For example: considering chord inversions, the harmony in general, the scales and the intervals and also for example the notes on the fretboard of the guitar we can have different permutations, some calculus and mathematics could clarify and improve the understanding of all the hypotheses and possibilities from a certain point of view. Can you fundamentally define some of these concepts mathematically? Is combinatorics a field with application in music?
You certainly can consider permutations in music. There's a branch of music theory called neo-Riemannian theory, and it looks at what we call "parsimonious voice leading." ("Parsimonious" basically means "most efficient.")
Let's say we have a C major triad that moves to an A minor triad. Here's one way:
G E (down three semitones) E C (down four semitones) C A (down three semitones)
Between these two root-position triads, then, we have a net movement of 10 semitones (!). Hardly parsimonious. So let's move to an A minor triad in first inversion instead!
G E (down three semitones) E A (down seven semitones, or up five) C C (no movement!)
This is actually equivalent in a sense; if the E moves down to A, we again have a net movement of 10 semitones. Even if the E moves up, we're looking at 8 total semitones traversed.
So here's where the permutations come into play, because if we clarify the first-inversion A minor triad as
C E A, we get:
G A (up two semitones) E E (no movement!) C C (no movement!)
Here, only one voice is moving, and it by just two semitones!
If you're really interested in the article discussing this concept, you can access it here.
There's another recent article here which is a tutorial on combinatorics in music and discusses other ways that this can be helpful.
Studying both maths and music at university level, I have asked myself very similar questions - in fact, when I was looking for a topic for my bachelor thesis in maths I was actively searching for literature going in this direction.
Long story short, I didn't find anything useful.
Obviously, since music is not entirely random, you will have some patterns emerging, and as large parts of mathematics (eg Algebra) were invented to describe patterns, you will find some applications for mathematics. And there's a good amount of books/articles doing exactly that. You can quite easily define interval jumps as operations on a set of 12 semitones, be proud that you have yourself what is called a group mathematically, and then you go on to a lot of nice little theorems about what you would usually do, and pack them into neat little "theorems". Thing is, from what I've read so far, that doesn't have much to do with maths per se, but more with what you could call common musical sense. Just described unnecessarily complicated with some basic math notation.
If, for instance, you have a progression from one chord to another, both of which share a note, well, it's not that far-fetched to leave that note the same and change the other ones. Of course, if you wish, you could also define some kind of functional on the set of chords, which you then try to minimize, and only then discover that you should not randomly jump around on your keyboard - but I don't quite see the point there. Using mathematics does not make this any simpler. Just noticing, that is actually precisely what Richard mentions in his reply - so, as stated, yes, there is this kind of theory, I just don't deem it particularly useful.
Same goes with permutations in the sense of how many/what combinations there are. It's not that hard to count, say, the number of melodies you can create using a given amount of notes, rhythmical values, articulations, dynamics and so on. But, unless you go to hardcore serial music, where those patterns were sometimes really just mechanically applied, this doesn't really help you, musically speaking. (Okay, there's a bit more to serial music than that. But it would be the rough general idea.) Obviously, you can also always just count the number of possibilities for something without corporating it into a piece, but, honestly, what's the musical point there? Ignoring for a moment that you might be just curious. Of course, you could count the number of possible finger combinations (within physical limitations) on your guitar, practise every one of them, then practise all connections between all possibilities, and if you can do this at any speed then, theoretically, you could play all pieces. But you will find that there's many books with technical exercises who do the same based on experience, having already sorted out with chords you need frequently, or almost never.
Since you mentioned calculus specifically, I'd like to add that calculus is something where, spontaneously, I couldn't think of any application to music, or only if you take the detour via physics. Calculus by definition works with limits, mostly via differentiation and integration, and since the amount of notes is mostly assumed to be discrete, as is rhythm, calculus seems very out of place. Sure, you could build it into a composition, but that would likely be something very modern-sounding, and be a piece where you use calculus for the sake of using calculus, and not for the sake of resulting music.
