Compute prime form from pitch-class input with rotation tests. Review inversion, transposition, and ordering instantly. Study advanced modulo twelve set structure with confidence today.
| Input Set | Cleaned Set | Prime Form | Note |
|---|---|---|---|
| 0, 4, 7 | {0, 4, 7} | {0, 3, 7} | Major and minor triads share this class. |
| 0, 1, 2 | {0, 1, 2} | {0, 1, 2} | This is already the most packed form. |
| 0, 2, 4, 6 | {0, 2, 4, 6} | {0, 2, 4, 6} | Symmetry keeps the same prime form. |
| 0, 2, 7 | {0, 2, 7} | {0, 2, 7} | An open trichord with a stable compact form. |
Transposition: Tn(x) = (x + n) mod 12
Inversion: In(x) = (-x + n) mod 12
Normal order: rotate the ordered set and choose the form with the smallest span.
Tie rule: if spans match, choose the arrangement that is more packed to the left.
Prime form: compare the transposed normal order of the set and its inversion. Then keep the more compact T0 version.
A set theory prime form calculator helps reduce a pitch-class set to one compact pattern. This makes comparison easier. Different transpositions can look different at first. Their prime form reveals shared structure. That is useful in mathematical music analysis, pattern study, and classroom work.
The calculator begins by cleaning the input. It removes duplicates. It wraps every value with modulo twelve arithmetic. Then it sorts the remaining pitch classes. Next, it builds every circular rotation of the ordered set. Each rotation is tested for span. The smallest span wins. If two rotations share the same span, the tool chooses the version packed furthest to the left.
Normal order is only part of the process. Prime form also checks inversion. Inversion flips each pitch class around zero. After inversion, the calculator repeats the same normal order test. Both candidates are then transposed so the first value becomes zero. This creates two T0 forms. The more compact arrangement becomes the final prime form.
This topic is strongly connected to finite sets and modular arithmetic. It is a clear example of equivalence classes. Two sets may differ by transposition or inversion, yet still belong to the same class. Prime form gives one standard label for that class. That saves time during analysis and prevents inconsistent notation.
Students can test trichords, tetrachords, and larger collections. Teachers can build fast examples for lectures. Analysts can compare motivic shapes across passages. Researchers can document set relations in a consistent way. The export tools also help when results must be shared, printed, or placed into notes and worksheets.
Enter only integers from zero through eleven. Negative values are accepted, then wrapped correctly. Repeated values do not change the set class. This page shows the cleaned set so you can verify the input before using the prime form in further work.
Prime form is the most compact transpositionally normalized version of a pitch-class set. It lets you compare related sets under transposition and inversion using one standard form.
Yes. Duplicate pitch classes are removed before analysis. Prime form depends on set membership, not repeated entries, so duplicates do not affect the final class.
Yes. Negative integers are wrapped into the 0 to 11 range with modulo twelve arithmetic. For example, -1 becomes 11 and -13 also becomes 11.
Normal order is the arrangement of a set with the smallest outer span. If several rotations tie, the calculator chooses the one packed furthest to the left.
Prime form must account for inversional equivalence. A set and its inversion can belong to the same class, so both candidates are checked before the final compact form is chosen.
It is most common in pitch-class set analysis, but it also illustrates modular arithmetic, equivalence classes, and compact ordering methods in a mathematical setting.
T0 means the selected ordering has been transposed so its first element becomes zero. This standardizes the form and makes comparison easier.
Yes. After you calculate, you can download a CSV file for data work or a PDF summary for printing, sharing, or saving.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.