Shouldn't technically all intervals have five names?
(Maybe with few exceptions where it goes below zero semitones but still I can't see why it can't go below zero semitone when it can be played)
With these five possible cases:
- ♯Note-♭Note ( n semitones )
- ♯Note-Note or Note-♭Note (n+1 semitones )
- Note-Note ( n+2 semitones )
- Note-Note♯ or ♭Note-Note ( n+3 semitones )
- ♭Note-♯Note ( n+4 semitones )