Circle of fifths
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In music theory, the circle of fifths (or cycle of fifths) is a geometrical space that depicts relationships among the 12 equal-tempered pitch classes comprising the familiar chromatic scale. The circle of fifths was originally published by Johann David Heinichen, in his 1728 treatise Der Generalbass in der Composition.
Structure and Use
If one starts on any equal-tempered pitch and repeatedly ascends by the musical interval of an equal-tempered perfect fifth, one will eventually land on a pitch with the same pitch class as the initial one, passing through all the other equal-tempered chromatic pitch classes in between.
The circle is commonly used to represent the relations 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 would have. Thus a major scale built on A will have three sharps in its key signature. The major scale built on F would have one flat. For minor scales, rotate the letters counter-clockwise by 3, so that e.g. A minor has 0 accidentals and E minor has 1 sharp. (See relative minor/major for details.)
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.
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, Z/12Z. The group [ Z_ ] 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 shown here.
Related concepts
Diatonic circle of fifths
The diatonic circle of fifths is the circle of fifths encompassing only members of the diatonic scale. As such it contains a diminished fifth, in C major between B and F. See structure implies multiplicity.Relation with chromatic scale
The circle of fifths, or fourths, may be mapped from the chromatic scale by multiplication, and vice versus. 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 (M5).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)
- (0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77)
- (0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5)
- (C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, F)
- (C, G, D, A, E, B, G♭, D♭, A♭, E♭, B♭, F)
See also
| Diatonic Scales and Keys | ||||||||||||||||||||||||||||||||||||||||||||||||||
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| lower case letters in the circle of fifths are minor | ||||||||||||||||||||||||||||||||||||||||||||||||||
| the table indicates the number of sharps or flats in each scale | ||||||||||||||||||||||||||||||||||||||||||||||||||
External links
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