In music theory, the circle of fifths (also called the cycle of fifths or circle of fourths) arranges the twelve notes of the chromatic scale in a pattern of perfect fifths. This same pattern can also be created using perfect fourths.
Starting with the note C and using the standard tuning system in Western music (12-tone equal temperament), the order of notes is: C, G, D, A, E, B, F♯/G♭, C♯/D♭, G♯/A♭, D♯/E♭, A♯/B♭, F, and C. This sequence shows how key signatures with similar relationships are placed next to each other.
If perfect fifths are tuned with an exact frequency ratio of 3:2 (a tuning method called just intonation), moving up twelve perfect fifths does not return to the original note. Instead, it moves slightly past it by a small interval called the Pythagorean comma. This causes the circle to not "close" and instead form an endless spiral, seen in modern notation between notes like C and B♯. To fix this problem, tuning systems such as well temperaments and 12-tone equal temperament were created. These systems slightly adjust the size of the perfect fifth to make the circle complete.
Definition
The circle of fifths is a tool that arranges musical notes in a sequence of perfect fifths, often shown as a circle with notes (and their related keys) listed in a clockwise order. It can also be viewed counterclockwise as a circle of fourths. In Western music, harmonic progressions often use neighboring keys from this system, making it helpful for composing music and understanding harmony.
At the top of the circle is the key of C Major, which has no sharps or flats. Moving clockwise, each key rises by a fifth. The key signatures change as follows: G Major has one sharp, D Major has two sharps, and so on. Moving counterclockwise from the top, each key descends by a fifth, and the key signatures change accordingly: F Major has one flat, B♭ Major has two flats, and so on. Some keys at the bottom of the circle can be written using either sharps or flats.
Starting at any note and moving up by a fifth creates all the tones before returning to the original note. A pitch class includes all notes with the same letter, such as all Cs, regardless of octave. Moving counterclockwise, notes descend by a fifth, but moving up by a fourth reaches the same note an octave higher (same pitch class). For example, moving counterclockwise from C is like going down a fifth to F or up a fourth to F.
When drawing the circle of fifths, an enharmonic substitution is used for certain notes. In the clockwise example, a perfect fifth above A♯ is E♯, but it is written as F, which sounds the same but is named differently. This creates a diminished sixth between A♯ and F. In the counterclockwise example, a perfect fifth below G♭ is C♭, but it is replaced with B, which also sounds the same but is named differently, creating a diminished sixth.
Structure and use
Each note can act as the starting point, or tonic, for a major or minor key. Each key has a scale made up of seven notes. A circle diagram shows how many sharps or flats are in each key’s signature. Major keys are written with capital letters, and minor keys are written with lowercase letters. Keys that share the same key signature are called relative major and relative minor keys.
In tonal music, the music may move to a new key that is only one sharp or flat away from the original key. These closely related keys are a fifth apart and are next to each other on the circle of fifths. Chord progressions often move between chords whose roots are a perfect fifth apart. This helps show the "harmonic distance" between chords.
The circle of fifths helps organize and describe how chords function in music. Chords can move in a pattern of ascending perfect fourths, which is the same as moving in descending perfect fifths. This pattern is shown by the circle of fifths, where the second scale degree is closer to the dominant chord than the fourth scale degree. In this view, the tonic, or main note, is the final point of a chord progression based on the circle of fifths.
According to Richard Franko Goldman’s Harmony in Western Music, the IV chord is farthest from the I chord in simple diatonic relationships. Using the circle of fifths, the IV chord moves away from the I chord instead of toward it. Goldman says that the chord progression I–ii–V–I (an authentic cadence) feels more complete than I–IV–I (a plagal cadence). He agrees with Nattiez, who notes that in the progression I–IV–vii–iii–vi–ii–V–I, the IV chord appears before the II chord and is farther from the tonic. In this system, uppercase Roman numerals show major chords, and lowercase numerals show minor chords.
Using a perfect fifth with a 3:2 frequency ratio (just intonation) does not return to the starting note after going around the circle of fifths. Equal temperament tuning creates fifths that return to a note exactly seven octaves above the starting note. In this system, the frequency ratio of a chromatic semitone is the same as a diatonic semitone. The standard tempered fifth has a frequency ratio of about 1.498307077:1, which is slightly smaller than a justly tuned fifth.
