My first thought is that there are loops in musical notation, but there is no way to write conditional branches, right?
I'm not a musician, so perhaps someone can help fill in the gaps?
Yes, if you admit a few instructions for transposition—uncommon but not unknown.
You can then interpret a piece as Choon, which is Turing-complete. The performer is the memory: they must remember the number of notes by which the piece is currently transposed, and all of the notes they have played thus far. Obviously it’s feasible only for a computer, or perhaps a savant.
From the Choon manual:
A John Cage is just a rest, a Tuning Fork is (roughly) dal segno, and a marker is a segno. I suppose the tuning fork could be played by an additional performer to whom the primary performer responds, but the principle is the same.
Turing completeness requires, at a minimum, three things: an infinite loop, a conditional jump (if-then), and a way to store the results of calculations to somewhere in memory. Even if musical notation had conditional jumps, it doesn't have state, so no, it's not Turing-complete.
The standard proof for a language to be Turing complete is to write a Turing machine in that language. This proves that there's an equivalence between the language (usually a subset of the language) and the Turing machine.
The notion of "Musical Notation" is a bit slippery. There is a lot of standardized engraving that is used. However. There are envelope-pushing composers who write all kinds of crazy stuff down on paper.
Let's pretend you want to focus on the subset of musical notation that is considered standard enough to be part of Finale or Sibelius or some main-stream engraving toolset.
For Python (or C or whatever) you define the symbols, the tape, the transition rules, and the various actions which update the tape to reflect state change and tape motion, reading and writing symbols on the tape.
Using "Musical Notation", we have to define symbols and the stateful tape, the transition rules and the various actions which update the tape.
What we lack is a stateful tape and rules that tell the musicians how how to responds to symbols on the tape and how to update that tape.
In a sense, the noises flowing around in the air might be the stateful tape. But. There's no easy way to rewind the tape. This lack of rewind means that the performer would have to keep a private "tape" of some kind.
This gets outside musical notation and into some other extra-musical instructions to the performer.
Much of the notation is open to interpretation, and natural language instructions are an accepted aspect of musical notation -- and have been throughout most if not all of the history of Western notated music.
Fermatas by definition depend on the performer's discretion, which means that it would depend on their own state, which is almost always altered by the music in conjunction to external factors -- so this raises some questions on the stateless nature of musical notation.
Canon a 2 per Tonus from Bach's Musical Offering is an infinitely looped piece whose tonality rises by a whole step each time through for as long as the piece is performed.
More recently, it's common to see instructions such as "repeat for each soloist" in, for instance, notated versions of Jazz pieces such as Dave Brubeck's Take Five.
That said, aside from inherently arbitrary aspects like the fermata, as the other answers state, musical notation with nothing but the general symbols is probably not Turing complete.
It is not related to Turing complete languages as it is a descriptive language. There are no commands in terms of calculation or modifying data, no states, no input, no output except for the result of the description itself.
Also there are no conditional jumps depending on the input. When you resolve all the jumps you get a linear structure, not a tree. So all "programs" which can be modelled by this language are linear without any loops or jumps at all.