Editing Circle of fifths

{{Short description|Relationship among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys;geometrical representation of relationships among the 12 pitch classes of the chromatic scale in pitch class space}}
[[Image:Circle of fifths deluxe 4.svg|thumb|right|400px|Circle of fifths showing major and minor keys]]
[[Image:Diletsky circle.jpg|thumb|right|[[Nikolay Diletsky]]'s circle of fifths in ''Idea grammatiki musikiyskoy'' (Moscow, 1679)]]

In [[music theory]], the '''circle of fifths''' is a way of organizing the 12 [[chromatic scale|chromatic]] [[Pitch (music)|pitches]] as a sequence of [[perfect fifth]]s. If C is chosen as a starting point, the sequence is: C, G, D, A, E, B (=C{{music|flat}}), F{{music|sharp}} (=G{{music|flat}}), C{{music|sharp}} (=D{{music|flat}}), A{{music|flat}}, E{{music|flat}}, B{{music|flat}}, F. Continuing the pattern from F returns the sequence to its starting point of C. This order places the most closely related [[key signature]]s adjacent to one another. It is usually illustrated in the form of a circle.

== 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 a clockwise progression. [[Musician]]s and [[composer]]s often use the circle of fifths to describe the musical relationships between pitches. Its design is helpful in [[Musical composition|composing]] and [[Harmony|harmonizing]] [[melodies]], building [[chord (music)|chords]], and modulating to different [[key (music)|keys]] within a composition.<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>

Using the system of [[just intonation]], a perfect fifth consists of two pitches with a frequency ratio of 3:2, but generating a twelve perfect fifths in this way does not result in a return to the [[pitch class]] of the starting note. To adjust for this, instruments are generally tuned with the [[equal temperament]] system. Twelve equal-temperament fifths lead to a note exactly seven [[octave]]s above the initial tone—this results in a perfect fifth that is equivalent to seven equal-temperament [[semitone]]s.

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 [[Perfect fifth|fifths]]. The key signatures associated with those pitches also change: the key of G has one sharp, the key of D has 2 sharps, and so on. Similarly, 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 twelve 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.

<gallery widths="250px" heights="50px" perrow="1">
Image:Circle of fifths ascending within octave.svg|{{audio|Circle of fifths ascending within octave.mid|Play circle of fifths clockwise within one octave}}
Image:Circle of fifths descending within octave.png|{{audio|Circle of fifths descending within octave.mid|Play circle of fifths counterclockwise within one octave}}
</gallery>

==Structure and use==

===Diatonic key signatures===
Each of the twelve pitches 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 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.

The circle of fifths is used to organize and describe the [[diatonic function|harmonic function]] of [[chord (music)|chords]]. 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 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 {{audio|Ii-V-I turnaround in C.mid|Play subdominant, supertonic seventh, and supertonic chords}} illustrating the similarity between them.]]
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."<ref name="Goldman 68">{{harvnb|Goldman|1965|p=68}}.</ref> 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<ref>{{harvnb|Goldman|1965|loc=chapter 3}}.</ref> 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.<ref name="Nattiez 226">{{harvnb|Nattiez|1990|p=226}}.</ref> (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. [[Equal temperament]] tuning produces fifths that return to a tone exactly seven [[octave]]s above the initial tone and makes the frequency ratio of each half step the same. An equal-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 at a ratio of 3:2.

Ascending by 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]]. In Pythagorean tuning, this problem is solved by markedly shortening 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. The [[quarter-comma meantone]] tuning system 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. Other tuning systems use up to [[53 equal temperament|53&nbsp;tones]] (the original 12&nbsp;tones and 42&nbsp;more between them) in order to close the circle of fifths.

