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| [[Image:Circle of fifths deluxe 4.svg|thumb|right|400px|Circle of fifths showing major and minor keys]]
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| [[Image:Diletsky circle.jpg|thumb|right|[[Nikolay Diletsky]]'s circle of fifths in ''Idea grammatiki musikiyskoy'' (Moscow, 1679)]]
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| In [[music theory]], the '''circle of fifths''' (or '''[[Circle of fifths#Structure and use|circle of fourths]]''') is a visual representation of the relationships among the 12 tones of the [[chromatic scale]], their corresponding [[key signature]]s, and the associated [[major and minor]] keys. More specifically, it is a [[Geometry|geometrical]] representation of relationships among the 12 [[pitch class]]es of the chromatic scale in [[pitch class space]].
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| == Definition ==
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| The term '[[Perfect fifth|fifth]]' defines an interval or mathematical ratio which is the closest and most [[Consonance and dissonance|consonant]] non-octave interval. The circle of fifths is a sequence of pitches or key tonalities, represented as a circle, in which the next pitch is found seven semitones higher than the last. [[Musician]]s and [[composer]]s use the circle of fifths to understand and describe the musical relationships among some selection of those pitches. The circle's 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>
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| At the top of the circle, the [[Key (music)|key]] of C Major has no [[Sharp (music)|sharps]] or [[Flat (music)|flats]]. Starting from the apex and proceeding clockwise by ascending [[Perfect fifth|fifths]], the key of G has one sharp, the key of D has 2 sharps, and so on. Similarly, proceeding counterclockwise from the apex by descending fifths, the key of F has one flat, the key of B{{music|flat}} has 2 flats, and so on. At the bottom of the circle, the sharp and flat keys overlap, showing pairs of [[enharmonic]] key signatures.
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| Starting at any [[Pitch (music)|pitch]], ascending by the [[interval (music)|interval]] of an [[Equal temperament|equal tempered]] [[perfect fifth|fifth]], one passes all twelve tones clockwise, to return to the beginning pitch class. To pass the twelve tones counterclockwise, it is necessary to ascend by [[perfect fourth]]s, rather than fifths. (To the ear, the sequence of fourths gives an impression of settling, or resolution. (see [[Cadence (music)|cadence]]))
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| <gallery widths="250px" heights="50px" perrow="1"> | |
| Image:Circle of fifths ascending within octave.png|{{audio|Circle of fifths ascending within octave.mid|Play circle of fifths clockwise within one octave}}
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| Image:Circle of fifths descending within octave.png|{{audio|Circle of fifths descending within octave.mid|Play circle of fifths counterclockwise within one octave}}
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| </gallery> | |
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| ==Structure and use==
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| Pitches within the chromatic scale are related not only by the number of semitones between them within the chromatic scale, but also related harmonically within the circle of fifths. Reversing the direction of the circle of fifths gives the '''circle of fourths'''. Typically the "circle of fifths" is used in the analysis of classical music, whereas the "circle of fourths" is used in the analysis of Jazz music, but this distinction is not exclusive. Since fifths and fourths are intervals composed respectively of 7 and 5 semitones, the [[circumference]] of a circle of fifths is an interval as large as 7 octaves (84 semitones), while the circumference of a circle of fourths equals only 5 octaves (60 semitones).
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| {{Wide image|Octaves versus fifths Cuisenaire rods.png|1600px|Octaves (7 × 1200 {{=}} 8400) versus fifths (12 × 700 {{=}} 8400), depicted as with [[Cuisenaire rods]] (red (2) is used for 1200, black (7) is used for 700).|400px|center}}
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| ===Diatonic key signatures===
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| The circle is commonly used to represent the relationship between [[diatonic scale]]s. Here, the letters on the circle are taken to represent the major scale with that note as tonic. The numbers on the inside of the circle show how many sharps or flats the [[key signature]] for this scale has. Thus a major scale built on A has 3 sharps in its key signature. The major scale built on F has 1 flat.
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| For [[minor scale]]s, rotate the letters counter-clockwise by 3, so that, e.g., A minor has 0 sharps or flats and E minor has 1 sharp. (See [[relative key]] for details.) A way to describe this phenomenon is that, for any major key [e.g. G major, with one sharp (F#) in its diatonic scale], a scale can be built beginning on the sixth (VI) degree (relative minor key, in this case, E) containing the same notes, but from E - E as opposed to G - G. Or, G-major scale (G - A - B - C - D - E - F# - G) is enharmonic (harmonically equivalent) to the e-minor scale (E - F# - G - A - B - C - D - E).
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| When notating the key signatures, the order of sharps that are found at the beginning of the staff line follows the circle of fifths from F through B. The order is F, C, G, D, A, E, B. If there is only one sharp, such as in the key of G major, then the one sharp is F sharp. If there are two sharps, the two are F and C, and they appear in that order in the key signature. The order of sharps goes clockwise around the circle of fifths. (The major key you are in is one half-step above the last sharp you see in the key signature. This does not work if you are in a minor key.)
