OPTIC-Equivalence

Set Theory

Octave + Permutation + Transposition + Involution + Cardinality Equivalence or OPTIC-Equivalence treats musical objects the same if they are transpositions and/or involutions of the same unordered collection of notes (see OPC-equivalence); that is, if they share the same forte number.

OPTC-Equivalent objects share a transposition-equivalent (rotationally-equivalent) pattern of intervals such that the order of the intervals cannot be involuted and remain OPTC-equivalent. Objects are OPTIC-Equivalent when they are OPTC-Equivalent and/or related by involution.

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

So for example, under OPTIC-Equivalence, these six patterns of intervals are one forte number, 3-11, containing all transpositions of the major triad and the minor triad:

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

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