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#REDIRECT [[Key signature#Conventions]]
{{Short description|Musical concept}}
{{refimprove|date=December 2018}}
In [[music theory]], a '''theoretical key''' is a [[key (music)|key]] whose [[key signature]] would have at least one [[Flat (music)|double-flat]] ({{Music|bb}}) or [[Sharp (music)|double-sharp]] ({{Music|##}}).


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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{{Music|#}} major requires eight sharps, and, since there are only seven scale tones, one tone requires a double sharp. The enharmonically equivalent key of A{{music|flat}} only requires four flats, making it clearer to read.
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==Enharmonic equivalence==
{{R to section}}
{|class="wikitable floatright" style="max-width:360px;"
}}
|<score>{ \magnifyStaff #3/2 \omit Score.TimeSignature \set Staff.keyAlterations = #`((3 . ,DOUBLE-SHARP)(0 . ,SHARP)(4 . ,SHARP)(1 . ,SHARP)(5 . ,SHARP)(2 . ,SHARP)(6 . ,SHARP)) s^"" }</score>||<score>{ \magnifyStaff #3/2 \omit Score.TimeSignature \key f \minor s^"" }</score>
|-
|G{{music|#}} major, a key signature with a double-sharp||A{{music|b}} major, equivalent key
|}

{|class="wikitable floatright"
|G{{Music|#}} major:||G{{music|#}}||A{{music|#}}||B{{music|#}}||C{{music|#}}||D{{music|#}}||E{{music|#}}||F{{Music|doublesharp}}
|-
|A{{music|b}} major:||A{{music|b}}||B{{music|b}}||C||D{{music|b}}||E{{music|b}}||F||G
|}
The key of G{{music|#}} major is a theoretical key because its [[key signature]] has an F{{Music|doublesharp}}, giving it eight sharps. An equal-tempered [[Scale (music)|scale]] in G{{music|#}} major contains the same pitches as the A{{music|b}} major scale, making the two keys [[enharmonic|enharmonically equivalent]]. In the absence of other factors, this key would generally be notated as [[A-flat major|A{{music|b}} major]].

===Modulation===
[[File:Circle of fifths deluxe 4.svg|thumb|Circle of fifths showing major and minor keys]]
While a piece of Western music generally has a [[Tonic (music)|home key]], a passage within it may [[modulate (music)|modulate]] to another key, which is usually [[closely related key|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:
{| class="wikitable"
!Major (minor) || Key signature || &nbsp; || Major (minor) || Key signature
|-
|[[B major|B]] ([[G-sharp minor|g{{Music|sharp}}]]) || 5 [[sharp (music)|sharps]] || &nbsp; || [[C-flat major|C{{Music|flat}}]] ([[A-flat minor|a{{Music|flat}}]]) || 7 [[flat (music)|flats]]
|-
|[[F-sharp major|F{{Music|sharp}}]] ([[D-sharp minor|d{{Music|sharp}}]]) || 6 sharps || &nbsp; || [[G-flat major|G{{Music|flat}}]] ([[E-flat minor|e{{Music|flat}}]]) || 6 flats
|-
|[[C-sharp major|C{{Music|sharp}}]] [[A-sharp minor|(a{{Music|sharp}}]]) || 7 sharps || &nbsp; || [[D-flat major|D{{Music|flat}}]] ([[B-flat minor|b{{Music|flat}}]]) || 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 key|parallel]] major/minor keys of those above.

