The simplest way to define set class is “a group of pitch-class sets related by transposition or inversion.” This may initially seem confusing, but it’s just another kind of class. As you have learned in other chapters, “class” is another name for “group.” Recall the other kinds of classes you have already learned about.

• Pitch vs. pitch class: A pitch occurs at a specific octave, and often we conceive of it with a specific spelling. A pitch class is a group of pitches that is related by transposition or inversion.
• Interval vs. interval class: An interval has a specific distance in semitones, while an interval class is a group of intervals that are inversions of each other or related by octaves.

Introduction

Pitch-class set vs. set class (pitch-class set class) is the topic of this chapter. The reason the definition of “set class” may seem more confusing is that it involves two kinds of groups: classes and sets.

• A class is a group that is related in some way.
• A set is a group that is not necessarily related in any specific way.

As an analogy, consider biology and the way different living things are categorized. Plants in the same class are are all biologically related in a specific way: Angiospermae is a class of plants that produce flowers. But we can group together plants for other reasons: the group of plants in someone’s front yard, for example. That would be a set of plants, but not a class of plants.

So a pitch-class set is a group of pitches that the analyst decides to put together for some reason. The pitch-class set class—a term that is very unwieldy, so theorists have agreed to shorten it to the last two words, set class—is the group of groups of pitches that are all related by transposition or inversion.

Why transposition and inversion?

One way of analyzing a lot of post-tonal music is by studying the transpositional and inversional relationships between pitch-class sets. Take the short example below: two passages from Béla Bartók’s “Subject and Reflection” (Example 1). Comparing across the two passages, the two sets that comprise the right hand, [10, 0, 2, 3, 5] and [3, 5, 7, 8, 10], are related by T₅. The two left-hand sets are also related in the same way._.Now looking within each passage, the right and left hands are related to each other by inversion. In the first passage, they are related by I₈; in the second, by I₆.

Example 1. T_n and I_n relationships between passages in “Subject and Reflection.”

To quickly explain why these snippets of notes all sound the same, we can say they are all members of the same set class.

Major and minor triads may be a helpful and more familiar example. Major and minor triads all sound “the same” compared to tone clusters, quartal harmonies, or even augmented or diminished triads. The reason for that is that all major and minor triads are transpositionally or inversionally related to one another.

• Triads of the same quality are transpositionally related.
  • T₂ of a C major triad [0, 4, 7] is a D major triad [2, 6, 9]
  • T₂ of an A minor triad [9, 0, 4] is a B minor triad [11, 2, 6].
• Triads of opposite quality (major vs. minor) are inversionally related.
  • I₀ of a C major triad [0, 4, 7] is an F minor triad [5, 8, 0].
  • I₂ of a C major triad [0, 4, 7] is a G minor triad [7, 10, 2].

Prime Form

Just as pitch class sets are named by their normal form, set classes are named by their prime form: the version of the set that is transposed to zero and is most compact to the left (compared with its inversion).

Set classes are named by their prime form, just as.

Note that prime form is just a label for a set class. It does not have any special status—it’s not significant if a composer uses [0, 1, 4] as a pitch-class set just because it shares the same integers as the prime form (014).

Mathematical process

Here is the process to put a pitch-class set in prime form, with an example using the motive from Example 1.

Using the clock face

The video in Example 2 explains the differences between normal form and prime form and reviews how to find each by using the clock face.

Example 2. Video lesson on prime form.

The Set Class Table

There are a finite number of set classes and prime forms. The table below arranges the set classes by the cardinality of the set classes. You’ll notice another number-based name for each set class in the format X–X; this is the Forte number of the set. Set class tables also pair sets together by their complements (the set which, together with the original set, will complete the twelve-tone collection). Other features of the set class table, such as the interval class vector, are discussed in later chapters.

The table has two vertical halves. The smaller set classes are in the left half, and the larger in the right. Together, the two set classes are complementary (together, they create the chromatic set).

• Blank clock faces (integer notation)
• Blank clock faces (letter names)
• Set Theory Quick Reference Sheet: summarizes the definitions of pitch vs. pitch class, intervals vs. interval classes, and sets vs. set classes.

1. Set Class Composition prep worksheet (.pdf, .docx). Prepares students for the set class composition by asking them to find sets and transformations.
2. Set Class Composition (.pdf, .docx). Builds on the prep worksheet. Asks students to compose and analyze a 24-bar ABA form piece for unaccompanied solo instrument using set classes.