In musical set theory, a Z-relation, also called isomeric relation, is a relation between two pitch-class sets in which the two sets have the same intervallic content (i.e. they have the same interval vector), but they are of different Tn-type and Tn/TnI-type. That is to say, one set cannot be derived from the other through transposition or inversion.
For example, the two sets {0,1,4,6} and {0,1,3,7} have the same interval vector (<1,1,1,1,1,1>) but they are not transpositionally or inversionally related.
In the case of hexachords each may be referred to as a Z-hexachord. Any hexachord not of the "Z" type is its own complement while the complement of a Z-hexachord is its Z-correspondent, for example 6-Z3 and 6-Z36. See: 6-Z44 and Forte number.
The term, for "zygotic" (yoked or the fusion of two reproductive cells), originated with Allen Forte in 1964, but the notion seems to have first been considered by Howard Hanson. Hanson termed this the isomeric relationship, defining two such sets to be isomeric. According to Michael Schuijer (2008), "the discovery of the relation," was, "reported," by David Lewin in 1960.
Though it is commonly observed that Z-related sets always occur in pairs, David Lewin noted that this is a result of twelve-tone equal temperament (12-ET). In 16-ET, Z-related sets are found as triplets. Lewin's student Jonathan Wild continued this work for other tuning systems, finding Z-related tuplets with up to 16 members in higher ET systems.
Straus argues, " in the Z-relation will sound similar because they have the same interval content." Some argue that the "relation" is often so remote as to be imperceptible, but certain composers have exploited the Z-relation in their work. For instance, the play between {0,1,4,6} and {0,1,3,7} is clear in Elliot Carter's second string quartet.
Read more about this topic: Interval Vector