RealMusicTheory

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Interval-Class Vectors

The interval-class vector (ICV) is a six-place tally that summarizes the interval content of a pc set. Each position records how many times each of the six interval classes appears among all pairs of pitch classes in the set. The vector is written in angle brackets: ⟨f1, f2, f3, f4, f5, f6⟩, where f_n is the count of pairs separated by interval class n.

To compute the ICV, list every unordered pair of pitch classes in the set and calculate the interval class for each pair (min of the directed interval and its complement). Then tally the counts for IC 1 through IC 6. For a major triad {0, 4, 7}: the pair (0,4) gives IC 4, the pair (0,7) gives IC 5, and the pair (4,7) gives IC 3 — yielding the ICV ⟨0,0,1,1,1,0⟩. All sets belonging to the same set class share the same ICV.

The ICV encodes the sonority's "color" in measurable terms. Sets rich in IC 5 (fourths/fifths) sound open and consonant; those rich in IC 1 or IC 2 (semitones and whole tones) sound more dissonant or ambiguous. The special case where two different set classes share the same ICV is called Z-relation — such pairs are Z-related sets. The ICV thus provides an efficient way to compare the interval content of any two collections, regardless of transposition or inversion.

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