Stahl Algebraic Music Theory
Content
ALGEBRAIC MUSIC THEORY
JOSEPH STAHL
1. Introduction
There are many different ways to analyze a piece of music. Some of the philosophies are mostly selfcontained in the realm of music, but there are many methods that utilize a decent amount of mathematics.
Because human beings like symmetries, group theory provides us a valuable, natural tool with which we can
talk about the symmetries and themes of music. In this talk, we will look at some of the different ways group
theory is used to analyze music, focusing on Hugo Riemann’s “Neo-Riemannian theory”, which is based on
the actions of two isomorphic groups (both of which are isomorphic to D12 ) on the set of major and minor,
or consonant, triads.
2. Group Theory
To begin, we must establish some preliminary group theory knowledge. So, it makes sense to start with
the definition of a group.
Definition 1. A group is a set G equipped with a binary operation ¨ that satisfies the following properties:
1) Associativity: @a, b, c P G, a ¨ pb ¨ cq “ pa ¨ bq ¨ c
2) Identity: De P G s.t. a ¨ e “ e ¨ a “ a @a P G
3) Inverses: @a P G, Da´1 P G s.t. a ¨ a´1 “ a´1 ¨ a “ e
In general, it is good to think of a group as describing the symmetries of an object, such as the symmetries
of a square. Note that elements of groups do not necessarily commute with each other. A very familiar
example of a group is the set Z under addition. For a slightly less familiar, more illustrative example, we
turn to the symmetries of a square.
Example. D4 “ tsymmetries of a squareu
We call this group the dihedral group of order 8, because it has eight elements (as we will see). The
elements of D4 are:
• e, the identity,
• r, rotation counterclockwise by 90 degrees,
• r2 , rotation counterclockwise by 180 degrees,
• r3 , rotation counterclockwise by 270 degrees,
• f , a flip across the vertical,
• rf , a flip across the lower right to upper left diagonal,
• r2 f , a flip across the horizontal, and
• r3 f , a flip across the lower left to upper right diagonal.
It isn’t hard to verify that this is a group (but this should be done in the privacy of one’s own room). As the
notation suggests, we really only need to specify two elements of this group to define it. r and f will suffice
to characterize D4 completely once we also specify some of their properties. Specifically, we can characterize
D4 as the group generated by r and f , where r4 “ f 2 “ e, and f rf ´1 “ f rf “ r´1 “ r3 . That is,
D4 “ xr, f | r4 “ f 2 “ e, f rf ´1 “ r´1 y
This contains all the information that D4 contains. What we have done is specified the order of the elements
of D4 (how many times you must compose it with itself to get the identity) and a commutative relation,
which tells us how to swap elements.
The main group that will concern us will be a group similar to D4 : D12 , the symmetries of a dodecagon. This
1
ALGEBRAIC MUSIC THEORY
2
group is characterized as being generated by two elements, s and t, such that s12 “ t2 “ e, and tst “ s´1 .
In the same notation as above,
D12 “ xs, t | s12 “ t2 “ e, tst´1 “ tst “ s´1 y
Now, we want to discuss the way that this group applies to music, so specifically we are concerned with the
way this group will operate on sets of musical notes. When a group acts on a set, this is called a group
action. We will quickly give the definition for completeness’ sake.
Definition 2. If G is a group and X is a set, a group action of G on X is a binary operator ˝ : G ˆ X ÝÑ X
such that
1) pghq ˝ x “ g ˝ ph ˝ xq @g, h P G, x P X
2) e ˝ x “ x @x P X
What is happening here, in a sense, is that G is moving around the elements of X. For example, D4 acts
on the set of vertices of a square. For example, r acts on the set of vertices by taking the upper right vertex
to the upper left vertex, the upper left vertex to the lower left vertex, and so on. (See the accompanying
figure for an idea of group actions). Note that a group can always act on itself.
Figure 1. Visual Representation of some Actions of D4
Before we move on to the music side, there is one more idea we should discuss, which is that of when two
groups are the same in some sense. If two groups encode the same information, we say they are isomorphic.
