Stahl Algebraic Music Theory

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ALGEBRAIC MUSIC THEORY

JOSEPH STAHL

1. Introduction

There are many different ways to analyze a piece of music. Some of the philosophies are mostly self-contained 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 cantalk about the symmetries and themes of music. In this talk, we will look at some of the different ways grouptheory is used to analyze music, focusing on Hugo Riemanns Neo-Riemannian theory, which is based onthe 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 withthe definition of a group.

Definition 1. Agroup is a set G equipped with a binary operation that satisfies the following properties:

1) Associativity: @a,b,cP G, a pb cq pa bq c2) Identity: DeP G s.t. a e e a a @aP G3) Inverses: @aP G, Da1 PG s.t. a a1 a1 a e

In general, it is good to think of a group as describing the symmetries of an object, such as the symmetriesof a square. Note that elements of groups do not necessarily commute with each other. A very familiarexample of a group is the set Z under addition. For a slightly less familiar, more illustrative example, weturn 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). Theelements ofD4 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, r2f, a flip across the horizontal, and r3f, a flip across the lower left to upper right diagonal.

It isnt hard to verify that this is a group (but this should be done in the privacy of ones own room). As thenotation suggests, we really only need to specify two elements of this group to define it. r andf will sufficeto characterize D4 completely once we also specify some of their properties. Specifically, we can characterizeD4 as the group generated by r and f, where r

4 f2 e, and frf1 f rfr1 r3. That is,

D4 xr, f |r4 f2 e,frf1 r1y

This contains all the information that D4contains. What we have done is specified the order of the elementsofD4 (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

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ALGEBRAIC MUSIC THEORY 2

group is characterized as being generated by two elements, s and t, such that s12 t2 e, and tst s1.In the same notation as above,

D12 xs, t| s12 t2 e, tst1 tst s1y

Now, we want to discuss the way that this group applies to music, so specifically we are concerned with theway this group will operate on sets of musical notes. When a group acts on a set, this is called a groupaction. We will quickly give the definition for completeness sake.

Definition 2. IfG is a group andXis a set, agroup actionofG on Xis a binary operator : GXXsuch that

1) pghq x g ph xq @g, hP G, x P X2) e x x @xP X

What is happening here, in a sense, is that G is moving around the elements ofX. For example, D4 actson the set of vertices of a square. For example, r acts on the set of vertices by taking the upper right vertexto the upper left vertex, the upper left vertex to the lower left vertex, and so on. (See the accompanyingfigure for an idea of group actions). Note that a group can always act on itself.

Figure 1. Visual Representation of some Actions ofD4

Before we move on to the music side, there is one more idea we should discuss, which is that of when twogroups are the same in some sense. If two groups encode the same information, we say they are isomorphic.

Definition 3. An isomorphismbetween two groups G and His a function : G Hwith the followingproperties:

1) pg1g2q pg1qpg2q @g1, g2P G2) is bijective

In property 1), it should be noted that the operation on the left is that ofG and the operation on theright is that ofH. An isomorphism gives us a structure preserving map fromG to Hthat is injective andsurjective, so we have a mapping between elements ofG and elements ofHthat 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 howwe think of those notes. Without giving formal definitions, we can say a few things about the set of allmusical notes. We call a sequence of notes starting at a certain note and ending on that note again an octavehigher a scale (an octave is the interval between notes that makes the notes sound the same, but one ishigher 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 equivalentif they are in the same spot of the sequence (for example, F 7is enharmonically equivalent to G5). We couldcontinue this sequence in either direction by repeating it on either side. Notes that are an octave apart aredenoted by the same letters. So we can identify all the As, Bs, etc. and wrap around in a way similar tomodular arithmetic to get a musical clock. The set containing a certain letter, say D, is called the pitchclass 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 givesus a way to relate mathematical operations to musical notes. Another common technique in tonal music isassociating a major or minor scale (specific subsequences of the chromatic scale) to Z7. These scales have


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Figure 2. The Musical Clock of Pitch Classes

seven different notes in them, and are the most commonly used scales. However, they are more restrictivethan 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 pitches2) Melodies, sequences of individual pitches3) Symmetries, which can be repetitions of phrases, ideas, chord patterns, etc.

