On-line app illustrating pitch constellations.Notenscheibe web application - pitch constellations of scales, triads, intervals and the circle of fifths, with basic audio.(1967), Music, Physics and Engineering, Dover Publications, ISBN 9-8 Kučinskas, Darius (2005), "Symmetry in creative work of Mikalojus Konstantinas Čiurlionis" (PDF), Menotyra, 38 (1): 42–46.(1967), The Physics of Musical Sound, Van Nostrand Company. ^ Yamaguchi, Masaya (2006), Symmetrical Scales for Jazz Improvisation, Masaya Music, ISBN 0-967.^ a b Glaser, Matt (1999), Ear Training for Instrumentalists (Audio CD), Homespun, ISBN 5-2.^ Lerdahl, Fred (2001), Tonal Pitch Space, Oxford University Press, ISBN 0-19-505834-8.The Psychology of Music., Academic Press, ISBN 0-12-213564-4. ^ a b Slonimsky, Nicolas (1947), Thesaurus of Scales and Melodic Patterns, Music Sales America, ISBN 0-8256-1449-X.McCartin, The College Mathematics Journal, Vol. ^ "Prelude to Musical Geometry", p.364, Brian J.One can represent the twelve equal-tempered pitch classes by the cyclic group of order twelve, or equivalently, the residue classes modulo twelve, Z/12Z. The circle of fifths drawn within the chromatic circle as a star dodecagon Larger motions on the piano (or in pitch space) can be represented in pitch class space by paths that "wrap around" the chromatic circle one or more times. One can also move down by eight semitones, corresponding to eight counterclockwise steps on the pitch class circle.
For instance, to move from any C on a piano keyboard to the nearest E, one must move up four semitones, corresponding to four clockwise steps on the chromatic circle. The chromatic circle is useful because it represents melodic distance, which is often correlated with physical distance on musical instruments.
Since the space is circular, it is also possible to descend by semitone. If one starts on any equal-tempered pitch and repeatedly ascends by the 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-tempered chromatic pitch classes in between. The chromatic circle is a clock diagram for displaying relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale on a circle.
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