Definition 3. An isomorphism between two groups G and H is a function φ : G ÝÑ H with the following
properties:
1) φpg1 g2 q “ φpg1 qφpg2 q @g1 , g2 P G
2) φ is bijective
In property 1), it should be noted that the operation on the left is that of G and the operation on the
right is that of H. An isomorphism gives us a structure preserving map from G to H that is injective and
surjective, so we have a mapping between elements of G and elements of H that “behave the same way.”
3. Music Theory Basics
Before we can talk about the way a group interacts with a set of musical notes, we must talk about how
we think of those notes. Without giving formal definitions, we can say a few things about the set of all
musical notes. We call a sequence of notes starting at a certain note and ending on that note again an octave
higher a scale (an octave is the interval between notes that makes the notes sound the same, but one is
higher or lower than the other). We say there are 12 notes in the chromatic scale, which we label A, A7/B5,
B, C, C7/D5, D, D7/E5, E, F, F7/G5, G, G7/A5. Each two notes in this sequence is separated by a semitone
(the smallest interval between standard musical notes), and we call two notes “enharmonically equivalent”
if they are in the same spot of the sequence (for example, F7 is enharmonically equivalent to G5). We could
continue this sequence in either direction by repeating it on either side. Notes that are an octave apart are
denoted by the same letters. So we can identify all the A’s, B’s, etc. and wrap around in a way similar to
modular arithmetic to get a “musical clock.” The set containing a certain letter, say D, is called the “pitch
class of D.” For short, we will often refer to a pitch class as just a pitch. Now, if we associate C with 0,
C7 with 1, and so on, we create a bijection between the notes of the chromatic scale and Z12 . This gives
us a way to relate mathematical operations to musical notes. Another common technique in tonal music is
associating a major or minor scale (specific subsequences of the chromatic scale) to Z7 . These scales have
ALGEBRAIC MUSIC THEORY
3
Figure 2. The Musical Clock of Pitch Classes
seven different notes in them, and are the most commonly used scales. However, they are more restrictive
than the chromatic scale, so we will be looking at Z12 so that we have more freedom.
In music analysis, we concern ourselves with a number of different things, including:
1) Harmonies, the interactions between different, simultaneously sounding pitches
2) Melodies, sequences of individual pitches
3) Symmetries, which can be repetitions of phrases, ideas, chord patterns, etc.
Let’s talk about harmonies. There are many different kinds (212 up to pitch class), so we’ll focus on certain
types of harmonies. Define a triad to be a set of three distinct pitch classes. The ones that will be of interest
to us are the major and minor triads. In these cases, a triad is given by the root, the third, and the fifth
of the triad. Thinking of a diatonic (major or minor) scale, the root note of a triad is given by the first
note of the scale, the third is given by the third note, and the fifth is given by the fifth note. Using our Z12
notation, we can say that a major triad with root n is given by the set rn, n ` 4, n ` 7s (the square brackets
are used to indicate that we are talking about the pitch class represented by the integer in any position of
the 3-tuple). Similarly, the minor triad with root n is given by rn, n ` 3, n ` 7s. So the major and minor
triads of a given root differ by a semitone in the third. The set S of all major and minor triads is called the
set of consonant triads. Note that there are 12 options for the root note of a triad, and 2 options for the
“modal parity” (that is, whether the triad is major or minor) and that triads are characterized completely
by their root and modal parity, so there are 24 elements of S.
4. Actions on S
Since we have associated the notes of the chromatic scale with Z12 , we can define different actions on
individual notes that permute them in a reasonable way, for example, moving each note up or down by a
fixed amount. Looking at our clock picture, there are a couple of obvious transformations on individual
notes.
1) We can move the notes around the circle clockwise or counterclockwise, like rotating a dodecagon.
2) We can flip the clock about a diagonal, like flipping a dodecagon.
These seem like judicious actions on notes. We will call a rotation by n ¨ 30 degrees clockwise Tn , and we
will call a flip about the pn, n ` 6q axis In . We define this set to be the T/I Group.