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| For notating flats, the order is reversed: B, E, A, D, G, C, F. This order runs counter-clockwise along the circle of fifths; in other words they progress by fourths. If you follow the major keys from the key of F to the key of C flat (B) counter-clockwise around the circle of fifths, you will see that as each key signature adds a flat, they are these seven flats always in this order. D flat in the key signature is always found in the fourth position among flats, and G is always in the fifth position. If there are three flats in the key signature, they are only B, E, and A, while the other positions are empty. (The major key you are in is the penultimate flat you find in the key signature; of course, there must be at least two flats in the key signature for this to work. It does not work if you are in a minor key.)
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| ===Modulation and chord progression===
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| [[Tonality|Tonal music]] often [[modulation (music)|modulates]] by moving between adjacent scales on the circle of fifths. This is because diatonic scales contain seven pitch classes that are contiguous on the circle of fifths. It follows that diatonic scales a perfect fifth apart share six of their seven notes. Furthermore, the notes not held in common differ by only a semitone. Thus modulation by perfect fifth can be accomplished in an exceptionally smooth fashion. For example, to move from the C major scale F – C – G – D – A – E – B to the G major scale C – G – D – A – E – B – F{{music|♯}}, one need only move the C major scale's "F" to "F{{music|♯}}".
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| In Western tonal music, one also finds chord progressions between chords whose roots are related by perfect fifth. For instance, root progressions such as D-G-C are common. For this reason, the circle of fifths can often be used to represent "harmonic distance" between chords.
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| [[Image:IV-V-I in C.png|thumb|left|IV-V-I, in C {{audio|IV-V-I in C.mid|Play}}]]
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| According to theorists including Goldman, [[diatonic function|harmonic function]] (the use, role, and relation of [[chord (music)|chords]] in [[harmony]]), including "functional succession", may be "explained by the circle of fifths (in which, therefore, [[scale degree]] II is closer to the [[Dominant (music)|dominant]] than scale degree IV)".<ref>Nattiez (1990).</ref>{{Page needed|date=May 2010}} In this view the tonic is considered the end of the line towards which a [[chord progression]] derived from the circle of fifths progresses.
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| [[Image:Ii-V-I turnaround in C.png|thumb|right|[[ii-V-I turnaround]], in C {{audio|Ii-V-I turnaround in C.mid|Play subdominant, supertonic seventh, and supertonic chords}} illustrating the similarity between them.]]
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| According to Goldman's ''Harmony in Western Music'', "the IV chord is actually, 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">Goldman (1965), p. 68.</ref> Thus 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>Goldman (1965), 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">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.)
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| [[File:IV frente ii7 sin raiz.png|thumb|left|IV vs. ii<sup>7</sup> with root in parenthesis, in C {{audio|Subdominant and supertonic similarity.mid|Play}}.]]
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| Goldman argues that "historically the use of the IV chord in harmonic design, and especially in [[cadence (music)|cadences]], exhibits some curious features. By and large, one can say that the use of IV in final cadences becomes more common in the nineteenth century than it was in the eighteenth, but that it may also be understood as a substitute for the ii chord when it precedes V. It may also be quite logically construed as an incomplete ii7 chord (lacking root)." <ref name="Goldman 68"/> The delayed acceptance of the IV-I in final cadences is explained aesthetically by its lack of closure, caused by its position in the circle of fifths. The earlier use of IV-V-I is explained by proposing a relation between IV and ii, allowing IV to substitute for or serve as ii. However, Nattiez calls this latter argument "a narrow escape: only the theory of a ii chord without a [[root (chord)|root]] allows Goldman to maintain that the circle of fifths is completely valid from [[Johann Sebastian Bach|Bach]] to [[Richard Wagner|Wagner]]", or the entire [[common practice period]].<ref name ="Nattiez 226"/>
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| ===Circle closure in non-equal tuning systems===
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| When an instrument is tuned with the [[equal temperament]] system, the width of the fifths is such that the circle "closes". This means that ascending by twelve fifths from any pitch, one returns to a tune exactly in the same pitch class as the initial tune, and exactly seven [[octave]]s above it. To obtain such a perfect circle closure, the fifth is slightly flattened with respect to its [[just intonation]] (3:2 [[interval ratio]]).
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| Thus, ascending by justly intonated 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 [[wolf fifth]] as a humorous metaphor of the unpleasant sound of the note (like a wolf trying to howl 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.
