Math concerns theorems, which are logically provable. A "theory" in math might be something like Category Theory or Set Theory, areas of study which then contain theorems.

I've never heard of a "theory of music" as in a different kind of theory besides the math or science kinds. I've heard of music theory, but that's more like the scientific theory about music, although I guess it's not quite that. Perhaps more the philosophical theory of music? (Philosophical theories are yet again different from mathematical or scientific ones).

I don't think the word "Theory" applied to music is used in quite the same way as in other uses. It is a sort of "right way" that musicians learn, but musical artists generally find ways to "break" theory rules once they understand them; so theory was never a true set of rules in the first place, especially considering the relative nature of music theory (there's Western music theory, and then there's Indian. I'm not sure the way African music is organized is considered a "theory"). Because music is a form of expression, whatever theories exist for it's organization tend to just be traditions, which are always evolving (or devolving).

Music theory is essentially aesthetic theory; what pleases the ear. There are more aesthetic and less aesthetic ways to form chord shapes, for instance, but even that is relative to a culture. For instance, in Western music you generally don't allow melodic lines to bisect each other; it doesn't sound that aesthetic to western ears, but that's not to say a way can't be found to make it sound aesthetically pleasing. Dunno if any of that helps; just off the top of my head.

Thanks for your thoughts! I'm also interested in the idea of a musical theory as something that's made to be broken, and even going beyond that, of theory as something that can be generated at the same time as the thing it's describing. For instance, Messiaen (a famous composer) invented "modes of limited transposition" as he was first using them. Do you think that relates to any mathematical or scientific use of the word "theory"?

If there is any sort of universal theory in arts then it's possibly coming from a neuro-aesthetics or evolutionary psychology pov. Currently any sort of theory would be a sort of folk science based on what worked in the past.
I think art theory would link more into phenomenology. So if we're going into positivism then it would be Newton VS Goethe on a theory of color. I'm thinking more of Color theory here as I know little about music.

Messiaen’s modes are a closed system, to my understanding; each one can only be transposed a very limited number of ways, in contrast to the seven traditional modes, which can be transposed to all 12 tones as key centers. Because transposition of Messiaen’s modes is so limited, they basically eliminate key centers, which was something that was already being developed before he did so. So yes, he certainly broke rules, but math is key to music theory; Messiaen still set up a new set of rules. Im not sure what you mean by a theory being generated at the same time as the thing it’s describing. Love Messiaen btw! Funny thing, I’ve actually been slowly trying to learn the modes myself. Not easy!

Does anyone know any good resources which talk about what a theory of music is in connection to what a theory in math is or a theory in science is? — Halley

No. But Aristotle included a Theory of Music in his Metaphysics, based on the science of his day. Yet, I doubt that we will make much headway in explaining the "power" of music in terms of Newtonian Physics --- maybe Quantum Physics??? Ari used the metaphysical concept of a Soul in his rationale. Music is definitely mathematical, as indicated by Pythagoras, but its aesthetic power seems to be due in some sense to a "harmony" between the mathematical structure of the world, and the logical structure of the Soul (Consciousness). Is there a theory of Harmonics between Matter & Mind?

I've recently seen a variety of videos on YouTube, showing animals, large & small, responding emotionally to music. What's going on in the "savage breast" might be related to the math of physical vibrations, but beyond that, empirical science seems to be at a loss to explain the aesthetics of oscillations --- without introducing some metaphysical notions. :smile:

Theory of Music : https://revije.ff.uni-lj.si/MuzikoloskiZbornik/article/view/5437

What I meant was that the idea of a "mode of limited transposition" didn't exist as something worth naming until exactly the moment when it was used by Messiaen (at least that's my understanding).

I don't doubt that there's math in music, but I think the process by which math becomes musically relevant is fascinating and potentially understudied (or I just haven't found the right resources).

If you want to explore musical theory you could do worse than exploring western music and its focus on 12 semitones in an octave, Why? Bach explores this is his 'well tempered clavicar'. Before then musical tones were simple fractions of the main tone 1/2, 1/3, 1/4 etc. I won't spoil your fun by telling you the answer. But I will hint that it relates to the 12th root of 2

Actually I've published academic papers in computational musicology, I'm very familiar with the math of music, rather a newcomer to philosophy and wondering what y'all have to add to the conversation - but I agree it's a good puzzle!

lol :)

I don't doubt that there's math in music, but I think the process by which math becomes musically relevant is fascinating and potentially understudied (or I just haven't found the right resources). — Halley

It's somewhat beside the point, but many mathematicians seem to have musical talents as well.