==History==
[[Image:Heinichen musicalischer circul.png|thumb|[[Johann David Heinichen|Heinichen]]'s musical circle (German: ''Musicalischer Circul'') (1711)]]

Some sources imply that [[Pythagoras]] invented the circle of fifths in the sixth century B.C. but there is no proof of this.<ref>{{cite web |title=The Circle of Fifths: A Brief History |author=Anon.|url=https://www.dummies.com/art-center/music/the-circle-of-fifths-a-brief-history/ |publisher=Dummies.com}}{{Failed verification|date=May 2020|reason=Unreliable source claims Pythagoras said this, but fails to verify lack of proof of the claim}}</ref><ref>{{cite web |url=https://www.classicfm.com/discover-music/music-theory/what-is-the-circle-of-fifths/}}{{Failed verification|date=May 2020|reason=Unreliable source merely says that the circle of fifths "all started out (probably) with Pythagoras"}}</ref><ref>{{cite web |title=The Circle of Fifths |author=Anon.|url=https://www.violinschool.com/circle-of-fifths/ |date=2016}}{{Failed verification|date=May 2020|reason=No claim that Pythagoras described the circle of fifths, nor that there is no proof of the claim}}</ref> Pythagoras was primarily concerned with the theoretical science of harmonics and is credited with having devised a system of tuning based upon the interval of a fifth, but did not tune more than eight notes, and left no written records of his work.<ref>{{cite journal |last1=Fraser |first1=Peter A. |title=The Development of Musical Tuning Systems |date=2001 |pages=9, 13 |url=https://web.archive.org/web/20130701202500/http://www.midicode.com/tunings/Tuning10102004.pdf |access-date=24 May 2020}} (archive from 1 July 2013).</ref>

In the late 1670s Ukrainian composer and theorist [[Nikolay Diletsky]] wrote a treatise on composition entitled ''Grammatika'', "the first of its kind, aimed at teaching a Russian audience how to write Western-style polyphonic compositions." It taught how to write ''kontserty'', [[polyphonic]] [[a cappella]] works usually based on [[liturgical]] texts and created by putting together musical sections with contrasting rhythm, meter, melodic material and vocal groupings. Diletsky intended his treatise to be a guide to composition using rules of [[music theory]]. The first circle of fifths appears in the Grammatika and it was used for students as a composition tool.<ref name="Jensen">{{harvnb|Jensen|1992|pp=306–307}}.</ref>

==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>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 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:[[File:Diatonic scale and the circle of fifths derived from it - major.wav|thumb|Diatonic scale and the circle of fifths derived from it – major]] [[File:Diatonic scale and the circle of fifths derived from it - major.png|thumb|center|500px|Diatonic scale and the circle of fifths derived from it – major]]

And from the (natural) minor scale:[[File:Diatonic scale and the circle of fifths derived from it - minor.wav|thumb|Diatonic scale and the circle of fifths derived from it – minor]] [[File:Diatonic scale and the circle of fifths derived from it - minor.png|thumb|center|500px|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:[[File:Circle of fifths chord progression - major.wav|thumb|Circle of fifths chord progression – major]] [[File:Circle of fifths chord progression - major.png|thumb|center|500px|Circle of fifths chord progression – major]]

And over the minor:[[File:Circle of fifths chord progression - minor.wav|thumb|Circle of fifths chord progression – minor]] [[File:Circle of fifths chord progression - minor.png|thumb|center|500px|Circle of fifths chord progression – minor]]

Adding sevenths to the chords creates a greater sense of forward momentum to the harmony:[[File:Circle of fifths chord progression - minor with added sevenths.wav|thumb|Circle of fifths chord progression – minor with added sevenths]] [[File:Circle of fifths chord progression - minor with added sevenths.png|thumb|center|500px|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."<ref>Taruskin, R. (2010, p. 184) ''The Oxford History of Western Music: Music in the Seventeenth and Eighteenth centuries''. Oxford University Press.</ref>

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:[[File:Bach from Cantata 51 bass line.wav|thumb|Bach from Cantata 51 bass line]][[File:Bach from Cantata 51 bass line.png|thumb|center|500px|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.

[[File:Handel Passacaille from Suite in G minor bars 1-4.wav|thumb|Handel Passacaille from Suite in G minor bars 1–4]][[File:Handel Passacaille from Suite in G minor bars 1-4.png|thumb|center|500px|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."<ref>Taruskin, R. (2010, p. 188) ''The Oxford History of Western Music: Music in the Seventeenth and Eighteenth centuries''. Oxford University Press.</ref> Striking passages that illustrate the use of sevenths occur in the aria "Pena tiranna" in [[Handel]]'s 1715 opera ''[[Amadigi di Gaula]]'' :[[File:Handel, aria "Pena tiranna" from Amadigi. Orchestral introduction. 01.wav|thumb|Handel, aria "Pena tiranna" from Amadigi. Orchestral introduction.]] [[File:Handel, aria "Pena tiranna" from Amadigi. Orchestral introduction.png|thumb|center|500px|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]]. [[File:Bach adagio BWV 974 (after Marcello).wav|thumb|Bach adagio BWV 974 (after Marcello)]][[File:Bach adagio BWV 975 (after Marcello).png|thumb|center|500px|Bach adagio BWV 974 (after Marcello)]]