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| ===In lay terms===
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| {{infobox|bodyclass=collapsible collapsed
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| |above = Playing the circle of fifths
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| |abovestyle = font-size:100%
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| |image1 =
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| {{multiple image|direction=vertical|align=none|width=220
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| | image1 = Circle of fifths within oc.png|caption1=1 octave, fourths, descending
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| | image2 = Circle of fifths ascend wi.png|caption2=1 octave, fifths, ascending
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| | image3 = Circle of fifths a 2 octave.png|caption3=2 octave, fifths, ascending
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| | image4 = Circle of fifths d 2 octave.png|caption4=2 octave, fourths, descending
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| | image5 = Circle of fourths 2 octave.png|caption5=2 octave, fourths, ascending
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| | image6 = Circle of fourths d 2 octa.png|caption6=2 octave, fifths, descending
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| | image7 = Circle of fifths on A.png|caption7=multi octave, fifths, ascending
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| | image8 = Circle of fifths d on A.png|caption8=*multi octave, fourths, descending*
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| | image9 = Circle of fourths on A.png|caption9=multi octave, fourths, ascending
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| }}
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| }}
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| A simple way to see the [[Interval (music)|musical interval]] known as a [[Perfect fifth|fifth]] is by looking at a [[Musical keyboard|piano keyboard]], and, starting at any key, counting seven keys to the right (both black and white) to get to the next note on the circle shown above. Seven [[half step]]s, the distance from the 1st to the 8th key on a piano is a "perfect fifth", called 'perfect' because it is neither major nor minor, but applies to both major and minor scales and chords, and a 'fifth' because though it is a distance of seven semitones on a keyboard, it is a distance of five steps within a major or minor scale.
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| A simple way to hear the relationship between these notes is by playing them on a piano keyboard. If you traverse the circle of fifths backwards, the notes will feel as though they fall into each other. This aural relationship is what the mathematics describe.{{Citation needed|date=July 2008}}
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| Perfect fifths may be [[Just intonation|justly tuned]] or [[Musical temperament|tempered]]. Two notes whose frequencies differ by a ratio of 3:2 make the interval known as a justly tuned perfect fifth. Cascading twelve such fifths does not return to the original [[pitch class]] after going round the circle, so the 3:2 ratio may be slightly detuned, or tempered. Temperament allows perfect fifths to cycle, and allows pieces to be [[Transposition (music)|transposed]], or played in any [[Key (music)|key]] on a piano or other fixed-pitch instrument without distorting their harmony. The primary tuning system used for Western (especially [[Keyboard instrument|keyboard]] and [[fret]]ted) instruments today, twelve-tone [[equal temperament]], uses an irrational multiplier, 2<sup>1/12</sup>, to calculate the frequency difference of a semitone. An equal-tempered fifth, at a frequency ratio of 2<sup>7/12</sup>:1 (or about 1.498307077:1) is approximately two cents narrower than a justly tuned fifth at a ratio of 3:2.
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| ==History==
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| [[Image:Heinichen musicalischer circul.png|thumb|[[Johann David Heinichen|Heinichen]]'s musical circle (German: ''musicalischer circul'')(1711)]]
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| In the late 1670s a [[treatise]] called ''Grammatika'' was written by the composer and theorist [[Nikolai Diletskii]]. Diletskii’s ''Grammatika'' is a treatise on composition, the first of its kind, which targeted Western-style polyphonic compositions. It taught how to write kontserty, [[polyphonic]] [[a cappella]], which were normally based on [[liturgical]] texts and were created by putting together musical sections that have contrasting [[rhythm]], meters, [[melodic]] material and [[vocal]] groupings. Diletskii intended his treatise to be a guide to composition but pertaining to the rules of [[music theory]]. Within the Grammatika treatise is where the first circle of fifths appeared and was used for students as a composer's tool.<ref name="Jensen">Jensen (1992), pp. 306–307.</ref>
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| ==Related concepts==
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| ===Diatonic circle of fifths===
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| {{Main|Circle progression}}
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| 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]].
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| The [[circle progression]] is commonly a circle through the diatonic chords by fifths, including one [[Diminished triad|diminished chord]] and one progression by diminished fifth:
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| [[Image:Progresión quintas.png|thumb|center|400px|I-IV-vii<sup>o</sup>-iii-vi-ii-V-I (in major) {{audio|Progresión quintas.mid|Play in C major}}]]
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| ===Chromatic circle===
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| {{Main|Chromatic circle}}
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| 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.
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| 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.
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| ===Relation with chromatic scale===
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| {{Main|Chromatic scale}}
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| [[Image:Pitch class space star.svg|thumb|right|The circle of fifths drawn within the chromatic circle as a [[star polygon|star]] [[dodecagram]].<ref>McCartin (1998), p. 364.</ref>]]
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| 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).
