Adding to the other replies, there's an off by one error in the interval names.
The unison is an interval of zero notes, the second an interval of one note, etc...
The unison is an interval of zero notes, the second an interval of one note, etc...
What I find interesting is that we double-count the end notes in a diatonic scale, but we don't double-count them in a chromatic scale. No one says there are 13 notes per octave in a chromatic scale, it's 12.
... well, up to the thirteenth, at least... I suppose one could continue to name diatonic intervals above that for really convoluted chords.
Without the off-by one error, one could do interval arithmetics in a way that intuitively makes sense... It's really weird such a bad convention caught on to begin with.
Edit: going down the rabbit hole... It looks like the off-by-one error for diatonic intervals goes back to the greek antiquity. The Romans named the fourth and fifth diatessaon and diapente, which have greek roots, but their chromatic intervals start a zero (tonus, ditonus, tritonus which gave the (semi-)tone and the tritone). The "semi" prefix means "subtract half a tone", not divide by two... https://en.wikipedia.org/wiki/Interval_(music)#Latin_nomencl...
Neither the Greeks nor the Romans had a concept of zero, so it starts to make sense as to why we started with a crooked convention. Inertia and habbit made it stick around.
There's another rabbit hole for microtonal interval names... https://en.wikipedia.org/wiki/List_of_pitch_intervals#List