for people to follow, the reality is that theorists simply observe trends in music. The most familiar type of theory taught at the undergraduate level and in the AP program deals with the observations made in music during the Baroque and Classical time period. The ‘rules’ for harmonic progressions, voice leading, and the basic phrase are all derived from common practice.

    So from this perspective, the purpose of music theory is to create a language that can describe music. Because certain commonalities can be found in music from the earlier time periods, music is very similar (although not the same) as language. It has a certain metaphorical syntax and grammar.

    But as we move into later time periods, the common practice expanded. What was once rooted in diatonic melo-harmony became rooted in chromaticism. Try using what is taught in AP class to analyze Strauss or Delius. So the new goal of music theorists had to be to find new commonalities to create a new (or expanded) language to describe what was happening. And the evolution continued… And continues.

    So the essence of music theory is really to create a language that people can use to describe music. The problem is that the language used to describe music is only loosely defined and often ambiguous, which can lead to mis-communication in the theory community.

    After some elaboration, the purpose of music theory is to create a language to describe music. Without the discipline, most harmonic ideas in music could not really be described. That is the ‘problem’ that music theory seeks to fix.

    2. I cannot go into much detail about where to find the axioms, but it seems to me that all music can be described in terms of contour. And axioms can be derived from that belief.

    3. It is not so much an implied metaphor: contour describes the “shape” of a musical phrase. This can be mathematically deduced, and you graph all kinds of relationships in music that create the “shape”. So my method is really loosely geometrically based… Which, by the way, also happens to be axiomatic.

    4. As described, I believe that music theory as a discipline of syntactical analysis (if we are to describe the interconnection of notes on paper that represent a collection of pitches through time as syntax) can be an axiomized.

    Let me know if you have questions or refutations. I bet you will.

  3. I certainly believe this is not only possible but vital for this sort of work to be, for several reasons which I won’t go into here.

    I do believe very possible to develop such a axiomatic system for music, one that is stronger even than that used in mathematics, and have in fact produced one based not on contour but the related concept of musical gesture for my Master’s dissertation.

    This seems to work pretty robustly and I have found it to have many pedagogical applications. More importantly, it can serve as a useful meta-theory for composition, something which most music theory proper is quite useless for (beyond typographical and terminological concerns).

    Most crucially it is amenable to empirical testing in a deep sense at almost every level and my next project is to do a concert or series of concerts at which such testing can be done with a live audience with music specially composed for this purpose.

    In closing, all I can say is I think you are on the right track and I, for one, would be very interested to see the results (and compare them to my own). But be aware that there are many who prefer music to be meaningless and mysterious and thus want theory to be purely descriptive. The ghost of Schenker will haunt us for a while yet.

    1. Hello, Doug. I can’t agree it’s possible to develop an axiomatic system for music that is stronger than one used in mathematics. An axiom is an assumption, and an axiomatic system is the set of theorems that are logically based on the assumptions. The assumptions in mathematics can hardly be questioned, but the assumptions in music are clearly open for debate. Even the most basic questions, like what sounds good, is subject to culture and aesthetic preference. But the theorems would have equal rigor to those of mathematics because logic is logic.

      I am not sure what you mean by the ghost of Schenker. Schenkerian analysis is a highly individualistic system that is used to express the way one person experiences a piece of music. In that light, I think it is more psychological than anything else.

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