Interval Vectors, Explained

What is an Interval Vector?

Every chord and scale has a characteristic sound, and a surprising amount of that sound comes down to one question: which intervals does it contain, and how many of each? The interval-class vector (also called the interval vector, or in Allen Forte's original term, the interval-class content) answers that question with a six-digit tally, written in angle brackets like <001110>.

Read left to right, the six digits count how many times interval classes 1 through 6 occur between every pair of notes in a set. It's a fingerprint: two sonorities with similar vectors tend to have a family resemblance, and two with very different vectors sound like different worlds — compare the crunch of a chromatic cluster (all ic1s) with the openness of a quartal chord (all ic5s).

If you're meeting post-tonal theory for the first time, the vector can look like intimidating math. In practice it requires nothing beyond careful, systematic counting, and by the end of this guide you'll be able to compute one for any set, then check yourself with our free interval vector calculator, which lets you click notes and watch the vector build pair by pair.

1 Start with Interval Classes (ic 1–6)

Post-tonal theory works with pitch classes: every C is the same C regardless of octave, and we number them C = 0, C♯/D♭ = 1, D = 2, … up to B = 11. Once octave doesn't matter, intervals simplify dramatically. A major 10th is just a major 3rd; a 12th is just a 5th.

One more folding step gets us to interval classes. Between two pitch classes you can always measure two distances — going up or going down. From C to A is 9 semitones one way, but only 3 the other. Post-tonal theory keeps the smaller of the two: the interval class (ic) is the shortest distance between two pitch classes, computed as the smaller of the interval and its inversion (i.e., of n and 12 − n).

Why does the list stop at 6? Because 6 is the halfway point of the octave. Any interval larger than a tritone is closer going the other way: 7 semitones up is 5 down, 11 up is 1 down. The tritone itself is perfectly balanced — 6 up equals 6 down — so it's the one interval that is its own inversion, and it caps the list. That's the whole reason the vector has exactly six digits.

2 Compute Your First Vector: The Major Triad

The recipe for any set: list every pair of notes, find each pair's interval class, and tally. Take the C major triad, {C, E, G} = {0, 4, 7}. Three notes make three pairs:

Now tally into six slots. Zero ic1s, zero ic2s, one ic3, one ic4, one ic5, zero ic6s:

Sanity check: a set of n notes always has n(n−1)/2 pairs, so the six digits must sum to that number. Three notes → 3 pairs, and 0+0+1+1+1+0 = 3. If your digits don't add up, you missed or double-counted a pair.

One further observation: compute the minor triad {0, 3, 7} and you'll get <001110> again. Inversionally related sets always share a vector — the mirror image contains the same distances — which is part of why major and minor triads feel like two faces of one object.

3 Scale It Up: The Major Scale

Bigger sets need a system, because the pair count grows fast. The major scale {0, 2, 4, 5, 7, 9, 11} has seven notes, so 7 × 6 ÷ 2 = 21 pairs. Listing them at random invites mistakes. Instead, count one interval class at a time, scanning the whole set for each:

Check the total: 2+5+4+3+6+1 = 21. All pairs accounted for. So:

Notice how much familiar theory is hiding in those six digits: exactly two half steps (mi–fa and ti–do), exactly one tritone (fa–ti, the engine of the dominant 7th chord), and fully six ic5s — the scale is practically built out of perfect fifths. Notice something subtler too: every digit is different. That detail turns out to be one of the most consequential facts in tonal music.

4 Read the Vector: What It Actually Tells You

So far this might feel like bookkeeping. Here's the payoff — three things the vector lets you do that would otherwise take real labor.

Compare sonorities at a glance

The vector is a summary of how a set sounds, independent of spacing or voicing. The whole-tone scale's vector <060603> announces its character instantly: no half steps, no perfect fifths, saturated with major seconds, major thirds, and tritones — that floating, rootless Debussy sound. The chromatic tetrachord {0,1,2,3} gives <321000>: front-loaded, all crunch. When you analyze a piece and want to say "these two chords belong to the same sound-world," matching or near-matching vectors are the evidence.

Predict common tones under transposition

This is the vector's most useful property. The entry for ic n tells you how many notes a set holds onto when you transpose it by n semitones — with one exception: the ic6 entry counts double, because a tritone maps onto itself in both directions.

Because every digit of <254361> is different, every transposition distance preserves a different number of tones — theorists call this the scale's unique multiplicity, or "maximal variety." It's the deep reason key distance works: closely related keys (a fifth away) overlap almost completely, distant keys barely at all, and each step around the circle of fifths changes the overlap by a predictable amount. In compressed form, the vector explains why tonal music can modulate the way it does.

Spot Z-related sets

Occasionally two sets share an identical vector yet are not transpositions or inversions of one another — you cannot map one onto the other, but they contain exactly the same intervals. Forte labeled these Z-related pairs (Z for "zygotic," twinned). The famous example is the pair of all-interval tetrachords: 4-Z15 {0,1,4,6} and 4-Z29 {0,1,3,7}. Each contains exactly one of every interval class — vector <111111> — yet no transposition or inversion turns one into the other. Composers from Berg to Carter treasured these sets precisely for that mix of sameness and difference. If two sets in your analysis have the same vector, check whether they're genuinely equivalent or a Z-pair before you claim they're "the same set."

Check your work: Interval Vector Calculator →

The calculator lets you click notes on a clock face and watch every pair get counted into the vector, step by step. Vector questions on exams reward speed and accuracy, and both come from doing the tally by hand until the six interval classes are thoroughly familiar — work through a handful of trichords and tetrachords first, then verify each answer with the calculator to find exactly where a hand tally went wrong.

Related guides and tools: how to build a twelve-tone matrix · interval vector calculator · twelve-tone matrix calculator · all-combinatorial hexachords · melodic inversion calculator.