Make twelve-tone matrix calculator part of your practice
The Twelve-tone matrix calculator takes one row of twelve pitch classes, each numbered 0 to 11 and used exactly once, and shows how that row relates to its inversion, retrograde and retrograde inversion. It is a numeric tool for studying pitch-class relationships, not an analysis of any particular piece. Use it to check a row, see its transpositions and hear your original input row as one finite example.
What the row and matrix mean
Use each number from 0 to 11 exactly once. For input and playback the numbers label chromatic pitch classes, with 0 played as C4 and 1 as C#4. The displayed matrix is normalized by subtracting the first input value, so its initial zero is a reference to that first row pitch, not a claim about a tonal tonic. Missing or repeated values are rejected.
Why prime, inversion and retrograde forms help
Prime forms read the normalized row left to right. Retrograde forms read the same row right to left, which lets you check whether a fragment reverses cleanly. Inversion forms are read downward in the first column: each value is subtracted from zero modulo twelve, so an upward interval becomes the same interval downward. Retrograde inversion reads that inverted column upward. Comparing these four shapes shows which intervals are preserved and which are mirrored, a useful way to see how a short fragment can be varied while keeping its identity.
Reading the matrix and its limits
Transposition preserves an ordered row’s interval pattern; inversion reverses interval direction and retrograde reverses order. The matrix organizes these relationships but does not choose register, rhythm or expression for a composition. Preview plays the original input row once. To hear a different matrix form, enter its complete twelve-number order or play it in another app. Keep any register and rhythm decisions separate from the pitch-class calculation.
Try it step by step
- Enter twelve numbers from 0 to 11 in the Pitch classes field, separated by spaces, with no duplicates.
- Confirm that each number from 0 to 11 appears exactly once.
- Set the Tempo (BPM) field, then use the preview to hear your original input row at that tempo.
- Swap two values in the input row to create a different valid permutation, then inspect the new matrix. Replacing just one value can create a duplicate and will be rejected.
A concrete example
For the sample row 0 1 4 2 7 5 8 6 11 9 3 10, the normalized first row begins 0, 1, 4, 2. Subtracting each value from zero modulo twelve gives an inversion column beginning 0, 11, 8, 10, because 12 minus 1 is 11, 12 minus 4 is 8 and 12 minus 2 is 10. The upward semitone from 0 to 1 becomes a downward semitone from 0 to 11.
A common wrong turn
Entering a row with a repeated or missing value, such as ending the sample row with 3 3 instead of 3 10.
Try this instead: Check that the twelve values are distinct and cover 0 through 11 exactly once. Count them and compare against the list 0 to 11 before reading the matrix.
Where to go next — and why
This guide was drafted with Claude Haiku and edited against this tool’s implemented controls. Report a confusing instruction or incorrect example.
