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VIII. 20th- and 21st-Century Techniques
Set Class and Prime Form

### Core Idea

A set class is a group of pitch-class sets related by transposition or inversion. 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).

### Explanation

A set class is a group of pitch-class sets related by transposition or inversion.

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).

You can find prime form mathematically or by using the clock face.

All possible set classes are summarized in the set class table, and are available on Wikipedia and many other websites.

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.

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. Tn and In 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.

Step

Example

1. Put the pitch class set in normal order.

[10, 0, 2, 3, 5]

2. Transpose the set so that the first pitch class is 0.

T2 = [0, 2, 4, 5, 7]

3. Invert the results from Step 2 and put the result in normal order.

I0 = [5, 7, 8, 10, 0]

4. Transpose the set from Step 3 so that the first pitch class is 0.

T7 = [0, 2, 3, 5, 7]

5. Compare the sets in Step 2 and Step 4.

Whichever set is most compact to the left is the prime form.

Write the prime form in parentheses with no commas.

(02357) is the prime form (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. Many resources, such as Wikipedia, have tables of these set classes, arranged 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 s

### Key Quote

A set class is a group of pitch-class sets related by transposition or inversion.

### Application Scenarios

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.

One way of analyzing a lot of post-tonal music is by studying the [transpositional and inversional relationships](https://viva.pressbooks.pub/openmusictheory/chapter/pc-sets-normal-order-and-transformations) 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. Tn and In relationships between passages in "Subject and Reflection."

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.

### Key Terminology

pitch-class set transposition pitch class prime form set class interval subject octave class tonal pitch form

### Related Links

available on Wikipedia Pitch vs. pitch class Interval vs. interval class transpositional and inversional relationships pitch class sets are named by their normal form Many resources, such as Wikipedia, have tables of these set classes interval class vector Blank clock faces (integer notation)Blank clock faces (letter names)Set Theory Quick Reference Sheet
Source: https://viva.pressbooks.pub/openmusictheory/chapter/set-class-and-prime-form/
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