Circle of fifths

Overview

In music theory, the circle of fifths (sometimes also cycle of fifths) organizes the twelve pitch classes of the chromatic scale in a cycle of ascending perfect fifths. Since the cycle ascends in the opposite direction by perfect fourths, it is sometimes also called the circle of fourths.

Starting on a C, the sequence is: C, G, D, A, E, B/C♭, F♯/G♭, C♯/D♭, A♭, E♭, B♭, F, and C. This order places the most closely related key signatures adjacent to one another.

Ascending by twelve justly tuned perfect fifths with a frequency ratio of 3:2 does not return exactly to the starting pitch class, but instead overshoots it by a small interval known as the Pythagorean comma. This leads to tuning issues when transposing or modulating, and tuning systems were developed to address these issues. By changing (tempering) the 3:2 ratio, these systems produce a sequence of perfect fifths that returns to the starting pitch class. Such tuning systems include historical well temperaments, and 12-tone equal temperament, which is the standard tuning system for Western music.

Definition

The circle of fifths organizes pitches in a sequence of perfect fifths, 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.

The top of the circle shows the key of C Major, with no sharps or 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♭ has 2 flats, and so on. Some keys (at the bottom of the circle) can be reasonably notated 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.

When notating a circle of fifths, an enharmonic substitution is made for one of the notes. In the clockwise example above, a perfect fifth above A♯ would be E♯, but is notated as F, which is enharmonically equivalent. This technically creates a diminished sixth between A♯ and F. In the counterclockwise example, a perfect fifth below G♭ should be C♭, but is replaced with the enharmonic note of B, also creating a diminished sixth.

Diatonic key signatures

Each pitch 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 the relative major and relative minor of one another.

Modulation and chord progression

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 or tonal 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)". In this view the tonic or tonal center is considered the end point of a chord progression derived from the circle of fifths.

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." 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 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. (In this and related articles, upper-case Roman numerals indicate major triads while lower-case Roman numerals indicate minor triads.)

History

The circle of fifths developed in the late 1600s and early 1700s to theorize the modulation of the Baroque era.

The first circle of fifths diagram appears in the Grammatika (1677) of the composer and theorist Mykola Pavlovych Dyletsky, who intended to present music theory as a tool for composition. 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), which he called the "Musical Circle" (German: Musicalischer Circul). 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.

Some sources imply that the circle of fifths was known in antiquity, by Pythagoras. This is a misunderstanding and an anachronism. Tuning by fifths (so-called Pythagorean tuning) dates to Ancient Mesopotamia, 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). 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.