Transformational Music Theory (19): metrical layers and meter networks

Last November, Jason Yust and I published a paper in Journal of Mathematics and Music about “meter networks”, a categorical construction for studying meter in music pieces via transformational networks. The paper is accessible here:

• Popoff, A. and Yust, J. Meter networks: a categorical framework for metrical analysis, Journal of Mathematics and Music, 2020,

In this blog post, I want to give a brief exposition of the main ideas we developed. In particular, it makes extensive use of a musically relevant 2-category (a monoid as a matter of fact). By “musically relevant”, I mean to say that both the 1-morphisms and the 2-morphisms in this category have musical meaning. At this point, if you have not yet read the previous entries in this blog about transformational networks and the use of relations in transformational music theory, I would encourage you to do so.

But first, let’s talk about meter in music ! As the wikipedia page about meter indicates, “the term meter is not very precisely defined”. The definition given on the openmusictheory website is perhaps my favorite:

“Meter involves the way multiple pulse layers work together to organize music in time.”

Let’s see an example. Here is Emile Waldteufel’s Estudiantina waltz :

Right after the beginning, at 0’07”, we get two measures of the classic background waltz rhythm before the melody itself begins. Here’s a simplified transcription of these measures:

Each quarter note marks the pulsation of the 3/4 rhythm, so we get one pulse layer represented with the gray circles above. In addition, we have an oscillation between D and A at each bar, so we also get one pulse layer represented with the yellow circles. Finally, we can clearly hear that it repeats every two bars, so we also get a pulse layer at this level, represented here with the blue circles. All these pulse layers (or metrical layers) are assembled to give the classic waltz rhythm.

In 2001, Richard Cohn published a paper in Music Analysis, in which he studied these metrical layers in detail, and introduced the concept of ski-hill graphs.

• Cohn, R. Complex Hemiolas, Ski-Hill Graphs and Metric Spaces, Music Analysis, 20 (3), 2001, pp. 295-326, accessible here.

To understand these concepts, let’s take Cohn’s example of Brahms’ Violin Sonata, Op. 78. Below is the score for bar 235:

The articulation of the different notes gives us different pulse layers for each part. For example, the violin part divides the duration of the bar (a dotted whole note) in three half notes, each one of these being divided in two quarter notes, and then further divided in eighth notes. The piano left hand divides the duration of the bar in two dotted half notes, each one being then divided in three quarter notes. The piano right hand divides the duration of the bar in two dotted half notes too, but then divides each one of them in two dotted quarter notes, which are then further subdivided in three eighth notes each. This bar gives us examples of hemiolas, i.e. the insertion of a ternary rhythmic pattern in a binary rhythmic pattern (and vice-versa).

Let’s represent this graphically, by showing how each note duration, starting with the dotted whole note, can be divided in either two or three:

Then, each part can be assigned a path in this graph, which corresponds to the way it divides the duration of the bar in the Brahms example above. We thus get the following paths:

This is what Cohn calls a ski-hill graph, because the representation of the different paths one can take in subdividing the initial bar duration (or in other words the different metrical layers) looks like a slalom track on a ski hill. Of course, one could conceive of more complex ski-hill graphs, if these note durations are further divided in two or three. Cohn does so in studying Dvořák’s Symphony n°7 in the article presented above.

As shown on the graphs above, the violin is in a state of metric dissonance with the piano right hand: the subdivisions of the bar duration each part induces give rise to incongruent metrical layers, i.e. pulses on one layer may not coincide with pulses at another level. In the Waldteufel example above, the metrical layers are in a state of metric consonance, because each pulse on a given metrical layer also occurs on the metrical layer below.

What we have seen is a particular example of metric dissonance called grouping consonance. But there also exists another type called displacement dissonance, in which two congruent layers may be displaced in time: pulses which would otherwise be aligned are thus not anymore. This terminology has been introduced by Krebbs in his book on metric dissonance in Schumann’s music

• Krebbs, H. Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. Oxford University Press. 1999

Now, since ski-hill graphs consider note durations and their subdivisions, they constitute an interesting tool for studying grouping dissonance. However, since displacement dissonance involves timepoints, ski-hill graphs are not quite appropriate. And it so happens that some pieces of music exhibit both grouping dissonance and displacement dissonance. When I approached Jason with some mathematical ideas regarding meter and metrical layers, he immediately found us an example to study: Brahms’s Lied Op.106 n°2 “Auf dem See”, which you can listen to below.

