Theoretical key

Impossible and theoretical keys are really one and the same; basically, they are keys that have no possible conventional key signature.

For example the key of D flat minor cannot logically exist, as its key signature would have to contain a double flat - an impossibility in conventional notation. In equal temperament, the (real) key of C sharp minor is the only possible way that this key can be notated, as C sharp logically equals D flat. In the same way, there are 6 keys that are duplicated by this relationship:

1. C sharp and D flat majors
2. D sharp and E flat minors
3. F sharp and G flat majors
4. G sharp and A flat minors
5. A sharp and B flat minors
6. B and C flat majors

In the same way that e.g. C sharp equals D flat, C sharp also sounds exactly the same as D flat; hence C sharp major sounds exactly the same as D flat major on the ear. The difference between these two is only distinguished by the obvious difference in appearance on a page of printed music, but also the implications of such a key as a secondary key to the primary key of the piece of music in question; that is to say that if a piece in F minor were to modulate to its sub-mediant, D flat major, no confusion would arise as whether to use C sharp or D flat majors: F minor and D flat major both have flats in their key signatures anyway.

As a product of these 6 duplicated keys, there are 6 keys which are only theoretical and cannot be notated properly in a conventional key signature - there are six perfectly common real keys that are their respective equals in pitch and mood:

1. D flat minor (=C sharp minor)
2. D sharp major (=E flat major)
3. G flat minor (=F sharp minor)
4. G sharp major (=A flat major)
5. A sharp major (=B flat major)
6. C flat minor (=B minor)

However, these keys, along with all keys for that matter, can exist in a different form using the 12 modes as a basis for their notation in a (modal) key signature e.g. the key of D flat minor could exist in the Dorian mode, using the B flat to start with in the key signature and taking the form of the key signature of A flat minor.

Further to this there are six other keys that are totally impossible to notate either in a key signature or using any of the modes - there are six perfectly common real keys that are their respective equals in pitch and mood:

1. B sharp major (=C major)
2. B sharp minor (=C minor)
3. E sharp major (=F major)
4. E sharp minor (=F minor)
5. F flat major (=E major)
6. F flat minor (=E minor)

References

The Computations of Edward Tambling