An Interval Class Vector (ICV) is a six-number code that captures the "interval fingerprint" of any collection of notes. It's essential for analyzing post-tonal and contemporary music—where traditional chord names don't apply—allowing you to compare and classify pitch-class sets by their sonic character.

Use the calculator below to build a set, see its ICV instantly, and explore how different chords share (or differ in) their interval content. The tutorial at the bottom walks through the calculation step-by-step.

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Interval Class Vector C - E - G

< 0 0 1 1 1 0 >

m2/M7 M2/m7 m3/M6 M3/m6 P4/P5 TT

All Intervals

Distribution

Common ICVs

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Notice: Major and minor triads share the same ICV—they're inversionally related. Sets with matching ICVs but no T/I relation are called Z-related.

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Quick Quiz

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What's the ICV for {D, F, A}

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How to Calculate an ICV (Complete Guide)

An Interval Class Vector (ICV) is like a fingerprint for a group of notes. It tells you exactly what kinds of intervals exist between all the notes in your set. This is incredibly useful for analyzing modern music that doesn't follow traditional key signatures.

Understand Pitch Classes

First, we convert note names to numbers. This makes the math easier and treats all octaves the same (C4 and C5 are both just "C").

C0 C#/Db1 D2 D#/Eb3

E4 F5 F#/Gb6 G7

G#/Ab8 A9 A#/Bb10 B11

Example: For a C major chord {C, E, G}, we get pitch classes {0, 4, 7}

Find All Intervals Between Notes

Now we measure the distance between every possible pair of notes. For 3 notes, that's 3 pairs. For 4 notes, it's 6 pairs.

Pairs = n(n-1) ÷ 2 where n = number of notes

To find the interval, subtract the smaller pitch class from the larger:

C to E 4 - 0 = 4 interval of 4

C to G 7 - 0 = 7 interval of 7

E to G 7 - 4 = 3 interval of 3

Convert to Interval Classes (The Key Step!)

Here's the crucial insight: we treat intervals and their inversions as the same thing. A minor 2nd (1 semitone) and major 7th (11 semitones) are considered equivalent because one is just the inversion of the other.

The Rule

If your interval is greater than 6, subtract it from 12

IC = 12 - interval (when interval > 6)

This gives us only 6 possible interval classes:

1 1 or 11 semitones minor 2nd / major 7th

2 2 or 10 semitones major 2nd / minor 7th

3 3 or 9 semitones minor 3rd / major 6th

4 4 or 8 semitones major 3rd / minor 6th

5 5 or 7 semitones perfect 4th / perfect 5th

6 6 semitones tritone (its own inverse!)

Continuing our example:
• C-E = 4 → stays 4 (IC 4)
• C-G = 7 → 12-7 = 5 (IC 5)
• E-G = 3 → stays 3 (IC 3)

Count Each Interval Class

Finally, count how many of each interval class you found. Write them in order from IC1 to IC6:

C Major Chord: {C, E, G}

IC1 0 none

IC2 0 none

IC3 1 E-G ✓

IC4 1 C-E ✓

IC5 1 C-G ✓

IC6 0 none

The ICV for C Major is: <001110>

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Why ICVs Matter

Transposition-invariant: C major and F# major have the same ICV—the shape of the chord matters, not the starting note.
Compare any sets: You can compare a jazz chord to an atonal cluster and see what interval content they share.
Identify sonorities: Sets with lots of IC1 sound "crunchy" while sets with IC5 sound more "open."
Find related sets: Two sets with the same ICV will sound similar even if they're not transpositions of each other.

🎹 Try It Yourself!

Use the calculator above to check your work. Try calculating the ICV for D minor {D, F, A} by hand, then enter it in the calculator to verify!

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Quick Summary

An ICV is a 6-digit code showing how many of each interval type exist between all notes in a set. Format: <IC1, IC2, IC3, IC4, IC5, IC6>

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Used By

Essential for analyzing works by Schoenberg, Bartók, Stravinsky, Webern, and contemporary composers who work outside traditional tonality.

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Related Concepts

Pitch-class sets, Forte numbers, prime form, set classes, and the 12-tone technique all connect to interval class vectors.