Here are bars 5-8 of this Lied: If we look at the voice part, we can see that the bar duration is divided in two dotted quarters, the pulses at A, B, and D being part of this metrical layer. Each dotted quarter is then subdivided in three eighth notes, a new metrical layer to which belong the pulse at C. Now, if we look at the piano right hand, we can see that the notes at A’, B’, and D’ are the same as the notes at A, B, and D, except that they are displaced by the duration of a sixteenth note. In other words, the metrical layer to which A, B, and D belong and the metrical layer to which A’, B’, and D’ belong are congruent (it’s the same dotted quarter pulsation), but are in displacement dissonance. This makes sense given the text of the Lied: this is a love affair, but one which is adressed to the lake that carries the boat, and not to the beloved. Thus, the lake acts as a reflection of the character’s ideas. Eventually, the piano part takes a direction of its own: it takes the duration between E and G and splits it in two. The notes at E’, E” and G’ now belong to a different metrical layer, in which the pulses are regularly spaced at intervals of a dotted eighth note. But there is more: though we are now considering distinct metrical layers, they nevertheless share a common timepoint at E”=F, emphasizing the fact that the lake is a distorted reflection of the narrator’s thoughts.

This examples illustrates the significance of timepoints in metrical analysis: ski-hill graphs are somehow inadequate to capture all these interrelationships. So let’s see how we can properly extend these ideas and formalize them mathematically.

Metrical relations

First of all, we define a timepoint to be an element of the set of rational numbers $\mathbb{Q}$. This is because $\mathbb{Q}$ is a natural choice for notated rhythm, though we have seen before in this blog examples of groups and monoids where timepoints are defined to be elements of $\mathbb{R}$.

Then, we want to relate timepoints in a given metrical layer. This is where the work on binary relations and their use in transformational music theory (which you may have read about in previous entries of this blog) becomes important. For a given positive rational number $d \in \mathbb{Q}_{\geq 0}$, we define a metrical relation $\mathcal{M}_d$ on the set of timepoints as the reflexive binary relation defined on $\mathbb{Q}$ such that for two timepoints $t$ and $t'$, we have $t \mathcal{M}_d t'$ whenever $t'-t=kd$ with $k \in \mathbb{Z}$.

So for example, if we take the bar duration in the Brahms example above to be the unit duration, we can say that B and D are related by $\mathcal{M}_{\frac{1}{2}}$. But A and D are also related by $\mathcal{M}_{\frac{1}{2}}$ ! In fact, even B and G are related by this metrical relation. This is very different from simply studying the time intervals between B and D, or B and G. Since all these timepoints belong to the same metrical layer, one in which the pulses are regularly spaced at intervals of a dotted quarter note, they are all related by the same metrical relation $\mathcal{M}_{\frac{1}{2}}$. Observe that B’ and D’ are also related by $\mathcal{M}_{\frac{1}{2}}$: this is precisely what we have said above when we commented on the displacement dissonance between these two congruent metrical layers.

Can you now guess by which metrical relation B and C are related ? And what about E’ and E” ?

If your conclusion is that B and C are related by $\mathcal{M}_{\frac{1}{6}}$, and that E’ and E” are related by $\mathcal{M}_{\frac{1}{4}}$, then congratulations ! In fact, there may be more than one metrical relation relating two timepoints: thus B and D are related by $\mathcal{M}_{\frac{1}{2}}$, but also by $\mathcal{M}_{\frac{1}{6}}$, as well as $\mathcal{M}_{\frac{1}{12}}$.

This is because binary relations can be included into each other. In categorical terms, we say the category $\mathbf{Rel}$ of sets and binary relations between them is a 2-category (in fact a 2-poset): the objects are sets, the 1-morphisms are binary relations, and the 2-morphisms are given by inclusion of binary relations.

In the specific case of metrical relations we have the following result: let $\mathcal{M}_{d_1}$ and $\mathcal{M}_{d_2}$ be two metrical relations. Then, the metrical relation $\mathcal{M}_{d_1}$ is included in $\mathcal{M}_{d_2}$ if and only if there exists a positive integer $u$ such that $d_1=ud_2$.

For example, $\mathcal{M}_{1}$ is included in $\mathcal{M}_{\frac{1}{2}}$ as shown graphically below: So these are inclusions of binary relations, i.e. the 2-morphisms. So far we haven’t really talked about how the metrical relations (the 1-morphisms) can be composed. You can easily prove that if we take two metrical relations $\mathcal{M}_{d_1}$ and $\mathcal{M}_{d_2}$, we can compose them just as we do with any binary relation, and we will obtain a metrical relation $\mathcal{M}_{d}$ where $d=\text{gcd}(d_1,d_2)$.

Here’s a graphical illustration of the composition of the two metrical relations $\mathcal{M}_{\frac{1}{2}}$ and $\mathcal{M}_{\frac{1}{3}}$. We get the metrical relation $\mathcal{M}_{\frac{1}{6}}$: this corresponds to the regular pulse of largest duration interval such that both the pulses corresponding to $\mathcal{M}_{\frac{1}{2}}$ and to $\mathcal{M}_{\frac{1}{3}}$ are included in it. And thus, if we consider the set of all metrical relations $\mathcal{M}_{d}$ with $d \in \mathbb{Q}_{\geq 0}$, they form a commutative monoid $M$ isomorphic to $\mathbb{Q}_{\geq 0}$ equipped with the gcd as the monoid operation (and assuming $\text{gcd}(0,x)=x$). The identity operation is the metrical relation $\mathcal{M}_{0}$, which relates a timepoint only to itself. Every metrical relation is idempotent since $\text{gcd}(x,x)=x$.