If you move up twelve justly tuned fifths, the circle does not close completely. This creates a small gap called the Pythagorean comma, about 23.46 cents, or a quarter of a semitone. If only twelve notes are used per octave, Pythagorean tuning shortens one fifth, making it very dissonant. This is called the wolf fifth. Other tuning systems, like non-extended quarter-comma meantone, use eleven slightly narrower fifths and require an even more dissonant wolf fifth to close the circle. More complex systems, such as 5-limit tuning, use up to eight just fifths and at least three non-just fifths to close the circle.
Today, electronic isomorphic keyboards allow equal temperament tunings with more than twelve notes per octave. These tunings can close the circle of fifths for other systems. For example, 31-tone equal temperament closely matches quarter-comma meantone, and 53-tone equal temperament closely matches Pythagorean tuning.
History
The circle of fifths was created in the late 1600s and early 1700s to help explain how to change keys in music during the Baroque era (see § Baroque era).
The first diagram of the circle of fifths appeared in the Grammatika (1677), written by the composer and theorist Mykola Pavlovych Dyletsky. He designed it to teach Russian audiences how to write Western-style music with multiple melodies. This was the first of its kind.
A similar diagram was created independently by the German composer and theorist Johann David Heinichen in his Neu erfundene und gründliche Anweisung (1711). He called it the "Musical Circle" (German: Musicalischer Circul). Heinichen also included this diagram in his Der General-Bass in der Composition (1728).
Heinichen placed the relative minor key next to the major key, which did not show how close these keys actually are. Later, Johann Mattheson (1735) and others tried to improve this. David Kellner (1737) suggested placing major keys on one circle and relative minor keys on a second, inner circle. This idea later led to the development of chordal space, which also included parallel minor keys.
Some sources claim the circle of fifths was known in ancient times by Pythagoras. This is incorrect. Tuning using fifths (called Pythagorean tuning) was used in Ancient Mesopotamia (see Music of Mesopotamia § Music theory), but they did not use it to create a twelve-note scale, only seven notes. The Pythagorean comma was calculated by Euclid and Chinese mathematicians (in the Huainanzi; see Pythagorean comma § History). This showed that a cycle of twelve fifths is almost equal to seven octaves. However, this was only theoretical knowledge and was not used to create a repeating twelve-note scale or to change keys. These developments, called meantone temperament and twelve-tone equal temperament, came much later in Europe, around 1500. Although called the "circle of fifths," its name originally came from the phrase "wheel of fifths" in Anglo-Saxon languages.
Use
In music from the Baroque and Classical eras, as well as in Western popular, traditional, and folk music, changing to a new key is often connected to the circle of fifths.
Most compositions do not use the entire circle of fifths. Instead, they often use a smaller or larger part of the circle, which represents the structure of musical tones. This structure is usually based on the seven notes of the diatonic scale, not the twelve chromatic notes. In the diatonic version of the circle, one interval is not a perfect fifth but a tritone (also called a diminished fifth). In the key of C, this happens between F and B natural, creating the following sequence in C major:
In the relative minor key of A minor, the sequence is:
These are the diatonic chords that can be built over this sequence in C major:
The diatonic chords in A minor, with an E major chord to create a leading tone to the tonic A:
According to Richard Taruskin, Arcangelo Corelli was the most influential composer to make the circle of fifths a standard harmonic pattern. "It was during Corelli’s time, in the late seventeenth century, that the circle of fifths was being studied as the main driver of harmonic movement. Corelli more than any other composer put this idea into practice."
The circle of fifths progression appears often in the music of J. S. Bach. In the piece "Jauchzet Gott in allen Landen," BWV 51, the solo bass line suggests the chords even when they are not clearly stated.
Handel used a circle of fifths progression as the foundation for the Passacaglia movement in his Harpsichord Suite No. 7 in G minor (HWV 432).
Baroque composers learned to strengthen the "driving force" of the harmony created by the circle of fifths by adding sevenths to most chords. These sevenths, which are dissonant, create a need for resolution, turning each progression into a source of tension and release. This technique was used for expressive purposes. Examples include the aria "Pena tiranna" from Handel’s 1715 opera Amadigi di Gaula, as well as Bach’s keyboard arrangement of Alessandro Marcello’s Concerto for Oboe and Strings.
Franz Schubert’s Impromptu in E♭ major, D 899, includes harmonies that move in a modified circle of fifths.
The Intermezzo movement from Mendelssohn’s String Quartet No. 2 has a short section with circle-of-fifths motion, where the ii° chord is replaced by the iv chord.
