Theoretical key

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

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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

Theoretical key
Theoretical key
G major, a key signature with a double-sharpA major, equivalent key
G major:GABCDEF DoubleSharp.svg
A major:ABCDEFG

The key of G major is a theoretical key because its key signature has an F DoubleSharp.svg , 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 Circle of fifths deluxe 4.svg
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 signatureMajor (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 Doubleflat.svg major (A major)9 flats (3 sharps)G minor (F minor)
E Doubleflat.svg major (D major)10 flats (2 sharps)C minor (B minor)
A Doubleflat.svg major (G major)11 flats (1 sharp)F minor (E minor)
D Doubleflat.svg major (C major)12 flats (no flats or sharps)B Doubleflat.svg minor (A minor)
G Doubleflat.svg major (F major)13 flats (1 flat)E Doubleflat.svg minor (D minor)
C Doubleflat.svg major (B major)14 flats (2 flats)A Doubleflat.svg 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 DoubleSharp.svg minor (G minor)
E major (F major)11 sharps (1 flat)C DoubleSharp.svg minor (D minor)
B major (C major)12 sharps (no flats or sharps)G DoubleSharp.svg minor (A minor)
F DoubleSharp.svg major (G major)13 sharps (1 sharp)D DoubleSharp.svg minor (E minor)
C DoubleSharp.svg major (D major)14 sharps (2 sharps)A DoubleSharp.svg 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 DoubleSharp.svg 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.

G-sharp major example 2.png

In very few cases, theoretical keys are used directly, with the necessary double accidentals in the key signature. The final pages of John Foulds' A World Requiem are written in G# major (with F DoubleSharp.svg 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 Doubleflat.svg in the key signature). [1] [2] Examples of theoretical key signatures are pictured below:

Theoretical key

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

Tunings other than twelve-tone equal-temperament

Tuning systems where the number of notes per octave is not a multiple of 12 can produce key signatures that have no equivalent in 12-tone equal temperament, in which case double-sharps, double-flats, or microtonal accidentals will be required. Additionally, keys such as G♯ major and F♭ major which 12-tone equal temperament and its multiples make redundant are distinguished in other tunings, and therefore, must be notated completely differently. For example, in 19-tone equal temperament, the key of A♯ major has 10 sharps, and is enharmonically equivalent to B Doubleflat.svg major, which has nine flats.

Other examples and considerations

See also

Related Research Articles

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In musical notation, an accidental is a symbol that indicates an alteration of a given pitch. Accidentals may alter the pitch associated with individual notes, or form a key signature that alters the pitches associated with the lines and spaces on a staff. In general, the names of pitch classes may be spelled with accidentals. In the Western tradition, the most common accidentals are the flat and the sharp which represent alterations of a semitone, as well as the natural which indicates no alteration, usually to explicitly "nullify" another accidental. Ultimately, the use of accidentals depends on both the musical system and the notation used to represent it—accidentals are often employed in adapting Western notation to non-Western music.

<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.

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<span class="mw-page-title-main">Enharmonic scale</span>

In music theory, an enharmonic scale is a very ancient Greek musical scale which contains four notes tuned to approximately quarter tone pitches, bracketed between four fixed pitches. For example, in modern microtonal notation, one enharmonic scale aligned with the conventional key of C major would be

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.

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, 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 222, 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, 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 flat or sharp oron 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. 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