Editing Circle of fifths

{{Short description|Relationship among tones of the chromatic scale}}
[[Image:Circle of fifths deluxe 4.svg|thumb|right|400px|Circle of fifths showing major and minor keys]]

In [[music theory]], the '''circle of fifths''' (sometimes also '''cycle of fifths''') is a way of organizing [[Pitch (music)|pitches]] as a sequence of [[perfect fifth]]s. Starting on a C, and using the standard system of tuning for Western music ([[12 equal temperament|12-tone equal temperament]]), the sequence is: C, G, D, A, E, B, F{{music|#}}/G{{music|flat}}, C{{music|#}}/D{{music|flat}}, G{{music|#}}/A{{music|flat}}, D{{music|#}}/E{{music|flat}}, A{{music|#}}/B{{music|flat}}, F, and C. This order places the most closely related [[key signature]]s adjacent to one another.

Twelve-tone equal temperament tuning divides each octave into twelve equivalent semitones, and the circle of fifths leads to a C seven octaves above the starting point. If the fifths are tuned with an exact frequency ratio of 3:2 (the system of tuning known as [[just intonation]]), this is not the case (the circle does not "close").

== Definition ==
The circle of fifths organizes pitches in a sequence of [[perfect fifth]]s, generally shown as a circle with the pitches (and their corresponding keys) in clockwise order. It can be viewed in a counterclockwise direction as a circle of fourths. Harmonic progressions in Western music commonly use adjacent keys in this system, making it a useful reference for musical composition and harmony.<ref>Michael Pilhofer and Holly Day (23 Feb 2009). [http://www.dummies.com/how-to/content/the-circle-of-fifths-a-brief-history.html "The Circle of Fifths: A Brief History"], www.dummies.com.</ref>

The top of the circle shows the [[Key (music)|key]] of C Major, with no [[Sharp (music)|sharps]] or [[Flat (music)|flats]]. Proceeding clockwise, the pitches ascend by fifths. The key signatures associated with those pitches change accordingly: the key of G has one sharp, the key of D has 2 sharps, and so on. Proceeding counterclockwise from the top of the circle, the notes change by descending fifths and the key signatures change accordingly: the key of F has one flat, the key of B{{music|flat}} has 2 flats, and so on. Some keys (at the bottom of the circle) can be notated [[Enharmonic|either in sharps or in flats]].

Starting at any pitch and ascending by a fifth generates all tones before returning to the beginning pitch class (a pitch class consists of all of the notes indicated by a given letter regardless of octave—all "C"s, for example, belong to the same pitch class). Moving counterclockwise, the pitches descend by a fifth, but ascending by a [[perfect fourth]] will lead to the same note an octave higher (therefore in the same pitch class). Moving counter-clockwise from C could be thought of as descending by a fifth to F, or ascending by a fourth to F.
{{col-begin}}
{{col-break}}
{{Image frame|content=<score sound="1"> {
\omit Score.TimeSignature
\relative c' { \set Score.tempoHideNote = ##t \tempo 1 = 60 \time 12/1
  c1 g' d a' e b' fis cis gis' dis ais' f | c! \bar "|."
} }
\layout { \context {\Score \omit BarNumber} line-width = #100 }
</score>|width=|align=center|caption=Circle of fifths clockwise within one octave}}
{{col-break}}
{{Image frame|content=<score sound="1">  {
\omit Score.TimeSignature
\relative c' { \set Score.tempoHideNote = ##t \tempo 1 = 60 \time 12/1
  c1 f bes ees, aes des, ges b e, a d, g | c, \bar "|."
} }
\layout { \context {\Score \omit BarNumber} line-width = #100 }
</score>|width=|align=center|caption=Circle of fifths counterclockwise within one octave}}
{{col-end}}

==Structure and use==
===Diatonic key signatures===
Each pitch can serve as the tonic of a major or [[Relative key|minor]] key, and each of these keys will have a [[diatonic scale]] associated with it. The circle diagram shows the number of sharps or flats in each [[key signature]], with the major key indicated by a capital letter and the minor key indicated by a lower-case letter. Major and minor keys that have the same key signature are referred to as [[Relative key|''relative major'' and ''relative minor'']] of one another.

===Modulation and chord progression===
[[Tonality|Tonal music]] often [[modulation (music)|modulates]] to a new tonal center whose key signature differs from the original by only one flat or sharp. These closely-related keys are a fifth apart from each other and are therefore adjacent in the circle of fifths. [[Chord progression]]s also often move between chords whose roots are related by perfect fifth, making the circle of fifths useful in illustrating the "harmonic distance" between chords.

[[File:The circle of fifths on umbilic torus surface.gif|thumb|left|Major 7th progressing on [[umbilic torus]] surface]]
The circle of fifths is used to organize and describe the [[diatonic function|harmonic or tonal function]] of [[chord (music)|chords]].<ref>{{Cite web |last=Broekhuis |first=Rogier |date=2022-10-11 |title=Wheel of Fifths – Harmonic Function |url=https://www.wheelof5ths.com/en-us/blogs/music-theory-lessons/harmonic-function |access-date=2023-10-05 |website=Wheel of Fifths |language=en}}</ref> Chords can progress in a pattern of ascending perfect fourths (alternately viewed as descending perfect fifths) in "functional succession". This can be shown "...by the circle of fifths (in which, therefore, [[scale degree]] II is closer to the [[Dominant (music)|dominant]] than scale degree IV)".{{sfn|Nattiez|1990|p=225}} In this view the tonic or tonal center is considered the end point of a [[chord progression]] derived from the circle of fifths.

[[File:Ii-V-I turnaround in C.png|thumb|right|[[ii–V–I progression]], in C, illustrating the similarity between them[[File:Ii-V-I turnaround in C.mid|thumb|Subdominant, supertonic seventh, and supertonic chords]]]]
According to [[Richard Franko Goldman]]'s ''Harmony in Western Music'', "the IV chord is, in the simplest mechanisms of diatonic relationships, at the greatest distance from I. In terms of the [descending] circle of fifths, it leads away from I, rather than toward it."{{sfn|Goldman|1965|p=68}} He states that the progression I–ii–V–I (an [[authentic cadence]]) would feel more final or [[resolution (music)|resolved]] than I–IV–I (a [[plagal cadence]]). Goldman{{sfn|Goldman|1965|loc=chapter 3}} concurs with Nattiez, who argues that "the chord on the fourth degree appears long before the chord on II, and the subsequent final I, in the progression I–IV–vii<sup>o</sup>–iii–vi–ii–V–I", and is farther from the tonic there as well.{{sfn|Nattiez|1990|p=226}} (In this and related articles, upper-case Roman numerals indicate major triads while lower-case Roman numerals indicate minor triads.)

===Circle closure in non-equal tuning systems===
Using the exact 3:2 ratio of frequencies to define a perfect fifth ([[just intonation]]) does not quite result in a return to the [[pitch class]] of the starting note after going around the circle of fifths. [[Twelve-tone equal temperament]] tuning produces fifths that return to a tone exactly seven [[octave]]s above the initial tone and makes the frequency ratio of the chromatic semitone the same as that of the diatonic semitone. The standard tempered fifth has a frequency ratio of 2<sup>7/12</sup>:1 (or about 1.498307077:1), approximately two cents narrower than a justly tuned fifth.

