

4. Pitchclass
sets and normal form 
Pitchclass sets
Suppose we are analyzing Webern's "Dies ist ein Lied für dich
allein," and we come upon the two segments presented in Example 41.
Example 41. from Webern op. 3, no. 1. a: m. 2, piano; b:
mm. 56 voice
These two fivenote segments share only one notated pitch, and they don't
project any clear tonal design. Might they nonetheless be related in pitch
structure?
If we apply the axioms of octave and enharmonic equivalence, we see
that segments a and b are indeed related in a basic way:
they comprise the same pitch classes: 2, 1, 3, 7, and 6. We can say, then,
that these segments represent the same pitchclass set, and we
can invent a set name for both segments that expresses their equivalent
status. It could be any name, as long as it's the same one for
both segments. We'll use a name that simply lists the set's pc content:
"[1,2,3,6,7]." Labeling any future segments that have the same pc content
"[1,2,3,6,7]" would express our perception that all such segments are
equivalent according to our two basic axioms.
(By the way, we're used to segment equivalence in tonal analysis. Applying
just the octave equivalence axiom, we routinely judge any segment that
embodies pitch classes C, E, and G to be a "Cmajor triad." This includes
the four different segments in Example 42, as well as countless others.
The label "Cmajor triad" is a pc set name, isn't it?)
Example 42. Some Cmajor triads
Naming sets: normal form
One of the abstractions of pc sets is that they are unordered,
that is, the pcs may be embodied in any order in the musical segments.
Segments in which pcs 1, 2, 3, 6, and 7 appear in the order 2, 1, 3, 7,
6 or 6, 3, 1, 2, 7 or 3, 1, 6, 1, 2, 3, 7, 2, or indeed simultaneously
are all classified as belonging to the same set. Since we usually name
a set by citing its pc content, it will help to have a standard order
for listing the pcs; then all examples of the set will have the same name.
This standard order is called normal form. By convention, the
normal form is the one that lists the pcs in ascending order and in
the intervallically most compact form. Here are the steps for finding
a set's normal form.
 Examining your segment, list its pc content, eliminating all pc repetitions.
 Arrange the pc integers in "ascending" (clockwise) orders.
Remember: these integers form a modulo 12 number group. Not only
can you "ascend" from 1 to 2, 2 to 3, 3 to 6, and 6 to 7;
you can also "ascend" from 7 to 1 (7 + 6 = 1). There are always
as many possible "ascending" orders as the set has pcs. For
pcs 1, 2, 3, 6, and 7, for example, the five orders are
1 2 3 6 7 2 3 6 7 1 3
6 7 1 2 6 7 1 2 3 7
1 2 3 6
 Now choose the most compact of the "ascending" orders: the
one whose interval span between the first and last pcs is smallest.
Do this by subtracting the first integer from the last.
"ascending"
orders

overall span

1 2 3 6 7 
7  1 = 6 semitones 
2 3 6 7 1 
1  2 = 11 semitones 
3 6 7 1 2 
2  3 = 11 semitones 
6 7 1 2 3 
3  6 = 9 semitones 
7 1 2 3 6 
6  7 = 11 semitones 
Here the most compact "ascending" ordering is 1 2 3 6 7. This
is the set's normal form, and the conventional name of the set will
be [1,2,3,6,7]. The square brackets, commas, and lack of spaces between
digits are also a convention in naming unordered sets (though some analysts
use other conventions).
 The above steps are often enough to find a set's normal form. With
some sets, however two or more orders tie for overall compactness. The
rule then is to choose the set that is most compact towards the left.
Measure the intervals from first to secondlast note. Still tied? Measure
from first to thirdlast note. Keep going until one set wins in compactness.
For example, name the set made up of pcs 1, 4, 7, 8, and 10.
"ascending"
orders

overall
span 
span
1st to 2ndlast pc 
span
1st to 3rdlast pc 
1 4 7 8 10 
10  1 = 9 
8  1 = 7 

4 7 8 10 1 
1  4 = 9 
10  4 = 6 
8  4 = 4 
7 8 10 1 4 
4  7 = 9 
1  7 = 6 
10  7 = 3 
8 10 1 4 7 
7  8 = 11 


10 1 4 7 8 
8  10 = 10 


As you can see, three of the five ascending orders are equally compact
overall, spanning 9 semitones. Two of these still tie for compactness
when we compare first to secondlast pcs. Only when we compare first
to thirdlast pcs does order 7 8 10 1 4 emerge as this set's normal
form, the one most compact towards the left. We shall call this set
[7,8,10,1,4].
 With sets of great intervallic regularity, no amount of interval measuring
will break the tie. Then choose the ordering that begins with the lowest
number. For example, name the set comprising pcs 2, 4, 8, and 10
"ascending"
orders

overall
span 
span
1st to 2ndlast pc 
span
1st to 3rdlast pc 
2 4 8 10 
10  2 = 8 
8  2 = 6 
4  2 = 2 
4 8 10 2 
2  4= 10 


8 10 2 4 
4  8 = 8 
2  8 = 6 
10  8 = 2 
10 2 4 8 
8  10= 10 


Since interval measurement here doesn't produce a single most compact
order, we simply choose [2,4,8,10] rather than [8,10,2,4] as the set's
name.
Key concepts on this page:
 pitchclass set
 normal form
 steps for determining normal form


Page last modified 20
July 2001 / GRT 

