(musth212) Pitch-Class Sets, Normal Form

The first thing to do with a collection of pitches, a pitch-class set, is to arrange the pitch classes into a form that can be used to compare one set to another. The first order of arrangement is Normal Form, which arranges the pitch classes in ascending order, starting with the pitch class that gives the most compact and compressed arrangement.

Let’s use the pitch classes C, G#, B, E, and A.

1. First, convert the pitch classes into integer notation, excluding any pitch-class doublings. (giving you 0,8,11,4,9)
2. Order the pitch classes into ascending order within an octave.
   0,4,8,9,11 is one possible order
3. Find the rotation that has the smallest interval from first to last pitch class.
   0,4,8,9,11 = 11
   4,8,9.11,0 = 8
   8,9,11,0,4 = 8
   9,11,0,4,8 = 11
   11,0,4,8,9 = 10
   Five pitch classes will give your five possible rotations. In this case, there is a tie between 4,8,9,11,0 and 8,9,11,0,4 (both at 8).
4. If there is a tie with rule 3, compare the interval size from first pc to next-to-last pitch class. (If there is a tie, keep moving left and comparing.)
   4,8,9,11,0: 4 to 11 = 7
   8,9,11,0,4: 8 to 0 = 4 (smallest!)
5. If there is a tie with rule 4, start with the smallest pc integer number. For example, comparing augmented triads will always yield ties through rule 4. If you have 1,5,9 you should start with 1.

From our rules above, the normal form of the collection is [8,9,11,0,4], which you will always write with brackets surrounding the pitch-class numbers.