So, when playing the piano, going up a ‘third’ means going up two notes and going up a ‘fifth’ is going up only four notes. Put together, the whole transition is a ‘seventh’, giving us 3 + 5 = 7. Counting the dividers and not the intervals means that the note between the transitions is double-counted. It is also why an ‘octave’ of seven notes (and seven intervals) is named ‘oct’ for eight. The upside is that I can blame my terrible lack of musical ability on the numbers not behaving normally.
