The trichord {0, 1, 4} appears within the tetrachord {0, 1, 4, 6}. How should this relationship be classified?
AA transposition relationship — {0, 1, 4} is a Tn-transform of {0, 1, 4, 6}
BA subset relationship — every pitch class in {0, 1, 4} is also contained in {0, 1, 4, 6}
CAn inversion relationship — {0, 1, 4} is the TnI-transform of the remaining element {6}
DA complement relationship — {0, 1, 4} and {0, 1, 4, 6} together fill a larger aggregate
Subset containment is a membership relationship: A ⊆ B if every element of A is also in B. Here {0, 1, 4} ⊆ {0, 1, 4, 6} because 0, 1, and 4 all appear in the larger set. Transposition and inversion are transformations between sets of the same cardinality that preserve interval structure — they cannot relate a trichord to a tetrachord. The common mistake is treating any structural relationship between sets as a form of Tn/TnI equivalence.
Question 2 Multiple Choice
In post-tonal analysis, a Hasse diagram is used primarily to:
AShow all possible transpositions and inversions of a pitch-class set under Tn and TnI
BDisplay which pitch-class sets share the same interval-class vector
CVisualize hierarchical subset containment, with sets ordered vertically by cardinality
DMap prime forms to show which set classes are most common in atonal repertoire
A Hasse diagram places pitch-class sets (usually in prime form) as nodes, stacked vertically by cardinality — dyads at the bottom, trichords above, tetrachords above those. An upward edge from A to B means A is a proper subset of B with no intermediate set between them. Reading the diagram reveals how small motivic cells nest inside larger structural collections, mapping the compositional logic of subset expansion and contraction. It is a tool for containment relationships, not transformation equivalences.
Question 3 True / False
A pitch-class set can simultaneously be a subset of a larger collection AND transpositionally equivalent to another set in the same piece — these are independent, non-contradictory relationships.
TTrue
FFalse
Answer: True
Subset containment (A ⊆ B) and transpositional equivalence (A = TnC for some C) are entirely independent claims. A trichord might be extracted from a governing hexachord (subset relationship) while also being a transposition of a trichord from a different section of the piece (equivalence relationship). Both can hold simultaneously without contradiction. Conflating them — e.g., assuming that shared pitch classes between two sets establish a structural connection — leads to overclaiming.
Question 4 True / False
If pitch-class set A is a subset of set B, then A should be transpositionally or inversionally equivalent to at least one other subset of B.
TTrue
FFalse
Answer: False
Subset containment makes no guarantee about transformation equivalences. Set B may contain subsets with distinct interval-class profiles, none of which are Tn or TnI related to each other. A can be a unique subset of B with no other subset of the same cardinality sharing its prime form. Equivalence relationships are about interval structure; subset relationships are about membership. They must be tracked separately.
Question 5 Short Answer
Explain why subset relationships and transposition/inversion equivalences must be tracked separately in post-tonal analysis.
Think about your answer, then reveal below.
Model answer: Subset containment is a membership claim: A ⊆ B means every pitch class in A appears in B, regardless of their interval structures or cardinalities. Transposition/inversion equivalence is a transformation claim: A and C are Tn-equivalent if C = A + n (mod 12), which requires equal cardinality and the same interval-class vector. A set can be contained in another without any transformation relationship holding, and two sets can be transformation-equivalent without one containing the other.
The distinction matters analytically: subset relationships reveal compositional derivation (how smaller cells are drawn from larger structural collections), while Tn/TnI relationships reveal motivic unity through transformation (the same interval structure recurring at different pitch levels). Conflating them produces false claims — e.g., concluding that a shared pitch class establishes a structural relationship between two sets.