Theoretical key

Last updated May 21, 2024

In music theory, a theoretical key is a key whose key signature requires double-flats ( ), or double-sharps ( ).

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 (F ). In 12-tone equal temperament, the standard tuning of Western music, the key of G♯ major is enharmonically equivalent to A♭ major (with four flats). In the absence of other factors, this key would generally be notated as A♭.


G♯ major, a key signature with a double-sharp	A♭ major, equivalent key

G♯ major:	G♯	A♯	B♯	C♯	D♯	E♯	F
A♭ major:	A♭	B♭	C	D♭	E♭	F	G

Modulation

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 theoretical keys would require up to seven double-sharps or double-flats. Six of these are the parallel major/minor keys of those above.

Major	Key signature	Minor
F♭ major (E major)	8 flats (4 sharps)	D♭ minor (C♯ minor)
B major (A major)	9 flats (3 sharps)	G♭ minor (F♯ minor)
E major (D major)	10 flats (2 sharps)	C♭ minor (B minor)
A major (G major)	11 flats (1 sharp)	F♭ minor (E minor)
D major (C major)	12 flats (no flats or sharps)	B minor (A minor)
G major (F major)	13 flats (1 flat)	E minor (D minor)
C major (B♭ major)	14 flats (2 flats)	A minor (G minor)
G♯ major (A♭ major)	8 sharps (4 flats)	E♯ minor (F minor)
D♯ major (E♭ major)	9 sharps (3 flats)	B♯ minor (C minor)
A♯ major (B♭ major)	10 sharps (2 flats)	F minor (G minor)
E♯ major (F major)	11 sharps (1 flat)	C minor (D minor)
B♯ major (C major)	12 sharps (no flats or sharps)	G minor (A minor)
F major (G major)	13 sharps (1 sharp)	D minor (E minor)
C major (D major)	14 sharps (2 sharps)	A minor (B minor)

A piece in a major key might modulate up a fifth to the dominant (a common occurrence in Western music), resulting in a new key signature with an additional sharp. If the original key was C-sharp, such a modulation would lead to the theoretical key of G-sharp major (with eight sharps) requiring an F in place of the F♯. This section could be written 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 from D-flat major to D-flat minor (eight flats) for a few measures but the passage is notated in C-sharp minor (four sharps); the same happens in the final movement, "Passepied", in which a G-sharp major section is 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 in measures 10-12.

While theoretical keys are sometimes notated with accidentals as in this example, the use of actual theoretical key signatures is very rare. The final pages of John Foulds' A World Requiem are written in G# major (with F 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 B in the key signature).^[1]^[2] Examples of theoretical key signatures are pictured below:

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 sharps or flats in the circle-of-fifths order before showing the double signs. This is the format used in John Foulds' A World Requiem, Op. 60, which uses the key signature of G# major as displayed above.^[3] The sharps in the key signature of G# major proceed C#, G#, D#, A#, E#, B#, F .
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 B at the end (with a double-flat symbol), going B♭, E♭, A♭, D♭, G♭, C♭, F♭, B .
Sometimes the double signs are written at the beginning of the key signature, followed by the single signs. For example, F♭ would be notated as B , E♭, A♭, D♭, G♭, C♭, F♭. This convention is used by Victor Ewald^[5] and by some theoretical works.
No. 18 of Anton Reicha's Practische Beispiele in B# major^[1] shows B♯, E♯, A , D , G , C , F .

Tunings other than twelve-tone equal-temperament

In tuning systems where the number of notes per octave is not a multiple of 12, keys such as G♯ and A♭ are not enharmonically equivalent. These tunings may also have keys with no analogue in 12-tone equal temperament, which can lead to situations where double sharps, double flats, or microtonal alterations need to be used in key signatures. For example, the key of A♯ major, with 10 sharps, is equivalent to B♭ major in 12-tone equal temperament, but in 19-tone equal temperament, it is equivalent to B major, with nine flats.

Other examples and considerations

The key of E appears in John Stump's "Prelude and the Last Hope" with multiple double flats,^[6] although it appears in a measure of whole rests with no pitches to read.
There can be countless such keys, using triple accidentals and beyond.
In just intonation, there are no enharmonic keys.

Related Research Articles

In Western musical notation, a key signature is a set of sharp, flat, or rarely, natural symbols placed on the staff at the beginning of a section of music. The initial key signature in a piece is placed immediately after the clef at the beginning of the first line. If the piece contains a section in a different key, the new key signature is placed at the beginning of that section.

In musical notation, an accidental is a symbol that indicates an alteration of a given pitch. The most common accidentals are the flat and the sharp, which represent alterations of a semitone, and the natural, which cancels a sharp or flat. Accidentals alter the pitch of individual scale tones in a given key signature; the sharps or flats in the key signature itself are not considered accidentals.

