Circle of fifths
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In music theory, the circle 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.
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Structure
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.
Since the space is circular, it is also possible to descend by fifths. In pitch class space, motion in one direction by fourth is equivalent to motion in the opposite direction by a fifth. For this reason the circle of fifths is also known as the circle of fourths.
A key difference between the chromatic circle and the circle of fifths is that the former is truly a continuous space: 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. A mathematician would say that the circle of fifths and the chromatic circle are not homeomorphic.
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 <math> Z_{12} </math> 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 fith gives rise to the circle of fifths shown here.
Use
The circle is commonly used to represent the relations between diatonic scales. The numbers on the inside of the circle show how many sharps or flats would be in the key signature for a major scale built on that note. Thus a major scale built on A will have three sharps in its key signature. For minor scales, rotate the numbers 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.)
Moving around the circle of fifths is a common way to modulate. This is because diatonic scales can be represented as contiguous seven-point segments of 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 C - D - E - F - G - A - B to the G major scale G - A - B - C - D - E - F#, one need only move the C major scale's "F" to "F#."
In Western tonal music, one often 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.
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)
representing the notes of the chromatic scale: 0 = C, 2 = D, 4 = E, 5 = F, 7 = G, 9 = A, 11 = B, 1 = C#, 3 = D#, 6 = F#, 8 = G#, 10 = A#. Now multiply the entire 12-tuple by 7:
- (0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77)
and then apply a modulo 12 reduction to each of the numbers (subtract 12 from each number as many times as necessary until the number becomes smaller than 12):
- (0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5)
which is equivalent to
- (C, G, D, A, E, B, F#, C#, G#, D#, A#, F),
which is the circle of fifths.
See also
External links
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