PermTriasIntervall

PermTriasIntervall is a concept in mathematical music theory that formalizes how the interval structure of a triad can appear under different internal permutations of its voices. Given a triad T consisting of three pitch classes in a 12-tone system, each permutation σ of the three voices yields an ordered triple (pσ(1), pσ(2), pσ(3)). One computes the ascending semitone intervals i1 = (pσ(2) − pσ(1)) mod 12 and i2 = (pσ(3) − pσ(2)) mod 12. The pair (i1, i2) characterizes the internal spacing of the triad in that voice order. The PermTriasIntervall(T) is the set of all six such interval pairs as σ runs over the symmetric group S3. This set is invariant under transposition of T and thus depends only on the interval class of the triad rather than its absolute pitch level.

Properties and interpretation: The construction encodes how triads can be realized in different voice-leading configurations. Major and minor triads produce distinct families of interval pairs, reflecting their different internal spacing. The concept provides a compact signature for comparing triads under permutations, complementing existing tools such as interval vectors and set-class analysis.

Applications: Used in analytical studies of chordal symmetry, voice-leading efficiency, and in computational musicology for classifying triadic sonorities and generating progressions with specific permutation-induced interval profiles.

See also: interval vector, set-class, triad, voice leading.

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