A fish may have more scales... ... but... you can't tune-a-fish. *
Lloccmttocs, it's a good exercise to think about how many major scales there are. Just like thinking about how many discreet augmented triads exist , and how many fully-diminished tetra-chords exist.
One can make an argument for the number of major scales on any number of grounds, even just semantics. Whether the number is right or wrong - who cares. The real question is: how useful is the information? This is what might get lost in a bad theory class or discussion:
Music theory is a collective methodology meant to serve the hearing of music.
It's not like "real science", which attempts to describe nature by a set of immutable laws. Nor is it a legal code where a technicality can trump common sense.
With that in mind, let's refer to Bach's Well-Tempered Clavier. He wrote two books, each with 24 preludes and fugues. The 24 is because he writes one in each major key and one in each minor key. 12 + 12 = ......
Now, how you feel about Bach is your business. Just 'cuz he does it one way doesn't mean you gotta bend over and take it from him. His way is not the only right way. But his way is amazing. And most of Western performers and musicologists agree with that, and have based their entire studies on the foundations of his tonal thinking. And if you want to understand them, study him. Other notable composers who have written etudes in groups of 12, out of homage to Bach:
debussy
liszt
chopin op10, op.15
scriabin op 8
Also imagine what other composers would say about the topic:
The 2nd viennese school virutually removes all enharmonic significance. Forte notation describes pitch classes with 12 numerals.
Messain, deb, ravel, and the french spectralists would not remotely consider there to be 15 major scales in 15 major keys.
Scriabin would say there are 12, and each one has a color.
Cage would hold up 12 car keys, jingle them, and say, "Listen!"
Ligeti would say there are 11.9999 major scales.
Anyway - want to understand the significance of the major scale as it's understood in western music? Start with Bach's WTC - it's more valuable than any arguement on 12 vs. 15.
PS:
Temperments cannot be ignored, but usually play a more local role, as in: where to place the third of a chord. Temperments are not generally useful when talking about large scale harmonic organization. At least, according to bruckner, mahler, or wagner.
Don't argue all this with your band teacher - it's likely not to elucidate the matter, and it will be harder to skip class later on. If he says 15, just write 15 on the quiz.
btw, the *functional* answer is 12.
If one wants to count enharmonic scales on a technicality, that's fine with me. But the number 15 makes no sense to me. It has to be unbounded if you want to play fair. From the key of G#, if you modulate to the V, you're in D#. Modulate again to the V and you're in A#, again to the V you're in E#, B#, F##, C## - so where does it stop? (based on the premise that, due to tempermants, "E# is not the same note as F-natural", etc..)
Oh wait, ok. Let me say one more thing. One plausible reason for notating something in F-double-sharp Major (a "silly" key signature), would be to clue the performer into the idea that the composer is writing that passage as the secondary dominant (fifth) of the fifth of E#. (E# -> B# -> F##) This is entirely reasonable if you were in the development/variation/B-section of a piece in G# Major (a SERIOUS key signature
). The F-double-sharp implies to the performer the context of the scale. For all intents and purposes, in western music practiced by bach/beethoven/mozart/brahms/etc/etc/etc/etc, this is the exact same scale as G Major. The context (for example, where it is leading: building tension vs releasing tension, etc), and thus the _expressive_ value might be different, but it's the same scale.
*Trombipulation
12 vs 15? I know this guy, JS Bach, and he says... 
