Music: Practice & Theory Stack Exchange is a question and answer site for musicians, students, and enthusiasts. It only takes a minute to sign up.
Sign up to join this community
A double-sharp (x) raises a natural (♮) note by two semitones.
A double-sharp (x) raises a sharp (#) by one semitone.
A double-flat (♭♭) lowers a natural note by two semitones.
A double-flat (♭♭) lowers a flat by one semitone.
A double-sharp would raise a flat note by a tone and a half (three semitones).
A double-flat would lower a sharp note by a tone and a half (three semitones).
Having said both these things, if you have a piece that is changing a flat note to a double-sharp, or changing a sharp note to a double flat, it is likely that you could use a much better enharmonic spelling. For example, it seems far more likely that you would move from G# to E# (or F, depending upon context), than G# to Gbb, even if you wanted to use, for instance, Fb, Eb and D close to these notes too. In fact, if music is very chromatic, it often makes sense to use fewer double-sharps and double-flats, even if this requires using a few more accidentals (naturals, for instance). This is because very chromatic music can start to lose its connection with major or minor scale degrees, this chromatic "alteration" of scale degrees being a good reason to use double-sharps and double-flats in the first place (either as chromatic alteration within a particular key, or chromatic alteration which produces a modulation to another key). In the example above, it would make sense to use the "simplest" accidentals (Ab F E Eb D, G# F E Eb D, Ab F E D# D or G# F E D# D), to make it easy to read. Which of these would be used, would depend upon whether these are heard within a diatonic context or not, and if so within which key. (If, of course, these are used during a passage in F# Minor, it's likely that an E# would be used - context is everything.)
I suppose you could also adjust a double-sharp note to a double-flat (and vice versa), in theory, which would be a a movement down of two tones (four semitones). I can't think of a reason why you would want to do this, apart from as a theoretical exercise, though! If these two notes were next to each other, I guess this would be the interval of a quadruply augmented unison!! (Thanks for correction, @PatMuchmore.)
EDIT: as @Guidot points out in his answer, this situation is basically theoretical; it is very unlikely to actually appear in practice. Double-sharps are nearly always used to extend a sharp by a further semitone (usually where the sharp already appears in the key signature); double-flats are nearly always used to extend a flat by a further semitone (usually where the flat already appears in the key signature). For this reason, in practice it is very unlikely that there would be a need to move from flat to double-sharp, or sharp to double-flat. (As I point out above, if one ever finds the need to do either of these things in a piece of music, the enharmonic spelling is almost certainly incorrect.)
Also, @CarlWitthoft has asked me to be clear: any accidental (sharp, flat, natural, double-sharp, double-flat) "cancels" the associated key-signature sharp or flat for the same note. So, in theory, this could be a way that a note gets "changed" by three semitones by using a double-sharp or double-flat. An example would be using an A double-sharp where the key-signature has an Ab (Eb Major for example). But, again, this is extremely unlikely to be necessary, for three reasons:
As you mentioned double accidentals start working from the natural tone and shift by two semitones, therefore if the natural tone was shifted towards the other direction before, this has to be neutralized first. Actually this is a quite academic issue, since more than 99% of the double accidentals I encountered in classical music, occured in pieces, where the key signature had already shifted the tone by a semitone in the same direction. I would expect, that contemporary pieces requiring such wild shifts have no key signature at all.
