Significance in Music Theory

Why do we learn about Interval Class Vectors? They serve several critical functions:

  • Characterizing Sound: The IC vector helps describe the inherent consonance or dissonance of a set. A set with many seconds and tritones (IC 1, 2, and 6) will sound more dissonant than one with many thirds, fourths, and fifths (IC 3, 4, and 5).
  • Analytical Tool: It allows musicians and analysts to make connections between sets that might not appear related on the surface but share similar intervallic content, a concept called Z-relation.
  • Compositional Aid: Composers use the vector to make pre-compositional decisions about the "sound world" of a piece, choosing sets that provide a desired balance of intervals. It can also indicate how many common tones a set will share with its own transpositions. (Wikipedia)

Connection to Rhythm

You've already done this.

Remember calculating the full interval content of timelines in Weeks 2-3? That was an interval vector.

For the clave son rhythm, you calculated: (0,1,2,2,0,3,2,0)

Today we're applying the same concept to pitch classes.

Just as timelines have characteristic interval patterns that create their distinctive sound, pitch class sets have interval content that defines their sonic quality.

From Rhythm to Pitch

The process is identical:

  • Rhythm: Count distances between onsets in pulses
  • Pitch: Count distances between pitch classes in semitones

Both tell us about the intervallic structure and contribute to what we perceive as the overall sound.

Interval Classes (IC)

Interval class represents the shortest distance measured in semitones between two pitch classes.

Think of it as a class or general type of interval, easily visualized on a clock face diagram.

The Six Interval Classes

There are only six interval classes we care about (we ignore IC 0, which is unison/doubling):

IC Semitones Traditional Names Example
1 1 or 11 Minor 2nd / Major 7th C–C♯ or C–B
2 2 or 10 Major 2nd / Minor 7th C–D or C–B♭
3 3 or 9 Minor 3rd / Major 6th C–E♭ or C–A
4 4 or 8 Major 3rd / Minor 6th C–E or C–A♭
5 5 or 7 Perfect 4th / Perfect 5th C–F or C–G
6 6 Tritone (Augmented 4th / Diminished 5th) C–F♯

Why "shortest distance"?

On a clock face, C to F♯ could be 6 semitones clockwise OR 6 semitones counter-clockwise. The tritone (IC 6) is special because it's the same distance either way.

For all other intervals, we count the shorter path. C to B♭ is 10 semitones up but only 2 semitones down, so it's IC 2.

Interval Classes on Clock Face

Click an interval class to see examples on the clock face.

The Interval Class Vector

An interval class vector summarizes all the interval classes present in a pitch class set.

Format: <abcdef> where:

  • a = number of IC 1 intervals
  • b = number of IC 2 intervals
  • c = number of IC 3 intervals
  • d = number of IC 4 intervals
  • e = number of IC 5 intervals
  • f = number of IC 6 intervals (tritones)

Important Note on Tritones (IC 6): Tritones are counted only once per pair. Since IC 6 divides the octave exactly in half, it maps onto itself when inverted. Therefore, the number in slot 6 of the vector is exactly half the number of tritones in the set. For example, a tritone interval itself like [0,6] maps to 0 and 6, which is just one pair, so its ICV is <000001>.

Number of Intervals in a Set

Before calculating, it is helpful to know exactly how many total intervals a set contains based on its size (cardinality). You can visualize this graphically: if each pitch class is a point, how many lines does it take to connect every point to every other point?

PCS Size Total Intervals
21
33 (2 + 1)
46 (3 + 3)
510 (4 + 6)
615 (5 + 10)
721 (6 + 15)
828 (7 + 21)
936 (8 + 28)
1045 (9 + 36)
1155 (10 + 45)
1266 (11 + 55)

Notice the pattern: The number of intervals is always the number of pitch classes in the previous size plus the number of intervals in the previous size. For example, a 5-note set has 4 + 6 = 10 intervals.

In mathematical set theory pedagogy, these are called Triangular Numbers.

