The musical scales are A B C D E F G A, or Do Re Mi Fa So La Ti Do. If you do it that way, yes, there are eight. But the last note is the same as the first but at a higher pitch. If this were math, we’d basically be dealing with a base-7 system, with A or Do playing the part of zero.
So why is the set of scales called an octave if it’s all based on a base-7 system?
EDIT: Many of the first answers from when I originally asked were helpful but now I’m getting a lot of wrong answers from people who don’t seem to understand how numbers work. In a base-8 system, there are eight unique numbers, 0 through 7, after which it goes to 10. If you translate the notes into numbers, you don’t get 0 through 7, though. You get 0 through 6, after which it goes to 10 at the second Do. That’s why I was trying to reconcile that with the term “octave.”
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If you measure the distance around a circular track and label the starting line zero meters, you must return to the starting line and count that point again to measure the distance travelled. We don’t call it a 399 meter track.
If you call the starting line “one” and stop when you are at 400, the other racers will pass you as you stand one meter away from the finish line. It doesn’t mean we should call the race the “401 meter”.
Music can be analogous to running around a circular track as the starting line and finish line may be considered to be unique, yet may be the same point and be required to be counted twice.
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