Ascending by twelve justly tuned fifths fails to close the circle by an excess of approximately 23.46 [[Cent (music)|cents]], roughly a quarter of a [[semitone]], an interval known as the [[Pythagorean comma]]. If limited to twelve pitches per octave, Pythagorean tuning markedly shortens the [[interval (music)|width]] of one of the twelve fifths, which makes it severely [[consonance and dissonance|dissonant]]. This anomalous fifth is called the [[wolf fifth]] – a humorous reference to a wolf howling an off-pitch note. Non-extended [[quarter-comma meantone]] uses eleven fifths slightly narrower than the equally tempered fifth, and requires a much wider and even more dissonant wolf fifth to close the circle. More complex tuning systems based on just intonation, such as [[5-limit tuning]], use at most eight justly tuned fifths and at least three non-just fifths (some slightly narrower, and some slightly wider than the just fifth) to close the circle.

===Equal-tempered tunings with more than twelve notes===
Nowadays, with the advent of electronic [[isomorphic keyboard]]s, equal temperament tunings with more than twelve notes per octave can be used to close the circle of fifths for other tunings. For example, [[31-tone equal temperament]] closely approximates quarter-comma meantone, and [[53-tone equal temperament]] closely approximates Pythagorean tuning.

==History==
[[Image:Diletsky circle.jpg|thumb|left|200px|Circle of fifths in ''Idea grammatikii musikiyskoy'' (Moscow, 1679)]]
[[Image:Heinichen musicalischer circul.png|thumb|[[Johann David Heinichen|Heinichen]]'s musical circle (German: ''Musicalischer Circul'') (1711)]]

The circle of fifths developed in the late 1600s and early 1700s to theorize the modulation of the Baroque era (see {{slink||Baroque era}}).

The first circle of fifths diagram appears in the ''Grammatika'' (1677) of the composer and theorist [[Nikolay Diletsky]], who intended to present [[music theory]] as a tool for composition.<ref name="Jensen">{{harvnb|Jensen|1992|pp=306–307}}</ref> It was "the first of its kind, aimed at teaching a Russian audience how to write Western-style polyphonic compositions."  

A circle of fifths diagram was independently created by German composer and theorist [[Johann David Heinichen]] in his ''Neu erfundene und gründliche Anweisung'' (1711),<ref>Johann David Heinichen, ''Neu erfundene und gründliche Anweisung'' (1711), [https://books.google.com/books?id=4O1CAAAAcAAJ&pg=PA260-IA1 p. 261]</ref> which he called the "Musical Circle" (German: ''Musicalischer Circul'').{{sfn|Barnett|2002|p=444}}{{sfn|Lester|1989|pp=110-112}} This was also published in his ''Der General-Bass in der Composition'' (1728).

Heinichen placed the relative minor key next to the major key, which did not reflect the actual proximity of keys. [[Johann Mattheson]] (1735) and others attempted to improve this—[[David Kellner]] (1737) proposed having the major keys on one circle, and the relative minor keys on a second, inner circle. This was later developed into [[chordal space]], incorporating the parallel minor as well.<ref>{{cite book |last1=Lerdahl |first1=Fred |title=Tonal Pitch Space. |date=2005 |publisher=Oxford University Press |location=New York |isbn=0195178297 |pages=42 |url=https://books.google.com/books?id=6bxFrgVMDpsC&pg=PA42}}</ref>

Some sources imply that the circle of fifths was known in antiquity, by [[Pythagoras]].<ref>{{Cite web|url=https://jadebultitude.com/the-circle-of-fifths-complete-guide/#:~:text=From%20what%20we%20can%20see,known%20as%20the%20Pythagorean%20circle.|title = The Circle of Fifths Complete Guide!|date = 17 January 2021}}</ref><ref>{{Cite web|url=https://www.linkedin.com/pulse/circle-fifths-made-clear-mark-newman|title = The Circle of Fifths made clear}}</ref><ref>{{Cite web|url=https://www.dummies.com/article/academics-the-arts/music/music-theory/the-circle-of-fifths-a-brief-history-201561|title = Dummies - Learning Made Easy}}</ref> This is a misunderstanding and an anachronism.<ref>{{citation|last1=Frazer |first1=Peter A. |title=The Development of Musical Tuning Systems |date=2001 |pages=9, 13 |url=http://www.midicode.com/tunings/Tuning10102004.pdf |archive-url=https://web.archive.org/web/20130701202500/http://www.midicode.com/tunings/Tuning10102004.pdf |access-date=24 May 2020|archive-date=1 July 2013}}</ref> Tuning by fifths (so-called [[Pythagorean tuning]]) dates to Ancient Mesopotamia;<ref>{{cite book |last1=Dumbrill |first1=Richard J. |title=The archaeomusicology of the Ancient Near East |date=2005 |location=Victoria, B.C. |isbn=978-1412055383 |pages=18}}</ref> see {{slink|Music of Mesopotamia|Music theory}}, though they did not extend this to a twelve-note scale, stopping at seven. The [[Pythagorean comma]] was calculated by [[Euclid]] and by Chinese mathematicians (in the ''[[Huainanzi]]''); see {{slink|Pythagorean comma|History}}. Thus, it was known in antiquity that a cycle of twelve fifths was almost exactly seven octaves (more practically, alternating ascending fifths and descending fourths was almost exactly an octave). However, this was theoretical knowledge, and was not used to construct a repeating twelve-tone scale, nor to modulate. This was done later in [[meantone temperament]] and [[twelve-tone equal temperament]], which allowed modulation while still being in tune, but did not develop in Europe until about 1500. Although popularized as the circle of fifths, its Anglo-Saxon etymological origins trace back to the name "wheel of fifths."

==Use==
In musical pieces from the [[Baroque music]] era and the [[Classical era of music]] and in Western [[popular music]], [[traditional music]] and [[folk music]], when pieces or songs modulate to a new key, these modulations are often associated with the circle of fifths.

