1. Quick Review
Before we get to set classes, let's make sure the vocabulary is fresh. In set theory, the word "class" always means "group of equivalent things."
A specific note in a specific register — A4 (the A above middle C) is a different pitch from A3 or A5, even though they share the same letter name.
All pitches that share the same letter name across all octaves — plus their enharmonic equivalents — belong to the same pitch class. Every A in every octave, along with G𝄪 (G double-sharp) and B𝄫 (B double-flat) in every octave, all belong to pitch-class 9.
An unordered collection of pitch classes. It tells you which notes are present, not which octave or which order. A minor triad A–C–E is always pitch-class set {9, 0, 4}, no matter how it's voiced.
Play A–C–E in root position, in close position, or spread across several octaves — the pitch-class content is always the same three notes: {9, 0, 4}. The interval distances between them are also always the same:
- A to C: interval class 3 (a minor third)
- C to E: interval class 4 (a major third)
- A to E: interval class 5 (a perfect fifth)
- Interval-class vector: <001110>
The ICV is a fingerprint — it summarizes all the intervals in a set. Transpose or rearrange the set all you like, the fingerprint stays the same.
2. What Is a Set Class?
Here's the central idea: two pitch-class sets belong to the same set class if one can be turned into the other by transposition or by the operation TnI. Think of it like saying two melodies are "the same tune" even when one is in a different key or flipped upside-down — the interval relationships are preserved.
The complete family of all pitch-class sets that are T/I equivalent to each other. Every member of the family has the same interval-class vector and the same prime form label. For the full catalog, see the List of Set Classes.
| Collection | Notes | PC Set | ICV |
|---|---|---|---|
| C major | C–E–G | [0, 4, 7] | <001110> |
| A minor | A–C–E | [9, 0, 4] | <001110> |
| F major | F–A–C | [5, 9, 0] | <001110> |
| D minor | D–F–A | [2, 5, 9] | <001110> |
All four collections look different on the page, but every pair has the same interval profile. What does that tell us?
Key Consequences
- Major and minor triads are in the same set class — they are T/I equivalent and share prime form (037).
- All members of a set class share exactly the same interval-class vector.
- 220 distinct unordered trichords collapse into just 12 trichordal set classes.
- Every set class gets one canonical label: its prime form, which starts on 0.
Explore: Minor and Major Triads on the Same Root
Pick any root. The minor triad and the major triad built on that same root both belong to set class (037) — same prime form, same ICV. Use the selector to see any of the 12 roots.
Minor triad (blue)
Major triad (orange)
3. Finding Prime Form
Every set class needs one agreed-upon label. That label is the prime form: the most compact, left-packed version of the set, always starting on 0. Think of it as the set in its most "economical" shape.
The algorithm has four steps. The worked example below runs through each one live — try different sets to build intuition.
- Input set:
- Normal form:
- Candidate A (transpose to 0):
- Apply In to Candidate A:
- Candidate B — normal form of inverted set:
- Prime form:
Input set on clock
Prime form on clock
4. Try It Yourself: Prime-Form Calculator
Type in any pitch-class set — integers 0–11, comma- or space-separated — and the tool will walk through every step of the algorithm. This is the same process you'll use in analysis: input a chord or motive, get its prime form and ICV instantly.
Tip: duplicates are removed automatically. Values outside 0–11 are reduced mod 12.
Input set
Prime form
5. The 12 Trichord Families
There are 220 distinct unordered trichords you can build from 12 pitch classes — but once you account for T/I equivalence, all 220 collapse into just 12 families. Every three-note sonority in any piece of music — tonal, post-tonal, jazz, folk — belongs to exactly one of these twelve.
Need to look up other cardinalities (tetrachords, pentachords, hexachords, etc.)? Use the full List of Set Classes.
chromatic cluster
half-step + whole-step
half-step + minor third
half-step + major third
half-step + tritone
two whole-steps
whole-step + minor third
whole-step + tritone
whole-step + perfect fifth
diminished triad
major/minor triad
augmented triad
C major (C–E–G), A minor (A–C–E), F major (F–A–C), G minor (G–B♭–D), and all other major and minor triads — 24 sets in total — all reduce to the same prime form (037). That's 24 sonorities that sound different tonally, but are structurally identical from a set-theory perspective.
This is exactly why set class is such a powerful analytical tool: it reveals structural similarities that tonal labels — "major," "minor," "root position," "key of G" — would keep hidden from you.
6. Full Set-Class Reference
The 12 trichord families above are just the beginning. There are 208 distinct set classes in total — spanning dyads (2 notes) through decachords (10 notes). The full reference table lists every set class with its Forte number, prime form, interval-class vector, and Z-relation partner where applicable.
All 208 set classes organized by cardinality — trichords through nonachords — with prime forms and ICVs.
7. Check Your Understanding
Work through these on your own, then reveal the answer. If something doesn't click, try running the set through the calculator in section 4.
Q1. Find the prime form of A minor: A–C–E (pitch classes 9, 0, 4).
Normal form is [9, 0, 4] (span of 7). Transpose to 0: [0, 3, 7]. Inversion: [3, 0, 8] → normal form [8, 0, 3] → transpose to 0: [0, 4, 7]. Comparing [0,3,7] vs [0,4,7]: position 1 is 3 vs 4 — smaller first, so prime form is (037).
Q2. Do C major and A minor belong to the same set class?
Yes. C major (0–4–7) and A minor (9–0–4) both reduce to prime form (037). They are T/I equivalent, so they are members of set class 3-11.
Q3. Two sets share the same interval-class vector. What does that tell us?
Sharing an ICV is a necessary condition for set-class equivalence — all members of a set class have identical ICVs. However, two sets can share an ICV without being T/I-related (these are called Z-related pairs, a topic for a future lesson). So: same ICV → possibly same set class; but same set class → definitely same ICV.
Q4. Find the prime form of 1, 2, 6.
Sorted: [1, 2, 6]. Normal form: [1, 2, 6] (span 5). Candidate A: [0, 1, 5]. Inverted: [11, 10, 6] → normal form [6, 10, 11] → Candidate B: [0, 4, 5]. Compare: at position 1, 1 < 4, so prime form is (015).
Q5. How many trichordal set classes are there? Why isn't it more?
There are exactly 12. Although you can form 220 distinct unordered trichords from 12 pitch classes, transposition alone reduces many of those to the same family, and TnI equivalence reduces some further. The prime form algorithm identifies one representative for each equivalence class, leaving 12 distinct trichord families. You can verify this in the List of Set Classes (cardinality 3).