23e Lesson - Using Pitch Class Sets in Analysis

Class discussion

How do you decide normal form with a semetrical chord?

• they are all normal form, but in reference to music, whichever occurence is more common is normal form

Ascending Order vs. Normal Form

The order a PC set goes through is:

1. Unorder
   • (e,6,0)
2. Ascending order
   • (e,0,6), (0,6,e), (6,e,0)
3. Normal form
   • (6,e,0)
   • this is normal form because the distance between the two outer intergers is smaller than the other ascending orders.
4. Prime form
   • (056)
   • you take normal form and orient the intervals on zero.
   • even though we have a new set of numbers, the intervals between each interger is the same.

Ascending order is not normal form, but normal form occurs in ascending order. Ascending order takes the unordered PC set and puts it in a number of orders, one of which is normal form. Normal form is a PC set in ascending order with the notes as close together as possible.

• one way to find normal form is to find the smallest interval between intervals.
  • this is after you have found the ascending orders that are our options for normal form.

With ascending order, you have different options to order a PC set into, like inversions in tonal harmony.

Structure for each form

1. Unordered PC set (x,x,x)
2. Ascending order (x,x,x)
3. Normal order [x,x,x]
4. Prime form (xxx)
   • the only form that uses brackets is normal form
   • the only form that doesn’t use commas is prime form

Allen Forte

Allen Forte created a numbering system for each kind of PC set.

• cardinality: the number of pitch classes in a PC set. The cardinality is how Forte labels PC sets x-x. For example, 5-1, like (0,4,7), is a major triad.
  • the problem with refurring to this system is unlike (0,4,7), writing 5-1 gives you no information about what kind of chord you have unless you look up the Forte table.

The reason we use numbers is to represent all enharmonic equivilants and narrow the idea down to how it exists sonically: as 1 note. 1 note gets 1 number.

Using Pitch Class Sets in Analysis

How many possible combinations of (015) are there?

• this includes any PC set that shares the same intervals, or the inversion and its shared intervals
• this gives us 24 combinations.

Ex: (015)

• could be [1,2,6], [2,3,7], [3,4,8], [4,5,9] etc…
• inversion: (0,e,7), normal form [7,e,0]
  • could be [8,0,1], [9,1,2], [t,2,3] etc…

Finding pitch class set lets you know the basis for an entire piece of music.