Editing Chromatic circle (section)

==Explanation==
If one starts on any equal-tempered pitch and repeatedly ascends by the [[interval (music)|musical interval]] of a [[semitone]], one will eventually land on a pitch with the same pitch class as the initial one, having passed through all the other [[equal temperament|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]]) can be represented in pitch class space by paths that "wrap around" the chromatic circle one or more times.

[[Image:Pitch class space star.svg|thumb|right|The [[circle of fifths]] in [[12 equal temperament|12-tone equal temperament]] drawn within the chromatic circle as a [[star polygon|star]] [[dodecagon]]<ref>"Prelude to Musical Geometry", p.364, Brian J. McCartin, ''The College Mathematics Journal'', Vol. 29, No. 5 (Nov., 1998), pp. 354-370. [http://www.maa.org/pubs/cmj_Nov98.html (abstract)] [https://www.jstor.org/stable/2687250 (JSTOR)]</ref>]]

For any positive integer ''N'', one can represent all of the equal-tempered pitch classes of ''N''-tone equal temperament by the [[cyclic group]] of order ''N'', or equivalently, the [[modular arithmetic|residue classes]] modulo twelve, Z/NZ. For example, in twelve-tone equal temperament, the group <math> Z_{12} </math> has four [[Generating set of a group|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]], 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]].