Definition 4. The T/I Group is defined to be the set of Tn and In such that 0 ď n ă 12. That is,
T/I Group “ tTn , In | 0 ď n ă 12u
Then we can define the way that these motions change specific elements of Z12 :
Tn : x ÞÑ x ` n
pmod 12q
In : x ÞÑ n ´ x
pmod 12q
ALGEBRAIC MUSIC THEORY
4
I claim this is a group (under the operation of function composition) that acts on Z12 . With a couple of
easy (i.e. check it yourself) computations, we see that
Tn “ pT1 qn
In “ Tn I0
Tm Tn “ Tm`n
Tm In “ Im`n
Im Tn “ Im´n
Im In “ Tm´n
Where the subscripts are always taken mod 12. These computations show closure of the T {I group. T0 is
obviously the identity, and associativity follows after a few calculations. T12´n “ pTn q´1 , and In “ pIn q´1 .
Thus, the T/I group is, in fact, a group. From the relations above or by the way the T/I presents itself as
symmetries of a dodecagon, we can see that the T {I group is isomorphic to D12 . With a few calculations,
we can also come to the conclusion that the elements that take the roles of s and t as generators of the T/I
group are T1 and I0 (note that there are other choices of elements in T/I that also satisfy the generating
relations). This group acts on S by extending the notion of its action on Z12 : an element X P T {I acts on
rn, m, n ` 7s (where m “ n ` 3 or n ` 4) as such: Xprn, m, n ` 7sq “ rXpnq, Xpmq, Xpn ` 7qs, so elements
of T/I act on elements of S by acting on each component of an element individually. There is another
Figure 3. Action of I0 on the C Major Triad
musically interesting way to operate on the elements of S. Earlier, we said a triad was determined solely by
its root and modal parity. So we might want a set of operations that take major triads to minor triads in
specific ways. As remarked, one way to do this is to take a major triad to a minor triad on the same root.
That is, we take rr, r ` 4, r ` 7s to rr, r ` 3, r ` 7s and vice versa. This action, which we will call P , changes
the modal parity of a triad by changing the middle note by a semitone. Similarly, we can change the parity
by moving the other two notes: L takes rr, r ` 4, r ` 7s to rr ´ 1, r ` 4, r ` 7s and vice versa, and R takes
rr, r ` 4, r ` 7s to rr, r ` 4, r ` 9s and vice versa. This is our motivation for defining the PLR-group.
ALGEBRAIC MUSIC THEORY
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Figure 4. The PLR-group acting on the C major triad
Definition 5. The PLR-group is the group of three functions P, L, R : S ÞÑ S defined by
P prx1 , x2 , x3 sq “ Ix1 `x3 rx1 , x2 , x3 s
Lprx1 , x2 , x3 sq “ Ix2 `x3 rx1 , x2 , x3 s
Rprx1 , x2 , x3 sq “ Ix1 `x2 rx1 , x2 , x3 s
One can check that the two definitions given for P , L, and R are equivalent when r¨, ¨, ¨s is interpreted
as an unordered set. These operations are called P , L, and R for musical reasons: P stands for parallel,
L stands for leading tone exchange, and R stands for relative. These functions are fundamentally different
from the Tn and In functions, because they do not act on a triad component-wise. Rather, they depend on
the triad as a whole to decide how the transformation is made. Because the P , L, and R actions appear
so different from the Tn and In , it is a surprising claim that the PLR-group is isomorphic to D12 . In fact
it is, and the main idea of the proof is due not to a mathematician, but to Beethoven. A particular chord
progression in Beethoven’s ninth symphony runs through 19 out of 24 different chords in S, and if we extend
this chain, we can use it to show that certain elements of the PLR-group are distinct, which greatly helps
us in establishing the isomorphism.
Theorem 1. The PLR-group is isomorphic to D12 and is generated by L and R.
