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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.

    Circle of fifths
        Structure and Use
            In Laymans Terms
            Diatonic circle of fifths
            Relation with chromatic scale
            Infinite Series
        See also

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Structure and Use

If one starts on any equal-tempered pitch and repeatedly ascends by the musical interval of a 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 a 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.

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.

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In Laymans Terms
A simple way to see the relationship between these notes is by looking at a piano keyboard, and starting at any key and counting 7 keys to the right(both black and white) to get to the next note on the circle above - which is a perfect fifth. 7 half steps or the distance from the 1st to the 8th key on a piano is a perfect fifth.

The frequency of two notes that are a perfect fifth apart differs by a factor of 3:2 (this depends on the temperament of the scale). Each half-step on an equal-tempered scale differs by a factor of the 12th root of 2. Since the 12th root of 2, raised to the 7th power, is 1.498 it should be noted that an equal-tempered scale does not include a note that is precisely a perfect fifth from any other note.

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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.

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Relation with chromatic scale
The circle of fifths, or fourths, may be mapped from the chromatic scale by multiplication, and vice versa. 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.
Note that this is enharmonically identical to:

(C, G, D, A, E, B, G♭, D♭, A♭, E♭, B♭, F)

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Infinite Series
The “bottom keys” of the circle of fifths are often written in flats and sharps, as they are easily interchanged using enharmonics. For example, the key of B, with five sharps, is enharmonically equivalent to the key of C♭, with 7 flats. But the circle of sharps doesn’t stop at 7 sharps (C♯) nor 7 flats (C♭). Following the same pattern, one can construct a circle of fifths with all sharp keys, or all flat keys.

After C♯ comes the key of G♯ (following the pattern of being a fifth higher, and, coincidently, enharmonically equivalent to the key of A♭). The “8th sharp” is placed on the F♯, to make it F♯♯. The key of D♯, with 9 sharps, has another sharp placed on the C♯, making it C♯♯. The same for key signatures with flats is true; The key of E (four sharps) is equivalent to the key of F♭ (again, one fifth below the key of C♭, following the pattern of flat key signatures. The double-flat is placed on the B♭)

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