In practice, compositions rarely make use of the entire circle of fifths. More commonly, composers make use of "the compositional idea of the 'cycle' of 5ths, when music moves consistently through a smaller or larger segment of the tonal structural resources which the circle abstractly represents."<ref>[[Arnold Whittall|Whittall, A.]] (2002, p. 259) "Circle of Fifths", article in Latham, E. (ed.) ''The Oxford Companion to Music''. Oxford University Press.</ref> The usual practice is to derive the circle of fifths progression from the seven tones of the diatonic scale, rather than from the full range of twelve tones present in the chromatic scale. In this diatonic version of the circle, one of the fifths is not a true fifth: it is a tritone (or a diminished fifth), e.g. between F and B in the "natural" diatonic scale (i.e. without sharps or flats). Here is how the circle of fifths derives, through [[permutation]] from the diatonic major scale:

{{Image frame|content=<score sound="1">  {
\omit Score.TimeSignature
\relative c' { \clef F \time 8/4
  c b a g f e d c \bar "||" c' f, b e, a d, g c, \bar "||"
} }
\layout { \context { \Score \override SpacingSpanner.base-shortest-duration = #(ly:make-moment 1/16) } }
</score>|width=|align=center|caption=Diatonic scale and the circle of fifths derived from it – major}}

And from the (natural) minor scale:

{{Image frame|content=<score sound="1">  {
\omit Score.TimeSignature
\relative c' { \key c \major \clef bass \time 8/4
  a g f e d c b a \bar "||" a' d, g c, f b, e a, \bar "||"
} }
\layout { \context { \Score \override SpacingSpanner.base-shortest-duration = #(ly:make-moment 1/16) } }
</score>|width=|align=center|caption=Diatonic scale and the circle of fifths derived from it – minor}}

The following is the basic sequence of chords that can be built over the major bass-line:

{{Image frame|content=<score sound="1">  {
\omit Score.TimeSignature
\new PianoStaff << 
 \new ChordNames \chordmode {
    c f b,:dim e:m a,:m d:m g, c
 }
 \new Staff \relative c' { \time 8/4
  <g' c e> <a c f> <f b d> <g b e> <e a c> <f a d> <d g b> <e g c> \bar "||"
 }
 \new Staff \relative c' { \clef F \time 8/4
  c f, b e, a d, g c,
 } >> }
\layout { \context { \Score \override SpacingSpanner.base-shortest-duration = #(ly:make-moment 1/128) } }
</score>|width=|align=center|caption=Circle of fifths chord progression – major}}

And over the minor:  
{{Image frame|content=<score sound="1">  {
\omit Score.TimeSignature
\new PianoStaff << 
 \new ChordNames \chordmode {
    a,:m d:m g, c f, b,:dim e, a,:m
 }
 \new Staff \relative c' { \time 8/4
  <e a c> <f a d> <d g b> <e g c> <c f a> <d f b> <b e gis> <c e a> \bar "||"
 }
 \new Staff \relative c' { \clef F \time 8/4
  a d, g c, f b, e a,
 } >> }
\layout { \context { \Score \override SpacingSpanner.base-shortest-duration = #(ly:make-moment 1/128) } }
</score>|width=|align=center|caption=Circle of fifths chord progression – minor}}

Adding sevenths to the chords creates a greater sense of forward momentum to the harmony:
{{Image frame|content=<score sound="1">  {
\omit Score.TimeSignature
\new PianoStaff << 
 \new ChordNames \chordmode {
   a,:m d:m7 g,:7 c:maj7 f,:maj7 b,:m7.5- e,:7 a,:m
 }
 \new Staff \relative c' { \time 8/4
  <e a c> <f a c> <d f b> <e g b> <c e a> <d f a> <b d gis> <c e a> \bar "||"
 }
 \new Staff \relative c' { \clef F \time 8/4
  a d, g c, f b, e a,
 } >> }
\layout { \context { \Score \override SpacingSpanner.base-shortest-duration = #(ly:make-moment 1/128) } }
</score>|width=|align=center|caption=Circle of fifths chord progression – minor with added sevenths}}

===Baroque era===
According to [[Richard Taruskin]], [[Arcangelo Corelli]] was the most influential composer to establish the pattern as a standard harmonic [[Trope (literature)|"trope"]]: "It was precisely in Corelli's time, the late seventeenth century, that the circle of fifths was being 'theorized' as the main propellor of harmonic motion, and it was Corelli more than any one composer who put that new idea into telling practice."{{sfn|Taruskin|2010|p=184}}

The circle of fifths progression occurs frequently in the music of [[J. S. Bach]]. In the following, from [[Jauchzet Gott in allen Landen, BWV 51|''Jauchzet Gott in allen Landen'', BWV 51]], even when the solo bass line implies rather than states the chords involved:

{{Image frame|content=<score sound="1"> 
{ \new Staff \relative c' { 
   \set Staff.midiInstrument = #"cello" \clef F \time 12/8
   r r8 << {a( b c) | d,( e f) g( a b) | c,( d e) f( g a) | b,( c d) e( d) e | a, c e}
   \new Staff = "chords" \with {
      \omit TimeSignature
      \magnifyStaff #2/3
      firstClef = ##f
    } \relative c' 
 { \set Staff.midiInstrument = #"cello" \clef F
 {\hide Stem a8 s s d, s s g s s c, s s f s s b, s s e s s a, s s}}
\addlyrics \with { alignAboveContext = "chords" } { \override LyricText.font-size = #-1.5 Am \markup{\concat{Dm\super{7}}} \markup{\concat{G\super{7}}} \markup{\concat{C\super{maj7}}} \markup{\concat{F\super{maj7}}} \markup{\concat{B\super{7(♭5)}}} \markup{\concat{E\super{7}}} Am
   } >>
 s1 s8
} }
</score>|width=|align=center|caption=Bach from Cantata 51}}

[[Handel]] uses a circle of fifths progression as the basis for the [[Passacaglia]] movement from his Harpsichord suite No. 6 in G minor.

{{Image frame|content=<score sound="1"> {
\new PianoStaff << 
 \new Staff = "chords" << \magnifyStaff #3/4
  \new Voice \relative c' { 
   \set Staff.midiInstrument = #"harpsichord" \key g \minor
   \mark \markup { \abs-fontsize #10 { \bold { Passacaille } } }
   <bes' d>8. <c ees>16 <a c>8. <bes d>16 <c ees>8. <c ees>16 <bes d>8. <c ees>16 |
   <a c>8. <a f'>16 <g ees'>8. <a f'>16 <bes d>8. <bes d>16 <a c>8. <bes d>16 |
   <g bes>8. <a c>16 <fis a>8. <g bes>16 <a c>8. <a c>16 <g bes>8. <a c>16 |
   <fis a>8. <fis d'>16 <e c'>8. <fis d'>16 bes8 a g4 \bar "||"
} 
  \new Voice \relative c' { 
   s1 s s s2 \stemDown g'2
} 
 \new Staff \relative c' { \magnifyStaff #3/4
   \set Staff.midiInstrument = #"harpsichord" \key g \minor \clef F
   <g bes d>2 <ees g c> <f a c> <bes, d f bes> <ees g bes> <c ees a> d4 d, g2
} 
 \addlyrics \with { alignAboveContext = "chords" } { \override LyricText.font-size = #-1.5 Gm Cm F B♭ E♭ \markup{\concat{Am\super{(♭5)}}} D _ Gm }
>> >> }
</score>|width=|align=center|caption=Handel Passacaille from Suite in G minor bars 1–4}}

[[Baroque music|Baroque]] composers learnt to enhance the "propulsive force" of the harmony engendered by the circle of fifths "by adding sevenths to most of the constituent chords." "These sevenths, being dissonances, create the need for resolution, thus turning each progression of the circle into a simultaneous reliever and re-stimulator of harmonic tension... Hence harnessed for expressive purposes."{{sfn|Taruskin|2010|p=188}} Striking passages that illustrate the use of sevenths occur in the aria "Pena tiranna" in [[Handel]]'s 1715 opera ''[[Amadigi di Gaula]]'':