Proof. (Beethoven) We remark that one can use the description of P , L, and R in terms of Tn to show
that P T1 “ T1 P , LT1 “ T1 L, and RT1 “ T1 R. If we start with the C major triad and alternately apply R
and L, we obtain the following sequence of triads (the first 19 of which occur in order in Beethoven’s ninth
symphony):
C, a, F, d, B5, g, E5, c, A5, f, D5, b5, G5, e5, B, g7, E, c7, A, f 7, D, b, G, e, C
(Note: we use a lowercase letter to denote a minor triad and a capital letter to denote a major triad.) This
tells us that the 24 bijections R, LR, RLR, . . ., RpLRq11 , and pLRq12 “ e are distinct, that the PLR-group
has at least 24 elements, and that LR has order 12. Further, RpLRq3 pCq “ c, and since RpLRq3 has order
2 and commutes with T1 , we see that RpLRq3 “ P , and the PLR-group is generated by L and R alone.
ALGEBRAIC MUSIC THEORY
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If we set s “ LR and t “ L, then s12 “ e, t2 “ e, and
tst “ LpRLqL “ RL “ s´1
All that remains to be shown is that the PLR-group has order 24, which we will omit here.
Now that we have shown this surprising isomorphism, we shall discuss an interesting way to interpret
actions of the PLR-group: motions on a torus.
5. Musical Paths on Tori
As mentioned above, we can construct a torus on which the PLR-group acts visibly. The specific torus
we look at is called the “tonnetz ” or “tone network.” If we position pitch classes in triangles such that
each triangle is surrounded by three pitch classes that form either a minor or major triad, then we get the
Figure 5. We can identify the lower edge with the upper edge and the the two sides with each other to
Figure 5. Action of the PLR-group on the Tonnetz
get a torus. The PLR-group acts visibly on the tonnetz by flipping a triangle over an edge, depending on
which transformation it is (as indicated by the figure). Taking this interpretation, we can view Beethoven’s
19-chord progression in the second movement of his ninth symphony as a path around the tonnetz, going
from the r0, 4, 7s triangle to the r7, 10, 2s triangle in a semi-horizontal way.
Figure 6. Viewing the Tonnetz as a Torus
References
[1] Alissa S. Crans, Thomas M. Fiore, Ramon Satyendra, arXiv:0711.1873v2 [math.GR].
JOSEPH STAHL
1. Introduction
There are many different ways to analyze a piece of music. Some of the philosophies are mostly selfcontained in the realm of music, but there are many methods that utilize a decent amount of mathematics.
Because human beings like symmetries, group theory provides us a valuable, natural tool with which we can
talk about the symmetries and themes of music. In this talk, we will look at some of the different ways group
theory is used to analyze music, focusing on Hugo Riemann’s “Neo-Riemannian theory”, which is based on
the actions of two isomorphic groups (both of which are isomorphic to D12 ) on the set of major and minor,
or consonant, triads.
2. Group Theory
To begin, we must establish some preliminary group theory knowledge. So, it makes sense to start with
the definition of a group.
Definition 1. A group is a set G equipped with a binary operation ¨ that satisfies the following properties:
1) Associativity: @a, b, c P G, a ¨ pb ¨ cq “ pa ¨ bq ¨ c
2) Identity: De P G s.t. a ¨ e “ e ¨ a “ a @a P G
3) Inverses: @a P G, Da´1 P G s.t. a ¨ a´1 “ a´1 ¨ a “ e
In general, it is good to think of a group as describing the symmetries of an object, such as the symmetries
of a square. Note that elements of groups do not necessarily commute with each other. A very familiar
example of a group is the set Z under addition. For a slightly less familiar, more illustrative example, we
turn to the symmetries of a square.
Example. D4 “ tsymmetries of a squareu
We call this group the dihedral group of order 8, because it has eight elements (as we will see). The
elements of D4 are:
• e, the identity,
• r, rotation counterclockwise by 90 degrees,
• r2 , rotation counterclockwise by 180 degrees,
• r3 , rotation counterclockwise by 270 degrees,
• f , a flip across the vertical,
• rf , a flip across the lower right to upper left diagonal,
• r2 f , a flip across the horizontal, and
• r3 f , a flip across the lower left to upper right diagonal.
