Pitch
A specific sounding frequency in register (for example, A4).
Lesson Goals
- Define a set-class using Tn/In equivalence.
- Find normal form and prime form for any small pitch-class set.
- Relate prime form notation to familiar tonal examples.
- Connect interval-class vector invariance to set-class membership.
- Recognize why 220 trichords compress into 12 trichordal set-classes.
- Apply this framework in short analytical prompts.
1. Fast Review: Pitch, Pitch-Class, and ICV
Pitch-Class (PC)
A family of pitches equivalent by octave and enharmonic spelling.
IC Vector
A six-number profile counting interval classes 1 through 6.
Example: A minor triad
[9, 0, 4] => intervals 3, 4, 5 => ICV <001110>
When you transpose or invert this set, interval distances remain the same. So the ICV stays the same.
2. What Is a Set-Class?
A set-class is the collection of all pitch-class sets related by transposition and inversion. If two sets are T/I-related, they belong to the same set-class and share one prime form label.
Key Consequences
- Major and minor triads are members of the same set-class.
- All members of a set-class share the same ICV.
| Collection | PC Set | Prime Form | ICV |
|---|---|---|---|
| C major | [0,4,7] | (037) | <001110> |
| A minor | [9,0,4] | (037) | <001110> |
| F major | [5,9,0] | (037) | <001110> |
Minor-Major Transformation Grid
Converted from your worksheet: left side shows all transpositions of [0,3,7] (minor), right side shows all transpositions of [0,4,7] (major), with inversion labels in the center.
| Minor Family Label | PC 1 | PC 2 | PC 3 | Inversion | Reference | PC 1 | PC 2 | PC 3 | Major Family Label |
|---|---|---|---|---|---|---|---|---|---|
| C minor | 0 | 3 | 7 | I7 (T7I) | from [0,3,7] | 0 | 4 | 7 | C Major |
| T1 (from 0,3,7) | 1 | 4 | 8 | I8 | from [0,3,7] | 1 | 5 | 8 | T1 |
| T2 | 2 | 5 | 9 | I9 | from [0,3,7] | 2 | 6 | 9 | T2 |
| T3 | 3 | 6 | 10 | I10 | from [0,3,7] | 3 | 7 | 10 | T3 |
| T4 | 4 | 7 | 11 | I11 | from [0,3,7] | 4 | 8 | 11 | T4 |
| T5 | 5 | 8 | 0 | I0 | from [0,3,7] | 5 | 9 | 0 | T5 |
| T6 | 6 | 9 | 1 | I1 | from [0,3,7] | 6 | 10 | 1 | T6 |
| T7 | 7 | 10 | 2 | I2 | from [0,3,7] | 7 | 11 | 2 | T7 |
| T8 | 8 | 11 | 3 | I3 | from [0,3,7] | 8 | 0 | 3 | T8 |
| T9 | 9 | 0 | 4 | I4 | from [0,3,7] | 9 | 1 | 4 | T9 |
| T10 | 10 | 1 | 5 | I5 | from [0,3,7] | 10 | 2 | 5 | T10 |
| T11 | 11 | 2 | 6 | I6 | from [0,3,7] | 11 | 3 | 6 | T11 |
Relationship Explorer
Pick a row, and compare the two triads on clockfaces. The selected row is highlighted in the table.
I7: [0,3,7] -> [0,4,7], and T1([0,3,7]) = [1,4,8].
Minor side (Tn of [0,3,7])
Major side (Tn of [0,4,7])
3. Finding Prime Form
Prime form is a canonical representative of a set-class. It begins on 0 and is the most left-packed arrangement when compared against its inversion.
This section now uses the same normal-form algorithm pattern as Week 7 (getNormalForm + tie-break rules), so the workflow is consistent across lessons.
1Normal form - order the set to its most compact clockwise arrangement.
2Transpose to 0 - subtract the first pitch-class from all elements (mod 12).
3Invert and normalize - invert each member around 0, then find normal form and transpose to 0.
4Choose the more left-packed form - compare candidate interval strings from left to right.
Worked Example (JS)
- Input set: [4,2,5]
- Normal form: [2,4,5]
- Transpose to zero: [0,2,3]
- Invert original set: [8,10,7]
- Normalize inversion: [7,8,10]
- Transpose inversion to zero: [0,1,3]
- Prime form: (013)
Input clock
Prime-form clock
This uses the same Week 7-style normal-form function chain, computed live.
4. Interactive Prime-Form Lab
Enter integers 0-11 (comma or space separated). The tool computes normal form, both prime candidates, and the final prime form.
Tip: duplicates are removed automatically.
Normalized set: [2,4,5]
Normal form: [2,4,5]
Candidate A: [0,2,3]
Candidate B (inversion): [0,1,3]
Prime form: (013)
ICV: <100110>
Input set clock
Prime form clock
5. The 12 Trichord Set-Classes
There are 220 distinct unordered trichords, but only 12 trichordal set-classes after T/I equivalence is applied.
3-1: (012)
3-2: (013)
3-3: (014)
3-4: (015)
3-5: (016)
3-6: (024)
3-7: (025)
3-8: (026)
3-9: (027)
3-10: (036)
3-11: (037)
3-12: (048)
6. Notation and Listening Check
Both excerpts below are transpositionally related and keep the same interval content.
Example A
Example B (T4 of A)
If your browser blocks audio, click once anywhere on the page and try again.
7. Quick Practice
Q1. Find the prime form of [0,3,7].
Prime form is (037). It is already in prime form.
Q2. Do [0,4,7] and [2,6,9] belong to the same set-class?
Yes. [2,6,9] is T2 of [0,4,7], so both are members of (037).
Q3. What does sharing an ICV suggest about two sets?
They share the same interval content profile. This is necessary for set-class equivalence and usually signals T/I relation in this context.
Q4. Prime form of [1,2,6]?
Normal form is [1,2,6], candidate A [0,1,5], candidate B [0,4,5], so prime form is (015).