{{Image frame|content=<score sound="1"> {<<
 \new Staff \with{ \magnifyStaff #3/4 instrumentName = "Bassoon" } \relative c' { 
   \set Staff.midiInstrument = #"bassoon" \key f \major \clef tenor \time 3/4
   \mark \markup { \abs-fontsize #10 { \bold { Largo } } }
   f2. ~ | f | e ~ | e | \break
   d ~ | d | cis8 a d4 \clef F d, | a a'8 g f e
} 
 \new ChoirStaff <<
  \new Staff = "chords" \with{ \magnifyStaff #3/4 instrumentName = "Violins" } <<
    \new Voice \relative c' { 
    \set Staff.midiInstrument = #"violin" \key f \major \clef treble
    <d' f a>4 <d, f a>4. r16 <d' f a> | <d f bes>8.-. <d, f bes>16 <d f bes>4. r 16 <d f bes> |
    <c e bes'>8.-. <g bes>16 <g bes>4. r16 <e'' g bes> | <e f a>4 <e, f a>4. r16 <c f a> |
    <d f a>8.-. a16 a4. r16 <d' f a> | <d e g>4 <d, e g>4. r16 <g e' g> |
    \stemUp <e' g>8.-. <f a>16 <d f>4. <cis e>8 | <cis e>4 <cis e> r
} 
    \new Voice \relative c' { 
    s2. s s s s s | \stemDown cis'4 a4. a8 | a4 a s
} >>
  \new Staff \with{ \magnifyStaff #3/4 instrumentName = "Violas"} \relative c' { 
    \set Staff.midiInstrument = #"viola" \key f \major \clef C
    a'4 a,4. r16 a' | g8.-. g,16 g4. r16 g | g8.-. e16 e4. r16 c'' | c4 c,4. r16 c |
    bes8.-. f16 f4. r16 bes' | bes4 bes,4. r16 e | e4 f e | e e r
} 
  \new Staff \with{ \magnifyStaff #3/4 instrumentName = "Bassi"} \relative c' { 
    \set Staff.midiInstrument = #"cello" \key f \major \clef F
    d4 d,4. r16 d | g4-. g, r | c-. c,4. r16 c' | f4 f, r8 f' |
    bes4 bes, r | e e, r | a' d, d, | a' a r
} 
 \addlyrics \with { alignAboveContext = "chords" } { \override LyricText.font-size = #-1.5 Dm  _ _ \markup{\concat{Gm\super{7}}} _ \markup{\concat{C\super{7}}} _ _ \markup{\concat{F\super{maj7}}} _ _ \markup{\concat{B♭\super{maj7}}} _ \markup{\concat{Em\super{7(♭5)}}} _ \markup{\concat{A\super{7}}} Dm _ A }
>> >> }
\layout {
  indent = 1.0\cm
  line-width = #150
}
</score>|width=|align=center|caption=Handel, aria "Pena tiranna" from ''Amadigi'', orchestral introduction}}

– and in Bach's [[Weimar concerto transcriptions (Bach)|keyboard arrangement]] of [[Alessandro Marcello]]'s [[Oboe Concerto (Marcello)|Concerto for Oboe and Strings]]. 

{{Image frame|content=<score sound="1"> { <<
 \new PianoStaff <<
  \new Staff = "chords" << \magnifyStaff #2/3
   \new Voice \relative c' { 
   \key f \major \set Score.tempoHideNote = ##t \tempo 4 = 40 \time 3/4
   \mark \markup { \abs-fontsize #10 { \bold { Adagio } } }
   d8 d d d d d | e e e e e e | g g g g g g | \stemUp d'( f) \stemNeutral f( a) a( c16 bes) | bes2 \mordent r4 | \break
   c,8( e16 d) e8( g16 f) g8( bes16 a) | a2 \mordent r4 | bes,16( c32 a bes16 d32 cis) d16( e32 cis d16 f32 e) f16( g32 e f16 a32 g) | \break
   g2 \mordent r4 | a,32( gis a b a b cis b) cis( d cis d e d e f e f g! f g f g e) | f4 \mordent s4
  }
   \new Voice \relative c' { 
   s2. | s | \stemDown e8 e e e e e | f8 
  }
  \new Staff << \magnifyStaff #2/3
   \new Voice \relative c' { 
   \key f \major \clef F \time 3/4
   R2. | d8 d d d d d | \stemUp cis cis cis cis cis cis | d <d f>[ <d f> <d f> <d f> <d f>] | <d f> <d f> <d f> <d f> <d f> <d f> | e e e e e e | <c e> <c e> <c e> <c e> <c e> <c e> | d d d d d d | <bes d> <bes d> <bes d> <bes d> <bes d> <bes d> | cis cis cis cis cis cis | d[ d]  s4
  } 
   \new Voice \relative c' { \clef F 
   s2. | s | \stemDown a8 a a a a a | d, r r4 r | g8 g g g g g | c8 c c c c c | f, f f f f f | bes bes bes bes bes bes | e, e e e e e | a a a a a a | d,[ d] s4
  }
 \addlyrics \with { alignAboveContext = "chords" } { \override LyricText.font-size = #-1.5 _ _ _ _ _ _ Dm \markup{\concat{Gm\super{7}}} _ _ _ _ _ C _ _ _ _ _ \markup{\concat{F\super{maj7}}} _ _ _ _ _ B♭ _ _ _ _ _ \markup{\concat{Em\super{7(♭5)}}} _ _ _ _ _ \markup{\concat{A\super{7}}} _ _ _ _ _ Dm }
>> >> >>
\new Staff \with {
      \omit TimeSignature
      \magnifyStaff #2/3
      firstClef = ##f
    } \relative c' 
 { \hide Staff.KeySignature \key f \major \clef bass
 {\stopStaff s2. s s \startStaff \hide Stem d8 s s s s s g, s s s s s c s s s s s f, s s s s s bes s s s s s e, s s s s s a s s s s s d,}} 
 >> }
\layout { line-width = #150 }
</score>|width=|align=center|caption=Bach adagio BWV 974 (after Marcello)}}