It isn’t hard to verify that this is a group (but this should be done in the privacy of one’s own room). As the
notation suggests, we really only need to specify two elements of this group to define it. r and f will suffice
to characterize D4 completely once we also specify some of their properties. Specifically, we can characterize
D4 as the group generated by r and f , where r4 “ f 2 “ e, and f rf ´1 “ f rf “ r´1 “ r3 . That is,
D4 “ xr, f | r4 “ f 2 “ e, f rf ´1 “ r´1 y
This contains all the information that D4 contains. What we have done is specified the order of the elements
of D4 (how many times you must compose it with itself to get the identity) and a commutative relation,
which tells us how to swap elements.
The main group that will concern us will be a group similar to D4 : D12 , the symmetries of a dodecagon. This
1
ALGEBRAIC MUSIC THEORY
2
group is characterized as being generated by two elements, s and t, such that s12 “ t2 “ e, and tst “ s´1 .
In the same notation as above,
D12 “ xs, t | s12 “ t2 “ e, tst´1 “ tst “ s´1 y
Now, we want to discuss the way that this group applies to music, so specifically we are concerned with the
way this group will operate on sets of musical notes. When a group acts on a set, this is called a group
action. We will quickly give the definition for completeness’ sake.
Definition 2. If G is a group and X is a set, a group action of G on X is a binary operator ˝ : G ˆ X ÝÑ X
such that
1) pghq ˝ x “ g ˝ ph ˝ xq @g, h P G, x P X
2) e ˝ x “ x @x P X
What is happening here, in a sense, is that G is moving around the elements of X. For example, D4 acts
on the set of vertices of a square. For example, r acts on the set of vertices by taking the upper right vertex
to the upper left vertex, the upper left vertex to the lower left vertex, and so on. (See the accompanying
figure for an idea of group actions). Note that a group can always act on itself.
Figure 1. Visual Representation of some Actions of D4
Before we move on to the music side, there is one more idea we should discuss, which is that of when two
groups are the same in some sense. If two groups encode the same information, we say they are isomorphic.
Definition 3. An isomorphism between two groups G and H is a function φ : G ÝÑ H with the following
properties:
1) φpg1 g2 q “ φpg1 qφpg2 q @g1 , g2 P G
2) φ is bijective
In property 1), it should be noted that the operation on the left is that of G and the operation on the
right is that of H. An isomorphism gives us a structure preserving map from G to H that is injective and
surjective, so we have a mapping between elements of G and elements of H that “behave the same way.”
3. Music Theory Basics
Before we can talk about the way a group interacts with a set of musical notes, we must talk about how
we think of those notes. Without giving formal definitions, we can say a few things about the set of all
musical notes. We call a sequence of notes starting at a certain note and ending on that note again an octave
higher a scale (an octave is the interval between notes that makes the notes sound the same, but one is
higher or lower than the other). We say there are 12 notes in the chromatic scale, which we label A, A7/B5,
B, C, C7/D5, D, D7/E5, E, F, F7/G5, G, G7/A5. Each two notes in this sequence is separated by a semitone
(the smallest interval between standard musical notes), and we call two notes “enharmonically equivalent”
if they are in the same spot of the sequence (for example, F7 is enharmonically equivalent to G5). We could
continue this sequence in either direction by repeating it on either side. Notes that are an octave apart are
denoted by the same letters. So we can identify all the A’s, B’s, etc. and wrap around in a way similar to
modular arithmetic to get a “musical clock.” The set containing a certain letter, say D, is called the “pitch
class of D.” For short, we will often refer to a pitch class as just a pitch. Now, if we associate C with 0,
C7 with 1, and so on, we create a bijection between the notes of the chromatic scale and Z12 . This gives
us a way to relate mathematical operations to musical notes. Another common technique in tonal music is
associating a major or minor scale (specific subsequences of the chromatic scale) to Z7 . These scales have
ALGEBRAIC MUSIC THEORY
3
Figure 2. The Musical Clock of Pitch Classes
seven different notes in them, and are the most commonly used scales. However, they are more restrictive
than the chromatic scale, so we will be looking at Z12 so that we have more freedom.
