What Is "The Matrix"?

Alternate Notation for Prime Form

Here I have changed the procedure slightly to conform to the method espoused by some theorists (including Solomon). Occasionally the prime form of a set is in disagreement between the two methods, but the identity of the set and its unique Forte Number remains the same.

Procecure for finding Normal Form:

Excluding doublings, write the pitch classes ascending within an octave. There will be as many different ways of doing this as there are pitch classes in the set, since an ordering can begin on any of the pitch classes in the set.
Choose the ordering that has the smallest interval from first to last (from the lowest to highest).
If there is a tie under Rule 2, choose the ordering that is most packed to the left. To determine which is most packed to the left, compare the intervals between the first and second notes. If there is still a tie, compare the intervals between the first and third notes, and so on.
If the application of Rule 3 still results in a tie, then choose the ordering beginning with the pitch class represented by the smallest integer. For example, (A, C#, F), (C#, F, A), and (F, A, C#) are in a three-way tie according to Rule 3. So we select [C#, F, A] as the normal form since its first pitch class is 1, which is lower than 5 or 9.

Procedure for finding Prime Form:

Put the set into normal form (see above).
Transpose the set so that the first element is 0.
Invert the set and repeat steps 1 and 2.
Compare the results of step 2 and step 3; whichever is more packed to the left is the prime form. (Use the procedure for determining "left-packedness" given above in Rules 3 and 4 of finding Normal Form.)

The difference between the two methods is in the determination of "left-packedness" (Normal Form, Rule 3). The method espoused by Straus is probably the most consistent because you begin with the outside interval of the set, and then check next-to-outside, and so on until the prime form is determined. However, when Forte first introduced the procedure for arriving at prime form, his method was to pack it to the left starting from the left and working to the right (once the smallest outside interval was found). Because this was the method first chosen by Forte, and because it inherently has no less merit than any other method, many theorists wish simply to retain Forte's original method.

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