# An introduction to neo-Riemannian theory (3)

In the previous posts (here and here), we have seen some examples of groups that can arise in music theory. However, I haven’t explained why we need groups in the first place, so now is the time for that. In this post, I will be talking about the main idea in David Lewin‘s work, and probably one of the most revolutionary one too in the field of music theory during the past 50 years. The classic reference is the following seminal book by Lewin :

Lewin also wrote two books, in which he applies his findings for music analysis :

But before we examine Lewin’s work, a short music example first :

This is the opening of the fugue n°14 in F# minor (BWV 859), from Book 1 of the Well-Tempered Clavier of J.S. Bach. There is a peculiar feature in first three bars of this passage which can be easily heard: the fugue opens with a small motif of three notes (F#-G#-A) which then seems to repeat twice until one reaches C, at which point the melody goes down to the initial F# (the second voice of the fugue then enters). If we forget about the embellishments in the melody, this passage can be reduced to the following form

emphasizing the repeated motif in the ascending part.

(Disclaimer: I am drawing this reduced analysis from the following two articles, which analyze this fugue in detail using tools from transformational theory… which we haven’t explained yet ! Here are the references anyway in case you’re interested :

)

One could ask the following question about this part: how come we recognize the repeated motif, even though the notes are all different ? To answer this question, I would ask another one: if the whole fugue was played a semi-tone lower, would you recognize it anyway ?

The answer is probably yes, and the reason would be that even though the notes are transposed a semi-tone lower, the intervals between the notes remain the same. In the same view, our initial fugue motif is made up of three notes separated by 2 (F#-G#) and 1 (G#-A) semitones. Though the subsequent motif is transposed, the intervals are the same for the couple G#-A# (2 semitones) and the couple A#-B (1 semitone), or for the couple A#-B#(=C, remember that Bach used an equal tempered tuning) and B#-C#.

Thus our point of view has changed: instead of analyzing the notes themselves, we are now analyzing the intervals between them. But why limit ourselves to notes ? We could also do the same for chords, which are sets of notes. Indeed, to analyze chords, one can use Roman numeral analysis, which assigns to each chord a Roman numeral, usually denoting the scale degree, which allows to quickly understand the progression of chords. However, when the music is highly chromatic, as it was the case in the late XIXth century, Roman numeral analysis breaks down as it becomes more difficult to assign unambiguously a label to some chords. Neo-Riemannian theory is the result of the same shift of point of view as for notes: instead of analyzing chords by themselves, one analyzes the possible transformations between chords. We will see in later posts what are those transformations.

So, in the general case, we have musical objects, and we want to assign some sort of label for each couple of musical objects, which will act as a generalized interval. This generalized interval reflects the transformation needed for going from a musical object to another. It could be numbers reflecting transposition as in the case of notes, but not necessarily. Indeed, you would have a hard time defining intervals between chords as numbers, except in some special cases, not mentionning intervals between percussion sounds for example. So how can we define these generalized intervals ? Lewin’s big idea was to jump to an abstract setting and to realize that the set of generalized intervals actually has much more structure than just being a set. Indeed :

• Generalized intervals can composed: if we need the interval $I_1$ to go from the musical object $O_1$ to the musical object $O_2$, and interval $I_2$ to go from the musical object $O_2$ to the musical object $O_3$, then the interval between $O_1$ and $O_3$ would be $I_1 \circ I_2$ (or $I_2 \circ I_1$, I’ll explain the difference shortly).
• For each musical object $O$, the interval between $O$ and $O$ should be special, a kind of “null” interval, which reflects that going from $O$ to $O$ is actually the identity transformation.
• If the interval between objects $O_1$ and $O_2$ is $I$, then there also exists an interval between $O_2$ and $O_1$ which should act as the inverse of $I$. In other words, the composition of these two intervals should be the “null” interval, as we would go from $O_1$ to $O_2$ to $O_1$, i.e we would end up on the same musical object.

Does all of this ring a bell ? It certainly does look like a mathematical group ! Of course, we need to throw in associativity too, not mentionning that the definitions above are mathematically messy. So, I’ll turn directly to Lewin’s definition to make it clearer.

Lewin starts with a set of musical objects $S$, a mathematical group of intervals $IVLS$ and defines a Generalized Interval System (GIS) as such :

Definition (Lewin):  A Generalized Interval System (GIS) is an ordered triple (S,IVLS, int) where S, the space of the GIS, is a family of elements, IVLS, the group of intervals for the GIS, is a mathematical group, and $int$ is a function mapping $S \times S$ into IVLS, all subject to the two conditions (A) and (B) following.

• (A): $\forall r, s, t \in S$, $int(r,s) \circ int(s,t) = int(r,t)$
• (B): $\forall s \in S, i \in IVLS$, there is a unique $t \in S$ which lies the interval $i$ from $s$, that is a unique $t$ which satisfies the equation $int(s,t)=i$.

In Lewin’s definition of a GIS, the group structure plays a central role, as it conveniently allows to define null intervals (the identity element), inverse intervals (group inverses) and composition of intervals.