In music analysis, we concern ourselves with a number of different things, including:
1) Harmonies, the interactions between different, simultaneously sounding pitches
2) Melodies, sequences of individual pitches
3) Symmetries, which can be repetitions of phrases, ideas, chord patterns, etc.
Let’s talk about harmonies. There are many different kinds (212 up to pitch class), so we’ll focus on certain
types of harmonies. Define a triad to be a set of three distinct pitch classes. The ones that will be of interest
to us are the major and minor triads. In these cases, a triad is given by the root, the third, and the fifth
of the triad. Thinking of a diatonic (major or minor) scale, the root note of a triad is given by the first
note of the scale, the third is given by the third note, and the fifth is given by the fifth note. Using our Z12
notation, we can say that a major triad with root n is given by the set rn, n ` 4, n ` 7s (the square brackets
are used to indicate that we are talking about the pitch class represented by the integer in any position of
the 3-tuple). Similarly, the minor triad with root n is given by rn, n ` 3, n ` 7s. So the major and minor
triads of a given root differ by a semitone in the third. The set S of all major and minor triads is called the
set of consonant triads. Note that there are 12 options for the root note of a triad, and 2 options for the
“modal parity” (that is, whether the triad is major or minor) and that triads are characterized completely
by their root and modal parity, so there are 24 elements of S.
4. Actions on S
Since we have associated the notes of the chromatic scale with Z12 , we can define different actions on
individual notes that permute them in a reasonable way, for example, moving each note up or down by a
fixed amount. Looking at our clock picture, there are a couple of obvious transformations on individual
notes.
1) We can move the notes around the circle clockwise or counterclockwise, like rotating a dodecagon.
2) We can flip the clock about a diagonal, like flipping a dodecagon.
These seem like judicious actions on notes. We will call a rotation by n ¨ 30 degrees clockwise Tn , and we
will call a flip about the pn, n ` 6q axis In . We define this set to be the T/I Group.
Definition 4. The T/I Group is defined to be the set of Tn and In such that 0 ď n ă 12. That is,
T/I Group “ tTn , In | 0 ď n ă 12u
Then we can define the way that these motions change specific elements of Z12 :
Tn : x ÞÑ x ` n
pmod 12q
In : x ÞÑ n ´ x
pmod 12q
ALGEBRAIC MUSIC THEORY
4
I claim this is a group (under the operation of function composition) that acts on Z12 . With a couple of
easy (i.e. check it yourself) computations, we see that
Tn “ pT1 qn
In “ Tn I0
Tm Tn “ Tm`n
Tm In “ Im`n
Im Tn “ Im´n
Im In “ Tm´n
Where the subscripts are always taken mod 12. These computations show closure of the T {I group. T0 is
obviously the identity, and associativity follows after a few calculations. T12´n “ pTn q´1 , and In “ pIn q´1 .
Thus, the T/I group is, in fact, a group. From the relations above or by the way the T/I presents itself as
symmetries of a dodecagon, we can see that the T {I group is isomorphic to D12 . With a few calculations,
we can also come to the conclusion that the elements that take the roles of s and t as generators of the T/I
group are T1 and I0 (note that there are other choices of elements in T/I that also satisfy the generating
relations). This group acts on S by extending the notion of its action on Z12 : an element X P T {I acts on
rn, m, n ` 7s (where m “ n ` 3 or n ` 4) as such: Xprn, m, n ` 7sq “ rXpnq, Xpmq, Xpn ` 7qs, so elements
of T/I act on elements of S by acting on each component of an element individually. There is another
Figure 3. Action of I0 on the C Major Triad
musically interesting way to operate on the elements of S. Earlier, we said a triad was determined solely by
its root and modal parity. So we might want a set of operations that take major triads to minor triads in
specific ways. As remarked, one way to do this is to take a major triad to a minor triad on the same root.
