23a Lesson - Basic PC Set Manipulation

Class discussion

Refresher: what is a pitch class (pc)?

• A number to represent all enharmonic equivalents of a pitch. We use 0 to e
• Remember 10 and 11 are represented as t and e

Refresher: what is a pitch class set (pcs)?

• A collection of pitch classes. Each group size has a different name (refer to the chart in 22b)

Notating pitch class sets

• Many of the examples on your homework will be unordered pitch class sets. Despite this name, we need a standard system for writing these so we can easily make them ordered later
• Undordered pc sets are notated with parantheses around the set and commas between each pc. Examples: (0,1,5,t) and (Bb,C,D,G). Like hwo points on a graph are notated before you actually graph them

Modulo 12 arithmetic

• 12 is the magic number in this system! Everything is based around 12. Just like how you know that the clock resets when you go from noon to 1 P.M., the number above e/11 in this notation roll back around to 0, 1, 2, etc.
• If you have a number above 12, you subtract 12 from it to get the number that’s actually within the bounds of our system

Notating transposition

• We are solely in fixed zero now
• We have the pitch class set (E,F,G,Bb). Using traditional transposition methods, transpose up a P4 to get (A,Bb,C,Eb)
• If we convert our first pcs to integer notation, we get (4,5,7,t). Add 5 to each number (subtracting 12 from any numbers over e), and we have (9,t,0,3). Convert the second set back to note names, and you will find it’s the same as the second set the first time we transposed in our traditional method
• The numbers in integer notation represent intervals from 0. 5 = P4 because F is a P4 (5 half steps) from C (0)
• You would notate this process like this: T5(E,F,G,Bb) = (A,Bb,C,Eb) OR T5(4,5,7,t) = (9,t,0,3). T5 just means that we are transposing by adding 5, and it is fixed to the front of the original set to show that it is the one we’re manipulating

Transposing downward

• Same process as above, but instead of adding numbers we subtract them!
• In Mod12, you can also add 12 to a negative number to get a pc within the bounds of our notation system

Normal form/normal order

• Normal form/order is an ascending arrangement of a pcs where the outside interval is smallest. This is very different from tertian harmony, where we’re concerned with stacking in thirds
• Let’s use (1,7,t). Think about this process like using a clock face: you can only go clockwise, which means you can shuffle which nubmer is first in the set as long as the set remains in the same order. You can have (7,t,1) but you can’t have (1,t,7) because t is out of ascending order
• We can have (1,7,t,), (7,t,1), or (t,1,7). We’re looking for the smallest outside interval, which we do by subtracting the last integer from the first one in the set. t-1=9, 1-7=6, and 7-t=9. Since the middle one has the smallest interval, the normal form for (1,7,t) is [7,t,1].
• Square brackets and commas denote normal form
• Always put things in ascending form before you put them in normal or prime form. This form allows us to go through the process of finding the smallest outside interval, so it’s important that we start out using it

Breaking ties for ordering pitch class sets

• What do we do if we do all our subtracting and end up with two of the same result??
  • Go one more number in from the right with the remaining possibilities to break the tie! Obviously there was no tie to break with (1,7,t), but if we wanted to break the tie between the two 9 answers we would go 7-1=6 and 1-t=3 respectively

Developing a method for normal form using (3,e,5,2,8)

• Put in ascending order: (2,3,5,8,e). Any ascending order is fine, but I usually start with the smallest number on the left because it makes sure I don’t get mixed up
• Get all our different ascending forms:
  • (2,3,5,8,e)
  • (3,5,8,e,2)
  • (5,8,e,2,3)
  • (8,e,2,3,5)
  • (e,2,3,5,8)
• Subtract the outside numbers (last number - first number):
  • e-2=9
  • 2-3=e
  • 3-5=t
  • 5-8=9
  • 8-e=9
• Subtract the next smallest intervals for the tiebreakers (second to last number - first number):
  • 8-2=6
  • 3-8=7
  • 5-e=6
• Subtract the next smallest intervals for the second tiebreakers (third to last number - first number):
  • 5-2=3
  • 3-e=4 -The normal form for this pentachord is [2,3,5,8,e]. Remember that we’re subtracting the same number every time–we’re just changing what we’re subtracting it from

Completely symmetrical pitch class sets don’t have a normal form, so we would just write it in ascending form with the smallest number at the bottom

Further reading

From Open Music Theory

Transposition

In post-tonal music, transposition is often associated with motion: Take a chord, motive, melody, and when it is transposed, the aural effect is of moving that chord, motive, or melody in some direction. That’s the effect here, in two disconnected passages from Debussy’s, La cathédrale engloutie:

The opening motive — comprising the notes B, D, E, or {11, 2, 4} — is transposed four semitones higher in m. 18, representing the cathedral’s slow ascent above the water. Transposing something preserves its intervallic content, and not only that, it preserves the specific arrangement of that thing’s intervals. When we hear the passage at m. 18 above, we recognize its relationship to the passage in m. 1 because the same intervals return, but starting on a different pitch.

Transposition is an operation — something that is done to a pitch, pitch class, or collection of these things — or alternatively a measurement — representing the distance between things. We represent it as Tn, where n represents the ordered pitch-class interval between the two things. To transpose something by Tn, add n to every element in that thing (mod 12). Given the collection of pitch classes in m. 1 above and transposition by T4:

The result is the pitch classes in m. 18. T4 {11, 2, 4} = {3, 6, 8}.

Alternatively, to determine the transpositional relationship between two things, subtract the first thing from the second. If the numbers that result are all the same, the two things are related by that Tn.

This is how I arrived at the T4 arrow label in the musical example above, by “subtracting” the pitch class integers of m. 1 from the pitch-class integers in m. 18.

Normal Form

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.

1. Write as a collection of pitch classes (eliminating duplicates) in ascending order and within a single octave. There are many possible answers.
2. Duplicate the first pitch class at the end.
3. Find the largest ordered pitch-class interval between adjacent pitch classes.
4. 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}:

1. Write as a collection of pitch classes (eliminating duplicates) in ascending order and within a single octave. {8,9,3}
2. Duplicate the first pitch class at the end. {8,9,3,8}
3. Find the largest ordered pitch-class interval between adjacent pitch classes. In this case, the largest interval is between “9” and “3.”
4. 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.