Given for example a triad how many interpretations can there be? Does the number of interpretations depend on the notes or is there a general rule? How does this change when we add another note? etc.
--edit-- Sorry for the very loose term "interpretations". I was thinking both as a function of a key and more along the lines of C,E,G is Cmaj but then again perhaps it could be seen as Emin with a flat 6 and no 5th.
I mostly know common-practice theory, and only the broad strokes of jazz and rock harmony. Be advised that the interpretation of a C major chord as potentially an Em6 is not generally accepted or discussed in classical terms. However, a move from a root position chord to a first inversion chord with a root a third away by moving the fifth up a step is quite common. For example, a C major chord (C–E–G) often becomes an "a minor chord" in first inversion (C–E–A), especially as the setup to a 7–6 sequence. The move is often called 5–6 (named from the typical figured bass symbols), and many common-practice theorists, especially Schenkerians, do NOT consider the second chord to truly be a different harmony than the first. If my example happened in the key of C major, they generally wouldn't label them as two chords, I–vi6, but just as a prolongation of I, I5–6. Although the terminology is different, that actually isn't all that different from jazz and pop ideas about 6 chords, wherein E–G–C might indeed still be interpreted, in a sense, as an Em harmony.
At any rate, more specifically responding to your question, every triad in first inversion could theoretically have a second "interpretation" as a 6 chord in root position. The second inversion is even shiftier: in most classical situations the second inversion of any triad—due to the dissonant interval of a fourth above the bass—functions very differently from its root-position counterpart. For example, in the key of C major, a C major triad in second inversion could function as a passing chord (IV–I6/4–IV6), a pedal or neighboring chord (V–I6/4–V) or as a dissonant embellishment of V called a Cadential 6/4 (V6/4–5/3). Every diatonic chord could function as a passing or pedal 6/4, but only the "I" chord can be Cadential.
So, in the sense that changing inversion changes a chord's interpretation, every diatonic harmony has at least 4 interpretations. It can be itself, a 6th chord in the jazz sense, a passing 6/4 or a pedal 6/4. The I chord has a fifth standard interpretation as a Cadential 6/4. I suppose for completeness sake we could also say that every diatonic harmony has one more possible interpretation as a mere link in a sequential chain.
The other half of your question is about functions within a key. Aside from the subtler shifts wrought by inversion discussed above, most diatonic and chromatic chords within a single key have only one standard function. V chords tend to always behave like dominant-function harmonies, ii6 tends to always be pre-dominant, etc. The most common exception among diatonic harmonies is vi, which can often have weak pre-dominant function—tending to move to a stronger pre-dom like IV or ii—or tonic function as the goal of a deceptive cadence (V–vi). iii is a protean harmony (and not very common in major keys) that occasionally has weak dominant or tonic function depending on context. As part of a circle of fifths sequence it can even be a weak pre-dominant.
Otherwise, chords tend to only have one standard functional interpretation (again, ignoring the complexities discussed in my second paragraph) so long as you stay within a single key. When modulating however, it is quite common to use two different interpretations of some pivot chord to effect the transition. For example, if I'm in the key of C major, a C major triad is my I chord. If I suddenly start treating it like a IV chord in the key of G major, I can make a sneaky modulation into the new key. Re-interpretation of the triad is precisely how I'm able to make the smooth shift.
This question is a little bit strange because there are no hard and fast rules to naming chords and that's going to come up a lot in my answer. Nonetheless:
A chord in isolation has infinite possible interpretations, all of which are equally valid:
The reason all names are equally valid is because we don't know what the relationship of the notes in the chord being named is to the other notes. While C E G pretty much always sounds like C major, sometimes it doesn't; sometimes what's happening around it makes it sound like something else. What really restricts you, then, is function. Chords with a function -- in other words, chords that are part of a progression -- generally have only one correct name. As soon as a chord is ascribed a function, factors #1 and #3 mentioned above no longer become relevant - there will only be one correct letter name for the chord (since its function is related to its assigned letter relative to the tonic) and there will subsequently only be one way to spell it.
Let's say you play C G7 C F. This is a fairly simplistic progression. Looking specifically at the second C here, you can ascertain pretty easily that any attempt to spell that chord differently is wrong: Calling it Emib6no5/C implies that it's anything other than the tonic chord. Calling it Dbb is wrong because the next chord is F, and therefore this spelling (which implies a key change albeit to a key that is functionally identical to the first key) does not exhibit the functional relationship between C and F.
The second factor continues to exist, however, and some chords really will have multiple names. Note that this is pretty rare for triads and other chords built off of thirds, and things like calling something Ami7b5 versus Cmi6 is not an example of this; context dictates which answer is correct or not (Ami7b5 D7b9 Gmi, not Cmi6 D7b9 Gmi, even if the Ami7b5 for some reason has a C root). Chords that legitimately do have multiple valid names tend to be built out of a single repeated interval. Augmented triads, particularly successive ones, can sometimes take multiple correct labels. Again, however, it's rare: An augmented chord that takes or implies a specific variety of 7th (like a 7#5) often has only one real correct name because, while the augmented triad stays the same when the root changes to another note that's in the chord, a 7#5 does not. You might expect that diminished chords would also be beholden to this, but they're generally not, since diminished chords are often dominant-functioning and therefore need to take the letter name of a dominant-functioning chord. The very obscure double diminished chord probably bucks this since it, by design, is completely non-functional, but this chord is so rare that it isn't really worth mentioning.
Often in those cases logic simply carries the day. And again, there isn't really anything stopping you from being wrong since theory isn't codified and there is a certain degree of subjectivity and idiomatic variance to function (See the many names for Ab C Eb Gb/F# in the key of C minor), so technically the answer to this question, even in the context of function, is "infinity". But if you were to say "how many justifiable names are there for a chord in a progression", the answer is nearly always one and, when it's not, it's nearly always the number of notes in the chord.
I don't think looking at chords without conext is very useful.
What is the harmonic motion of the music in context? Where does it seem to resolve and feel final? THat's typically the music's root. So, C-E-G appears in a variety of contexts and can be interpreted according to the context. Odds are it will be a C Major chord 99% of the time.
If an eminor-sharp 5 chord fits in the context of the 'James Bond Theme' progression:
E-G-B (Em) E-G-C (Em#5) E-G-C# (Em6) E-G-C E-G-B
Then that's NOT a C Major effect in context. It has resolution by that 1/2 step rise and fall of the e minor 5th to 6th sequence. You can apply this chromatic motion technique to the 3rd or b7th in the minor chord as well. The odd chords created up and down are just tension and resolution effects.
