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So a perfect fifth on B natural is F♯, an augmented 5th on B natural is F♯♯, would that then mean that an augmented 5th on B♯♯ would be F♯♯♯♯?

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Technically yes, but you would almost never see B♯♯ as B♯♯ is an enharmonic equivalent to C♯ which makes much more senses in most cases.

Likewise I've never seen more than 3 accidentals applied to a note so a quadrupled sharped F you would never see. Going back to C♯, the equivalent interval would be G♯♯ or Gx better known as A.

So yes B♯♯ to F♯♯♯♯ is an augmented 5th, but you would most likely see it written as C♯ to Gx for everyone's sake.

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Yes, as Dom said, it would be F#### indeed, despite how cumbersome that seems. The reason why is because if instead you wrote G## or A, it would no longer be a fifth (of any type). It would instead become either a 6th or 7th, because of the choice of note letter.

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Whilst technically there may be a key called B##, there would never be any reason for it. There's not even a good reason for the key of B#, which from a writing and reading perspective has no credibility over the enharmonic C. Here, we're talking of intervals rather than keys, but B## will almost never exist, so an augmented 5th on it would be rare too.

Sensibility would make a B## to be called C#,(or even Db) making the aug 5 Gx,(or even A) thus reasonably writable and readable for all concerned. Apart from that, there is no sign for triple or quadruple sharp, as there has been no perceptible need.

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  • actually in music from the 1800's to now the flat and sharp has been extended to double sharps and double flats and more recently triple sharps and triple flats as part of standard accidental notation. B# is there for a good reason. It would not make sense to have a key signature with one of the sharps being C, so they use B# instead, same goes for Cb, Fb, and E# as well as all double sharps. So C## Major does exist and it is D major written in terms of C. This might be done for several reasons. – Caters Jan 17 '15 at 5:14
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    @caters Double and triple accidentals had a surge in popularity in the early 20th century, but now most musicians see them as impractical and unnecessary. C## Major does not exist. – jjmusicnotes Jan 17 '15 at 5:58
  • @caters ever heard of C Lydian? It's got one sharp with C as a tonic. Also why would every want your tonic represented by a different letter name instead of the natural one? – Dom Jan 17 '15 at 6:26
  • basicmusictheory.com/c-double-sharp-major-scale This proves my point that the C## major scale actually exists. – Caters Jan 17 '15 at 7:43
  • And the same website has things about notes like F### and Bbbb – Caters Jan 17 '15 at 7:45

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