Hexachords

The Guidonian hexachord plays an extraordinary role in several music-theoretical regards. Here we start with some elementary combinatorial considerations on hexachords within the diatonic scale. The cardinality 6 refers to the number of tones. But note that the number of steps is just 5. The periodic step interval pattern ... TTSTTTSTTSTTTS... of the diatonic scale contains six different hexachordal step interval patterns, namely TTSTT, TSTTT, STTTS, TTTST, STTTS and TTTST. This circumstance is a manifestation of the cardinality equals variety property (the refers to the tone cardinality). Within the repeating sequence of the seven consecutive hexachordal factors TTSTT, TSTTT, STTTS, TTTST, TTSTT, TSTTS, STTST exactly one pattern occurs twice: the Guidonian hexachord TTSTT.

Each diatonic species of the octave contains three different hexachords.