Glossary, OTC-Equivalence

Set Theory

Octave + Transposition + Cardinality Equivalence or OTC-Equivalence treats musical objects the same if they have the same intervals relative to a root. Chord inversion types and mode types are examples of OTC-equivalent objects.

In OC-equivalence, permutation or order still matters, especially which note is on the bottom (the root). In OTC-equivalence, the actual root note does not matter, but the order of intervals above the root does matter. So for example, all major chords in first inversion are treated as one OTC-Equivalent object, but root position major chords are not OTC-equivalent to first inversion major chords.

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

Tymoczko 2011 (section 10.4) offers a table similar to the following:

The following tables show every cluster-free OTC-equivalent object which is a subset of the four scales (left column) and the possible modes they are subsets of, with every possible formula spelled out.

Tables are sorted by evenness, so more common chords are at the top of each table. Modes which are actually compatible with a given chord are shown with an underline, and these links can be tapped to learn about that chord type. Intervals which are avoid notes are shown with a strike-through. Formulas which do not yield useful chord types are shown grayed out.

For an explanation of why anyone might need the following tables, see The Game.