Introduction to Groups
Mathematics and Music
Congratulations! You’ve made it to the last topic of the semester: symmetry in music! 
We have already seen that transposition and inversion are important operations that musicians
and composers perform on chords and scales. (Quick reminder: transposition means shifting all
pitches in a chord or scale up by the same amount, and inversion means reversing the order of
intervals in a chord or scale.)
Transposition and inversion are easily described as symmetries of a dodecagon—that is, ways
you can rotate or flip over a dodecagon so that it looks essentially the same.
Let’s solve this problem systematically. There are two types of
operations that leave this diagram unchanged: rotations and flips.
Describe the “legal” rotations precisely:
Describe the “legal” flips precisely:
Now number the nodes on the circle from 0 to 11, as we did of pitch classes. Each symmetry can
be completely described by how it alters the node numbers. For example, 60 rotation means
that node 0 ends up in position 2, node 1 ends up in position 3, etc. Although we could write
something like 02, 13, 24, …, 100, 111, the following notation is more compact:
0 1 2 3 4 5 6 7 8 9 10 11


2 3 4 5 6 7 8 9 10 11 0 1 
Of course, we can also write this particular symmetry as a function: f (n)  n 2 (mod 12), or
just describe it as “60 rotation,” which is probably the best way to think of it.

 notation” used above.
Try writing the flip over the axis from 2.5 to 8.5 in the “array
Try writing that flip as a function of n.
The symmetries of a dodecagon are examples of a very important mathematical structure: a
group. I’ll give you a formal definition of a group later. First, let’s investigate what happens if
you perform multiple symmetry operations (it’s sort of like figuring out how symmetries
“multiply”).
Mathematicians often like to use a simpler case to study a more complex one. Let’s figure out
the symmetries of a hexagon! It helps to have some good notation. The “legal” rotations of a
hexagon are 60 rotations, and the “legal” flips are flips in a diameter that either connects two
nodes or the midpoints between two nodes. I’m going to say that, for any integer n,
rn = rotation by 60n
fn = flipping in the axis connecting n/2 (mod 6) with n/2 + 3 (mod 6).
Under what circumstances does rn = rm, where n and m are integers?
Under what circumstances does fn = fm?
Why is it sufficient to study these operations for n = 0, 1, …, 5, and not all integers?
We’re going to make a group table for the symmetries of the hexagon. This table describes what
happens if you perform two symmetry operations, with the one at the top first.
r0
r0
r1
r2
r3
r4
r5
f0
f1
f2
f3
f4
f5
r1
r2
r3
r4
r5
f0
f1
f2
f3
f4
f5
More about Groups
Mathematics and Music
The symmetries of a hexagon are just one example of a group. Groups appear many places in
mathematics, so mathematicians have developed ways of talking about them abstractly. This is
especially useful because the same groups often appear in different contexts, and we don’t want
to have to create the theory from scratch each time. For example, the group of symmetries of a
dodecagon is “the same” as the group of transpositions and inversions of the pitch class circle.
Definition
Suppose X is a set and * is some “operation” defined on any pair of elements in the set (that is,
“ x  y is defined for every pair x, y in X”). Then X is a group if and only if it has all the
following properties:
 Closure: If x and y are elements of X, x  y is also an element in X.
 Associativity: If x, y, and z are elements of X, then x  (y  z)  (x  y)  z .
 Existence of an identity element: there is some element e in X so that for all x in X,
xe  e x  x . The elemente is called the “identity” of the group.
 Existence of inverses: For each element 
x in X, there exists an element y such that
x  y  y x  e. The element y is called the “inverse” of x and often written y  x 1 .


Examples of groups
 The integers with addition. Check that


o If m and n are integers, m + n is an integer.
o If m, n, and p are integers, m + (n + p) = (m + n) + p.
o The identity element is 0: n + 0 = 0 + n = n.
o The inverse of n is –n, because n + (-n) = 0. The inverse of every integer is an
integer.
 The positive real numbers with multiplication. (What is the identity? If x is a positive
real number, how do you find its inverse?)
 The integers modulo 12 with addition. (Notice that the identity is addition of any number
that equals 0 (mod 12), and n and 12 – n are inverses.)
 The symmetries of a hexagon. (In fact, the symmetries of any geometric object form a
group.)
 Transpositions and inversions of the pitch class circle.
 The set of sequences of “legal” change ringing moves.
Examples of things that look like groups but aren’t
 The positive integers with multiplication. The problem is that not every element has an
inverse (in fact, 1 is the only element that has an inverse).
 The positive real numbers with division (This one is interesting—see if you can find
which condition(s) fail).
Generators of a group
Often, all the elements in a group can be described by sequences of relations (for example,
3  111). If every group element is equal to a sequence of products of elements belonging to
a subset of X and the inverses of these elements, then that subset is called a set of generators of
X.
 For example, every integer can be described as a sum of 1’s or (-1)’s ( 3  111,

