Some Questions that the Music Mathematicians Forget to Ask

11 August, 2006
There are some people out there investigating the mathematics of music using quite sophisticated concepts from group theory and category theory. But is this mathematics really telling us anything interesting about what music is? I highlight some questions that the music mathematicians forget to ask.

Week 234

In week234 of "This Week's Finds in Mathematical Physics", John Baez discusses and links to various ideas about the mathematics of music.

Many deep and powerful mathematical concepts are discussed, including symmetry groups, centralizers, torsors and Mathieu groups.

However, a few rather basic questions about music are completely missing from the discussion.

The Mathematics of Music or the Science of Music?

Music can be regarded as a natural phenomenon: it is something that happens in the real world. Thus the study of music, like the study of anything else in the real world, is part of science. Which means that the mathematics of music must be connected to the science of music.

If physics is science, and the study of music can be science, then we might expect some analogy between mathematical physics and the mathematics of music, and we might expect an expert on mathematical physics such as John Baez to shed some light on which are the most interesting topics in "music mathematics".

However, I think there are some important differences between physics and music science, which lay traps for the unwary.

For example, we can compare the mathematics of Quantum Gravity and the mathematics of music. Solving the "Quantum gravity problem" is one of the holy grails of theoretical physics. However, a curious thing about quantum gravity is that almost everything we know about the real world can be ignored when studying it. This is because almost everything known about the real world that matters for the purpose of studying quantum gravity is contained in the following two theories:

And each of these theories can be expressed as a very small set of precise mathematical equations. Anything else that we know about reality, for the purpose of studying quantum gravity, is subsumed by these two theories. So "reality", to the student of quantum gravity, is just a set of equations.

Unfortunately this doesn't work for music. There are some aspects of our knowledge of music which are very mathematical, but there are many aspects which aren't, and there are many things about the world which might or might not be relevant to the scientific study of music, and which (so far) have not been described by precise mathematical equations. For example:

If one is inclined to study the profound mathematical nature of music, one might decide to ignore all this "messy" stuff, and only look at the mathematical aspects of music represented in musical notation. But, like the legendary drunk man looking for his keys only where it is well lit, or perhaps even like a string theorist, one's intellectual efforts could become somewhat disconnected from reality, or at least unconstrained by the nitty-gritty details of reality. One could construct mathematical objects representing the known mathematical features of music, and construct new constructions based on those constructions, and calculate abstract characteristics of those constructions, and so on, and none of those constructions would be of any use in helping to solve the basic mystery (i.e. what is music?).

My suggestion to those studing music mathematically, some of who know much more mathematics than I ever will, is to think a little bit about the science. And they could start by considering the following "missing" questions.

The Missing Questions

First Missing Question: What is Music?

To me this seems like a fairly basic question, indeed I would say it is the question. When we talk about music, we are talking about something which we don't really understand what it is. However, I do not detect any reference to this question, be it direct or implied, in John Baez's discussion.

Second Missing Question: How Does Music Lead to More Grandchildren?

Music is something that people make and respond to, and people are living organisms, so the study of music is ultimately part of biology. And everything in biology must be explained with reference to evolution by natural selection. If music diverts significant resources to its creation and consumption, then there must be some major reason why this cost has not selected against the genes of those people who appreciate music more than other people.

And there is barely a hint of this issue in any of the mathematical analyses of music – at least those that I have been able to find on the Internet. The scientists who do consider natural selection generally commit the opposite sin – they ignore all the specific details of music, and they try to explain music as something that is beneficial because people make music and other people listen to music and this somehow leads to better "social bonding" or something. The problem is that these theories typically say nothing about why music should have the specific features that it has, because their logic is completely independent of the actual content of music.

Third Missing Question: What Kind of Set is the Set of Musical Items?

When I first heard about "Musical Set Theory", I thought it was going to be about a set that only music belonged to (and non-music didn't belong to). But this turned out not to be the case.

So, what sort of set is the set of musical items? How sparse is it (if we consider it embedded in the set of all possible sounds)? Is it connected (in a topological sense)? Is it a fuzzy set? (Which leads to the next "missing" question.)

Fourth Missing Question: What is the Dimensionality of Musicality?

The definition of "musicality" that I am referring to here is the measurement of how "good" or "strong" an item of music is. This definition recognises the fact that some music affects us more than other music. It implies that musicality lies on a one-dimensional scale, with the best music at one end, not-so-good music towards the middle, and non-musical sounds right at the other end. Is this assumption of one-dimensional musicality valid? Or should musicality be represented by some more complex mathematical object?

The concept of "musicality" is almost non-existent in most theoretical discussions of music. Many theoreticians seem unwilling even to consider "non-music" as something that contrasts with music, just in case someone accuses them of not being open-minded enough to treat John Cage's infamous 4'33'' as "music".

Fifth Missing Question: What are the Symmetries of Music?

The discussion of musical groups in John Baez's article leads on to discussion of symmetries. But for the most part, the symmetries that he discusses are symmetries of the components from which music is composed, within modern Western music, such as sets of three note chords on the chromatic scale. They are not the symmetries of music itself.

This is one question that has a very strong precedent in mathematical physics, where the symmetries of the Lagrangian determine conservation laws as a consequence of Noether's theorem. So it should seem natural to ask what the basic symmetries of music are, and to ask if there is any common principle that applies to one or more of these symmetries. (For an example of which see my grand unified theory of pitch-translation invariance and time-scaling invariance.)

When you start to think about the symmetries of music as symmetries of music perception, then you realise that their existence has profound implications for how the brain processes musical information. Whatever invariances there are in the perception of musicality, those invariances must be intrinsic to the mechanisms that the brain uses to "calculate" said musicality. And if the invariant calculations are non-trivial to implement, then they impose a cost on the organism (i.e. ourselves), so there must be some good reason why they exist and they must serve some important biological purpose.

Sixth Missing Question: Why Is Music Constructed From Scales?

There are many mathematical theories and hypotheses about the construction of musical scales, and why the diatonic scale has been adopted as the "best" scale, and what all the relationships are between notes and chords constructed on various scales. Yet almost all these theories ignore a more basic question, which is why do we have scales at all? Could musical melodies have been constructed without any use of fixed sets of pitch values? And if not, why not?

Seventh Missing Question: Does Any Music Theory Make Predictions?

One can invent systems of musical "analysis", and use them to analyse various items of music. But if an analysis system doesn't result in some means of making predictions, at least in principle, then we have to ask whether that system has any scientific meaning at all. Is much of musical "theory" actually a form of pseudo-science? Academic music theorists "analyse" music, and publish their analyses in music theory journals and music theory books, but never in such a manner that their analyses can lead to testable predictions.

The implication is that music "theory" is still very "pre-scientific". We know that Freudian psychoanalysis lasted for decades as a supposedly "scientific" enterprise, without anyone bothering to ask the hard questions. Does something similar happen in the ivory towers of musical academia?

Following The References

It's not just John Baez who ignores or avoids these "missing" questions. They are also conspicuously absent from the various websites and articles that he references.

In particular Thomas Fiore's site has slides and a draft paper, none of which give any hint of asking the questions I have raised here. To give an idea of the "pre-scientific-ness" of Fiore's analyses, the following is his explanation (from page 3 of the first set of slides) of what "music theory" is:

In other words music theory provides us with a good way of hearing a work of music.
No mention of biology, or falsifiable predictions, or wanting to know what music actually is.