Theoretical key

Theoretical keys or impossible keys are keys which exist in music theory, but are not actually used in music because they have an impractical key signature. In music that uses the equal temperament, which includes most Western music, an impossible or theoretical key is one whose key signature contains one or more double-flats or double-sharps - with the exception of sharp minor keys whose raised leading note is a double-sharp. These keys are G-sharp minor, D-sharp minor, and A-sharp minor, whose leading notes are, respectively, Fx, Cx, and Gx. These notes are notated only as accidentals (as with the raised leading note in any minor key), and never incorporated into the key signature, and these keys do not fall under the definition of theoretical keys.

For example, the key of D-flat minor does not conventionally exist, because its key signature would have to contain a double-flat (B-double-flat). Double-flats and double-sharps are used in music as accidentals, but they are never placed in the key signature (in music that uses equal temperament), partly because it would be unnecessarily difficult to read such music. Since in an equal tempered scale D-flat is really the same note as C-sharp, the D-flat minor scale has the same notes as the C-sharp minor scale, but the (real) key signature of C-sharp minor is the only conventional way to notate this scale. Keys such as of C-sharp minor and D-flat minor which have the same notes in an equal tempered scale are said to be enharmonic. The scales sound almost the same, and using equal temper exactly the same, but they are notated differently, for example, C-sharp instead of D-flat, A instead of B-double-flat. Compare the corresponding notes in both scales:

C-sharp minor:	C#	D#	E	F#	G#	A	B
D-flat minor:	Db	Eb	Fb	Gb	Ab	Bbb	Cb

The notation of D-flat minor is impractical to read, since it contains a double flat (it notates the sixth scale degree as Bbb instead of simply calling it A). Hence, the "real" key of C-sharp minor is used instead of the theoretical key of D-flat minor.

For a pair of enharmonic keys (like C-sharp minor and D-flat minor used in the example above), it is not necessarily the case that one is theoretical and the other real. There are six pairs of enharmonic keys in which both are real keys (neither of the two are theoretical). Here they are:

• C-sharp and D-flat majors
• D-sharp and E-flat minors
• F-sharp and G-flat majors
• G-sharp and A-flat minors
• A-sharp and B-flat minors
• B and C-flat majors

The difference between two enharmonic keys is not 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, and D-flat major has a difference of only one accidental compared to F-minor, whereas C-sharp major has a difference of 11 accidentals compared to F minor.

The keys listed above, having 5, 6 or 7 sharps/flats in their key signatures, are the most "distant" keys to use a conventional key signature. In keys beyond these, double-sharps and double-flats have to be incorporated into the key signature; the following 6 keys require 1, 2 or 3 double-sharps or double-flats; these 6 keys are the parallel major/minor keys of the 6 duplicated keys above:

Key	Key Signature	Relative key
D-flat minor	8 flats (= C-sharp minor)	F flat major
G-flat minor	9 flats (= F-sharp minor)	B double-flat major
C-flat minor	10 flats (= B minor)	E double-flat major
G-sharp major	8 sharps (= A-flat major)	E sharp minor
D-sharp major	9 sharps (= E-flat major)	B sharp minor
A-sharp major	10 sharps (= B-flat major)	F double-sharp 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.

With regular 12-tone tuning, these keys, and all other "impossible" ones, can be re-spelled as conventional keys. However, in a different tuning system (for example, 19 tone equal temperament), several keys do require a double-sharp or double-flat in the key signature, and no longer have conventional equivalents: where there are 19 tones in the scale, the key of B-double-flat major (9 flats) is equivalent to A-sharp major (10 sharps). In 17 and 31 tone equal temperament, keys that are enharmonic in a 12 tone system (for example, C-sharp and D-flat minor) may be notated completely differently.

Adding or subtracting 12 accidentals (or the same number as the tuning system) to or from a key signature results in the same key enharmonically respelled. Here are the six simplest keys respelled as "impossible" keys (with 11, 12 or 13 sharps or flats), each in two different ways:

• C major = B-sharp major or D-double-flat major
• D minor = C-double-sharp minor or E-double-flat minor
• E minor = D-double-sharp minor or F-flat minor
• F major = E-sharp major or G-double-flat major
• G major = F-double-sharp major or A-double-flat major
• A minor = G-double-sharp minor or B-double-flat minor