Piston Words

We mathematize the notion of a “Piston word” (i.e. a diatonic chord sequence that uniquely determines key) and seek a formal language method of constructing all triadic Piston words. Since these triadic words are (simplified) superstrings of atomic triadic words, it suffices to just construct the language \(\mathcal {L}\) of the atomic triadic words. To this end, we construct a minimal DFA (deterministic finite automaton) for this language \(\mathcal {L}\) . Interestingly, the states of this DFA, which are the Myhill-Nerode classes of \(\mathcal {L}\) , correspond to classical harmonic building blocks: silence, the tonic, the subdominant, the dominant, the diminished triad, the perfect cadence, the plagal cadence, the deceptive cadence, and a fourth type of cadence that we call the “deceptive plagal cadence”.