As @user45266 says 'tetrad' etc are logical choices, but they aren't commonly used.
Common practice classical music on the whole just used triad, seventh chords, and to some degree ninth chords. In terms of quantity concepts like arity or sets I think the traditional musical terms is voice or part as in four part harmony, etc. Bach's famous two and three part inventions are an example of the usage. I think it would be perfectly acceptable to say something like 'four voice chord' or 'five part chord.' The meaning should be clear.
However, traditionally a large number of parts - like five part harmony - would often involve octave doubling of tones. This loops us back around to common practice harmony. The triad really is the 'default' chord is some seventh chords and fewer ninth chords. If you want to un-ambiguously convey n-voices of unique pitch classes, four-voice, five-voice, etc. will not be absolutely clear.
You may be interested in these mathy music terms: tertian for chords built of thirds, and quartal and quintal for chords built of 4ths and 5ths.
Tetrachord is a common term, but paradoxically it typically refers to a scale-like set of tones rather than a tertian chord.
Pitch set is a musical term and it has all the quantifiers you are looking for, the only potential down side to that term is its association with atonal music. Not that association with atonal music is a negative (that's a joke) but the only musicians who are likely to be familiar with set theory terminology are those with a lot of academic training.