How is Middle C generally defined and located on any instrument?
After reading into some music theory, it seems that a lot of the definitions depend on specific octaves i.e. Treble Clef on G4 and Middle C on C4.
Now these definitions seem self-explanatory on an 88-key piano because it's where it is. But how can a musician know where Middle C is on any other instrument or even vocally? What is the actual definition of numbered octaves?
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But how can a musician know where Middle C is on any other instrument...?
When you learn an instrument, this is one of the things you learn. Actually, it may be two of the things you learn. It's two things (for some instruments) because you normally want to know what note to play on your instrument when you read middle C in notation, and if you have a transposing instrument, you also normally want to know what note to play on your instrument to create the sound of middle C.
On transposing instruments, what you play when you read middle C on the staff and what note sounds like middle C are two different things.
You might be someone who was taught piano at an early age, so it seems obvious to you. I say that because you wrote, "...these definitions seem self-explanatory on an 88-key piano because it's where it is." I don't think middle C on a piano is self-explanatory at all! For example, one of the first things I was shown when I was taught piano is that the C under the brand/logo for the piano is called "middle C" and looks like this on the page (the ledger line between the treble and bass clefs on the grand staff). That's also something I have taught piano students. Otherwise, how would you know?
A similar teaching process happens for all instruments. Now perhaps middle C itself is not taught on the first day of other instruments, but "home base" notes will be taught on day one. The open strings of a guitar or violin will be matched with the sheet music. And simple finger positions (e.g., all valves open) and their notated notes on wind and brass instruments will be taught very early on. From there, students will sooner or later play up or down the range of their instrument to middle C when it's notated.
Middle C is C4, 9 semitones below A4. A4 is commonly defined as 440Hz, so C4 is 2(−9/12) × 440 Hz, about 261.625565302 Hz. That's it, regardless of instrument.
Some clefs are transposing and this is not always clearly indicated by small numbers attached to the clef (tenors tend to sound an octave lower than standard violin clef would indicate, soprano recorders an octave higher, other transposing instruments have fixed offsets).
But regardless of how it is written, C4 is the same pitch for all instruments.
To address the "what is the actual definition of numbered octaves?" question:
Referring to pitches via note-names with octave numbers, like C4, is scientific pitch notation. Leaving aside some complications related to pitch standards and temperament, these indicate specific note frequencies (physics) and thus pitches (psychoacoustics), as indicated in the table in the linked Wiki page.
You've been asking about history on other answers, so I wanted to address this.
The first historical answer that could be given would be "they didn't care." If you're just playing solo, you often don't need to line up "perfectly" with an particular pitch. Instruments could vary accordingly. You'd simply make the instrument play one note that's "roughly" in the right place, and then tune the rest of the instrument from there.
This process is great until you have to have two instruments play together. When they have to play together, they have to agree on which pitches they're going to play. One approach to this is to have every pitch variable. Many old stringed instruments could retune strings or even move frets.
Now, to take the step towards "middle C," you have to play this way for a bunch of years. Over time musicians will start to realize that it's cheaper and easier to play together if all of the instruments "agree" on a pitch. Musical instrument makers started standardizing their instruments to suit. It might be a local standard, or a worldwide size and shape. All that mattered was that they agreed enough.
Eventually we standardized on a tonal system which centered around the pitch of the note we now call A4. It's currently specified to be 440Hz (it's varied through the years a bit). We can then define A5 A6, etc. by doubling that frequency (880Hz 1.7KHz respectively). Any instrument that wants to play well with others will be judged by how well it matches those reference frequencies. Of course they didn't phrase it that way at first. However, there are many shapes of resonators which can be constructed to a fixed size and thus resonate at a given frequency. The numeric frequencies came later -- reproducible shapes and sizes came first.
Now days, even with the ubiquitous standard of A4=440Hz, we still tune the same way they did in the days of yore. Due to atmospheric and temperature changes, instruments change tuning all the time. In an orchestra, everyone adjusts their tuning to that of the oboe because the oboe has the least room to adjust their pitch (due to the design of the instrument). The one exception is if there is a piano on stage, in which case the piano is even harder to re-tune than the oboe, so everyone tunes to the piano instead).
Middle C is the note with frequency 262Hz (or thereabouts, we don't need to split hairs here). It's MIDI note #60 . It's usually called C4, there is also a system that calls it C3.
A side issue - some instruments are 'Transposing instruments'. They read a middle C, the pitch that sounds is really something else. For a Bb Trumpet it's Bb. Are we talking about the true middle C or that instrument's middle C? Whatever you feel about which is 'correct', sometimes it will need to be made clear.
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