Diatonic circle of fifths The diatonic circle of fifths is the circle of fifths encompassing only members of the diatonic scale. Therefore, it contains a diminished fifth, in C major between B and F. See
structure implies multiplicity. The
circle progression is commonly a circle of fifths through the diatonic chords, including one
diminished chord. A circle progression in C minor with chords i–iv–VII–III–VI– iio–V–i is shown below. : { \new PianoStaff > \new Staff > >> }
Chromatic circle The circle of fifths is closely related to the
chromatic circle, which also arranges the equal-tempered pitch classes of a particular tuning 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 arranged through distinct
intervals, 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, for any positive integer
N, the
pitch classes in
N-tone equal temperament can be represented by the
cyclic group of order
N, or equivalently, the
residue classes modulo equal to
N, \mathbb{Z}/N\mathbb{Z} . In twelve-tone equal temperament, the group \mathbb{Z}_{12} 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 fourth and perfect fifth give rise to the circle of fifths. In most other tunings, such as in
31 equal temperament, many more intervals can be used as the generator, and many more circles are possible as a result.
Relation with chromatic scale dodecagram. 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 (P5). In twelve-tone equal temperament, one can 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. This is
enharmonically equivalent to: : (C, G, D, A, E, B, G, D, A, E, B, F).
Enharmonic equivalents, theoretical keys, and the spiral of fifths fail to close (the size of the gap is the
Pythagorean comma), resulting in a "broken" circle of fifths.
Equal temperament tunings do not use the exact 3:2 ratio of frequencies that defines a perfect fifth, whereas
just intonation uses this exact ratio. Ascending by fifths in equal temperament leads to a return to the starting pitch class—starting with a C and ascending by fifths leads to another C after a certain number of iterations. This does not occur if an exact 3:2 ratio is used (just intonation). The adjustment made in equal temperament tuning is called the
Pythagorean comma. Because of this difference, pitches that are enharmonically equivalent in equal temperaments (such as C and D in 12-tone equal temperament, or C and D in
19 equal temperament) are not equivalent when using just intonation. In just intonation the sequence of fifths can therefore be visualized as a spiral, not a circle—a sequence of twelve fifths results in a "
comma pump" by the Pythagorean comma, visualized as going up a level in the spiral. See also . Without enharmonic equivalences, continuing a sequence of fifths results in notes with double accidentals (double sharps or double flats), or even triple or quadruple accidentals. In most equal temperament tunings, these can be replaced by enharmonically equivalent notes. Key signatures with double or triple sharps and flats are redundant in 12-tone equal temperament, so their use is extremely rare. They may become more relevant in other tuning systems. Notation in these cases is not standardized. \relative c' { \key gis \major \set Staff.printKeyCancellation = ##f \bar "||" \key dis \major \bar "||" \key fes \major \bar "||" \key beses \major } The default behaviour of
LilyPond (pictured above) writes single sharps or flats in the circle-of-fifths order, before proceeding to double sharps or flats. This is the format used in
John Foulds'
A World Requiem, Op. 60, which ends with the key signature of G major, as displayed above. The sharps in the key signature of G major here proceed C, G, D, A, E, B, F. Single sharps or flats in the key signature are sometimes repeated as a courtesy, e.g.
Max Reger's
Supplement to the Theory of Modulation, which contains D minor key signatures on pp. 42–45. These have a B at the start and also a B at the end (with a double-flat symbol), going B, E, A, D, G, C, F, B. The convention of LilyPond and Foulds would suppress the initial B. Sometimes the double signs are written at the beginning of the key signature, followed by the single signs. For example, the F key signature is notated as B, E, A, D, G, C, F. This convention is used by Victor Ewald, by the program
Finale, and by some theoretical works. ==Use in computational musicology==