But there is more. In his definition, Lewin also asks that for any couple of musical objects, the interval between them should be unique. This actually makes a lot of sense, as the analysis of relationships between musical objects will be unambiguous. However, adding condition (B) to the definition turns any GIS into a structure which is very well known by mathematicians under the name of principal homogeneous space, or G-torsor. Before we introduce the definition of a G-torsor, we need to introduce the mathematical definition of a group action, which comes in two flavors as left- or right- group action:

Definition (Left group action):  If $G$ is a group, and $S$ is a set, a left group action of $G$ on $S$ is a function $G \times S \to S$ which associates to any couple $(g,s)$ an element $s'=g.s \in S$ such that

• $(g \circ h).s = g.(h.s)$, $\forall g,h \in G, s \in S$
• $1_G.s=s, \forall s \in S$, where $1_G$ denotes the identity element of $G$

Definition (Right group action):  If $G$ is a group, and $S$ is a set, a right group action of $G$ on $S$ is a function $G \times S \to S$ which associates to any couple $(g,s)$ an element $s'=s.g \in S$ such that

• $s.(g \circ h) = (s.g).h$, $\forall g,h \in G, s \in S$
• $1_G.s=s, \forall s \in S$, where $1_G$ denotes the identity element of $G$

Note that the only difference between a left- and a right- group action is the order in which a product $g \circ h$ acts on an element $s \in S$. We say that a (left) group action is free if $g.s=h.s \implies g=h$ (idem for a right group action). We say that a (left) group action is transitive if $\forall s,s' \in S$ there exists $g \in G$ such that $g.s=s'$. Now for the definition of a G-torsor :

Definition :  A G-torsor, or principal homogeneous space for a group $G$, is a non-empty set $S$ equipped with a free and transitive action of $G$ on $S$.

In other words, a G-torsor is a set equipped with a group action of $G$ such that for any two elements $s,s' \in S$, there exists a group element $g \in G$ such that $s'=g.s$ (or $s'=s.g$), and this group element is unique. Observe in particular that it follows directly from the definition that the cardinality of $S$ and the order of $G$ are the same.

The interesting part is that Lewin’s definition of a GIS is equivalent to the definition of a G-torsor ! Try to prove it ! Or you can check this article by Kolman, where the correspondence between the two concepts is made explicit:

• “Transfer Principles for Generalized Interval Systems”, O. Kolman, Perspectives of New Music, Vol. 42, N°1 (Winter, 2004), pp. 150-190.

By the way, you can also check that condition (A) in Lewin’s definition is actually equivalent to a G-torsor with a right group action. Had Lewin instead defined condition (A) as

• (A): $\forall r, s, t \in S$, $int(s,t) \circ int(r,s) = int(r,t)$

he would have obtained G-torsors with left group actions, though this alternative definition does not seem to be present. The difference between left and right group actions does not matter if the group is abelian, as elements commute, but it can have very different results in the general case, as we will see in a later post about transformations of temporal structures.

G-torsors are very interesting mathematical structures. I urge you to read the introduction to torsors written by John Baez, which explains quite a lot of stuff about them. One of most interesting fact from torsors is that you can divide elements but you can’t multiply them. How’s that ? Say you have elements $s, s' \in S$. The “quotient” $s' / s$ of $s'$ by $s$ is the unique group element $g \in G$ such that $s'=g.s$. This is exactly how we define intervals for musical objects ! However, you cannot define the product $s.s'$ because, although you can obtain the relative interval, or difference, between two elements, there is no absolute correspondence between elements of $S$ and elements of $G$ which would allow you to multiply elements of $S$ like you would do in $G$. Of course, you can fix a correspondence, but this is an additional information that you have to introduce, and an arbitrary one as there is no canonical way of doing so. It can readily be seen that fixing a bijection between $G$ and $S$ is equivalent to choosing a particular element of $S$ as the group identity (since all other elements results from the action of the group elements). Hence, it is often said that a G-torsor is like a group “which has forgotten its identity”: you will be able to multiply elements of $S$ as in $G$ only if you fix one particular element of $S$ as $1_G$.

Can you provide an example of a musical G-torsor ? In fact we have seen one in our last post: the circle of semi-tones. The group we consider here is the group of semi-tone transpositions $\mathbb{Z}/\mathbb{Z}_{12} = < t | t^{12}=1 >$ (which we’ll abbreviate as $\mathbb{Z}_{12}$) and the action of the generator $t$ on the set of semi-tones {C, C#, D, ….} is $t.C = C\#$$t.C\# = D$$t.D = D\#$, etc. We can also define this action in another way by identifying each semi-tone with a number in the integers modulo 12, like on a clock: C=[0], C#=[1], D=[2], etc.. (wait ! what ?). Then the action of a group element $t^n \in \mathbb{Z}_{12}$ on a semi-tone element $[m]$ is $t^n.[m]=[m+n] \pmod {12}$.

Wait… have you seen what I’ve just done ? When I assigned a number to each element of the semi-tones set, I actually turned this set into a group by choosing one particular element as the identity ! In that case I chose to identify C with the identity element of $\mathbb{Z}_{12}$ but it could have been any other semi-tone.

One word of caution: the definition of intervals through a GIS, or equivalently in a G-torsor, does not define a notion of shortest distance between musical objects. Indeed, consider for example the interval between C and G in the circle of semi-tones. This is the unique element $t^n \in \mathbb{Z}_{12}$ such that $G=t^n.C$. It can readily be seen that we have $n=7$. However, in terms of shortest distance, G is only five semi-tones away from C. We have $t^7=t^{-5}$ and in the general case one can define the shortest distance between two semi-tones as $min(|n \pmod {12}|, |-n \pmod {12}|)$ if $t^n$ is the interval between these semi-tones. Hence we see that the notions are completely different from each other.

In the next post, we will talk about GIS, or G-torsors, for major and minor chords, introducing the famous $D_{24}$ dihedral group.