In music theory, a theoretical key is a key whose key signature would have at least one double-flat (double flat) or double-sharp (double sharp).

Some musical keys are not normally used because they would require a double sharp or double flat in the key signature. For example, G♯ major requires eight sharps, and, since there are only seven scale tones, one tone requires a double sharp. The enharmonically equivalent key of A♭ only requires four flats, making it clearer to read.

Enharmonic equivalence

{ \new Staff \with{ \magnifyStaff #3/2 } << \time 1/4 \override Score.TimeSignature #'stencil = ##f { \clef treble \set Staff.keyAlterations = #`((3 . ,DOUBLE-SHARP)(0 . ,SHARP)(4 . ,SHARP)(1 . ,SHARP)(5 . ,SHARP)(2 . ,SHARP)(6 . ,SHARP)) s16 ^\markup "" } >> }
{ \new Staff \with{ \magnifyStaff #3/2 } << \time 1/4 \override Score.TimeSignature #'stencil = ##f { \clef treble \key f \minor s16 ^\markup "" } >> }
G♯ major, a key signature with a double-sharpA♭ major, equivalent key
G♯ major:G♯A♯B♯C♯D♯E♯Fdouble sharp
A♭ major:A♭B♭CD♭E♭FG

The key of G♯ major is a theoretical key because its key signature has an Fdouble sharp, giving it eight sharps. An equal-tempered scale in G♯ major contains the same pitches as the A♭ major scale, making the two keys enharmonically equivalent. In the absence of other factors, this key would generally be notated as A♭ major.

Modulation

Circle of fifths showing major and minor keys

While a piece of Western music generally has a home key, a passage within it may modulate to another key, which is usually closely related to the home key (in the Baroque and early Classical eras), that is, close to the original in the circle of fifths. When the key has zero or few sharps or flats, the notation of both keys is straightforward. But if the home key has many sharps or flats, particularly if the new key is on the opposite side, double sharps or flats may be necessary, or an enharmonically equivalent key may be used to avoid double sharps or flats.

In the bottom three places on the circle of fifths the enharmonic equivalents can be notated with single sharps or flats and so are not theoretical keys:

Major (minor)Key signature Major (minor)Key signature
B (g♯)5 sharps C♭ (a♭)7 flats
F♯ (d♯)6 sharps G♭ (e♭)6 flats
C♯ (a♯)7 sharps D♭ (b♭)5 flats

The need to consider theoretical keys

When a parallel key ascends the opposite side of the circle from its home key, theory suggests that double-sharps and double-flats would have to be incorporated into the notated key signature. The following keys (six of which are the parallel major/minor keys of those above) would require up to seven double-sharps or double-flats:

MajorKey signatureMinor
F♭ major (E major)8 flatsD♭ minor (C♯ minor)
Bdouble flat major (A major)9 flatsG♭ minor (F♯ minor)
Edouble flat major (D major)10 flatsC♭ minor (B minor)
Adouble flat major (G major)11 flatsF♭ minor (E minor)
Ddouble flat major (C major)12 flatsBdouble flat minor (A minor)
Gdouble flat major (F major)13 flatsEdouble flat minor (D minor)
Cdouble flat major (B♭ major)14 flatsAdouble flat minor (G minor)
G♯ major (A♭ major)8 sharpsE♯ minor (F minor)
D♯ major (E♭ major)9 sharpsB♯ minor (C minor)
A♯ major (B♭ major)10 sharpsFdouble sharp minor (G minor)
E♯ major (F major)11 sharpsCdouble sharp minor (D minor)
B♯ major (C major)12 sharpsGdouble sharp minor (A minor)
Fdouble sharp major (G major)13 sharpsDdouble sharp minor (E minor)
Cdouble sharp major (D major)14 sharpsAdouble sharp minor (B minor)

For example, pieces in the major mode commonly modulate up a fifth to the dominant; for a key with sharps in the signature this leads to a key whose key signature has an additional sharp. A piece in C-sharp that performs this modulation would lead to the theoretical key of G-sharp major, requiring eight sharps, meaning an Fdouble sharp in place of the F♯ already present. Such a section could be recast using the enharmonically equivalent key signature of A-flat major instead. Claude Debussy's Suite bergamasque does this: in the third movement "Clair de lune" the key shifts for a few measures from D-flat major to D-flat minor (eight flats), but the passage is notated in C-sharp minor (four sharps); the same happens in the final movement "Passepied", which reaches theoretical G-sharp major written as A-flat major.

