Questions: The Tonnetz and Pitch Space Visualization
3 questions to test your understanding
Score: 0 / 3
Question 1 Multiple Choice
On the standard Tonnetz, what interval is encoded by horizontal adjacency (moving one step to the right)?
AMajor third (4 semitones)
BMinor second (1 semitone)
CPerfect fifth (7 semitones)
DMinor third (3 semitones)
The horizontal axis of the Tonnetz maps the circle of fifths: each step right adds a perfect fifth (7 semitones). The upper-left diagonal encodes major thirds (4 semitones) and the lower-left diagonal encodes minor thirds (3 semitones). Each triangle in the lattice represents a triad built from these three intervals.
Question 2 True / False
Moving from one triad to an adjacent triad on the Tonnetz typically requires changing most three pitch classes.
TTrue
FFalse
Answer: False
Adjacent triangles on the Tonnetz share an edge, meaning they share two pitch classes. Only one pitch class changes — it moves to the new vertex across the shared edge. This is precisely parsimonious voice leading: the minimum possible change (one voice, typically by semitone or whole step). This geometric adjacency is why neo-Riemannian operations feel smooth to listeners.
Question 3 Short Answer
Why do neo-Riemannian operations P, L, and R correspond to reflections on the Tonnetz rather than translations or rotations?
Think about your answer, then reveal below.
Model answer: Each PLR operation holds two pitch classes fixed (the shared edge between two adjacent triangles) and moves the third. Geometrically, holding a line (edge) fixed and flipping across it is exactly a reflection. Translations would move all three pitch classes by the same interval (that is transposition), while rotations around a point have no simple musical meaning. Reflections across edges are the natural geometric expression of parsimonious voice leading.
The algebraic structure reinforces this: each operation is an involution (its own inverse), which is the defining property of a reflection. A translation (transposition) is not its own inverse unless the transposition interval is 0 or 6. The match between musical parsimony and geometric reflection is one of the deepest results of neo-Riemannian theory.