Two pitches played together is a dyad, three together is a triad. There may be words for four or more pitches together but I just call them chords. The term interval only makes sense to describe distance between two elements, whether pitches or two marks on a ruler.

Monad Diad Triad Tetrad Pentad Sextad Septad Octad Nontad Dectad Monodectad Didectad/Bidectad (?)

Or,

1 Note/Unison

2 Interval/Diad

>3 Chord

And, I agree an interval is essentially a distance. Distance between three points makes no sense as they might very well lay outside of one straight line. Even they are on the same line.. are we measuring the distance between each distance?

It's ambiguous what that might even mean, but the original poster might think of a collection of intervals which is 0 or more notes with intervals relative to a given root.

For example if you think in integers (pitch set notation):

  m6   { 0 3 7 9 } Minor 6
  5    { 0 7 }     Power Chord
  ma13 { 0 2 4 5 7 9 11 } Ionian
  N.C  { } No Chord/Rest
  °7   { 0 3 6 9 11 } Diminished Seventh/Dim Seven

  .. etc

Also note that 13 means two different things, either the septad above or a pentad of the form 1 3 5 7 13 aka 7(6) "dominant add six"

So in set notation it's:

  ma13 { 0 4 7 9 11 }
  13   { 0 4 7 9 10 }

  Etc..