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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Numerical bases aren’t really the right way to think about pitch. Assigning pitches to counting numbers implies that they are increasing linearly in some fundamental measure (they’re not) and makes it clunky or impossible to talk about the cyclicality of the pitches and their ratios. All of this comes from the fact that sound is produced by waves, and the math of waves has lots of cycles and ratios.
This means, assuming we’re sticking to some standard western scale rather than a chromatic scale or some non-western pitch system, we “count” the notes: 1 2 3 4 5 6 7 1 2 3 4 5 6 7… etc. We *could* start counting at 0, but who realistically ever starts counting at 0? The first note is… the first note. This counting system gives us an easy way to refer to the relationship between the first note and any subsequent note. Playing the first and third note is “a third”. Playing the first and fifth note is “a fifth” and so on.
Logically, this means that playing the first note along with the eighth note (which is just the first note again) could be called “an eighth,” but we use a fancy latinized term for “eighth” instead: octave. So there’s a good reason to use something related to “eight” but not necessarily a great reason to use that particular term – Enlightenment-era Europeans just love them some Latin. It has actually worked out nicely for American musicians, who refer to a half-beat rhythmic unit as an “eighth note”. It would be quite confusing if “eighth” referred to both a common tonal idea and a common rhythmic idea.
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