Circle of fifths

Entry for EPMOW by Philip Tagg

View correct PDF file
Why did I land here?

circle of fifths: central concept of tonality in Western music theory since the advent of equal tone tuning (c. 1700), its main functions are (i) to visualise the system of keys and key signatures used in much music of the Western world; (ii) to facilitate understanding of harmonic progressions found frequently in such music (see harmony).

Since ancient times (China, Greece, etc.) it has been known that an interval of twelve fifths is, with a minimal margin of error (the Pythagorean comma or 0.24% of one semitone per octave), equal to an interval of eight octaves, i.e. that the frequences of pitches one fifth apart are separated by a factor of 12:8 or 3:2 (×1.5) when ascending and of 2:3 (×0.67) when descending (see octave). The concept also assumes that the interval of a fourth (4:3 or ×1.33 up and 3:4 or ×0.75 down) is complementary to that of the fifth within an octave, so that ascending a fourth and then descending an octave (e.g. from c3 to f 3 to f 2 ) will land on the same pitch as just descending a fifth (e.g. c3 to f 2 ) and, conversely, that ascending a fifth and then descending an octave (e.g. g3 to g3 to g 2 ) will end up on the same pitch as just descending a fourth (e.g. c3 to g 2 ). Hence, a series of alternately falling fifths and rising fourths, running anticlockwise round the complete circle of fifths (e.g. c3 - f 2 - b3 - e2 - a2 - d2 - g2/f 2- b2 - e2 - a2 - d2 - g1 -c2, see fig. 1 [only in PDF]) visits every note in the twelve-tone chromatic scale within a relatively restricted range. The same applies to a series of alternately rising fifths and falling fourths running clockwise (e.g. c2 - g2 - d2 - a - e2 - b2 - f 2/g2 - d3 - a3 - e3 - b3 - f3 - c3 ). The fact that the circle of fifths also constitutes a ‘circle of fourths’ but is never referred to as such probably stems from the notion’s development in the European classical tradition where chords constructed on the fifth degree of any scale (V) are understood and referred to as ‘dominant’, those on the fourth degree (IV) as ‘subdominant’.

The circle of fifths is a tonal concept applied to harmony rather than to melody, not least because progressions based on fourths and fifths are much more common in the former than in the latter. It is of particular use in the theoretical and practical study of popular music in most jazz idioms as well as in other styles influenced by European traditions of tertial harmony.

Fig. 1: Circle of fifths [only in PDF]

Keys and their signatures are arranged as the twelve figures of an analogue clock with C major and its relative A minor (no sharps and no flats) on the hour, and F/G major with their relative D/E minor (six sharps or six flats) at half past. Moving clockwise, the number of sharps in each key signature increases (one for G major at five past, two for D major at ten past and so on) or the number of flats decreases (five for D major at twenty-five to, four for A major at twenty to, etc.). Since movement clockwise round the circle is by ascending fifths and since an increase in sharps or a decrease in flats implies upward movement, this tonal direction ‘sharpwards’ towards the dominant (from I to V, e.g. C to G) can be referred to as rising, while anticlockwise tonal movement ‘flatwards’ towards the subdominant (from V to I or from I to IV, e.g. from G to C or from C to F) can be referred to as falling (see harmony).

Harmonic progressions based on the circle of fifths are extremely common in popular music. Those running anticlockwise (‘flatwards’, falling, see fig. 1) are particularly common in styles using the tertial harmonic practices of jazz or classical music. Two basic types of such progression exist: (i) the real or modulatory circle of fifths; (ii) the virtual or key-specific circle of fifths. Both these types of anticlockwise progression involve the same two-stage V- I cadence (e.g. G7-C) because all unaltered notes in the dominant seventh chord (V7, e.g. g b d f in G7) are contained in the major scale of the tonic (I, e.g. C major, containing c d e f g a b). However, as soon as an anticlockwise circle-of-fifths progression contains more than two stages it will become either real/modulatory, for example VI7- II7- V7- I (A7- D7- G7- C in C, see ex. 1), or virtual/key-specific, e.g. vi7- ii7- V7- I (Am7- Dm7- G7- C in C). The former constitutes a real circle of fifths because A7 (VI – the chord on the sixth degree) is the real dominant seventh of D (II, on the second degree), D7 (II) the real dominant seventh of G (V); it can also be termed modulatory because A7 and D7 both contain notes foreign to the tonic key of C major (c and f respectively). The virtual circle-of-fifths progression is key-specific because all notes in all chords belong to the same tonic key (e.g. C major, see example 1) and can be termed virtual because neither Am7 (on the sixth degree) nor Dm7 (second degree) are real dominant sevenths of subsequent chords in the progression.

Ex. 1 [only in PDF]

A certain predilection for real circles of fifths in US popular song from the nineteen tens and twenties was superseded by preference for more virtual variants in standards and evergreens of the thirties and forties (see figure 2). The virtual or key-specific circle-of-fifths is moreover a distinctive trait of the baroque style (Corelli, Vivaldi, J.S. Bach, etc., see ex. 2) and is also quite common in European popular song showing classical influences (fig. 2). Many well-known popular songs use a mixture of real and virtual circle-of-fifths progressions.

Anticlockwise circle-of-fifths progression are, as shown in example 1 and figure 2, frequently constructed as a chain of seventh chords (sometimes also ninths, elevenths or thirteenths). Example 2 illustrates one common way of playing such chains as key-specific circles in (i) C major, (ii) D major, (iii) G minor. (This example assumes the presence of each chord’s root in the bass part.) To effectuate any complete key-specific circle-of-fifths one step in the bass line will be a diminshed fifth (between vii and IV in the major key, between ii and V in the minor, e.g. from FD7 to Bm75 in C major or A minor), each of the remaining seven steps either falling by a perfect fifth or rising by a perfect fourth.

Fig. 2: Anticlockwise circle-of-fifth progression types in well-known English-language songs [only in PDF]

Ex. 2 Seventh chords in key-specific (virtual) sequence anti-clockwise round the circle of fifths: (i) C major; (ii) D major; (iii) G minor [only in PDF]

Playing anticlockwise circle-of-fifth progressions demands a minimum of physical effort because: (i) stringed bass instruments are tuned in fourths, facilitating leaps of the fourth, fifth and octave (see above); (ii) fifths, fourths and octaves are easy to pitch on brass instruments playing a bass line; (iii) the constituent notes of any two contiguous seventh chords in a circle-of-fifths progression are, with the exception of the root, either immediately adjacent or the same (see ex. 2), this rendering them amenable to hand and finger positioning for keyboard players and guitarists. (See also harmony, turnaround).

Clockwise (‘rising’) circle-of-fifths progressions may be less common than their anticlockwise counterparts but do occur in pop and rock styles using certain types of modal harmony (see harmony). For example, the mixolydian turnaround VII IV I runs clockwise (e.g. B F C), as do all progressions listed in figure 3.

Fig. 3: Examples of clockwise circle-of-fifth progressions in English-language rock music [only in PDF]

Ex. 3 Rolling Stones: Brown Sugar (1971). Clockwise circle-of-fifths progression through plagal ornamentation of aeolian cadence. [only in PDF]

Bibliography

Musical references