A late Romantic passage cycles through C major → E major → Ab major → C minor → E minor → Ab minor → C major via PL operations. How should this be analyzed?
AAs a diatonic progression that returns to C major as tonic
BAs a modulation through three keys (C, E, Ab) and back
CAs a cycle through a single hexatonic system, producing tonal ambiguity because no single pitch center governs all six triads
DAs a sequence of secondary dominants since each chord is a major third from the previous
The PL cycle is closed within one hexatonic system — it never leaves that system. The three major triads are spaced four semitones apart (augmented triad spacing), so no single pitch class asserts itself as tonic. This is precisely the tonal ambiguity hexatonic systems create. Option B (modulation) implies arrival at new tonal centers, but the smooth neo-Riemannian motion within the system never establishes a new key — it suspends tonal gravity entirely.
Question 2 Multiple Choice
Why does a hexatonic system produce tonal ambiguity rather than implying a single tonal center?
ABecause it uses all twelve pitch classes, preventing any single pitch from dominating
BBecause the three major triads within the system are spaced four semitones apart — augmented triad spacing — so no single root stands out as the tonic
CBecause hexatonic systems are atonal by definition and have no key signatures
DBecause PLR operations are undefined in tonal music
The three major triads in any hexatonic system (e.g., C, E, Ab in the Northern system) each sound like they could be a local tonic, but they are spaced symmetrically a major third apart — the same spacing as an augmented triad. Symmetric divisions of the octave like this have no natural resting point; every root is equally privileged (or equally deprivileged). Option A is wrong: hexatonic systems use only six of the twelve pitch classes, not all twelve.
Question 3 True / False
The four hexatonic systems partition all twelve pitch classes into four disjoint groups of six, with no pitch class appearing in more than one system.
TTrue
FFalse
Answer: True
Cohn showed that the four hexatonic systems (Northern, Western, Southern, Eastern) are disjoint and together cover all twelve pitch classes exactly once. This is structurally analogous to how the twelve pitch classes are partitioned by whole-tone scales or diminished seventh chords — symmetric divisions that create equal-interval spacing and harmonic ambiguity. Each system is a closed, self-contained harmonic region.
Question 4 True / False
The hexatonic pole progression (e.g., C major to Ab minor) sounds smooth because the two chords share several pitch classes in common.
TTrue
FFalse
Answer: False
C major {C, E, G} and Ab minor {Ab, Cb, Eb} share no pitch classes whatsoever. The hexatonic pole pair is the most distant pair within a system precisely because of this zero-common-tone relationship. The smoothness comes from the near-semitonal voice leading enabled by the P transformation within the system — each voice moves by a small interval — not from common tones. This is what makes the hexatonic pole progression so striking: it is harmonically jarring (no shared notes) yet tonally fluid (smooth voice leading).
Question 5 Short Answer
What makes hexatonic systems analytically useful for late 19th-century music, and what does a passage staying within one hexatonic system tell us about that music?
Think about your answer, then reveal below.
Model answer: Hexatonic systems explain smooth harmonic progressions with no diatonic logic — triads with no common tones or diatonic relationship can be adjacent steps in a PL cycle within one system. A passage staying within a single system is harmonically suspended: it moves through a tonally ambiguous region where no pitch center is established, yet the voice leading is smooth and the motion is internally coherent. This helps explain why late Liszt and Wagner can feel simultaneously fluid and destabilizing — the logic operates in neo-Riemannian space, not in tonal key relationships.
Traditional tonal analysis requires identifying keys, modulations, and dominant-tonic motion. But many late Romantic passages resist this — they move between triads with no common-tone or diatonic basis. Hexatonic analysis provides an alternative framework: these passages are navigating a single harmonic region defined by neo-Riemannian closure, not by key. Progressions that cross hexatonic system boundaries mark genuine harmonic ruptures; motion within a system is fluid but tonally weightless.