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VIII. 20th- and 21st-Century Techniques
Pitch-Class Sets, Normal Order, and Transformations

### Core Idea

A pitch-class set (pc set) is a group of pitch classes. The standard way of naming a pitch class set is by its normal order: the smallest possible arrangement of pitch classes, in ascending order.

### Explanation

A pitch-class set (pc set) is a group of pitch classes.

The standard way of naming a pitch class set is by its normal order: the smallest possible arrangement of pitch classes, in ascending order.

To transpose a set (Tn), add n to each integer of the set.

To invert a set (In), first invert the set (take each integer's complement mod 12), then transpose by n. Alternately, subtract each integer of the set from n (n–x=y).

The clock face may help you perform any of these tasks.

Pitch-Class Sets

When we talk about a group of pitch classes as a unit, we call that group a pitch-class set, often abbreviated "pc set." Any group of pitch classes can be a pitch-class set.

Normal Order

Normal order is the most compressed way to write a given collection of pitch classes, in ascending order. Normal order has a lot in common with the concept of root position. Root position is a standard way to order the pitch classes of triads and seventh chords so that we can classify and compare them easily. Normal order does the same, but in a more generalized way so as to apply to chords containing a variety of notes and intervals.

Following are a mathematical and a visual method for determining normal order.

Mathematical method

Process

Example set:

G♯4, A2, D♯3, A4

1. Write as a collection of pitch classes (eliminating duplicates) such that they would fit within a single octave if played in ascending order. There are multiple possible options.

8,9,3

2.Duplicate the first pitch class at the end.

8,9,3,8

3. Find the largest ordered pitch-class interval between adjacent pitch classes.

8 to 9: 1

9 to 3: 6

3 to 8: 5

9 to 3 is the biggest interval.

4. Rewrite the collection beginning with the pitch class to the right of the largest interval and write your answer in square brackets.

[3,8,9]

Occasionally you’ll have a a tie for largest interval in step 3. In these cases, the ordering that is most closely packed to one side or the other is the normal form. If there is still a tie, choose the set most closely packed to the bottom.

Visual method (clock face method)

If you don't like the process described above, the video in Example 1 clearly explains how to use the clock face to quickly find normal order.

Example 1. Using the clock face to find normal form.

Transposition

In post-tonal music, transposition is often associated with motion: take a chord, motive, melody, and when it is transposed, the aural effect is of moving that chord, motive, or melody in some direction. In two disconnected passages from Claude Debussy’s La cathédrale engloutie, the opening motive <D, E, B> or <2, 4, 11> is transposed four semitones higher to <F♯, G♯, D♯> or <6, 8, 3> in m. 18 (both in blue in Example 2), representing the cathedral’s slow ascent above the water. When we hear the passage at m. 18, we recognize its relationship to the passage in m. 1 because the same set of ordered intervals return, but starting on a different pitch. Transposition preserves intervallic content, and not only that, it preserves the specific arrangement of the intervals.

Example 2. Transposition of a motive, ordered and unordered.

The same motive is preserved in a more obscured fashion going from m. 18 into m. 19 (in red in Example 2). Here, the unordered pitch class set is transposed, but not the ordered set from before. This may not look as much like a transposition as the first two sets of motives, but if the pitches are put into normal order, the transposed intervallic relationship becomes clear.

Transposition is often abbreviated Tn, where n, the index number of a transformation, represents the ordered pitch-class interval between the two sets. Transposition is an operation—something that is done to a pitch, pitch class, or collection of these things. Alternatively, transposition can also be a measurement—representing the distance between things.

Transposing a set

To transpose a set by Tn, add n to every integer in that set (mod 12, meaning numbers wrap around after reaching 12, like on a clock face).

Given the collection of pitch classes in m. 1 above and transposition by T4:

\begin{alignat*}{2}&& [11, 2, 4] \\ {}+ && 4\ 4\ 4\ \\ \hline && [3, 6, 8] \\ \end{alignat*}

The result is the pitch classes in m. 18.

T4 [11, 2, 4] = [3, 6, 8]

Identifying transpositions and calculating the index number

To determine the transpositional relationship between two sets, subtract the first set from the second. If the numbers that result are all the same, the two sets are related by that Tn. For example, to label the arrows in Example 1, an analyst would “subtract” the pitch class integers of m. 1 from the pitch-class integers in m. 18. Note that both sets should be in normal order.

\begin{alignat*}{2} && [3, 6, 8] \\ {}- && [11, 2, 4] \\ \hline && 4\ 4\ 4\ \\ \end{alignat*}

[3, 6, 8] and [11, 2, 4] are related by T4.

Inversion

Inversion, like transposition, is often associated with motion that connects similar objects. The passage in Example 3 from Chen Yi's Duo Ye (2000) is an example: just as in the transpositionally related passages, these two gestures have the same intervallic content, so our ears recognize them as very similar. Unlike transposition, however, the interval content of these two gestures is not arranged in the same way: both have the same intervals, but the [1, 4, 6] set has the interval 3 on the bottom instead of on the top (Example 4).

Example 3. [2, 4, 7] is inverted to become [1, 4, 6].

[](https://viva.pressbooks.pub/app/uploads/sites/12/2019/05/inversion-2-1.png) Example 4. These two sets both have the same intervals (a 2, a 3, and a 5), but the intervals are arranged differently.

General inversion

If you are asked to invert a set and are not given an index number, assume you are inverting the set mod 12. This means taking the complement of each number mod 12. The complement of each integer x mod 12 is the number y that is the difference between x and 12. For example, the complement of 4 is 8: 4+8=12. The complement of 6 is 6: 6+6=12. The complement

### Key Quote

Normal Order Normal order is the most compressed way to write a given collection of pitch classes, in ascending order.

### Application Scenarios

Example set:

If you don't like the process described above, the video in Example 1 clearly explains how to use the clock face to quickly find normal order.

Example 1. Using the clock face to find normal form.

In post-tonal music, transposition is often associated with motion: take a chord, motive, melody, and when it is transposed, the aural effect is of moving that chord, motive, or melody in some direction. In two disconnected passages from Claude Debussy’s La cathédrale engloutie, the opening motive <D, E, B> or <2, 4, 11> is transposed four semitones higher to <F♯, G♯, D♯> or <6, 8, 3> in m. 18 (both in blue in Example 2), representing the cathedral’s slow ascent above the water. When we hear the passage at m. 18, we recognize its relationship to the passage in m. 1 because the same set of ordered intervals return, but starting on a different pitch. Transposition preserves intervallic content, and not only that, it preserves the specific arrangement of the intervals.

### Key Terminology

ordered pitch-class interval complement mod 12 pitch-class set transposition root position normal order pitch class collection beginning interval octave mod 12

### Related Links

https://viva.pressbooks.pub/app/uploads/sites/12/2019/05/inversion-2-1.png https://viva.pressbooks.pub/app/uploads/sites/12/2019/05/Inversional-pairs.png https://viva.pressbooks.pub/app/uploads/sites/12/2019/05/Cross-addition-for-inversion.png Blank clock faces (integer notation)Blank clock faces (letter names)Set Theory Quick Reference Sheet pdf docx pdf docx
Source: https://viva.pressbooks.pub/openmusictheory/chapter/pc-sets-normal-order-and-transformations/
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