On the Tonnetz
The Tonnetz ("tone-net") is an old idea — Euler sketched
one in 1739, Oettingen and Riemann gave it harmonic life in the
late 19th century. Each triangle is a major or minor triad; edges
run along perfect fifths and major and minor thirds. Adjacent
triangles share two pitches.
David Lewin's Generalized Musical Intervals and
Transformations (1987) reframed harmony as a network of
group-theoretic moves: Parallel, Leittonwechsel,
and Relative generate the dihedral group on the 24 triads.
Richard Cohn (Music Analysis 15.1, 1996; JMT
41.1, 1997) showed that these are precisely the
parsimonious motions — single-semitone voice leadings —
and traced their cycles through the late-Romantic chromatic
repertoire (Wagner, Liszt, Schubert, Bruckner) where roman-
numeral function-theory loses its grip.
Click P, L, R, N, S to traverse. Press ? for
keyboard shortcuts. Run a cycle to watch a parsimonious
orbit close on itself.