Philip Tagg (2014-08-10)
The interval between c and f is a fourth (4) because, starting on c as 1, d counts as 2 (a second), e as 3 (a third) and f as 4.
There are, however, only 3 steps —three intervals— between 1 and 4:  from 1 to 2;  from 2 to 3; and  from 3 to 4. The interval between c and f should therefore more logically be a third (3) because ‘interval’, when applied to music, means difference in pitch and because the difference between 4 and 1 is 4-1 = 3.
Counting confusion continues with the octave (octava = ‘eighth’). That 8th pitch is boundary for the heptatonic or 7-note scale, e.g. c=1, d=2, e=3, f=4, g=5, a=6, b=7. The heptatonic scale then starts again an octave higher.
Take the note c in the second octave as an example: it’s c² (65.4 Hz., 2 octaves below middle c or c4). Counting c² as note nº 1 and ascending one note at a time, c³ is the 8th note reached —it’s the octave above (c³=130.8 Hz). With c as tonic (I, ‘one’), c³ is just as much scale degree 1 as is c².Now, the interval between c² and d² is a second (2nd); so is the interval between c³ and d³. However, the interval between c² and d³ —an octave plus a second— is not a tenth (8+2=10) but the seven notes of the heptatonic octave (see above) plus two, i.e. 7+2=9 —it’s a ninth (9th). Similarly, a tenth (10th) is an interval of the 7-note (not 8-note) octave plus 3, i.e. the third one octave higher.
Tip. If you’re confused by 11ths, 13ths and 15ths, just subtract 7 (the heptatonic octave). An eleventh chord contains the fourth (11-7=4) and a thirteenth chord the sixth (13-7=6). Th twelfth is an interval of one octave plus a fifth (12-7=5) and the fifteenth an interval of two octaves (15-7=8).
As if all that weren’t enough, the octave (8) can also be equal to nine (9). That's because the octave’s three internal complementary interval pairs add up to nine —the second and the seventh (2+7=9), the third and the sixth (3+6=9), and the fourth and fifth (4+5=9). In other words watch out: the octave (8) can ‘equal’ both 7 and 9.
All these inconsistencies are inevitable because we use no zero as starting point for counting intervals and because we mix cardinals with ordinals. We are dealing with the same anomaly as when we count years and centuries. Just as we had no year zero, starting instead in the year 1 —only 99 years in the first century but 100 in all the others— we call an interval of zero (0) steps (no difference of pitch, total absence of interval) a prime (1), as if it were an interval of one step. We then call a 1-step interval a second (2), a 2-step interval a third (3) and so on. There’s no point in trying to bring order into this ingrained confusion but it’s certainly worth bearing in mind.