The chromatic circle is a
clock
A clock or chronometer is a device that measures and displays time. The clock is one of the oldest Invention, human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month, a ...
diagram for displaying relationships among the
equal-tempered pitch class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...
es making up a given equal temperament tuning's
chromatic scale
The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...
on a
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
.
Explanation
If one starts on any equal-tempered pitch and repeatedly ascends by the
musical interval
In music theory, an interval is a difference in pitch (music), pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and v ...
of a
semitone
A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between ...
, one will eventually land on a pitch with the same pitch class as the initial one, having passed through all the other
equal-tempered chromatic pitch classes in between. Since the space is circular, it is also possible to descend by semitone.
The chromatic circle is useful because it represents melodic distance, which is often correlated with physical distance on musical instruments. For instance, assuming 12-tone equal temperament, to move from any C on a keyboard to the nearest E, one must move up four semitones, corresponding to four clockwise steps on the chromatic circle. One can also move ''down'' by eight semitones, corresponding to eight counterclockwise steps on the pitch class circle.
Larger motions (or in
pitch space
In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches farther apart. Depe ...
) can be represented in pitch class space by paths that "wrap around" the chromatic circle one or more times.
For any positive integer ''N'', one can represent all of the equal-tempered pitch classes of ''N''-tone equal temperament by the
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
of order ''N'', or equivalently, the
residue classes modulo twelve, Z/NZ. For example, in twelve-tone equal temperament, the group
has four
generators, which can be identified with the ascending and descending semitones and the ascending and descending perfect fifths. In other tunings, such as
31 equal temperament
In music, 31 equal temperament, which can also be abbreviated (31 tone ) or (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equally-proportioned steps (e ...
, many more generators are possible.
The semitonal generator gives rise to the chromatic circle, while the perfect fourth and perfect fifth give rise to the
circle of fifths
In music theory, the circle of fifths (sometimes also cycle of fifths) is a way of organizing pitches as a sequence of perfect fifths. Starting on a C, and using the standard system of tuning for Western music (12-tone equal temperament), the se ...
.
Comparison with circle of fifths
A key difference between the chromatic circle and the
circle of fifths
In music theory, the circle of fifths (sometimes also cycle of fifths) is a way of organizing pitches as a sequence of perfect fifths. Starting on a C, and using the standard system of tuning for Western music (12-tone equal temperament), the se ...
is that the former is truly a continuous space: every point on the circle corresponds to a conceivable
pitch class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...
, 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.
References
Further reading
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External links
Notenscheibe web application- pitch constellations of scales, triads, intervals and the circle of fifths, with basic audio
On-line app illustrating pitch constellationsScaleTapper- IPhone app which utilizes pitch constellations.
PDF of musical scales
{{Pitch space
Pitch space
Post-tonal music theory