===Nineteenth century===

During the nineteenth century, composers made use of the circle of fifths to enhance the expressive character of their music. [[Franz Schubert]]'s poignant Impromptu in E flat major, D899, contains such a passage:[[File:Schubert Impromptu in E flat 01.wav|thumb|Schubert Impromptu in E flat]] [[File:Schubert Impromptu in E flat 02.png|thumb|center|500px|Schubert Impromptu in E flat]] – as does the [[Intermezzo]] movement from [[Mendelssohn]]'s [[String Quartet No. 2 (Mendelssohn)|String Quartet No.2]]:[[File:Mendelssohn String Quartet 2, 3rd movement (Intermezzo).wav|thumb|Mendelssohn String Quartet 2, 3rd movement (Intermezzo)]][[File:Mendelssohn String Quartet 2, 3rd movement (Intermezzo).png|thumb|center|450px|Mendelssohn String Quartet 2, 3rd movement (Intermezzo)]]Robert Schumann's evocative "Child falling asleep" from his ''[[Kinderszenen]]'' springs a surprise at the end of the progression: the piece ends on an A minor chord, instead of the expected [[Tonic (music)|tonic]] E minor.[[File:Schumann, Kind im Einschlummern (Child falling asleep) 01.wav|thumb|Schumann, Kind im Einschlummern (Child falling asleep)]] [[File:Schumann, Kind im Einschlummern.png|thumb|center|500px|Schumann, Kind im Einschlummern]]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>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:[[File:Wagner, Gotterdamerung, transition between the end of the Prologue and Act 1 scene 1.wav|thumb|Wagner, Götterdämmerung, transition between the end of the Prologue and Act 1 scene 1]] [[File:Wagner, Gotterdamerung, transition between the end of the Prologue and Act 1 scene 1.png|thumb|center|500px|Wagner, Götterdämmerung, transition between the end of the Prologue and Act 1 scene 1]] [[Ravel]]'s [[Pavane pour une infante défunte|"Pavane for a Dead Infanta"]], uses the cycle of fifths to evoke Baroque harmony to convey regret and nostalgia for a past era. The composer described the piece as "an evocation of a [[pavane]] that a little princess ([[infanta]]) might, in former times, have danced at the Spanish court.":<ref>Andres, Robert, "An introduction to the solo piano music of Debussy and Ravel", BBC Radio 3, accessed 17 November 2011</ref>[[File:Ravel Pavane pour une infante defunte.png|thumb|center|700px|Ravel Pavane pour une Infante Défunte]]

===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.
* [[Bart Howard]], "[[Fly Me to the Moon]]"
{{quote|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.<ref>Gioa, T. (2012, p.115) The Jazz Standards; a Guide to the repertoire. Oxford University Press.</ref>}}
* [[Jerome Kern]], "[[All the Things You Are]]"<ref>Gioa, T. (2012, p.16) The Jazz Standards; a Guide to the repertoire. Oxford University Press.</ref>
* [[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>
* Kosmo, Prevert and Mercer, "[[Autumn Leaves (1945 song)|Autumn Leaves]]"<ref name="Kostka23846">{{cite book|last1=Kostka|first1=Stefan|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, p1099-1100, bars 1-16) ''The Beatles Complete Scores''. Hal Leonard.</ref>
*[[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>
*[[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>
*[[Donna Summer]], "[[Love to Love You Baby (song)|Love to Love you, Baby]]"<ref>Moroder, G., Bellote, P. and Summer, D. (1975, bars 11-14) "Love to Love you, Baby" copyright 1976, Bulle Music Inc.</ref>

==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 lang="lilypond" %vorbis="1"%%T257066%>
    {
      \new PianoStaff <<
        \new Staff <<
            \new Voice \relative c' {
                \clef treble \time 4/4
                \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 bass \time 4/4
                \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 twelve equal-tempered pitch classes 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, 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, the twelve equal-tempered [[pitch class]]es can be represented by the [[cyclic group]] of order twelve, or equivalently, the [[modular arithmetic|residue classes]] modulo twelve, <math> \mathbb{Z}/12\mathbb{Z} </math>. 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 fifth gives rise to the circle of fifths.

