6.6: Set Theory - Normal Order
Normal order (sometimes called normal form) has a lot in common with the concept of triad “root position.” Among other things, root position is a standard way to order the pitch-classes of triads and seventh chords so that we can classify and compare them easily. Normal order does the same, but in a more generalized way so as to apply to chords containing a variety of notes and intervals.
Normal order is the most compressed way to write a given collection of pitch classes. Often, you’ll be able to determine normal order intuitively using a keyboard or a clockface, but it’s good to learn a process that will always give you the correct answer.
- Write as a collection of pitch classes (eliminating duplicates) in ascending order and within a single octave. There are many possible answers.
- Duplicate the first pitch class at the end.
- Find the largest ordered pitch-class interval between adjacent pitch classes.
- Rewrite the collection beginning with the pitch class to the right of the largest interval and write your answer in square brackets.
For example, given {G-sharp4, A2, D-sharp3, A4}:
- Write as a collection of pitch classes (eliminating duplicates) in ascending order and within a single octave. {8,9,3}
- Duplicate the first pitch class at the end. {8,9,3,8}
- Find the largest ordered pitch-class interval between adjacent pitch classes. In this case, the largest interval is between “9” and “3.”
- Rewrite the collection beginning with the pitch class to the right of the largest interval and write your answer in square brackets. [3,8,9]
Occasionally you’ll have a tie in step 3. In these cases, write the ordering implied by each tie and calculate the interval from the first to the penultimate pitch class. The ordering with the smallest interval is the normal order.