Circle of Fifths

In music theory, the circle of fifths (or circle of fourths) shows the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys. More specifically, it is a geometrical representation of relationships among the 12 pitch classes of the chromatic scale in pitch class space.



Background
Since the term 'fifth' defines an interval or mathematical ratio which is the closest and most consonant non-octave interval, then the circle of fifths is a circle of closely related pitches or key tonalities. Musicians and composers use the circle of fifths to understand and describe those relationships. The circle's design is helpful in composing and harmonizing melodies, building chords, and moving to different keys within a composition. At the top of the circle, the key of C Major has no sharps or flats. Starting from the apex and proceeding clockwise by ascending 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♭ has 2 flats, and so on. At the bottom of the circle, the sharp and flat keys overlap, showing pairs of enharmonic key signatures. Starting at any pitch, ascending by the interval of an equal tempered 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 fourths, rather than fifths. (To the ear, the sequence of fourths gives an impression of settling, or resolution.)

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).

Diatonic key signatures
The circle is commonly used to represent the relationship between diatonic scales. 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. For minor scales, 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, (i.e. G major, with one sharp in its diatonic scale, F#) 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).

Modulation and chord progression
Tonal music often 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♯, one need only move the C major scale's "F" to "F♯". 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.