===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).

Here is a demonstration of this procedure. 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.
Note that 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 and theoretical keys===
{{main|Theoretical key}}
The key signatures found on the bottom of the circle of fifths diagram, such as D{{music|flat}} major, are often written one way in flats and in another way using sharps. These keys are easily interchanged using enharmonic equivalents. Enharmonic means that the notes sound the same, but are written differently. For example, the key signature of D{{Music|flat}} major, with five flats, contains the same sounding notes, enharmonically, as C{{music|sharp}} major (seven sharps).

After C{{music|sharp}} comes the key of G{{music|sharp}} (following the pattern of being a fifth higher, and, coincidentally, enharmonically equivalent to the key of A{{music|flat}}). The "eighth sharp" is placed on the F{{music|sharp}}, to make it F{{music|doublesharp}}. The key of D{{music|sharp}}, with nine sharps, has another sharp placed on the C{{music|sharp}}, making it C{{music|doublesharp}}. The same is true for key signatures with flats; The key of E (four sharps) is equivalent to the key of F{{music|flat}} (again, one fifth below the key of C{{music|flat}}, following the pattern of flat key signatures). The last flat is placed on the B{{music|flat}}, making it B{{music|doubleflat}}. Such keys with double accidentals in the key signatures are called [[theoretical key]]s: the appearance of their key signatures is extremely rare, but they are sometimes tonicised in the course of a work (particularly if the home key was already heavily sharped or flatted).
<!-- <score %vorbis="1"%%T257066%>
\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>
-->

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 (flats) in the circle-of-fifths order, before proceeding to the double sharps. 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, [https://imslp.simssa.ca/files/imglnks/usimg/5/59/IMSLP527245-PMLP756657-JFoulds_A_World_Requiem,_Op.60_vocalscore.pdf pp. 153ff.]) 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}}.
*The single sharps or flats 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 [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, by the program [[Finale (software)]],<ref>https://www.hickeys.com/music/brass/brass_ensembles/brass_quintets/products/sku035994-ewald-victor-quintet-no-4-in-ab-op-8.php</ref> 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]]

==Notes==
{{Reflist}}

==References==
* {{cite book|last=Goldman|first=Richard Franko|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=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 (Summer)|pages=305–331}}
* {{cite journal|last=McCartin|first=Brian J.|date=1998|title=Prelude to Musical Geometry|journal=The College Mathematics Journal|volume=29|issue=5 (November)|pages=354–370|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}}<!--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, NJ: 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}}).}}

==Further reading==
* D'Indy, Vincent (1903). ''Cours de composition musicale''. Paris: A. Durand et fils.
* Lester, Joel. [https://books.google.com/books?id=t2xAAIK7jd0C&pg=PA110&vq=heinichen&dq=intitle:modes+intitle:and+intitle:keys+inauthor:lester&lr=&as_brr=0&source=gbs_search_r&cad=1_1&sig=ACfU3U00ygBnXabE0oXQKUHQ2SazAHjFRw#PPA110,M1 ''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.

==External links==
* [http://mdecks.com/graphs/mcircle.php Decoding the Circle of Vths]
* [https://www.fachords.com/tools/circle-of-fifths/ Interactive Circle Of Fifths with guitar fretboard]
* [http://guitardashboard.com/ An open-source interactive circle of fifths for guitarists]
* [https://newt.phys.unsw.edu.au/~jw/music/circle/ Circle of Fourths: an orchestral overture using the circle as a chord and a melodic element]

{{Key (music)}}
{{Pitch space}}
{{Tonality}}
{{Authority control}}

{{DEFAULTSORT:Circle Of Fifths}}
[[Category:Circle of fifths| ]]
[[Category:Harmony]]
[[Category:Musical keys]]
[[Category:Tonality]]