2 Notes
1 Interval
3 Notes
3 Intervals
4 Notes
6 Intervals
5 Notes
10 Intervals

How to Calculate (The Manual Way)

Count all intervals between all pairs of pitch classes (not just adjacent ones!):

Example: [0,4,7] (C Major Triad)

Step 1: List all pairs of pitch classes:

  • 0 to 4 = 4 semitones → IC 4
  • 0 to 7 = 7 semitones → IC 5 (shorter than 5 going backwards)
  • 4 to 7 = 3 semitones → IC 3

Step 2: Count occurrences of each IC:

IC 1: 0, IC 2: 0, IC 3: 1, IC 4: 1, IC 5: 1, IC 6: 0

IC Vector: <001110>

Major and Minor Triads

Both major [0,4,7] and minor [0,3,7] triads have the same IC vector: <001110>

This is because they contain the same interval classes (just in different order). The IC vector tells us about interval content, not order.

Canaries Example

[11,1,4,5] (Canaries)

Calculate all intervals between pairs:

  • 11 to 1 = 2 → IC 2
  • 11 to 4 = 5 → IC 5
  • 11 to 5 = 6 → IC 6
  • 1 to 4 = 3 → IC 3
  • 1 to 5 = 4 → IC 4
  • 4 to 5 = 1 → IC 1

IC Vector: <111111>

Every interval class appears exactly once. This is a very balanced, even distribution.

Using Calculators: The Smart Way

We Use Tools, Not Formulas

In this course, we focus on musical relationships, not manual calculation.

Just as you use calculators for arithmetic, you'll use set theory calculators for analysis.

Quick IC Vector Calculator

Enter a pitch class set (e.g., "0,4,7" or "11,1,4,5"):

For detailed analysis, use the PC Set Calculator (link above).

External Set Theory Calculator

This tool is indispensable for analysis:

PC Set Calculator

For the rest of this course: Use this calculator freely. Your job is to understand what the IC vector tells you about the music, not to compute it by hand.

Step-by-Step: C Major Scale IC Vector

Let's calculate the IC vector for the complete diatonic collection (major scale): [0,2,4,5,7,9,11]

C Major Scale

Pitch classes: [0,2,4,5,7,9,11]

Count All Intervals

We count intervals between every pair of pitch classes. There are 21 pairs total (7 choose 2).

From 0:

  • 0→2 = 2 → IC 2
  • 0→4 = 4 → IC 4
  • 0→5 = 5 → IC 5
  • 0→7 = 7 → IC 5
  • 0→9 = 9 → IC 3 (9 or 3, take shorter)
  • 0→11 = 11 → IC 1 (11 or 1, take shorter)

Running count: IC1:1, IC2:1, IC3:1, IC4:1, IC5:2, IC6:0

From 2:

  • 2→4 = 2 → IC 2
  • 2→5 = 3 → IC 3
  • 2→7 = 5 → IC 5
  • 2→9 = 7 → IC 5
  • 2→11 = 9 → IC 3

Running count: IC1:1, IC2:2, IC3:3, IC4:1, IC5:4, IC6:0

From 4:

  • 4→5 = 1 → IC 1
  • 4→7 = 3 → IC 3
  • 4→9 = 5 → IC 5
  • 4→11 = 7 → IC 5

Running count: IC1:2, IC2:2, IC3:4, IC4:1, IC5:6, IC6:0

From 5:

  • 5→7 = 2 → IC 2
  • 5→9 = 4 → IC 4
  • 5→11 = 6 → IC 6

Running count: IC1:2, IC2:3, IC3:4, IC4:2, IC5:6, IC6:1

From 7:

  • 7→9 = 2 → IC 2
  • 7→11 = 4 → IC 4

Running count: IC1:2, IC2:4, IC3:4, IC4:3, IC5:6, IC6:1

From 9:

  • 9→11 = 2 → IC 2

Final count: IC1:2, IC2:5, IC3:4, IC4:3, IC5:6, IC6:1

C Major Scale IC Vector: <254361>

What this tells us about tonal music:

  • 6 perfect fourths/fifths - the core harmonic progression of a cadence
  • 5 major seconds - the main melodic stepwise motion
  • 4 minor thirds and 3 major thirds - the core harmonic components of tonality
  • 2 half steps - determining the cadence in tonal music. Must be only a few, otherwise there would be more than one perceived tonic. This way just the tonic and the major third are led to strongly in upper voices
  • 1 tritone - between the two unstable tones that resolve by half steps

The Power of Set Theory

Notice what just happened: we derived fundamental principles of tonal harmony directly from the IC vector.