Robert Schumann’s "Child Falling Asleep" from Kinderszenen uses the progression but ends on an A minor chord instead of the expected tonic E minor.
In Wagner’s opera Götterdämmerung, a circle of fifths progression appears in the music that transitions from the end of the prologue into the first scene of Act 1, set in the Gibichungs’ hall. "Status and reputation are shown through the motifs assigned to Gunther," the leader of the Gibichung clan.
The circle of fifths remains popular as a tool for building musical forms and expressing emotions. It is used in many standard popular songs from the twentieth century and is favored by jazz musicians for improvisation, as it helps them understand intervals, chord relationships, and progressions.
Examples include:
– Bart Howard, "Fly Me to the Moon"
– Jerome Kern, "All the Things You Are"
– Ray Noble, "Cherokee"
– Kosma, Prévert, and Mercer, "Autumn Leaves"
– The Beatles, "You Never Give Me Your Money"
– Mike Oldfield, "Incantations"
– Carlos Santana, "Europa (Earth's Cry Heaven's Smile)"
– Gloria Gaynor, "I Will Survive"
– Pet Shop Boys, "It's a Sin"
– Donna Summer, "Love to Love You, Baby"
Related concepts
The diatonic circle of fifths includes only notes from the diatonic scale. It contains a diminished fifth, such as the interval between B and F in C major. The term "structure implies multiplicity" refers to the idea that the circle of fifths can represent multiple musical patterns. A common circle progression uses diatonic chords, including one diminished chord. An example in C minor includes the chords i–iv–♭VII–♭III–♭VI–ii–V–i.
The circle of fifths is related to the chromatic circle, which arranges all pitches in a tuning system in a circular order. The chromatic circle is a continuous space where every pitch corresponds to a point on the circle. The circle of fifths, however, is a discrete structure made up of distinct intervals, and its points do not directly correspond to specific pitches. This makes the two circles mathematically different.
In any tuning system with N equal parts per octave, the pitches can be represented using a mathematical structure called the cyclic group of order N, or Z/NZ. In twelve-tone equal temperament, the group Z12 has four generators: semitones and perfect fifths. The semitone generator creates the chromatic circle, while the perfect fifth and fourth generate the circle of fifths. Other tunings, like 31 equal temperament, allow more generators and more possible circles.
The circle of fifths can be connected to the chromatic scale using multiplication. To move from the chromatic scale to the circle of fifths, multiply by 7 (M7). To move from the chromatic scale to the circle of fourths, multiply by 5 (P5).
In twelve-tone equal temperament, the chromatic scale can be written as a 12-tuple of integers: 0 (C), 2 (D), 4 (E), 5 (F), 7 (G), 9 (A), 11 (B), 1 (C♯), 3 (D♯), 6 (F♯), 8 (G♯), 10 (A♯). Multiplying this sequence by 7 and reducing each number modulo 12 gives the circle of fifths. This sequence is enharmonically equivalent to another arrangement of notes.
Equal temperament uses a slightly adjusted version of the perfect fifth (3:2 ratio) to allow all keys to function equally. In just intonation, the exact 3:2 ratio is used, but ascending by fifths never returns to the starting pitch. This difference is called the Pythagorean comma. In equal temperament, enharmonically equivalent notes (like C♯ and D♭) are the same, but in just intonation, they are not.
In just intonation, the sequence of fifths forms a spiral rather than a circle because of the Pythagorean comma. Without enharmonic equivalences, sequences of fifths eventually produce notes with double or triple accidentals. In most equal temperament systems, these are replaced with simpler enharmonic equivalents.
Keys with double or triple sharps or flats are called theoretical keys. They are rarely used in 12-tone equal temperament but may appear in other tunings. Notation for these keys is not standardized.
In LilyPond, sharps and flats in key signatures are listed in the order of the circle of fifths. For example, G♯ major includes sharps in the order C♯, G♯, D♯, A♯, E♯, B♯, F. Some composers, like Max Reger, use repeated sharps or flats in key signatures for clarity.
The "signature of fifths" is a way to represent the frequency of notes in a musical piece. It uses the circle of fifths to show how often each pitch class appears. In symbolic formats like MIDI, this is shown as the number of times each note is played. In audio formats, it uses measurements like volume or power for each pitch class.
The "trajectory of fifths" connects multiple "signatures of fifths" from different parts of a piece. Each signature is represented as a point on the circle of fifths, and these points are connected to show how the music’s pitch focus changes over time. This method helps analyze and classify music in both symbolic and audio formats.