On the Tonnetz, C major and A♭ major are only two transformation steps apart. What does this tell us about their relationship?
AThey have a standard diatonic functional relationship
BThey share common tones and require small voice movements, even though they lack a diatonic functional relationship
CThey belong to the same tonal center and key area
DThey are enharmonically equivalent triads
Tonnetz proximity describes voice-leading efficiency, not functional harmony. C major and A♭ major share the pitch class C (and with enharmonic spelling, E♭), and only one voice moves by semitone. The progression is characteristic of Romantic chromatic mediant writing precisely because smooth voice leading disguises the functional distance. Confusing Tonnetz proximity with functional relationship is the central misconception to avoid.
Question 2 Multiple Choice
Why do the P (parallel), L (leading-tone exchange), and R (relative) transformations each preserve exactly two common tones between the source and target triad?
ABecause they operate within the same diatonic scale, sharing scale tones
BBecause each corresponds to flipping a Tonnetz triangle across a shared edge, where the two endpoints of the edge are the two retained pitch classes
CBecause all reversible transformations preserve two common tones by definition
DBecause triads within the same key always share exactly two pitch classes
On the Tonnetz, each triangle (triad) shares an edge with adjacent triangles. That shared edge represents two pitch classes — the common tones. The P, L, R operations are precisely the flips across these edges: one voice moves (by semitone or whole tone) while the two endpoints of the shared edge remain. This geometric picture directly explains the common-tone structure of each transformation.
Question 3 True / False
A Tonnetz path of minimal length (fewest P/L/R steps) between two triads always corresponds to the most voice-leading-efficient progression between them.
TTrue
FFalse
Answer: True
This is the parsimony principle. Each P, L, R step preserves two common tones and moves at most one voice by a small interval. The shortest path through the Tonnetz accumulates the least total voice movement, making it the most parsimonious route between the two triads. This equivalence between Tonnetz distance and voice-leading efficiency is the central insight of neo-Riemannian theory.
Question 4 True / False
Because the Tonnetz encodes voice-leading efficiency, composers who navigate it with minimal steps are necessarily following the syntax of functional harmony.
TTrue
FFalse
Answer: False
Tonnetz proximity describes geometric proximity in voice-leading space, not functional harmonic syntax. A chain of chromatic mediant progressions can traverse remote regions of the Tonnetz with impeccably smooth voice leading while completely bypassing the dominant-tonic progressions that define functional grammar. Efficient Tonnetz paths and functional harmonic syntax are independent dimensions of musical structure.
Question 5 Short Answer
Explain why the Tonnetz can reveal voice-leading proximity between two triads that seem 'harmonically distant' in traditional tonal theory.
Think about your answer, then reveal below.
Model answer: Traditional tonal theory measures harmonic distance by diatonic relationships — how far two chords are from a shared tonal center or diatonic scale. The Tonnetz instead measures voice-leading efficiency: how much the individual voices must move. Two triads can be adjacent on the Tonnetz (sharing common tones, requiring small voice movements) even if they have no diatonic functional connection. Chromatic mediants are the paradigm: C major to A♭ major involves moving only one voice by semitone despite lacking a standard functional relationship, explaining the characteristic smoothness of such progressions in Romantic music.
The Tonnetz makes visible a geometric logic in voice leading that is orthogonal to functional syntax — it reveals structure that functional theory cannot capture.