That is, we take rr, r ` 4, r ` 7s to rr, r ` 3, r ` 7s and vice versa. This action, which we will call P , changes
the modal parity of a triad by changing the middle note by a semitone. Similarly, we can change the parity
by moving the other two notes: L takes rr, r ` 4, r ` 7s to rr ´ 1, r ` 4, r ` 7s and vice versa, and R takes
rr, r ` 4, r ` 7s to rr, r ` 4, r ` 9s and vice versa. This is our motivation for defining the PLR-group.
ALGEBRAIC MUSIC THEORY
5
Figure 4. The PLR-group acting on the C major triad
Definition 5. The PLR-group is the group of three functions P, L, R : S ÞÑ S defined by
P prx1 , x2 , x3 sq “ Ix1 `x3 rx1 , x2 , x3 s
Lprx1 , x2 , x3 sq “ Ix2 `x3 rx1 , x2 , x3 s
Rprx1 , x2 , x3 sq “ Ix1 `x2 rx1 , x2 , x3 s
One can check that the two definitions given for P , L, and R are equivalent when r¨, ¨, ¨s is interpreted
as an unordered set. These operations are called P , L, and R for musical reasons: P stands for parallel,
L stands for leading tone exchange, and R stands for relative. These functions are fundamentally different
from the Tn and In functions, because they do not act on a triad component-wise. Rather, they depend on
the triad as a whole to decide how the transformation is made. Because the P , L, and R actions appear
so different from the Tn and In , it is a surprising claim that the PLR-group is isomorphic to D12 . In fact
it is, and the main idea of the proof is due not to a mathematician, but to Beethoven. A particular chord
progression in Beethoven’s ninth symphony runs through 19 out of 24 different chords in S, and if we extend
this chain, we can use it to show that certain elements of the PLR-group are distinct, which greatly helps
us in establishing the isomorphism.
Theorem 1. The PLR-group is isomorphic to D12 and is generated by L and R.
Proof. (Beethoven) We remark that one can use the description of P , L, and R in terms of Tn to show
that P T1 “ T1 P , LT1 “ T1 L, and RT1 “ T1 R. If we start with the C major triad and alternately apply R
and L, we obtain the following sequence of triads (the first 19 of which occur in order in Beethoven’s ninth
symphony):
C, a, F, d, B5, g, E5, c, A5, f, D5, b5, G5, e5, B, g7, E, c7, A, f 7, D, b, G, e, C
(Note: we use a lowercase letter to denote a minor triad and a capital letter to denote a major triad.) This
tells us that the 24 bijections R, LR, RLR, . . ., RpLRq11 , and pLRq12 “ e are distinct, that the PLR-group
has at least 24 elements, and that LR has order 12. Further, RpLRq3 pCq “ c, and since RpLRq3 has order
2 and commutes with T1 , we see that RpLRq3 “ P , and the PLR-group is generated by L and R alone.
ALGEBRAIC MUSIC THEORY
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If we set s “ LR and t “ L, then s12 “ e, t2 “ e, and
tst “ LpRLqL “ RL “ s´1
All that remains to be shown is that the PLR-group has order 24, which we will omit here.
Now that we have shown this surprising isomorphism, we shall discuss an interesting way to interpret
actions of the PLR-group: motions on a torus.
5. Musical Paths on Tori
As mentioned above, we can construct a torus on which the PLR-group acts visibly. The specific torus
we look at is called the “tonnetz ” or “tone network.” If we position pitch classes in triangles such that
each triangle is surrounded by three pitch classes that form either a minor or major triad, then we get the
Figure 5. We can identify the lower edge with the upper edge and the the two sides with each other to
Figure 5. Action of the PLR-group on the Tonnetz
get a torus. The PLR-group acts visibly on the tonnetz by flipping a triangle over an edge, depending on
which transformation it is (as indicated by the figure). Taking this interpretation, we can view Beethoven’s
19-chord progression in the second movement of his ninth symphony as a path around the tonnetz, going
from the r0, 4, 7s triangle to the r7, 10, 2s triangle in a semi-horizontal way.
Figure 6. Viewing the Tonnetz as a Torus
References
[1] Alissa S. Crans, Thomas M. Fiore, Ramon Satyendra, arXiv:0711.1873v2 [math.GR].
