Glossary
Complement
Glossary
Set Theory
Complements are set classes with the same Lewin-Quinn FC-components, which are not related by transposition or involution. So for example the pentatonic scale and the diatonic scale are complements—the “holes” in one are the set class for the other: the black keys (pentatonic scale) are the holes in the white key (diatonic) scale, and vice versa.
See Grouping Clocks for more on this.
Complements have the additional relation that sets with lower cardinality are always a subset of some transposition of their higher cardinality complement or are involutions of their complement. For example, the pentatonic scale is a subset of the diatonic scale; the augmented chord is a subset of the enneatonic (nine-note) scale; the diminished chord is a subset of the octatonic diminished scale; the whole-tone and double augmented hexatonic scales are their own complements, etc.
Complementarity also preserves evenness ordering within set classes of the same cardinality. For example, all of the set classes of cardinality three have a certain evenness ordering, and their complements, all of the set classes of cardinality nine, are ordered exactly the same. This is easy to see over on the Set Classes tables page. (There is a simple mathematical relationship between the Distance values, which will be left as an exercise to the reader.)
Ian Quinn’s masters thesis on Q-space (see References) describes in more detail the importance of complementarity and the mathematics behind set classes which are complements.
Set Theory
Complements are set classes with the same Lewin-Quinn FC-components, which are not related by transposition or involution. So for example the pentatonic scale and the diatonic scale are complements—the “holes” in one are the set class for the other: the black keys (pentatonic scale) are the holes in the white key (diatonic) scale, and vice versa.
See Grouping Clocks for more on this.
Complements have the additional relation that sets with lower cardinality are always a subset of some transposition of their higher cardinality complement or are involutions of their complement. For example, the pentatonic scale is a subset of the diatonic scale; the augmented chord is a subset of the enneatonic (nine-note) scale; the diminished chord is a subset of the octatonic diminished scale; the whole-tone and double augmented hexatonic scales are their own complements, etc.
Complementarity also preserves evenness ordering within set classes of the same cardinality. For example, all of the set classes of cardinality three have a certain evenness ordering, and their complements, all of the set classes of cardinality nine, are ordered exactly the same. This is easy to see over on the Set Classes tables page. (There is a simple mathematical relationship between the Distance values, which will be left as an exercise to the reader.)
Ian Quinn’s masters thesis on Q-space (see References) describes in more detail the importance of complementarity and the mathematics behind set classes which are complements.