A set class is a collection of all the sets with the same intervallic recipe, the intervals of which are realized either upward or downward. A set type is also defined by an intervallic recipe, but the intervals are always realized upward from a “root.” Given a set type of (037) we can realize a pitch class set of that type by specifying a root and then supplying the notes 0, 3, and 7 semitones above that note. With a “root” of G, for instance, we get the set {G B-flat D}, or {7 t 2}, or a G minor triad.

Now compare the set class [037]. Now the intervals can be read as upward intervals +0, +3, and +7 semitones or as downward intervals -0, -3, and -7. Let’s make G the referential pitch class, the “zero” from which we build the interval pattern. Going upward from G produces the set {G B-flat D}. As noted above, this is a G minor triad. Going downward produces another set. Going down 3 and 7 semitones from G produces {G E C}, a C major triad. Major and minor triads, set types (047) and (037), belong to the same set class because they both can be described by the same intervallic recipe: [037]. Their intervallic content is the same.

In other words, two different set types will belong to a single set class. The set type (037), a minor triad, and the set type (047), a major triad, both belong to the same set class [037]. We call (037) the **prime form** of this set class. We call (047) the **inversion**. Of these two set types, the prime form is the one that has the smaller interval in second-to-last last place (3 instead of 4). The prime form is more compact, and it is used to name the set class. Thus, [037] is the name of the set class for major and minor triads–for set types (037) and (047). The square brackets, [ ], around the prime form indicate that the intervals are realizable in an upward or a downward direction. The parentheses, ( ), used for set types contain intervals that are realized upward only.

Here are simple procedures for naming sets, including their set class.