What are Pitch Class Sets?

Pitch class sets are unordered collections of pitch classes.

Think of them as containers for pitch classes without regard to order or repetition.

When we analyze post-tonal music, we often want to identify which pitch classes are being used in a section (or sometimes an entire piece). A pitch class set helps us organize this information.

Key Characteristics of Pitch Class Sets

Remember these three important properties:

  1. Pitch Class Sets contain pitch classes (not pitches!) We reduce all pitches to their pitch class equivalents (0-11).
  2. No repetition. Each pitch class appears only once in the set, even if it occurs multiple times in the music.
  3. No other pitch class belongs in the set. The set contains exactly the pitch classes used, nothing more.

Example: A Simple Chord

Consider a C major triad: C-E-G

The pitch class set would be: [0,4,7]

It doesn't matter if the chord is C-E-G, G-C-E, or E-G-C-E-G-C. The pitch class set is the same because we're only concerned with which pitch classes are present, not their order or how many times they appear.

Notation Convention

Unordered pitch class sets use square brackets with commas: [0,4,7]

The Canaries Example

You've already encountered pitch class sets in this course! Remember Elliott Carter's Canaries for Timpani (which we analyzed for metric modulation in Week 4)?

Canaries uses only four pitch classes throughout the entire piece:

C♯, E, F, B

or in pitch class integers: [1,4,5,11]

Canaries Pitch Classes on Clock Face

Hear It in Action

Watch from this timestamp to hear the pitch classes in the score:

This set of four pitch classes creates the distinctive sound world of the piece. By limiting the pitch material, Carter creates unity while exploring different rhythmic relationships.

Normal Form

There are many ways to write the same pitch class set. For example, [1,4,5,11] could also be written as [4,5,11,1] or [11,1,4,5] or [5,11,1,4].

Normal form (also called normal order) is a standardized way of writing a pitch class set that presents the most compact and simple version.

Rules for normal form:

  • Always written in ascending order (moving clockwise on the clock face)
  • The ordering that has the smallest interval span from first to last pitch class

Why Normal Form?

Normal form allows us to:

  • Compare different pitch class sets consistently
  • Identify relationships between sets more easily
  • Communicate clearly about musical structures

Finding Normal Form: Step by Step

Let's find the normal form of the Canaries set using a systematic process.

Step 1: List All Possible Rotations

For the set [1,4,5,11], write all possible ascending orderings (rotations):

Rotation Pitch Classes First PC Last PC Span (clockwise)
1 [1,4,5,11] 1 11 11 - 1 = 10
2 [4,5,11,1] 4 1 (next octave) 1 - 4 = -3 → 9 (mod 12)
3 [5,11,1,4] 5 4 (next octave) 4 - 5 = -1 → 11 (mod 12)
4 [11,1,4,5] 11 5 (next octave) 5 - 11 = -6 → 6 (mod 12)

Step 2: Find the Smallest Span

The winner: [11,1,4,5] has the smallest span of 6 semitones.

This is the normal form of the Canaries pitch class set.

Visualizing Normal Form

Notice the large gap between 5 and 11 - this often indicates where the normal form should start.

What If There's a Tie?

Sometimes two or more rotations will have the same span. When this happens:

  1. Compare the interval between the first and second-to-last pitch class
  2. If still tied, compare first to third-to-last, and so on
  3. If tied all the way to the second pitch class, prefer the rotation that begins with the smallest number

Tie-Breaking Example 1: The Major 7th Chord

Consider the set [0,4,7,11] (e.g., a C major 7th chord: C, E, G, B):

[0,4,7,11] Span: 11
[4,7,11,0] Span: 8 1st to 2nd-to-last (4 to 11) = 7
[7,11,0,4] Span: 9
[11,0,4,7] Span: 8 1st to 2nd-to-last (11 to 4) = 5

We have a tie for the smallest span (8 semitones) between [4,7,11,0] and [11,0,4,7]. To break the tie, we check the interval from the first to the second-to-last pitch class. Since 5 is smaller than 7, the winner is [11,0,4,7].

Tie-Breaking Example 2: The Diatonic Collection

Consider the diatonic set [0,2,4,5,7,9,11] (the "white keys"). Finding the normal form of the diatonic collection requires using the first tie-breaker.

The smallest span is 10 semitones. Out of 7 rotations, 5 rotations share this same span of 10! To break the tie, we check the distance from the first note to the second-to-last note:

[2,4,5,7,9,11,0] Span: 10 1st to 2nd-to-last (2 to 11) = 9
[4,5,7,9,11,0,2] Span: 10 1st to 2nd-to-last (4 to 0) = 8
[7,9,11,0,2,4,5] Span: 10 1st to 2nd-to-last (7 to 4) = 9
[9,11,0,2,4,5,7] Span: 10 1st to 2nd-to-last (9 to 5) = 8
[11,0,2,4,5,7,9] Span: 10 1st to 2nd-to-last (11 to 7) = 8

Still a three-way tie! We move to the third-to-last note:

[4,5,7,9,11,0,2] 1st to 3rd-to-last (4 to 11) = 7
[9,11,0,2,4,5,7] 1st to 3rd-to-last (9 to 4) = 7
[11,0,2,4,5,7,9] 1st to 3rd-to-last (11 to 5) = 6

Winner: [11,0,2,4,5,7,9]! Notice this has the distinct half-step sequence shape of the Locrian mode (e.g., B to C, E to F).

Tie-Breaking Example 3: Absolute Tie

What if there's a tie all the way down, such as with [0,2,6,8] (Subset of the whole tone scale, or a French augmented sixth chord)?

[0,2,6,8] Span: 8 1st to 2nd-to-last (0 to 6) = 6 1st to 3rd-to-last (0 to 2) = 2
[2,6,8,0] Span: 10
[6,8,0,2] Span: 8 1st to 2nd-to-last (6 to 0) = 6 1st to 3rd-to-last (6 to 8) = 2
[8,0,2,6] Span: 10

The interval gaps inside [0,2,6,8] and [6,8,0,2] are identical all the way to the second pitch class! In this case, we choose the one that starts on a lower number. Since 0 is lower than 6, our completely tied normal form starts on 0.

Winner: [0,2,6,8]

A Visual Shortcut

Look for the largest gap on the clock face. The normal form usually starts right after this gap (moving clockwise).

In the Canaries example, the largest gap is between 5 and 11, so normal form starts at 11.

Practice Examples

Interactive Normal Form Finder

Click through different pitch class sets and see their normal form:

Try It Yourself

For each set below, find all rotations and determine the normal form:

  1. [0,1,4,6]
  2. [2,5,8,11] (augmented triad + 1 note)
  3. [0,2,4,6,8,10] (whole tone scale)