===Nineteenth century===
[[Franz Schubert]]'s Impromptu in E-flat major, D 899, contains harmonies that move in a modified circle of fifths:
{{Image frame|content=<score sound="1"> { <<
 \new PianoStaff <<
  \new Staff << \magnifyStaff #2/3
   \new Voice \relative c' {
   \key es \major \set Score.tempoHideNote = ##t \tempo 4 = 120 \time 3/4
   \mark \markup { \abs-fontsize #10 { \bold { Allegro } } }
   \times 2/3 {ges'8_\markup{\dynamic pp} bes aes} \times 2/3 {ges f ees} \times 2/3 {des ces bes} |
   \times 2/3 {ces ees aes} \stemDown \times 2/3 {ces ees, aes} \times 2/3 {ges ees ges} |
   \stemNeutral \times 2/3 {f ces' bes} \times 2/3 {aes ges f} \times 2/3 {ees des ces} |
   \times 2/3 {bes des ges} \stemDown \times 2/3 {bes des, ges} \times 2/3 {f des f} |
   \stemNeutral \times 2/3 {ees bes' aes} \times 2/3 {ges f ees} \times 2/3 {des ces bes} |
   \times 2/3 {aes ces f} \stemDown \times 2/3 {aes ces, f} \times 2/3 {ees ces ees} |
   \stemNeutral \times 2/3 {d aes' ges} \times 2/3 {f ees d} \times 2/3 {ces bes aes} |
   \times 2/3 {ges bes ees} \stemDown \times 2/3 {ges bes, ees} \times 2/3 {des bes des} |
   }
   \new Voice \relative c' { \stemUp
   s2. | s4 ces'2 | s2. | s4 bes2 |
   s2. | s4 aes2 | s2. | s4 ges2 |
   }
  \new Staff << \magnifyStaff #2/3
   \new Voice \relative c' {
   \key es \major \clef F \time 3/4 \stemUp \slurUp
   ees,,4( ees'2) | aes,4( aes'2) | des,,4( des'2) | ges,4( ges'2)
   ces,,4( ces'2) | f,4( f'2) | bes,,4( bes'2) | ees,4( ees'2)
   } 
   \new Voice \relative c' { \stemDown
   ees,,2. | aes2. | des,2. | ges2.
   ces,2. | f2. | bes,2. | ees2.
   } 
 >> >> >>
 \new Staff = "chords" \with {
      \omit TimeSignature
      \magnifyStaff #2/3
    } \relative c' 
 { \hide Staff.KeySignature \key ees \major \clef bass \hide Staff.Clef
 {\stopStaff s4 \startStaff \hide Stem ees,8 s s s s s aes s s s s s des, s s s s s ges s s s \stopStaff s s \startStaff ces, s s s s s f s s s s s bes, s s s s s ees}} 
\addlyrics \with { alignAboveContext = "chords" \override VerticalAxisGroup.nonstaff-relatedstaff-spacing =
      #'((basic-distance . 3)) } { \override LyricText.font-size = #-2 E♭m \markup{\concat{A♭m\super{7}}} \markup{\concat{D♭\super{7}}} \markup{\concat{G♭\super{maj7}}} \markup{\concat{C♭\super{maj7}}} \markup{\concat{Fm\super{7(♭5)}}}  \markup{\concat{B♭\super{7}}} E♭m }
 >>
 }
\layout { line-width = #150 }
</score>|width=|align=center|caption=}}

The [[Intermezzo]] movement from [[Mendelssohn]]'s [[String Quartet No. 2 (Mendelssohn)|String Quartet No.2]] has a short segment with circle-of-fifths motion (the ii° is substituted by iv):

{{Image frame|content=<score sound="1"> { <<
 \new ChoirStaff <<
  \relative c' { \magnifyStaff #3/4 \set Staff.midiInstrument = #"violin" \p
   \set Score.tempoHideNote = ##t \tempo 4 = 60 \time 2/4
   \mark \markup { \abs-fontsize #10 { \bold { Allegretto con moto} } }
    \repeat volta 2 { 
    e'4( f8. e32 d | c8) d-. e4 | a, d8.( c32 b | a8) b-. e,4 |
    e'( f8. e32 d | c8) d-. e4 | a, d8.( c32 b | e8) e, a, r }
   }
  \relative c' { \magnifyStaff #3/4 \set Staff.midiInstrument = #"pizzicato strings" 
   \time 2/4
    <e c'>8_"pizz." r <f d'> r | <g, f' b> r <g e' c'> r | <c f a> r <d a'> r | e f b, r |
    <c e a> r <d a'> r | <g, f' b> r <g e' c'> r | <c f a> r <d a'> r | e gis a r |
   }
  \relative c' { \magnifyStaff #3/4 \set Staff.midiInstrument = #"pizzicato strings" 
   \clef C \time 2/4
    <c e a>8_"pizz." r <d f a> r | <g, g'> r <g g'> r | <a f'> r <a f'> r | c d gis, r |
    <a e' c'> r <a f' d'> r | <g d' b'> r <g e' c'> r | <a f' a> r <a f'> r | c d c r |
   }
  \relative c' { \magnifyStaff #3/4 \set Staff.midiInstrument = #"pizzicato strings" 
   \clef F \time 2/4
    a8_"pizz." r d, r | g r c, r | f r d r | e e e, r |
    a' r d, r | g r c, r | f r d r | e, e' a, r |
   }
 >> 
 \new Staff = "chords" \with {
      \omit TimeSignature
      \magnifyStaff #3/4
    } \relative c' 
 { \hide Staff.KeySignature \clef F
 { \hide Stem a8 s d, s g s c, s f s }} 
\addlyrics \with { alignAboveContext = "chords" \override VerticalAxisGroup.nonstaff-relatedstaff-spacing =
      #'((basic-distance . 3)) } { \override LyricText.font-size = #-2 Am Dm \markup{\concat{G\super{7}}} C F }
>>
 }
\layout { line-width = #150 }
</score>|width=|align=center|caption=}}

Robert Schumann's "Child falling asleep" from his ''[[Kinderszenen]]'' uses the progression, changing it at the end—the piece ends on an A minor chord, instead of the expected [[Tonic (music)|tonic]] E minor.

{{Image frame|content=<score sound="1"> { <<
  \new Dynamics {
    s2 s s4 s \tempo 8 = 87 s-"rit." \tempo 8 = 82 s \tempo 8 = 77 s \tempo 8 = 72 s
  }
 \new PianoStaff <<
  \new Staff \with{ \magnifyStaff #3/4 } << 
   \new Voice \relative c' {
   \key e \minor \set Score.tempoHideNote = ##t \tempo 8 = 92 \time 2/4 \stemUp \slurUp \tieUp
    r4 b'8( fis'16 b, | b g'8.) b,8( fis'16 b, | b b'8.) c,8( b'16 c, | c a'8.^\startTextSpan) b,8( a'16 b, | b g'8.) a,8( g'16 a, | a fis'8.) e4 ~ \stopTextSpan e2 \fermata ~ e4 r \bar "|."
   }
   \new Voice \relative c' {
   \stemDown \tieDown
    s4 b'4 b b b c c b b a a a ~ a2 ~ a4 s4
   }
  \new Dynamics {
    \p
  }
  \new Staff \with{ \magnifyStaff #3/4 } <<
   \new Voice \relative c' { 
    \key e \minor \clef F \time 2/4 \stemUp \slurUp \tieUp
    g8( c16 b b fis'8.) | b,8( c16 b b fis'8.) | b,8( c16 b c b'8.) | c,8( d16 c b a'8.) | b,8( c16 b a g'8.) | a,8( b16 a ~ <a c e>4 ~ <a c e>2 \fermata ~ <a c e>4) r | 
   }
   \new Voice \relative c' { 
    \stemDown \tieDown
    e,4 <dis fis> | <e g> <dis fis> | <e g> a | <d, fis> g | <c, e> fis | <b, dis> e4 ~ e2 ~ <a, e'>4 s
   }
  \new Dynamics {
    s2 s s s\> s s s s \!
  }
 >> >> >>
 \new Staff = "chords" \with {
      \omit TimeSignature
      \magnifyStaff #2/3
    } \relative c' 
 { \hide Staff.KeySignature \hide Staff.Clef \key e \minor \clef bass
 { \hide Stem \stopStaff \once \hide \p s2 s \startStaff e,8 s a s d, s g s c, s fis s b, s e s \stopStaff}} 
\addlyrics \with { alignAboveContext = "chords" \override VerticalAxisGroup.nonstaff-relatedstaff-spacing = #'((basic-distance . 3)) } { \override LyricText.font-size = #-3 Em \markup{\concat{Am\super{9}}} \markup{\concat{D\super{7}}} \markup{\concat{G\super{maj9}}} \markup{\concat{C\super{maj7}}} \markup{\concat{F♯m\super{7♭5}}} \markup{\concat{B\super{7}}} Am}
>>
 }
\layout { line-width = #150 }
</score>|width=|align=center|caption=}}

