OPTC-Equivalence

Set Theory

Octave + Permutation + Transposition + Cardinality Equivalence or OPTC-Equivalence treats musical objects the same if they are transpositions of the same unordered collection of notes (see OPC-equivalence); that is, if they share the same prime form.

In OPTC-Equivalence, the order of the intervals matters, but not the absolute order relative to the root. So for example, these three patterns of intervals (OTC-equivalent) are one permutation-equivalent set class, the major triad:

• major third + minor third + perfect fourth
• minor third + perfect fourth + major third
• perfect fourth + major third + minor third

However, the following is not OPTC-Equivalent (minor triad inversions) to the major triad:

• minor third + major third + perfect fourth
• major third + perfect fourth + minor third
• perfect fourth + minor third + major third

See Grouping Clocks for more details about set classes, prime forms, and black and white clocks.

See Equivalence Groups for a tutorial that runs through some examples.