These double alterations are enharmonic names of other notes.
Double flats
Cbb = Bb or A#
Dbb = C
Ebb = D
Fbb = Eb or D#
Gbb = F
Abb = G
Bbb = A
Double sharps (The simbol of the double sharp is like a cross, here I'm going to use two numerals instead)
C## = D
D## = E
E## = F# or Gb
F## = G
G## = A
A## = G
B## = C# or Db
So now you can see that the double alterations are the same note but with another name. This theory is useful when you have a scale like G# major.
G# A# B# C# D# E# F## G#
This scale is exactly the same as Ab major
Ab Bb C Db Eb F G Ab
Usually you'll be using the Ab scale and avoid the G#, but that doesn't matter. The theory to make the G# uses a double alteration. Let's see other example but with a double flat: Fb major.
Fb Gb Ab Bbb Cb Db Eb Fb
This scale is exactly the same as E
E F# G# A B C# D# E
The explanation to the construction of chords with double alteration is simple too.
Imagine you want to build an augmented chord from C.
So I imagine everybody knows that the answer would be C- E- G# (root- major third- augmented fifth)
Now, I want to build an augmented triad but this time starting on C#.
The answer would be C#- E#- G## (root- major third- augmented fifth). The trick is here. Some of you could think "oh! but the G## is an A". That's true but the name A corresponds to the sixth of C so in that case it would be the minor sixth (C#- A) and not the augmented fifth (C#- G##). So the notes are the same but they are named different.
I wish that will help you to understand the double alterations.
I posted this information because Enforcer told me that there wasn't a post about this theme. I hope this serves to everybody.
Be free to ask any questions about.
Good luck.
