Circle of fifths

Each of the twelve pitches can serve as the tonic of a major or 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.

Tonal music often 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 progressions 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 harmonic function of 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 than scale degree IV)".[2] In this view the tonic is considered the end point of a chord progression derived from the circle of fifths.

ii–V–I progression, in C 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."[3] He states that the progression I–ii–V–I (an authentic cadence) would feel more final or resolved than I–IV–I (a plagal cadence). Goldman[4] 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^o–iii–vi–ii–V–I", and is farther from the tonic there as well.[5] (In this and related articles, upper-case Roman numerals indicate major triads while lower-case Roman numerals indicate minor triads.)

Perfect fifths with a frequency ratio 3:2 are said to have just intonation. Generating a series of twelve perfect fifths in this way does not result in a return to the pitch class of the starting note. Instruments are generally tuned with the equal temperament system, producing fifths that return to a tone exactly seven octaves above the initial tone. Equal temperament tuning makes the frequency ratio of each half step the same. An equal-tempered fifth has a frequency ratio of 2^7/12:1 (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 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 width of one of the twelve fifths, which makes it severely dissonant. This anomalous fifth is called 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 tones (the original 12 tones and 42 more between them) in order to close the circle of fifths.

According to Richard Taruskin, Arcangelo Corelli was the most influential composer to establish the pattern as a standard harmonic "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."[12]

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, even when the solo bass line implies rather than states the chords involved:

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.

Handel Passacaille from Suite in G minor bars 1–4

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."[13] Striking passages that illustrate the use of sevenths occur in the aria "Pena tiranna" in Handel's 1715 opera Amadigi di Gaula :

Handel, aria "Pena tiranna" from Amadigi. Orchestral introduction

– and in Bach's keyboard arrangement of Alessandro Marcello's Concerto for Oboe and Strings.

Bach adagio BWV 974 (after Marcello)

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:

Schubert Impromptu in E flat

– as does the Intermezzo movement from Mendelssohn's String Quartet No.2:

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 E minor.

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",[14] chief of the Gibichung clan:

Wagner, Gotterdamerung, transition between the end of the Prologue and Act 1 scene 1

Ravel's "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.":[15]

Ravel Pavane pour une Infante Défunte

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" 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"

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.[16]

• Jerome Kern, "All the Things You Are"[17]
• Ray Noble, "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 progressions that temporarily pass through several tonalities."[18]
• Kosmo, Prevert and Mercer, "Autumn Leaves"[19]
• The Beatles, "You Never Give Me Your Money"[20]
• Mike Oldfield, "Incantations"
• Carlos Santana, "Europa (Earth's Cry Heaven's Smile)"
• Gloria Gaynor, "I Will Survive"[21]
• Pet Shop Boys, "It's a Sin"[22]
• Donna Summer, "Love to Love you, Baby"[23]

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 chord. A circle progression in C major with chords I–IV–vii^o–iii–vi–ii–V–I is shown below.

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 classes can be represented by the cyclic group of order twelve, or equivalently, the residue classes modulo twelve, ${\displaystyle \mathbb {Z} /12\mathbb {Z} }$. The group ${\displaystyle \mathbb {Z} _{12}}$ 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.

The circle of fifths drawn within the chromatic circle as a star dodecagram.[24]

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 (M7), and for the circle of fourths multiply by 5 (P5).

Here is a demonstration of this procedure. Start off with an 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♯, 3 = D♯, 6 = F♯, 8 = G♯, 10 = A♯. 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 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♯, C♯, G♯, D♯, A♯, F)

which is the circle of fifths. Note that this is enharmonically equivalent to:

(C, G, D, A, E, B, G♭, D♭, A♭, E♭, B♭, F).

The key signatures found on the bottom of the circle of fifths diagram, such as D♭ 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♭ major, with five flats, contains the same sounding notes, enharmonically, as C♯ major (seven sharps).

After C♯ comes the key of G♯ (following the pattern of being a fifth higher, and, coincidentally, enharmonically equivalent to the key of A♭). The "eighth sharp" is placed on the F♯, to make it F. The key of D♯, with nine sharps, has another sharp placed on the C♯, making it C. The same is true for key signatures with flats; The key of E (four sharps) is equivalent to the key of F♭ (again, one fifth below the key of C♭, following the pattern of flat key signatures). The last flat is placed on the B♭, making it B. Such keys with double accidentals in the key signatures are called theoretical keys: 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).

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♯ major (exactly as displayed above, pp. 153ff.) The sharps in the key signature of G♯ major here proceed C♯, G♯, D♯, A♯, E♯, B♯, F.
• 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♭ minor key signatures on pp. 42–45. These have a B♭ at the start and also a B at the end (with a double-flat symbol), going B♭, E♭, A♭, D♭, G♭, C♭, F♭, B. The convention of LilyPond and Foulds would suppress the initial B♭.
• Sometimes the double signs are written at the beginning of the key signature, followed by the single signs. For example, the F♭ key signature is notated as B, E♭, A♭, D♭, G♭, C♭, F♭. This convention is used by Victor Ewald, by the program Finale (software),[25] and by some theoretical works.