<span class="mw-page-title-main">Chromatic scale</span> Musical scale set of twelve pitches

The chromatic scale is a set of twelve pitches used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the chromatic scale, while other instruments capable of continuously variable pitch, such as the trombone and violin, can also produce microtones, or notes between those available on a piano.

In music, two written notes have enharmonic equivalence if they produce the same pitch but are notated differently. Similarly, written intervals, chords, or key signatures are considered enharmonic if they represent identical pitches that are notated differently. The term derives from Latin enharmonicus, in turn from Late Latin enarmonius, from Ancient Greek $ἐναρμόνιος$ , from $ἐν$ ('in') and $ἁρμονία$ ('harmony').

In music, in English sharp – eqv. dièse or diesis – means higher in pitch. The sharp symbol, ♯, indicates that the note to which the symbol is applied is played one semitone higher. The opposite of sharp is flat, indicating a lowering of pitch. The ♯ symbol derives from a square form of the letter b.

In music, flat means lower in pitch. It may either be used generically, meaning any lowering of pitch, or refer to a particular size: lowering pitch by a chromatic semitone. A flat is the opposite of a sharp which raises pitch by the same amount that a flat lowers it.

In music theory, the circle of fifths is a way of organizing pitches as a sequence of perfect fifths. Starting on a C, and using the standard system of tuning for Western music, the sequence is: C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, E♯ (F), C. This order places the most closely related key signatures adjacent to one another. It is usually illustrated in the form of a circle.

B, also known as Si, Ti, or, in some European countries, H, is the seventh note and the twelfth semitone of the fixed-Do solfège. Its enharmonic equivalents are C♭ (C-flat) and A.

C-flat major is a major scale based on C♭, consisting of the pitches C♭, D♭, E♭, F♭, G♭, A♭, and B♭. Its key signature has seven flats. Its relative minor is A-flat minor. Its parallel minor, C-flat minor, is usually replaced by B minor, since C-flat minor's three double-flats make it impractical to use. The direct enharmonic equivalent of C-flat major is B major, a key signature with five sharps.

F-sharp major is a major scale based on F♯, consisting of the pitches F♯, G♯, A♯, B, C♯, D♯, and E♯. Its key signature has six sharps.

G-sharp minor is a minor scale based on G♯, consisting of the pitches G♯, A♯, B, C♯, D♯, E, and F♯. Its key signature has five sharps.

D-sharp minor is a minor scale based on D♯, consisting of the pitches D♯, E♯, F♯, G♯, A♯, B, and C♯. Its key signature has six sharps.

F minor is a minor scale based on F, consisting of the pitches F, G, A♭, B♭, C, D♭, and E♭. Its key signature consists of four flats. Its relative major is A-flat major and its parallel major is F major. Its enharmonic equivalent, E-sharp minor, has six sharps and the double sharp F, which makes it impractical to use.

In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps. Each step represents a frequency ratio of 2^1⁄53, or 22.6415 cents, an interval sometimes called the Holdrian comma.

In music, 22 equal temperament, called 22-TET, 22-EDO, or 22-ET, is the tempered scale derived by dividing the octave into 22 equal steps. Each step represents a frequency ratio of ²²√2, or 54.55 cents.

D-flat minor is a theoretical key based on D♭, consisting of the pitches D♭, E♭, F♭, G♭, A♭, B, and C♭. Its key signature has one double flat and six flats. Its relative major is F-flat major, which is usually replaced by E major. Its parallel major is D-flat major. Its direct enharmonic equivalent, C-sharp minor, is normally used.

G-sharp major is a theoretical key based on the musical note G♯, consisting of the pitches G♯, A♯, B♯, C♯, D♯, E♯, and F. Its key signature has one double sharp and six sharps.

F is a musical note, the fourth above C or fifth below C. It is the fourth note and the sixth semitone of the solfège. It is also known as fa in fixed-do solfège. It has enharmonic equivalents of E♯ (E-sharp) and G, amongst others.

F-flat major is a theoretical key based on F♭, consisting of the pitches F♭, G♭, A♭, B double flat, C♭, D♭, and E♭. Its key signature has one double flat and six flats.

In modern Western music notation, a natural (♮) is a musical symbol that cancels a previous sharp or flat on a note in the written music. The sharp or flat may be from a key signature or an accidental. The natural indicates that the note is at its unaltered pitch.

References

1 2 Anton Reicha: Practische Beispiele, pp. 52-53. : Scores at the International Music Score Library Project
↑ "Ewald, Victor: Quintet No 4 in A♭, op 8". imslp. Retrieved 14 February 2023.
↑ John Foulds: A World Requiem, pp. 153ff. : Scores at the International Music Score Library Project
↑ Max Reger (1904). Supplement to the Theory of Modulation. Translated by John Bernhoff. Leipzig: C. F. Kahnt Nachfolger. pp. 42–45.
↑ "Ewald, Victor: Quintet No 4 in A♭, op 8", Hickey's Music Center
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[:0-1] 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

[6] ↑

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