In traditional theory, understanding these concepts - cadential progression, voice leading, functional harmony, the role of the tritone - takes years of study across multiple courses.

Set theory operates at a meta-level. The IC vector gives us immediate analytical insight into any collection, scale, or chord without needing to construct a complete theoretical framework first.

This is why we can use set theory to understand not just tonal music, but also whole-tone scales, octatonic collections, jazz voicings, and post-tonal repertoire - all using the same analytical tool.

Try it yourself: Enter 0,2,4,5,7,9,11 in the calculator above to verify.

Musical Examples

Common Collections

Collection Pitch Classes IC Vector What It Tells Us
Major/Minor Triad [0,4,7] / [0,3,7] <001110> One each of IC 3, 4, 5
Diminished Triad [0,3,6] <002001> 2 minor 3rds, one tritone
Augmented Triad [0,4,8] <000300> All major 3rds
Whole Tone (6 notes) [0,2,4,6,8,10] <060603> All even numbers - no IC 1, 3, or 5
Octatonic (8 notes) [0,1,3,4,6,7,9,10] <448444> Very balanced distribution
Chromatic (all 12) [0,1,2,3,4,5,6,7,8,9,10,11] <12,12,12,12,12,6 12 of each IC 1-5, 6 tritones

Note on the Chromatic Scale: The interval class vector (ICV) of a chromatic scale (all 12 pitch classes) is <666663>. It represents the total count of each interval class (IC1-IC6) present in the collection.

Important Rule for Tritones: Notice that tritones are counted only once per pair. Since a tritone divides the octave exactly in half, it maps onto itself when inverted. Therefore, the number in slot 6 of the vector is actually half of the total number of tritones in the set. For example, the set [0,6] has two discrete tritones (0 to 6 and 6 to 0), but its ICV is <000001> since we only count the pair once. Following this rule, the chromatic scale has 6 of each IC 1-5, but only 3 in the tritone slot.

Practice: Try These

Use the calculator above or an external tool to find IC vectors for:

  1. [0,1,4,6] - What intervals are most common?
  2. [0,2,5,7,9] - Pentatonic collection
  3. [0,1,2,3,4,5] - Chromatic cluster

Coming Friday: We'll visualize pitch class sets and their relationships using clock faces and other graphical methods.

Analysis Example: The Z-Relation

One of the most powerful applications of the interval class vector is identifying the Z-relation. A Z-relation exists between two pitch-class sets if they share the same interval class vector but are not related by transposition or inversion. This means that despite having identical intervallic content (and therefore a very similar general "sound"), one set cannot be transformed into the other through simple translation or flipping.

Elliott Carter and the Z-Relation

Composers like Elliott Carter and Béla Bartók have exploited the Z-relation extensively.

For example, the sets 4-Z15 [0,1,4,6] and 4-Z29 [0,1,3,7] are Z-related. Let's look at their vectors:

  • 4-Z15 [0,1,4,6] → ICV: <111111>
  • 4-Z29 [0,1,3,7] → ICV: <111111>

Both 4-note sets are all-interval tetrachords—they contain exactly one instance of every interval class! Carter utilized this specific relationship prominently in his Second String Quartet, using these sets to provide structural unity while maintaining varied melodic surface contours.

Anton Webern

Anton Webern's twelve-tone compositions, such as his Symphony, Op. 21, are frequently analyzed using interval class vectors. In Op. 21, the 12-tone row breaks into two equivalent hexachords (both of set class 6-1, representing half of a chromatic scale), which themselves subdivide into carefully regulated trichords sharing specific intervallic profiles. The ICV provides the "fingerprint" allowing analysts to track these intervallic concentrations throughout the complex serial web.