Set Theory Primer
Atonal music hit the musical scene during the early 20th century under the leadership of Arnold
Schoenberg, and his “Second Viennese School” which is named as a con@nuing legacy that
Vienna had on classical music through composers such as Mozart, Haydn and Beethoven. The
music theory concept of Set Theory was created by theorist Alan Forte among other theorists as
a way to beHer understand this music. Sets are not how this music was composed, instead
composers used various shapes we now understand as sets so compose music where there
might not be a tonal center.
A Set is a collec@on of pitches organized by pitch class meaning that All pitches have the same
value, regardless of octave or spelling. So F# and Gb mean the exact same thing. There are
numerous ways to teach this music, but I have found that the best way to understand this is
through the use of a pitch wheel where the full chroma@c scale is represented from 0-11 like so
When trying to understands sets there are a few things you should be aware of which I’ll define
right now:
Normal Order (abr. NO): The order of the sets from lowest to highest that puts the smallest
intervals first
So lets say we are given the pitches D, F#, D#. we would say that this normal order is [2,3,6].
There are a few ways to visualize this but on a pitch wheel you will no@ce how the distance
between PC2 and PC3 is 1-half-step, then between PC3 and PC6 is 3-half-steps, and finally there
is a much larger distance between PC 6 and PC 2. This is how we know that the normal order is
[2,3,6]
The Next thing we like to know is what is the Prime Form
Prime Form (abr. PF): The most compact way of viewing the set star@ng on PC0.
So the previous example we would subtract 2 from each pitch from the Normal Order. (2-2 = 0),
(3-2 = 1) and (6-2 = 4) meaning that the PF of that set is PF (014) (note: we use (paratheses) for
prime form, and brackets for [normal order]. Prime form is used to classify all sets as its base
form so that when we look at musical examples we can say that any sets with the same Prime
form are mo@vically related.
NOW that probably all seems fairly straight forward. But sets have a few complica@ons that
make the previous example seem so simple. What if I was given the set NO [2,5,6]: Note that
the normal order is [2,5,6] and not [6,5,2], because normal order is always going clockwise or
from lowest to highest.
As you will no@ce the intervallic quality of this set is the exact same as the previous set, but
where the previous example looks like this 23||6 this new set is 2||56. No@ce the hash marks
happen in the opposite order. This implies that the set has been Inverted. The inverted set is the
same intervallic material but flipped on its side. The best way to figure out if something is
inverted is to see if you can re-order the set with the smallest intervals first going clockwise
around the pitch wheel. So in this case star@ng on PC6 going counter-clockwise it becomes 0,
PC5 becomes 1, and PC2 can becomes 4. This quickly gets us back to the prime form from an
inverted set.
This process will make finding sets much more simple. But what if I ask you to transform the
set?
This is actually quite simple.
Reminder that we go 0-11, so 12=0, 13=1 etc.
T(n) = Transposi@on (How much it is transposed by) = So T(9) from that same set just requires you to add 9 to
each number.
NO [236] (2+9 = 11), (3+9 = 12-12=0), and (6+9 = 15-12= 3) so T9 of NO [2,3,6] = [11,0,3]
What if I ask you to invert something. This takes another step. The first step is to find the base
12 complement of each pitch. The Base 12 complement is the addi@on required for 2 pitches to
equal 12. SO 0=0, 1=11, 2=10, 3=9, 4=8, 5=7 and 6=6
Note that understanding the Base 12 complement will make serialism much easier!
T(n)I = Transposi@on (How much it is transposed by) Inversion = So T(4)I from that same set
NO [2,3,6] inversion becomes [10,9,6] then like the previous one we just add the transposed
amount so in this case (10 + 4 = 14 – 12 = 2), (9 + 4 = 13 – 12 = 1), and (6+4 = 10)
SO T(4)I of NO [2,3,6] IS [10,1,2]
Sets can have up to 11 pitches, but are typically 3-5 pitches per set, as more than that becomes
imperceivable as far as shapes go. To show you one that is a bit more complicated lets say we
were given the set NO (4,5,10,11). I want you to no@ce something about this set
4,5||||10,11 OR 10,11||||4,5
The normal order of this set can be represented as [4,5,10,11] OR as [10,11,4,5] in our normal
order rules. With that in mind, I tend to like to think from the smaller number first, but more
importantly. The PF of this example is quite famous PF (0167) which is also featured on your
homework assignment. Because of these features, 0167 is not inver@ble.
Lets do one last one with a step by step guide so you can complete these on your own! We are
given the following pitches Bb, C#, D, E and F#.
STEP 1) Find the biggest interval between the Pitches.
1,2|4|6|||10|| In this case the biggest interval is between PC6 and PC10 which implies
that our PF is going to start from either 6 or 10 going clockwise or counter clockwise
STEP 2) Find the NO its going to be the clockwise version of this first
NO star@ng from 10 is NO [10,1,2,4,6]
STEP 3) Find the PF
This is where it gets tricky, is it more compact to go clockwise or counterclockwise. As
done in step 1, we wanted to find the biggest interval first and that told us which pitch the
NO/PF can start on. The next thing you need to do is find the next biggest interval which is
between 11 and 0. If you no@ce that one of the direc@ons has a smaller interval that than other,
that is the likely source of the prime form.
So the best way of find it is to count backwards star@ng from 0 on PC 6, skip 5 4
becomes 2, skip 3, 2 becomes 4, 1 becomes 5, skip 0 and 11, and 10 becomes 8. Our PF is
PF (02458)
STEP 4) Complete any transposi@ons or Inversions SO
T2 of PF(02458) is NO[2,4,6,7,10] by adding 2 to each number
And
T2I of PF(02458) is NO [6,9,10,0,2], which I got by inver@ng the PF by its Base 12 complement,
puong it in order clockwise, then finally adding 2 to each number
I have uploaded a pitch wheel for you on e-learning. Use them, I promise it makes it WAY easier.