If you ask whether you can "fundament some concepts mathematically", well, that is actually a bit tricky. In a way, yes, you can. Thing is, in order to fundament anything mathematically, what you need first are some ground-rules, some stuff so basic that you just define it to be true. Those are called axioms - as an example, one axiom used for creating the natural numbers (1,2,3,...) is "Every natural number has exactly one successor." If you do not assume that to be true, then there's no more sense talking about natural numbers, since this statement is integral to the very concept of what a number is. Now, you'd need the same things for music - problem is, what would you use for them?
Of course, you could go and use some basics about, say, chord progressions. Eg, if you have one chord succeeding another, and some notes occur in both of them, you will not jump around but just let them go on. As is usual in most of classical music - just, the problem is, there are exceptions. And a mathematical theory inherently assumes that all your axioms are always true. (In case you have heard of them, let me leave any incompleteness theorems out of here. They exist, but don't play into things at the level we are at.)
So then you'd need to put your exceptions on the same level of the axioms you already had - since else they would be a direct contradiction to those - in the sense of "Statement A is only true if case B does not apply", which is kind of ugly, considering they are be the most basic, fundamental fragments your music-theory construct is built on. And if your axioms already tell you in detail everything you need to know about music theory, well, then there's not much point in building theory around that, is there?
Also, while you could still do that, is it really much different from general rules about music theory? Or, to be more precise, do you gain any additional advantage? Since a lot of theory is built on "we do this because it tends to sound good", and a lot of what we find to sound good is dependant on what we are used to hearing, I'm not sure trying to find some general underlying mathematical rules is the right way to go. If you really wanted to do that, I'd suppose you'd need to go more into the psychology of hearing music, what we tend to find enjoyable, and why. But that's not something I'd be very familiar with.
TLDR; you can use maths to describe some of what's going on in music. But that's, in my opinion, not because of any inherent likeness of music to maths, or any cleverly hidden logic in music, but because a large part of maths was created to describe stuff, and I don't find it particularly helpful.
Richard wrote the answer I would prefer to read. It seems we have here a discussion going on about Mathematics and Music, arguments about if the connection between this two areas of knowledge is beneficial for Music.
I always heard the saying: "Music is Math". I also read about the applications of Fibonacci Sequences in Harmony (if I am not wrong), and again I was curious. I also think permutations can help to write a better "Hanon" for the guitar, an "Hanon" involves experience but might also seek the complete freedom of playing musical ideas in the instrument not conditioned by what is more natural to play on the fretboard of the instrument. Even simple concepts like "even and odd" might seem useful in guitar picking. So I find it could be useful to apply some mathematical concepts in Music in many ways, from the simplest to the more complicated too.
I am a beginner, true, but I want to be aware of any "keys that open" in Music. Can you imagine how is the sound of "randomness"? Or can we "draw" musically spirals in the harmonic movement? Is there an imaginary connection to geometry? I know there is a book by a Monk entitled Symmetrical Harmony... is not symmetry an concept from geometry? If all of us followed the same paths Music it would be boring, but Music is not strictly boring if we try to apply some Mathmatics, if we have it in the "pallette" or in the "bag of composing techniques". Mathematics does not "kill" emotion, I believe so.
So that`s why I have such basic questions about permutations and Mathematical application in Music. If Calculus is the study of "change and variations", a limit could express even a string bending or sliding of the fretboard, or an effect in the wham bar if we where modelling. This is not particularly Music but in another situations concepts can be closer, and to finish: calculus involves arithmetic, do not you use some arithmetic in music?
(sorry for any grammar error or incorrect spelling)
One of the things that confuses mathematicians when they go to learn music is that the numbers used to describe a lot of things in music are NOT being used as a mathematician would use them. For example, a note exactly seven note names higher than another note is called an octave. Since octave means eight, this makes no sense to the mathematician. The same principle applies to all other intervals.
This basic difference between the two realms would have to be taken into consideration in ANY attempt to relate the two in a sensible way.