Tone Systems

Certain musical styles are historically and geographically constrained to specific
tone systems. These pitch organizations, when plotted in the circle of fifths (hereafter COF), can reveal their generative algorithm.
Take pentatonic scales, for example, which can be either hemitonic or anhemitonic.
Hemitonic scales contain one or more semitones and anhemitonic scales do not contain any semitones at all. A hemitonic pentatonic scale, common in some areas of North and West Africa contains flatted 2nd, 3rd, and 6th degrees. So, if the scale begins off from a C, it will be followed by a D-flat, E-flat, an G-natural plus an A-flat).
Anhemitonic pentatonic scales, one of the oldest tone system we can recognize, can be constructed by taking five consecutive pitches from the circle of fifths; starting on C, these are C, G, D, A, and E. Transposing the pitches to fit into the interval of an octave, it reorders the pitches into the major pentatonic scale: C, D, E, G, A, C. It is typically associated with both non-Western traditional cultures and European music, serving as plainchant’s resonant backbone. From an algorithmic perspective, it can be generated by collecting any five consecutive pitches from the COF.
Heptatonic tone systems follow similar procedures. For example, the interval content of the lydian mode can be generated by collecting seven consecutive fifths in the COF, starting in C and moving clockwise to F#, the ionian mode starting from F to B, the mixolydian from Bb to  E, the dorian from Eb to A, the aeolian from Ab to D, the phrygian from Db to G, and the locrian, moving counterclockwise from C to F#.
The generative algorithm for the whole-tone scale, a trade mark for late 19th and early 20th century French music, uses one from each two consecutive fifths in the COF.
Twelve-tone rows, a type of tone organization associated with serial techniques, as initially developed by Arnold Schoenberg (1874-1951) during the first quarter of the 20th century, are generated under the rule that no tone of the equal-tempered chromatic scale can appear repeated. The original twelve-tone row is further systematized in a magic square of twelve rows, where each new row corresponds to an exact transposition of the original intervals, so that in each column all pitch classes appear only once.