In [[Wagner]]'s opera, ''[[Götterdämmerung]]'', a cycle of fifths progression occurs in the music which transitions from the end of the prologue into the first scene of Act 1, set in the imposing hall of the wealthy Gibichungs. "Status and reputation are written all over the motifs assigned to Gunther",<ref>[[Roger Scruton|Scruton, R.]] (2016, p. 121) ''The Ring of Truth: The Wisdom of Wagner's Ring of the Nibelung''. London, Allen Lane.</ref> chief of the Gibichung clan:

{{Image frame|content=<score sound="1"> {<<
 \new ChoirStaff <<
  \relative c' {
   \magnifyStaff #3/4 \set Score.tempoHideNote = ##t \tempo 4 = 60 \time 3/4 \set Staff.midiInstrument = #"trumpet" \transposition f'^"in F"
    \p \grace {s16 s} ees2( d4) | ees2( d4) | cis2. ~ cis4 r r | R2. | R | R | R | R | R
   }
  \new Staff \with{ \magnifyStaff #3/4 } <<
   \new Voice \relative c' { \override Hairpin.minimum-length = #3
    \set Staff.midiInstrument = #"trumpet" \transposition e'^"in E"
     \p \grace {s16 s} \hide \pp <g bes>2( <bes g>4) | <g bes>2( <bes g>4) | <bes d>4.( <g bes>8 <a c>4 | <bes d>2.) |<bes d>4.( <g bes>8 <a c>4 ) | <bes d>2 ees4-! | c2._"(marc.)" | s2. | R | R
    } 
   \new Voice \relative c' { \stemDown
    \hide \p \grace {s16 s} s2. | s | s | s | s | s2 ees4-! | c2. | d4-! bes2 | R2. | R2. \bar "|."
   } >> >>
  \relative c' {
   \magnifyStaff #3/4 \set Staff.midiInstrument = #"trombone" \transposition e^"in E"
   \p \grace {s16 s} \hide \pp g'2. ~ g ~ g ~ g | g ~ g2 bes4-! | g2._"(marc.)" | d'4-! bes2 | c2_"dim." r4 | \pp d2 r4 
   }
 \new ChoirStaff <<
  \relative c' {
   \magnifyStaff #3/4 \clef tenor \set Staff.midiInstrument = #"trombone"
    \p \grace {s16 s} aes2( g4) | aes2( g4) | fis2. ~ fis | fis ~ fis2 b4-! | g2._"(marc.)" | cis4-! b2 | b2_"dim." r4 | \pp ais2 r4
   }
  \relative c' {
   \magnifyStaff #3/4 \clef F \set Staff.midiInstrument = #"trombone"
    \p \grace {s16 s} \hide \pp <b, d>2. | <b d> | <b d> ~ <b d> | <b d> ~ <b d>2 <b g'>4-! | <e g>2._"(marc.)"  | <fis a>4-! <d fis>2 | <e g>_"dim." r4 | \pp <cis fis>2 r4
   } >>
  \relative c' {
   \magnifyStaff #3/4 \clef F \set Staff.midiInstrument = #"trombone"
    \p \grace {s16 s} \hide \pp f,,!2( g4) | f!2( g4) | gis2. ~ gis | g! ~ g2 e4 ~ e a8._"(marc.)"[ g16 fis8. e16] | d2 g4( | cis,2_"dim.") r4 | \pp fis2 r4
   }
  \relative c' {
   \magnifyStaff #3/4 \clef F \set Staff.midiInstrument = #"timpani"
    \p \grace {b,16 b} b4 r r | \grace {b16 b} b4 r r | \grace {b16 b} b4 r r | r b b | \grace {b16 b} b4 r r | r b f' | b, r r | R2. | R | R
   }
  \relative c' {
   \magnifyStaff #3/4 \clef F \set Staff.midiInstrument = #"tuba"
    \p \grace {s16 s} R2. | R | R | R | R | r4 r e,, ~ e a8._"(marc.)"[ g16 fis8. e16] | d2 g4( cis,2_"dim." ) r4 | \pp fis2 r4
   }
 >>  }
\layout { line-width = #150 }
</score>|width=|align=center|caption=}}