Lets talk about harmonies. There are many different kinds (2 12 up to pitch class), so well focus on certaintypes of harmonies. Define atriadto be a set of three distinct pitch classes. The ones that will be of interestto us are the major and minor triads. In these cases, a triad is given by the root, the third, and the fifthof the triad. Thinking of a diatonic (major or minor) scale, the root note of a triad is given by the firstnote of the scale, the third is given by the third note, and the fifth is given by the fifth note. Using our Z12notation, we can say that a major triad with root n is given by the set rn,n ` 4, n ` 7s (the square bracketsare used to indicate that we are talking about the pitch class represented by the integer in any position ofthe 3-tuple). Similarly, the minor triad with root n is given by rn, n`3, n`7s. So the major and minortriads of a given root differ by a semitone in the third. The set Sof all major and minor triads is called theset ofconsonant 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 completelyby their root and modal parity, so there are 24 elements ofS.

4. Actions on S

Since we have associated the notes of the chromatic scale with Z12, we can define different actions onindividual notes that permute them in a reasonable way, for example, moving each note up or down by afixed amount. Looking at our clock picture, there are a couple of obvious transformations on individualnotes.

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 n30 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. TheT/I Group is defined to be the set ofTn and In such that 0 n 12. That is,

T/I Group tTn, In| 0 n 12u

Then we can define the way that these motions change specific elements ofZ12:

Tn: x x ` n pmod 12q

In: x n x pmod 12q


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I claim this is a group (under the operation of function composition) that acts on Z12. With a couple ofeasy (i.e. check it yourself) computations, we see that

Tn pT1qn

In TnI0

TmTn Tm`n

TmIn Im`n

ImTn Imn

ImIn Tmn

Where the subscripts are always taken mod 12. These computations show closure of the T{I group. T0 isobviously the identity, and associativity follows after a few calculations. T12n pTnq

1, and In pInq1.

Thus, the T/Igroup 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{Igroup is isomorphic to D12

. With a few calculations,we can also come to the conclusion that the elements that take the roles ofs and t as generators of the T/Igroup are T1 and I0 (note that there are other choices of elements in T/Ithat also satisfy the generatingrelations). This group acts onSby extending the notion of its action on Z12: an element XPT{Iacts onrn,m,n`7s (where m n `3 or n`4) as such: Xprn,m,n`7sq rXpnq, Xpmq, Xpn`7qs, so elementsof T/Iact on elements of S by acting on each component of an element individually. There is another

Figure 3. Action ofI0 on the C Major Triad

musically interesting way to operate on the elements ofS. Earlier, we said a triad was determined solely byits root and modal parity. So we might want a set of operations that take major triads to minor triads inspecific 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 ` 7sto rr, r ` 3, r ` 7sand vice versa. This action, which we will call P, changesthe modal parity of a triad by changing the middle note by a semitone. Similarly, we can change the parityby moving the other two notes: L takes rr, r`4, r`7s to rr1, r`4, r`7s and vice versa, and R takesrr, r`4, r`7s to rr, r`4, r`9s and vice versa. This is our motivation for defining the PLR-group.


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If we set s LR and t L, then s12 e, t2 e, and

tst LpRLqL RL s1

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 interpretactions 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 toruswe look at is called the tonnetz or tone network. If we position pitch classes in triangles such thateach triangle is surrounded by three pitch classes that form either a minor or major triad, then we get theFigure 5. We can identify the lower edge with the upper edge and the the two sides with each other to

Figure 5. Action of thePLR-group on the Tonnetz

get a torus. ThePLR-group acts visibly on the tonnetz by flipping a triangle over an edge, depending onwhich transformation it is (as indicated by the figure). Taking this interpretation, we can view Beethovens19-chord progression in the second movement of his ninth symphony as a path around the tonnetz, goingfrom ther0, 4, 7s triangle to ther7, 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].

Stahl Algebraic Music Theory

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