Such passages may instead be notated with the use of double-sharp or double-flat accidentals, as in this example from Johann Sebastian Bach's Well-Tempered Clavier, which has this passage in G-sharp major, Bars 10-12.

In very few cases, theoretical keys are in fact used directly, putting the necessary double-accidentals in the key signature. The final pages of John Foulds' A World Requiem are written in G♯ major (with Fdouble sharp in the key signature), No. 18 of Anton Reicha's Practische Beispiele is written in B♯ major, and the third movement of Victor Ewald's Brass Quintet Op. 8 is written in F♭ major (with Bdouble flat in the key signature).[1][2] Examples of theoretical key signatures are pictured below:

\relative c' { \hide Staff.TimeSignature \set Staff.printKeyCancellation = ##f \key gis \major <gis' bis dis><eis gis bis>_\markup { \halign #0.2 "G# maj" }_\markup { \halign #0.2 "e# min" } \bar "||" \key dis \major <dis fisis ais><bis dis fisis>_\markup { \halign #0.2 "D# maj" }_\markup { \halign #0.2 "b# min" } \bar "||" \key fes \major <fes' as ces><des fes as>_\markup { \halign #0.2 "F♭ maj" }_\markup { \halign #0.2 "d♭ min" } \bar "||" \key beses \major <beses' des fes><ges beses des>_\markup { \halign #0.2 "B♭♭ maj" }_\markup { \halign #0.2 "g♭ min" } }

There does not appear to be a standard on how to notate theoretical key signatures:

  • The default behaviour of LilyPond (pictured above) writes all single signs in the circle-of-fifths order, before proceeding to the double signs. This is the format used in John Foulds' A World Requiem, Op. 60, which ends with the key signature of G♯ major exactly as displayed above.[3] The sharps in the key signature of G♯ major here proceed C♯, G♯, D♯, A♯, E♯, B♯, Fdouble sharp. This likely makes more sense than the last example because the notes represented in the key signature increase by a perfect fifth (or decrease by a perfect fourth) from left to right.
  • The single signs at the beginning are sometimes repeated as a courtesy, e.g. Max Reger's Supplement to the Theory of Modulation, which contains D♭ minor key signatures on pp. 42–45.[4] These have a B♭ at the start and also a Bdouble flat at the end (with a double-flat symbol), going B♭, E♭, A♭, D♭, G♭, C♭, F♭, Bdouble flat.
  • Sometimes the double signs are written at the beginning of the key signature, followed by the single signs. For example, the F♭ key signature is notated as Bdouble flat, E♭, A♭, D♭, G♭, C♭, F♭. This convention is used by Victor Ewald[5] and by some theoretical works.
  • However, no. 18 of Anton Reicha's Practische Beispiele in B♯ major,[1] it was written as B♯, E♯, Adouble sharp, Ddouble sharp, Gdouble sharp, Cdouble sharp, Fdouble sharp.

Tunings other than twelve-tone equal-temperament

Other tuning systems (such as 19-tone, 31-tone, or 53-tone equal temperament) produce key signatures that require double sharps, double flats, or microtonal accidentals because they do not correspond to any conventionally notated key. For example, in 19-tone equal temperament, the key of Bdouble flat major (9 flats) is equivalent to A♯ major (10 sharps). Keys that are enharmonic in a 12-tone system (for example, A-flat major and G-sharp major) must be notated differently when other tuning systems are used.

See also

  • Closely related key – distance measure for musical keys
  • Diatonic function – Musical term

References

  1. 1 2 Anton Reicha: Practische Beispiele, pp. 52-53.: Scores at the International Music Score Library Project
  2. ↑ "Ewald, Victor: Quintet No 4 in A♭, op 8". imslp. Retrieved 14 February 2023.
  3. ↑ John Foulds: A World Requiem, pp. 153ff.: Scores at the International Music Score Library Project
  4. ↑ Max Reger (1904). Supplement to the Theory of Modulation. Translated by John Bernhoff. Leipzig: C. F. Kahnt Nachfolger. pp. 42–45.
  5. ↑ "Ewald, Victor: Quintet No 4 in A♭, op 8", Hickey's Music Center
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