===Jazz and popular music===
The enduring popularity of the circle of fifths as both a form-building device and as an expressive musical trope is evident in the number of "[[Standard (music)|standard]]" popular songs composed during the twentieth century. It is also favored as a vehicle for improvisation by jazz musicians, as the circle of fifths helps songwriters understand intervals, chord-relationships and progressions.
* [[Bart Howard]], "[[Fly Me to the Moon]]"
{{blockquote|The song opens with a pattern of descending phrases – in essence, the hook of the song – presented with a soothing predictability, almost as if the future direction of the melody is dictated by the opening five notes. The harmonic progression, for its part, rarely departs from the circle of fifths.{{sfn|Gioia|2012|p=115}}}}
* [[Jerome Kern]], "[[All the Things You Are]]"{{sfn|Gioia|2012|p=16}}
* [[Ray Noble]], "[[Cherokee (Ray Noble song)|Cherokee]]." Many jazz musicians have found this particularly challenging as the [[middle eight]] progresses so rapidly through the circle, "creating a series of [[ii–V–I progression|II–V–I progressions]] that temporarily pass through several [[tonality|tonalities]]."<ref>Scott, Richard J. (2003, p. 123) ''Chord Progressions for Songwriters''. Bloomington Indiana, Writers Club Press.</ref>
* Kosma, Prévert and Mercer, "[[Autumn Leaves (1945 song)|Autumn Leaves]]"<ref>{{cite book|last1=Kostka|first1=Stefan|author1-link=Stefan Kostka|first2=Dorothy|last2=Payne|first3=Byron|last3=Almén|title=Tonal Harmony with an Introduction to Twentieth-century Music|edition=7th|year=2013|location=New York|publisher=McGraw-Hill|isbn=978-0-07-131828-0|pages=46, 238}}</ref>
* [[The Beatles]], "[[You Never Give Me Your Money]]"<ref>"You Never Give Me Your Money" (1989, pp. 1099–1100, bars 1–16) ''The Beatles Complete Scores''. Hal Leonard.</ref>{{Primary source inline|date=October 2021}}
*[[Mike Oldfield]], "[[Incantations (album)|Incantations]]"<ref>{{cite web|url=https://tubular.net/articles/1980_06/Mike-Oldfield-by-Tim-Oakes|title=Mike Oldfield|first=Tim|last=Oakes|date=June 1980|work=International Musician and Recording World|access-date=19 February 2021|via=Tubular.net}}</ref>
*[[Carlos Santana]], "[[Europa (Earth's Cry Heaven's Smile)]]"{{Citation needed|date=March 2020}}
*[[Gloria Gaynor]], "[[I Will Survive]]"<ref>Fekaris, D. and Perren, F. J. (1978) "I Will Survive". Polygram International Publishing.</ref>{{Primary source inline|date=October 2021}}
*[[Pet Shop Boys]], "[[It's a Sin]]"<ref>Tennant, N. and Lowe, C. (1987, bars 1–8) "It's a Sin." Sony/ATV Music Publishing (UK) Ltd.</ref>{{Primary source inline|date=October 2021}}
*[[Donna Summer]], "[[Love to Love You Baby (song)|Love to Love you, Baby]]"<ref>[[Giorgio Moroder|Moroder, G.]], [[Pete Bellotte|Bellote, P.]] and [[Donna Summer|Summer, D.]] (1975, bars 11–14) "Love to Love you, Baby" 1976, Bulle Music</ref>{{Primary source inline|date=October 2021}}

==Related concepts==
===Diatonic circle of fifths===
{{Main|Circle progression}}
The diatonic circle of fifths is the circle of fifths encompassing only members of the diatonic scale. Therefore, it contains a diminished fifth, in C major between B and F. See [[structure implies multiplicity]].
The [[circle progression]] is commonly a circle of fifths through the diatonic chords, including one [[Diminished triad|diminished chord]]. A circle progression in C major with chords I–IV–vii<sup>o</sup>–iii–vi–ii–V–I is shown below.
:<score sound="1">
    {
      \new PianoStaff <<
        \new Staff <<
            \new Voice \relative c' {
                \stemUp e2 f d e c d b c
                }
            \new Voice \relative c' {
                \stemDown c2 c b b a a g g
                }
            >>
        \new Staff <<
            \new Voice \relative c' {
                \clef F
                \stemUp g2 a f g e f d e 
                }
            \new Voice \relative c {
                \stemDown c f, b e, a d, g c,
                }
            >>
    >> }
</score>

===Chromatic circle===
{{Main|Chromatic circle}}
The circle of fifths is closely related to the [[chromatic circle]], which also arranges the equal-tempered pitch classes of a particular tuning in a circular ordering. A key difference between the two circles is that the [[chromatic circle]] can be understood as a continuous space where every point on the circle corresponds to a conceivable [[pitch class]], and every conceivable pitch class corresponds to a point on the circle. By contrast, the circle of fifths is fundamentally a ''discrete'' structure arranged through distinct [[Interval (music)|intervals]], and there is no obvious way to assign pitch classes to each of its points. In this sense, the two circles are mathematically quite different.

However, for any positive integer ''N'', the [[pitch class]]es in ''N''-tone equal temperament can be represented by the [[cyclic group]] of order ''N'', or equivalently, the [[modular arithmetic|residue classes]] modulo equal to ''N'', <math> \mathbb{Z}/N\mathbb{Z} </math>. In twelve-tone equal temperament, the group <math> \mathbb{Z}_{12} </math> has four generators, which can be identified with the ascending and descending semitones and the ascending and descending perfect fifths. The semitonal generator gives rise to the [[chromatic circle]] while the perfect fourth and perfect fifth give rise to the circle of fifths. In most other tunings, such as in [[31 equal temperament]], many more intervals can be used as the generator, and many more circles are possible as a result.

===Relation with chromatic scale===
{{Main|Chromatic scale}}

[[Image:Pitch class space star.svg|thumb|right|The circle of fifths drawn within the chromatic circle as a [[star polygon|star]] [[dodecagram]].{{sfn|McCartin|1998|p=364}}]]
The circle of fifths, or fourths, may be mapped from the [[chromatic scale]] by [[multiplication]], and vice versa. To map between the circle of fifths and the chromatic scale (in [[integer notation]]) multiply by 7 ([[twelve-tone technique|M7]]), and for the circle of fourths multiply by 5 (P5).

In twelve-tone equal temperament, one can start off with an [[order theory|ordered]] 12-tuple ([[tone row]]) of integers:
: (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
representing the notes of the chromatic scale: 0 = C, 2 = D, 4 = E, 5 = F, 7 = G, 9 = A, 11 = B, 1 = C{{music|sharp}}, 3 = D{{music|sharp}}, 6 = F{{music|sharp}}, 8 = G{{music|sharp}}, 10 = A{{music|sharp}}. Now multiply the entire 12-tuple by 7:
: (0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77)
and then apply a [[modulo operation|modulo]] 12 reduction to each of the numbers (subtract 12 from each number as many times as necessary until the number becomes smaller than 12):
: (0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5)
which is equivalent to
: (C, G, D, A, E, B, F{{music|sharp}}, C{{music|sharp}}, G{{music|sharp}}, D{{music|sharp}}, A{{music|sharp}}, F)
which is the circle of fifths.
This is [[enharmonic]]ally equivalent to:
: (C, G, D, A, E, B, G{{music|flat}}, D{{music|flat}}, A{{music|flat}}, E{{music|flat}}, B{{music|flat}}, F).

===Enharmonic equivalents, theoretical keys, and the spiral of fifths===
{{main|Theoretical key}}
[[File:Spiral of fifths.gif|thumb|right|If enharmonic notes are not equivalent, as in just intonation, fifths form a spiral, not a circle.]]
[[File:Pythagorean_tuning_geometric.svg|thumb|right|A sequence of twelve just fifths on a [[chromatic circle]] fail to close (the size of the gap is the [[Pythagorean comma]]), resulting in a "broken" circle of fifths.]]

[[Equal temperament]] tunings do not use the exact 3:2 ratio of frequencies that defines a perfect fifth, whereas [[just intonation]] uses this exact ratio. Ascending by fifths in equal temperament leads to a return to the starting pitch class—starting with a C and ascending by fifths leads to another C after a certain number of iterations. This does not occur if an exact 3:2 ratio is used (just intonation). The adjustment made in equal temperament tuning is called the [[Pythagorean comma]]. Because of this difference, pitches that are enharmonically equivalent in equal temperaments (such as C{{music|sharp}} and D{{Music|flat}} in 12-tone equal temperament, or C{{music|sharp}} and D{{Music|doubleflat}} in [[19 equal temperament]]) are not equivalent when using just intonation.