{| class="wikitable" style="margin-left: 2em;"
![[Major key|Major]]||Key signature||[[Minor key|Minor]]
|-
|[[F-flat major|F{{Music|flat}} major]] ([[E major]])|| 8 flats (4 sharps)
| [[D-flat minor|D{{Music|flat}} minor]] ([[C-sharp minor|C{{Music|sharp}} minor]])
|-
|B{{Music|doubleflat}} major ([[A major]])|| 9 flats (3 sharps)
| G{{Music|flat}} minor ([[F-sharp minor|F{{Music|sharp}} minor]])
|-
|E{{Music|doubleflat}} major ([[D major]])|| 10 flats (2 sharps)
| C{{Music|flat}} minor ([[B minor]])
|-
|A{{Music|doubleflat}} major ([[G major]])|| 11 flats (1 sharp)
| F{{Music|flat}} minor ([[E minor]])
|-
|D{{Music|doubleflat}} major ([[C major]])|| 12 flats (no flats or sharps)
| B{{Music|doubleflat}} minor ([[A minor]])
|-
|G{{Music|doubleflat}} major ([[F major]])|| 13 flats (1 flat)
| E{{Music|doubleflat}} minor ([[D minor]])
|-
|C{{Music|doubleflat}} major ([[B-flat major|B{{Music|flat}} major]])|| 14 flats (2 flats)
| A{{Music|doubleflat}} minor ([[G minor]])
|-
|[[G-sharp major|G{{Music|sharp}} major]] ([[A-flat major|A{{Music|flat}} major]])|| 8 sharps (4 flats)
| [[E-sharp minor|E{{Music|sharp}} minor]] ([[F minor]])
|-
|D{{Music|sharp}} major ([[E-flat major|E{{Music|flat}} major]])|| 9 sharps (3 flats)
| B{{Music|sharp}} minor ([[C minor]])
|-
|A{{Music|sharp}} major ([[B-flat major|B{{Music|flat}} major]])|| 10 sharps (2 flats)
| F{{Music|doublesharp}} minor ([[G minor]])
|-
|E{{Music|sharp}} major ([[F major]])|| 11 sharps (1 flat)
| C{{Music|doublesharp}} minor ([[D minor]])
|-
|B{{Music|sharp}} major ([[C major]])|| 12 sharps (no flats or sharps)
| G{{Music|doublesharp}} minor ([[A minor]])
|-
|F{{Music|doublesharp}} major ([[G major]])|| 13 sharps (1 sharp)
| D{{Music|doublesharp}} minor ([[E minor]])
|-
|C{{Music|doublesharp}} major ([[D major]])|| 14 sharps (2 sharps)
| A{{Music|doublesharp}} minor ([[B minor]])
|}
A piece in a major key might modulate up a [[perfect fifth|fifth]] to the [[dominant (music)|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{{Music|x}} in place of the F{{Music|#}}. 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.

[[File: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{{music|#}} major (with F{{music|x}} in the key signature), No. 18 of Anton Reicha's ''Practische Beispiele'' is written in B{{music|#}} major, and the third movement of [[Victor Ewald]]'s Brass Quintet Op. 8 is written in F{{music|b}} major (with B{{music|bb}} in the key signature).<ref name=":0">{{IMSLP|work=Practische Beispiele (Reicha, Anton)|cname=Anton Reicha: ''Practische Beispiele'', pp. 52-53.}}</ref><ref>{{cite web|url=http://www.enspub.com/pages/93/93503.htm|title=Ewald, Victor: Quintet No 4 in A{{music|b}}, op 8|author=<!--Staff writer(s); no by-line.-->|publisher=imslp|access-date=14 February 2023}}</ref> Examples of theoretical key signatures are pictured below:
<score>
\relative c' { \omit Staff.TimeSignature \omit Staff.KeyCancellation
\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" }
}
</score>