You could try looking up Pythagoras and music. You'll read about harmonic series and such. That's all I really know. Sorry couldn't be more helpful.

Not to be pedantic, but I would imagine it was in development for quite awhile, and then he eventually published that book outlining it. But since you were wondering about a philosophical bent, I would argue that the concept of the limited modes existed for as long as it existed in his head before he published the treatise (obviously in an evolving state, conceptually). So I'm still not grasping the idea of it coming into existence at the exact time that it...came into existence? Maybe I'm missing something.

Otherwise, as far as a more philosophical angle on this, I'm not well versed in math or science to add that angle; all I can point to is what I mentioned earlier; that music theory is essentially aesthetic theory. So if music theory is the technical information for musicians, then aesthetic theory is that information applied to philosophy.

EDIT: ...or rather, if music theory is the technical information for musicians, then aesthetic theory is the same information but viewed from a remote perspective, from which philosophical claims can be made. But it helps to know the information from the technical perspective first before removing oneself to a perspective from which the information is seen as remote.

What I meant was that the idea of a "mode of limited transposition" didn't exist as something worth naming until exactly the moment when it was used by Messiaen (at least that's my understanding).

I don't doubt that there's math in music, but I think the process by which math becomes musically relevant is fascinating and potentially understudied (or I just haven't found the right resources). — Halley

It's... difficult. I'm not a philosopher actually, but music theory has always reminded me somewhat of linguistic theory (something I know a little better). What they have in common is that we have lived systems we learn, and theories about how they work that feed into learning.

So basically, you have this theory and it can be used in different ways:

1. Describe what people are doing
2. Use the theory as a method of learning (which impacts what people are doing and creates a feedback loop)
3. Scour the theory for logical possibilities and see what people could be doing but aren't.
4. Use it as a set of rules to judge the value of the product

So there's a question in what ways a descriptive music theory is useful. For example, the following video discusses basically what theoretical framework is useful when describing the harmonics of "Sweet Home Alabama": Tonal Harmony? Modal Harmony? Or something else (with Tagg's tonic-outgoing-medial-incoming loop as an example)?



The problem here is that what sort of description you find most useful often depends on what you hear, and what you hear may be influenced by the theory you've been exposed to.

Similarly, you can write a piece of music with a particular theory in mind. And then you can make an adjustment because it sounds nice, but it's not entirely clear what you've done in terms of theory. Now, someone else comes along and doesn't like the adjustment you made. That person could easily try to use the obvious non-conformity as a legitimisation for why it doesn't work, but that would involve dismissing that it works for others, and dismissing the question of why it works for others as unimportant (since they're wrong).

So you have the same theory as a set of regularity interpreted as (a) a description of what's going on, (b) a generative aid, or provisory template, and (c) a system for judgement of good and bad. I see that all the time in both composition theory and linguistics.

Scientific theories for behaviour-external fields such as physics don't have (b) and (c). If you see an apple floating up into the sky you won't punish it for violating the laws of gravity; you'll try to figure out what's going on. And gravity, unlike language or music, isn't something you meaningfully participate in. (You do participate in gravity, obviously, to the extent that you're a physical object).

I'm not familiar with Messiaen. It seems to me, he's been scouring the formal properties of the theories for things that are logically possible, some of which have been done before (the wholetone scale is identical to his first mode as far as I can tell?), and some maybe haven't. You could, in theory, compose musics with a random scale (random variables: number of notes (1 -12), location of notes on the chromatic 12-tone scale). Even then, there a still elements we're taking for granted (for example the primacy of single notes a half-step apart), and they're ingrained in culture (for example in the form of instruments). That's not a problem if all you want is a generative method you can deviate from at pleasure, or if you want judgement standards. As a descriptive set of rules, though, you might encounter music you can't describe or will "misunderstand" if you use that particualr theory, and that's something you need to be aware of.

Not an expert in anything, but these are my thoughts about the topic.

Does anyone know any good resources which talk about what a theory of music is in connection to what a theory in math is or a theory in science is? Bonus points if it references logical positivism or formal first-order logic! — Halley

As an artist I would counter the idea that there can be a theory of aesthetics. I still don't know what art is. It is a mysterious thing. It happens but I don't know how. The poet Bukowski said he writes by a kind of magic. I think any serious attempt to squeeze art into a theory would be counter productive.

I'd imagine that the topology of 'good' music as it is formally represented, is a disconnected archipelago of irregular islands, each with many holes, quite unlike the densely connected forest of logic theorems.

Presumably, a composer is like a drunkard who awakens on one of these islands with no idea how he got there.

Superb post!