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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>So why is the set of scales called an octave if it’s all based on a base-7 system?
It’s not a counting system at all. I think that’s why you are getting confused. It’s a way of representing a pattern in how the frequency changes between notes. The Ancient Greeks discovered that if you take a vibrating string and add a node in the middle of it, the pitch changes but it sounds “the same.” What’s really happening here is the frequency doubles. They found that if they used that doubling of frequency as the beginning and end of a scale, that they could choose notes in a specific way between them, and then repeat them at higher and lower pitches and everything would sound good together. They used something called a 12-tone equal temperament scale, which means that there are 12 notes in between the root and the octave, and the frequency ratio between any two neighboring notes is the same. They found that to be the most natural way of dividing it up that led to repeatable patterns. They also decided that using 7 different pitches (including the root note) made the nicest patterns, if they followed certain rules about how the notes were spaced. Since there are seven “different” notes, you need seven letters to represent them (this came later, I am not sure how the Greeks labeled notes). If you play a scale without playing the last note, the one where the pitch doubles compared to the root note, it doesn’t sound good. So, a complete scale includes the root note, the octave, and the six other notes in between them. Since that’s 8 notes, we call it an octave.
If they used 8 letters to name the notes, then the system wouldn’t be as useful. As an example, let’s say you had A-H as your numbering scheme. Your scale in the first octave would be:
ABCDEFGH
then in the second, it would be:
HBCDEFG
That’s not a pattern that is easy to repeat. By choosing 7 named notes and repeating it starting a the 8th note, it matches up with the way that scales are defined in Western music.
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