Set Classes

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.

 

 Introduction to Set Classes

 

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

How to Name Sets.v2

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