What is Music, exactly?
In one direction it's an encoding of data inherent in a temporally sequenced, explicitly non-accidental series of sound waves of varying frequency, wavelength, waveform, duration and loudness striking the sound-wave-modulating apparatus ("ears") of ambulatory sacks of carbon-based matter ("people").
From another direction it's the vibration/oscillation of chunks of matter at and in the hands of those sacks of matter at least partially responsible for the inclusion of "non-accidental" in the preceding paragraph, ie "music-makers".
To keep it simple let's focus on painting / drawing: an encoding of data inherent in the non-accidental arrangement of molecules of matter upon a two-dimensional surface, such that light striking the surface is reflected into the photon-modulating apparatus ("eyes") of sacks of carbon-based matter, which apprehend the inherent encoding (look at the artwork) and explicitly catalyse paradigmatic information-content (and find meaning in it), yadda yadda yadda
In one sense it's any ensemble of frequency and duration over time of episodes of local proximity involving two sacks of carbon based matter; ie it's an ensemble that maximises the probability of additional, though smaller, sacks of carbon based matter "popping into" existence at time t in the future --- when a particular satellite has revolved around a particular ball of rock approximately 9 times.
In another sense it's simply the preceding paragraph minus the smaller sacks.
Non-dark energy processed into handy packets to maximise the convenience of the consumer, in this case all machines designed to thwart the second law of thermodynamics (SLOT), ie anti-slot machines.
The point of all of the above is to highlight that there's no place in a materialist/reductionist universe for emergent qualities such as "meaning", "purpose", "consciousness", "life", "love" even "quality" itself (not to mention "qualia", [you don't want to go there, trust me]). In other words, the materialist/reductionist universe cannot accommodate the multitude of warm and fuzzy sins that hide under the blanket of "encoding of data", per the discussion of music and visual arts at the start of this post. In other words, "encoding of data" is a perfect disguise for the sins of subjectivity, interpretation, consciousness, life, meaning, purpose and many others)
I believe that language is subject to "Godelian" incompleteness as much as are formal systems such as arithmetic or geometry. To mangle a complex set of thoughts and ideas: Godelian incompleteness refers to the idea that proofs of a system's completeness can only be found in the larger system in which the sub-system is embedded. In the case of Euclidean geometry, for instance, the axioms are "given", ie, cannot be shown within Euclidean geometry itself to be true (hence the incompleteness of the system known as Euclidean geometry). The problem is that everything that is not an axiom, depends on at least one axiom for its proof. So if the axioms cannot be proven, then neither can the truth or otherwise of all geometrical theorems, propositions and statements that depend on the axioms for their validity.
Re the formal system, Language, the metric or gauge for measuring the truths of that system is: meaning (whatever that is). To prove the truths of language (establish the true meaning of a word or language statement) within Language itself, is not possible. One has to step outside of Language to apprehend the "true" meaning of words or language statements. Using words to define the meaning of words is circular and therefore ultimately non-productive.
Every language statement, to a greater or lesser extent, is a metaphor. Language has no direct connection to bedrock Reality, whatever that is.