But what we have in fact is a 2-monoid, since we also have inclusion of metrical relations (i.e. 2-morphisms between the 1-morphisms). This 2-monoid, which is a submonoid of $\mathbf{Rel}$, generalizes Cohn’s ski-hill graphs since Cohn’s ski-hill graphs corresponds to sub-posets of the poset structure of the 2-morphisms between metrical relations. In particular, it isn’t limited to divisions by 2 or 3.

But more importantly, our 2-monoid $M$ comes naturally with a functor to $\mathbf{Sets}$. After all, metrical relations are defined on the set of timepoints: the image of the single object of $M$ is the set $\mathbb{Q}$ of timepoints itself. And thus, we overcome the problem of ski-hill graphs since we are now able to deal directly with timepoints.

In passing, we can see the monoid $M$ has even more structure: the set of metrical relations is closed under the operation of intersection of binary relations, which is not always the case. For any two metrical relations $\mathcal{M}_{d_1}$ and $\mathcal{M}_{d_2}$, we have that $\mathcal{M}_{d_1} \cap \mathcal{M}_{d_2} = \mathcal{M}_{\text{lcm}(d_1,d_2)}$. From a musical point of view, this corresponds to finding the regular pulse of smallest duration interval common to both pulses. From a mathematical point of view, the set of elements of the monoid $M$ has the structure of a join-semilattice, and the inclusion of metrical relations corresponds to the partial order induced by the lcm monoid operation of this join-semilattice.

So now that we have a perfect monoid for studying meter relations, what can we do with it ?

Meter networks

Well, as we have seen in a previous post, we can use the category $\mathbf{Rel}$ to form relational poly-Klumpenhouwer networks, or in other words transformational networks using sets as nodes and binary relations to label the arrows between these nodes. In this specific case, we will use our 2-monoid $M$ of metrical relations to label the arrows, and we will call the resulting networks meter networks. I won’t go into the details of their exact definition here: the structure of $M$ and $\mathbf{Rel}$ makes it a bit complicated if we want them to fullfill all the possibilities for analysis. All these details can be found in the reference given at the beginning of this post. Instead, let’s consider some examples to illustrate Brahms’s Lied Op.106 n°2.

As with any other transformational networks, we can relate timepoints (or sets of timepoints) by binary relations. For example, we can draw the following network which illustrates how the timepoints B and D are related by the metrical relation $\mathcal{M}_{\frac{1}{2}}$. This is a simplified representation of a more complex categorical construction, but never mind. In the same vein, we can draw the following network illustrating how the timepoints B, C, and D are related by the metrical relation $\mathcal{M}_{\frac{1}{6}}$.

Now, as we have seen previously, it is very important when we get networks to also study network morphisms, i.e. how networks can be transformed into each other. In the case of meter networks, their definition is complex because we have to take into account both the transformation of the network structure (the nodes and arrows) and the transformation of labels on arrows (the transformations of metrical relations themselves). For that, we make extensive use of the 2-morphisms in the monoid $M$, allowing inclusion of metrical relations. For example, the two networks above can be related by a network morphism which we represent (partially) in red below since the metrical relation $\mathcal{M}_{\frac{1}{2}}$ is included in $\mathcal{M}_{\frac{1}{6}}$. The labels in red marked $\mathcal{M}_{0}$ indicate that these timepoints do not change during the transformation (but they could).

Our definition of network morphisms is broad enough to allow us to consider displacement dissonance. Indeed, time translations and dilations are automorphisms of the functor $M \to \mathbf{Rel}$. Thus we can account for the displacement dissonance between timepoints E, G and E’, G’ in the Brahms example by drawing the following networks and the corresponding network morphism between them. By applying a time translation of a sixteenth note (t+1/12) on the upper network, we translate the timepoints E and G to E’ and G’ respectively. Since there is no time dilation in this morphism, the metrical relation $\mathcal{M}_{\frac{1}{2}}$ stays the same. Finally, we can combine all these network morphisms into one unifying diagram of meter networks, which shows both the grouping dissonance, the displacement dissonance, and the common timepoint at E”=F, as pictured below. Again, this is possible because $\mathcal{M}_{\frac{1}{2}}$ is included in both $\mathcal{M}_{\frac{1}{6}}$ and $\mathcal{M}_{\frac{1}{4}}$: this corresponds to a ski-hill graph in the style of Cohn. The displacement morphism between E,G and E’,G’ cannot however be expressed with ski-hill graphs.

In conclusion, we have used binary relations to study metrical layers, and in doing so we exhibited a 2-monoid where both 1-morphisms and 2-morphisms have musical relevance. This allowed us to define meter networks which can be used to describe both grouping dissonance and displacement dissonance in metrical analysis. Importantly, this shows that higher category theory can be relevant for mathematical music analysis. I’d be curious to see if there are other examples !