Group Actions in Musical Harmony and Modality

Chords and scales are basic building blocks of music. The prominence of diatonic modes throughout the centuries, and their chord inhabitants (triads and seventh chords) since the new era of harmonic tonality, raise theoretical curiosity in their structural constitution and interdependence. Transformational approaches with group actions have proved themselves to be fruitful since the 1980s. In this talk we highlight a duality of dihedral actions on the triads and extend these dihedral actions to voice leadings in a subgroup of SL(3, Z12) and discuss some music-theoretical consequences.

The step patterns of musical scales (or modes to be more precise) can be described as words in two or more letters. Diatonic modes are the backbones of the "modern" major and minor keys and their constitution can thereby be suitably described through automorphisms of the free group F_2. The tight connection between central palindromes and F_2 automorphisms illuminates a longstanding dispute in historic music theory about the role of the hexachords in connection to the modes.