Here's walkthrough of a calculation of the highest number of chord qualities that you could consider qualities (more on that later). Mathematically, there are a bunch of combinations of intervals that produce chords. There are 12 notes available for the 1st note in the chord, then 11 for the 2nd, then 10, etc. A note any multiple of 12 semitones (an octave) above or below another is the same note and doesn't change chord quality.
12! chord qualities.
However, note that this number is the number of chords that take all 12 chromatic notes. To find all the chords with any number of notes from 1 to 12 inclusive, we need to add in the number of chords with fewer notes. However, once you have that number, you must be careful of enharmonicity. Basically, that's like looking at every note, then considering each note the root, and checking for duplicates. Also, some combinations of intervals form an inversion of another different chord. That's the reason why there's only one chord quality for a 12 note chord (which makes sense, when you think about it).
So, in reality, here's what I can get from intuition (I'm not great at math).
- 1 12 note chord (like we said)
- 1 11 note chord (all 12 notes minus one note at any interval to the 1st, but reordering all of them makes them all the same)
- 1 one note chord (but don't count that, generally a note is not a chord)
- 11 two note chords (any note plus any other)
- 110 3 note chords (any note plus any of 11 others plus any of 10 others)
With lots of notes, checking that none of the notes can be considered the root of a different chord quality is extremely difficult. But mathematically, there are hundreds at least. Let me also note that depending on how you limit your horizons of chord qualities, the number of notes is extremely variable. If you consider only diatonic triads to major scales, there are only 3 qualities.