In just intonation the sequence of fifths can therefore be visualized as a spiral, not a circle—a sequence of twelve fifths results in a "[[comma pump]]" by the Pythagorean comma, visualized as going up a level in the spiral. See also {{slink||Circle closure in non-equal tuning systems}}.

Without enharmonic equivalences, continuing a sequence of fifths results in notes with double accidentals (double sharps or double flats), or even triple or quadruple accidentals. In most equal temperament tunings, these can be replaced by enharmonically equivalent notes.

Keys with double or triple sharps and flats in key signatures are called [[theoretical key]]s; they are redundant in 12-tone equal temperament, and so their use is extremely rare, but if the number of notes per octave is not a multiple of 12, they are distinguished. Notation in these cases is not standardized.
<score sound="1">
\relative c' {
 \key gis \major
\set Staff.printKeyCancellation = ##f
 <gis' bis dis>
\bar "||"
 \key dis \major
 <dis fisis ais>
\bar "||"
 \key fes \major
 <fes as ces>
\bar "||"
 \key beses \major
 <beses des fes>
}
</score>

The default behaviour of [[LilyPond]] (pictured above) writes single sharps or flats in the circle-of-fifths order, before proceeding to double sharps or flats. This is the format used in [[John Foulds]]' ''A World Requiem'', Op. 60,<ref>{{Cite web|url=https://imslp.org/wiki/A_World_Requiem%2C_Op.60_(Foulds%2C_John)|title=Foulds, John, ''A World Requiem'', Op. 60, pp. 153ff.}}</ref> which ends with the key signature of G{{music|#}} major, as displayed above. 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}}.

Single sharps or flats in the key signature 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 [https://archive.org/details/supplementtotheo00rege/page/42/mode/2up pp. 42–45]. 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}}. The convention of LilyPond and Foulds would suppress the initial B{{music|b}}.
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>{{Cite web|url=https://www.hickeys.com/music/brass/brass_ensembles/brass_quintets/products/sku035994-ewald-victor-quintet-no-4-in-ab-op-8.php|title=Ewald, Victor, ''Quintet No 4 in A{{music|b}}'', Op. 8 for Brass Quintet &#91;211.01&#93;}}</ref> by the program [[Finale (scorewriter)|Finale]], and by some theoretical works.

==See also==
* [[Approach chord]]
* [[Sonata form]]
* [[Well temperament]]
* [[Circle of fifths text table]]
* [[Pitch constellation]]
* [[Multiplicative group of integers modulo n]]
*[[Multiplication (music)]]
*[[Circle of thirds]]
*[[Music written in all major or minor keys]]

==Notes==
{{notelist}}

==References==
{{Reflist}}

===Sources===
* {{cite book |last=Barnett |first=Gregory |year=2002 |chapter=Tonal Organization in Seventeenth-century Music Theory. |title=The Cambridge History of Western Music Theory |editor=Thomas Christensen |pages=407–455 |location=Cambridge |publisher=Cambridge University Press}}
* {{cite book|last=Gioia|first=Ted|author-link=Ted Gioia|year=2012|title=[[The Jazz Standards: A Guide to the Repertoire]]|publisher=Oxford University Press|isbn=9780199769155}}
* {{cite book|last=Goldman|first=Richard Franko|author-link=Richard Franko Goldman|year=1965|title=Harmony in Western Music|url=https://archive.org/details/harmonyinwestern00gold|url-access=registration|location=New York|publisher=W. W. Norton}}
* {{cite journal|last=Jensen|first=Claudia R.|date=Summer 1992|jstor=831450|title=A Theoretical Work of Late Seventeenth-Century Muscovy: Nikolai Diletskii's "Grammatika" and the Earliest Circle of Fifths|journal=[[Journal of the American Musicological Society]]|volume=45|issue=2|pages=305–331|doi=10.2307/831450}}
* {{cite book |last=Lester |first=Joel |year=1989 |title=Between Modes and Keys: German theory 1592–1802 |location=Stuyvesant |publisher=Pendragon Press}}
* {{cite journal|last=McCartin|first=Brian J.|date=November 1998|title=Prelude to Musical Geometry|journal=[[The College Mathematics Journal]]|volume=29|issue=5|pages=354–370|doi=10.1080/07468342.1998.11973971|url=http://www.maa.org/pubs/cmj_Nov98.html|jstor=2687250|access-date=2008-07-29|archive-url=https://web.archive.org/web/20080517071420/http://www.maa.org/pubs/cmj_Nov98.html|archive-date=2008-05-17|url-status=dead|url-access=subscription}}<!--The link does not reach the article, but only the list of abstracts for that issue of the journal.-->
* {{wikicite|ref={{harvid|Nattiez|1990}}|reference=[[Jean-Jacques Nattiez|Nattiez, Jean-Jacques]] (1990). ''Music and Discourse: Toward a Semiology of Music'', translated by Carolyn Abbate. Princeton, New Jersey: Princeton University Press. {{ISBN|0-691-02714-5}}. (Originally published in French, as ''Musicologie générale et sémiologie''. Paris: C. Bourgois, 1987. {{ISBN|2-267-00500-X}}).}}
* {{cite book|last=Taruskin|first=Richard|author-link=Richard Taruskin|year=2010|title=The Oxford History of Western Music: Music in the Seventeenth and Eighteenth Centuries|publisher=Oxford University Press}}

==Further reading==
* [[Vincent d'Indy|D'Indy, Vincent]] (1903). ''Cours de composition musicale''. Paris: A. Durand et fils.
* Lester, Joel. [https://books.google.com/books?id=t2xAAIK7jd0C&q=heinichen&pg=PA110 ''Between Modes and Keys: German Theory, 1592–1802'']. 1990.
* Miller, Michael. [https://books.google.com/books?id=sTMbuSQdqPMC ''The Complete Idiot's Guide to Music Theory, 2nd ed'']. [Indianapolis, IN]: Alpha, 2005. {{ISBN|1-59257-437-8}}.
* Purwins, Hendrik (2005)."[http://ccrma.stanford.edu/~purwins/purwinsPhD.pdf Profiles of Pitch Classes: Circularity of Relative Pitch and Key—Experiments, Models, Computational Music Analysis, and Perspectives]". Ph.D. thesis. Berlin: [[Berlin Institute of Technology|Technische Universität Berlin]].
* Purwins, Hendrik, Benjamin Blankertz, and Klaus Obermayer (2007). "[http://www.ccarh.org/publications/cm/15/cm15-05-purwins.pdf Toroidal Models in Tonal Theory and Pitch-Class Analysis]". in: ''Computing in Musicology'' 15 ("Tonal Theory for the Digital Age"): 73–98.

{{Key (music)}}
{{Pitch space}}
{{Tonality}}
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[[Category:Circle of fifths| ]]
[[Category:Harmony]]
[[Category:Musical keys]]
[[Category:Tonality]]