IX. Post-tonal music

Pitch Class Sets, Normal Order, and Transformations

Brian Moseley and Megan Lavengood

Key Takeaways

  • A pitch class set (pc set) is a group of pitch classes.
  • Normal order is a way of naming a pitch class set.
  • Normal order is the smallest possible arrangement of pitch classes, in ascending order.
  • To transpose a set by Tn, add to each integer of the set.
  • To invert a set by In, first, invert the set (take each integer’s complement mod 12), then transpose by n.
  • 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 , often abbreviated pc set. Any group of pitch classes can be a pitch class set.

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 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) in ascending order and within a single octave. There are many possible answers. 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 tie 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 processes described above, this video clearly explains how to use the clock face to quickly find normal order.

Transposition

Example 1. [11, 2, 4] is transposed by 4 semitones.

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. That’s the effect in Example 1, in two disconnected passages from Claude Debussy’s La cathédrale engloutie. The opening motive [B, D, E] or [11, 2, 4] is transposed four semitones higher in m. 18, representing the cathedral’s slow ascent above the water.

Transposing something preserves its intervallic content, and not only that, it preserves the specific arrangement of that thing’s intervals. When we hear the passage at m. 18 above, we recognize its relationship to the passage in m. 1 because the same intervals return, but starting on a different pitch.

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.

Transposition is often abbreviated Tn, where n represents the ordered pitch-class interval between the two sets. is called the of this transformation.

Transposing a set

To transpose a set by Tn, add n to every integer in that set ().

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

    \begin{alignat*}{2} && [11, 2, 4]  \\ {}+ && 4\ 4\ 4\ \\ \cline{1-3} && [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 things are related by that Tn. For example, to label the arrow 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 .

    \begin{alignat*}{2} && [3, 6, 8] \\ {}- && [11, 2, 4] \\ \cline{1-3} && 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 2 from Chen Yi’s Duo Ye (2000) is an example: just as was the case in the transpositionally-related passages, these two gestures have the same intervallic content—and 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 3).

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

Example 3. These two sets both have the same intervals (a 2, a 3, and a 7), 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 of each number . The complement of each integer x is the number y that is the difference between and 12. For example, the complement of 4 is 8: 4+8=12. The complement of 6 is 6: 6+6=12. The complement of 0 is 0: 0+0=0, which is 12 mod 12.

Inverting [2, 4, 7] in this way would yield [5, 8, 10].

In: Invert-then-transpose method

But sometimes, sets are both inverted and then transposed, as in Example 2. The abbreviation for this is In.

In Example 2, the first set [2, 4, 7] is inverted by I8. To invert a set by I8 follow this process, in this order:

  1. Invert: [2, 4, 7] becomes [5, 8, 10].
  2. Transpose: adding 8 to every number in [5, 8, 10] yields [1, 4, 6].

In: Subtraction method

You can calculate the new set created by In by subtracting all the pitch classes of your first set from n.

What is I8 of [2, 4, 7]?

    \begin{alignat*}{2} && 8\ 8\ 8\ \\ {}- && [2, 4, 7] \\ \cline{1-3} && [6, 4, 1] \\ \end{alignat*}

I4 [0, 2, 7] = [1, 4, 6].

Identifying inversions and their index numbers

Any two pitches related by inversion can be added together to form the . This makes sense as a logical extension of the subtraction method above: if the inverted pitch is the result of n–x, then it is also true that n = x + y.

Another way to visualize this is on the clock face.If you have two sets that are 1) both in normal order and 2) related by inversion, the notes within each set will map onto one another in reverse order, as shown in Example 3 below. Write the two sets in normal form on top of one another, then add the opposing integers of each set together as illustrated in Example 4 to yield the index number of the I relation. If the sum of each number pair is 12 or more, subtract 12 so that your is in .

clock face illustration of paired integers adding to the same index number
Example 3. Two inversionally-related sets have integers that pair together in reverse order to form a mirror-like relationship. These paired intervals always sum to the same number, which is the index number of the inversion.
Illustration of crosswise addition
Example 4. For two 3-note sets, the leftmost integer in one set is added to the rightmost of the second; the middle numbers are added together; and the rightmost element of the first set is added to the leftmost of the second. This is the cross-addition method for finding the index number of inversions.

Using the clock face to transpose and invert

If you prefer a more visual method for transposing and inverting, watch the video below.

Further reading
  • Straus, Joseph Nathan. 2016. Introduction to Post-Tonal Theory. 4th ed. Upper Saddle River, NJ: Prentice Hall.
Assignments
  • Worksheet on normal form and transformations (.pdf, .docx). Asks students to calculate normal form of various sets, and to calculate Tn/In relationships in “Nacht” by Arnold Schoenberg.
  • Composition prep worksheet (.pdf, .docx). Prepares students for the set class composition by asking them to find sets and transformations.

Media Attributions

License

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Open Music Theory by Brian Moseley and Megan Lavengood is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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