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{{music|#}} major exactly as displayed above.<ref>{{IMSLP|work=A World Requiem, Op.60 (Foulds, John)|cname=John Foulds: ''A World Requiem'', pp. 153ff.}}</ref> The sharps in the key signature of G{{music|#}} major here proceed C{{music|#}}, G{{music|#}}, D{{music|#}}, A{{music|#}}, E{{music|#}}, B{{music|#}}, F{{music|x}}. 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{{music|b}} minor key signatures on pp. 42–45.<ref>{{cite book|author=Max Reger|translator=John Bernhoff|title=Supplement to the Theory of Modulation|location=Leipzig|publisher=C. F. Kahnt Nachfolger|year=1904|pages=[https://archive.org/details/supplementtotheo00rege/page/42/mode/2up 42–45]}}</ref> These have a B{{music|b}} at the start and also a B{{music|bb}} at the end (with a double-flat symbol), going B{{music|b}}, E{{music|b}}, A{{music|b}}, D{{music|b}}, G{{music|b}}, C{{music|b}}, F{{music|b}}, B{{music|bb}}.
*Sometimes the double signs are written at the beginning of the key signature, followed by the single signs. For example, the F{{music|b}} key signature is notated as B{{music|bb}}, E{{music|b}}, A{{music|b}}, D{{music|b}}, G{{music|b}}, C{{music|b}}, F{{music|b}}. This convention is used by Victor Ewald<ref>[https://www.hickeys.com/music/brass/brass_ensembles/brass_quintets/products/sku035994-ewald-victor-quintet-no-4-in-ab-op-8.php "Ewald, Victor: Quintet No 4 in A{{music|b}}, op 8"], Hickey's Music Center</ref> and by some theoretical works.
*However, no. 18 of Anton Reicha's ''Practische Beispiele'' in B{{music|#}} major,<ref name=":0" /> it was written as B♯, E♯, A{{music|x}}, D{{music|x}}, G{{music|x}}, C{{music|x}}, F{{music|x}}.

==Tunings other than twelve-tone equal-temperament==
[[Tuning system]]s 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 equal temperament|19-tone equal temperament]], the key of A♯ major has 10 sharps, and is enharmonically equivalent to B{{Music|bb}} major, which has nine flats.

== Etc. ==
* There is also an example of modulation in John Stump's Prelude and the Last Hope with a [[double flat]] in the [[key signature]].<ref>[https://www.flickr.com/photos/heatherlewin/6951090591]</ref> <score>{ \omit Score.TimeSignature { \omit Staff.KeyCancellation \set Staff.keyAlterations = #`((6 . ,DOUBLE-FLAT)(2 . ,DOUBLE-FLAT)(5 . ,DOUBLE-FLAT)(1 . ,FLAT)(4 . ,FLAT)(0 . ,FLAT)(3 . ,FLAT)) s^""}}</score>However, when using the tonality of this [[key signature]], the [[Measure (music)|measure]] is merely a measure of whole rests and no note at all.
* Going further theoretically, there may be countless such keys<ref>In other words, the '''theoretical keys''' can include not only the [[Double accidental|double accidentals]] but also after [[triple accidentals]] ([[triple flat]]/[[triple sharp]]).</ref>, and there may be other considerations as well. For reference, in [[equal temperament]] with [[enharmonics]], [[Enharmonic key signature|enharmonic keys]] can exist, but in the [[Just Intonation|just intonation]], there are no [[Enharmonic key signature|enharmonic keys]].<ref>For example, when considering [[Enharmonic key signature|enharmonic keys]], there is a simplest or most appropriate notation for each note. In addition, in ''n''-[[Equal temperament|TET]], which divides an [[octave]] into ''n-''equal parts, the [[Pitch (music)|pitches]] of [[Major (music)|major]] or [[Minor (music)|minor]] [[Key (music)|keys]] are limited to a total of ''n''-types per [[octave]], regardless of whether they are [[microtones]] or '''theoretical keys'''. For example, there will be 12 types in [[12 TET]] and 19 types in [[19 TET]].</ref>
* Some ask 'Why is there no [[G sharp major]]?', 'Why is there no [[D flat minor]]?' etc. have also been recognized.{{Citation needed|date=March 2024}} These keys are exactly the '''theoretical keys'''.
== See also ==
* {{Annotated link|Closely related key}}
* {{Annotated link|Diatonic function}}

==References==
{{reflist}}

{{Key (music)}}

{{DEFAULTSORT:Theoretical Key}}
[[Category:Musical keys]]
[[Category:Musical notation]]
[[Category:Music theory]]

Latest revision as of 15:00, 2 June 2024

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