3  (1)  (1)  (1), 0  1 (1) ). We would say that 1 generates the group of integers
with addition.
 2 generate?
 Explain why 2 does not generate the integers. What subset does

 Explain why 
no finite set generates the group of positive real numbers with
multiplication.
 The set {r1, f0} generates the group of symmetries of a hexagon! In order to show this,
you need to understand how they multiply.
o First, show that any rotation is the result of doing r1 a number of times:
n tim es
rn  r1  r1 
 r1  (r1 )n .
o Then—this is more difficult—show that any flip is the result of doing f0 and then
rotating some number of times: fn  (r1 )k  f0 (find k)

o You should also work out how r1 and f0 “multiply”: f0  r1  fm  (r1 )k  f0 (find
m and k).

o Finally, what are the inverses
of r1 and f0?
o Knowing this makes figuring out the result
of multiple rotations and reflections
much easier. For example,
f3  f4  (r1 )3  f0 (r1 )4  f0  (r1 )3 (r1 )4  f0  f0  (r1 )1.
Special types of groups
 A group is called abelian if its operation is commutative (this means that x  y  y x for
all x and y in the group).
o The integers with addition is abelian. (Explain.)
o The set of symmetries of a hexagon is not abelian. Can 
you explain why?
o Which of the other groups described above are abelian?
 A group is called cyclic if one element (and its inverse) generates the entire group.
o For example, the integers with addition are cyclic because they are generated by 1
(they’re also generated by -1).
o The integers mod 12 are also cyclic—they’re generated by 1. (BTW, they’re also
generated by 5, 7, or 11.)
o The subgroup of rotations of a hexagon is cyclic (what are its possible
generators?). The group of rotations and inversions of a hexagon is not cyclic.
o Every cyclic group is abelian (why?).
Groups in music
 The most common group used in music is the group of transpositions and inversions,
which is the same as the symmetries of a dodecagon (musicians write Tn instead of rn and
In instead of fn). Transpositions and inversions are usually applied to chords.
o For example, the D major triad (D) is the result of transposing the C major triad
(C) up by 2 pitches. We could write D = T2(C).
o The A minor triad (Am) is the result of inverting the C major triad so that the C
note is mapped to the E note (this corresponds to f4). We write Am = I4(C).
 The group of symmetries of an n-gon applies to periodic n-beat rhythm patterns, but in
this case, the rotation operation is called delay and the flip is called retrograde.
o For example, if P is the pattern in “Clapping Music”, then the second person plays
P, D-1(P), D-2(P), …, D-11(P), P; the first person just keeps playing P.
o The retrograde of the pattern x..x..x. is .x..x..x
 Transposition and inversion can also be applied to melodies. Melodies are both
sequences of pitches and rhythm patterns! So the group becomes pretty complicated.
You can transpose (shift the melody up or down a number of pitches), invert (turn the
melody “upside down”), delay (start playing some number of beats later), reverse (apply
the retrograde operation), or do any combination of these! Moreover, you can play
different versions of the melody simultaneously, creating a canon. (There are more—
augmentation (A) and compression (A-1), for example.)
 J.S. Bach used all of these operations in his “Goldberg Variations”!
 Let’s see how these apply to the melody “Twinkle Twinkle Little Star”. First, write the
melody as the sequence 0 0 7 7 9 9 7, then…
o T2(0 0 7 7 9 9 7) = 2 2 9 9 11 11 9
o I0(0 0 7 7 9 9 7) = 0 0 -7 -7 -9 -9 -7
o D4(0 0 7 7 9 9 7) = 9 9 7 0 0 7 7 (this is assuming the melody repeats periodically
with 4 beats per measure)
o R(0 0 7 7 9 9 7) = 7 9 9 7 7 0 0
T0
D3
A2
T2
I0
R
I0R
=
RI0
A1/2