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| Here is a demonstration of this procedure. Start off with an [[order theory|ordered]] 12-tuple ([[tone row]]) of integers
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| : (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
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| representing the notes of the chromatic scale: 0 = C, 2 = D, 4 = E, 5 = F, 7 = G, 9 = A, 11 = B, 1 = C{{music|♯}}, 3 = D{{music|♯}}, 6 = F{{music|♯}}, 8 = G{{music|♯}}, 10 = A{{music|♯}}. Now multiply the entire 12-tuple by 7:
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| : (0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77)
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| 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):
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| : (0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5)
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| which is equivalent to
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| : (C, G, D, A, E, B, F{{music|♯}}, C{{music|♯}}, G{{music|♯}}, D{{music|♯}}, A{{music|♯}}, F)
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| which is the circle of fifths. | |
| Note that this is [[enharmonic]]ally equivalent to:
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| : (C, G, D, A, E, B, G{{music|♭}}, D{{music|♭}}, A{{music|♭}}, E{{music|♭}}, B{{music|♭}}, F).
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| ===Enharmonics===
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| The “bottom keys” of the circle of fifths are often written in flats and sharps, as they are easily interchanged using enharmonics. For example, the key of B, with five sharps, is enharmonically equivalent to the key of C{{music|♭}}, with 7 flats. But the circle of fifths doesn’t stop at 7 sharps (C{{music|♯}}) or 7 flats (C{{music|♭}}). Following the same pattern, one can construct a circle of fifths with all sharp keys, or all flat keys.
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| After C{{music|♯}} comes the key of G{{music|♯}} (following the pattern of being a fifth higher, and, coincidentally, enharmonically equivalent to the key of A{{music|♭}}). The “8th sharp” is placed on the F{{music|♯}}, to make it F{{music|doublesharp}}. The key of D{{music|♯}}, with 9 sharps, has another sharp placed on the C{{music|♯}}, making it C{{music|doublesharp}}. The same for key signatures with flats is true; The key of E (four sharps) is equivalent to the key of F{{music|♭}} (again, one fifth below the key of C{{music|♭}}, following the pattern of flat key signatures). The last flat is placed on the B{{music|♭}}, making it B{{music|doubleflat}}.
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| ==See also==
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| * [[Approach chord]]
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| * [[Array system]]
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| * [[Sonata form]]
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| * [[Well temperament]]
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| * [[Circle of fifths text table]]
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| * [[Pitch constellation]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *D'Indy, Vincent (1903).{{Full|date=May 2010}}<!--Title of book or article, place, publisher needed. Otherwise, any inline citations should be to the source actually consulted (Nattiez 1990.--> Cited in Nattiez (1990).
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| *Goldman, Richard Franko (1965). ''Harmony in Western Music''. New York: W. W. Norton.
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| *Jensen, Claudia R. (1992). "[http://www.jstor.org/pss/831450 A Theoretical Work of Late Seventeenth-Century Muscovy: Nikolai Diletskii's "Grammatika" and the Earliest Circle of Fifths]". ''Journal of the American Musicological Society'' 45, no. 2 (Summer): 305–331.
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| * McCartin, Brian J. (1998). "Prelude to Musical Geometry". ''The College Mathematics Journal'' 29, no. 5 (November): 354–70. [http://www.maa.org/pubs/cmj_Nov98.html (abstract)] [http://links.jstor.org/sici?sici=0746-8342(199811)29%3A5%3C354%3APTMG%3E2.0.CO%3B2-Q (JSTOR)]
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| *[[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).
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| ==Further reading==
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| *Lester, Joel. [http://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.
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| *Miller, Michael. [http://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.
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| *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]].
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| *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.
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| ==External links==
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| * [http://www.circle-of-fifths.co.uk/ Interactive Circle of Fifths]
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| * [http://circleoffifths.com/ circleoffifths.com Poster]
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| * [http://randscullard.com/CircleOfFifths Interactive Circle of Fifths]
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| * [http://mdecks.com/graphs/mcircle.php Decoding the Circle of Vths]
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| * [http://www.larips.com/ Bach's Tuning by Bradley Lehman]
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| * [http://www.apassion4jazz.net/circle5.html Circle of Fifths – Diagram]
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| * [http://basssick.com/images/cof.jpg Circle of Fifths – In Bass Clef]
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| * [http://uk.youtube.com/watch?v=xkc_9Ql1HLY Major Keys: How to use the Circle of Fifths] A video showing how to use the Circle of Fifths for Major Keys
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| * [http://uk.youtube.com/watch?v=22s7Q6n87tU Minor Keys: How to use the Circle of Fifths] A video showing how to use the Circle of Fifths for Minor Keys
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| {{Pitch space}}
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| {{Tonality}}
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| {{DEFAULTSORT:Circle Of Fifths}}
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| [[Category:Harmony]]
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| [[Category:Musical keys| ]]
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| [[Category:Circle of fifths]]
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| [[Category:Tonality]]
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| {{Link FA|de}}
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