Last time, we’ve viewed major and minor triads (chords) as inscribed triangles in a regular $12$-gon.

If we move clockwise along the $12$-gon, starting from the endpoint of the longest edge (the root of the chord, here the $0$-vertex) the edges skip $3,2$ and $4$ vertices (for a major chord, here on the left the major $0$-chord) or $2,3$ and $4$ vertices (for a minor chord, here on the right the minor $0$-chord).

The symmetries of the $12$-gon, the dihedral group $D_{12}$, act on the $24$ major- and minor-chords transitively, preserving the type for rotations, and interchanging majors with minors for reflections.

Mathematical Music Theoreticians (MaMuTh-ers for short) call this the $T/I$-group, and view the rotations of the $12$-gon as transpositions $T_k : x \mapsto x+k~\text{mod}~12$, and the reflections as involutions $I_k : x \mapsto -x+k~\text{mod}~12$.

Note that the elements of the $T/I$-group act on the vertices of the $12$-gon, from which the action on the chord-triangles follows.

There is another action on the $24$ major and minor chords, mapping a chord-triangle to its image under a reflection in one of its three sides.

Note that in this case the reflection $I_k$ used will depend on the root of the chord, so this action on the chords does not come from an action on the vertices of the $12$-gon.

There are three such operations: (pictures are taken from Alexandre Popoff’s blog, with the ‘funny names’ removed)

The $P$-operation is reflection in the longest side of the chord-triangle. As the longest side is preserved, $P$ interchanges the major and minor chord with the same root.

The $L$-operation is refection in the shortest side. This operation interchanges a major $k$-chord with a minor $k+4~\text{mod}~12$-chord.

Finally, the $R$-operation is reflection in the middle side. This operation interchanges a major $k$-chord with a minor $k+9~\text{mod}~12$-chord.

From this it is already clear that the group generated by $P$, $L$ and $R$ acts transitively on the $24$ major and minor chords, but what is this $PLR$-group?

If we label the major chords by their root-vertex $1,2,\dots,12$ (GAP doesn’t like zeroes), and the corresponding minor chords $13,14,\dots,24$, then these operations give these permutations on the $24$ chords:

 P:=(1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) L:=(1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16) R:=(1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21) 

Then GAP gives us that the $PLR$-group is again isomorphic to $D_{12}$:

 gap> G:=Group(P,L,R);; gap> Size(G); 24 gap> IsDihedralGroup(G); true 

In fact, if we view both the $T/I$-group and the $PLR$-group as subgroups of the symmetric group $Sym(24)$ via their actions on the $24$ major and minor chords, these groups are each other centralizers! That is, the $T/I$-group and $PLR$-group are dual to each other.

For more on this, there’s a beautiful paper by Alissa Crans, Thomas Fiore and Ramon Satyendra: Musical Actions of Dihedral Groups.

What does this new MaMuTh info learns us more about our Elephant, the Topos of Triads, studied by Thomas Noll?

Last time we’ve seen the eight element triadic monoid $T$ of all affine maps preserving the three tones $\{ 0,4,7 \}$ of the major $0$-chord, computed the subobject classified $\Omega$ of the corresponding topos of presheaves, and determined all its six Grothendieck topologies, among which were these three:

Why did we label these Grothendieck topologies (and corresponding elements of $\Omega$) by $P$, $L$ and $R$?

We’ve seen that the sheafification of the presheaf $\{ 0,4,7 \}$ in the triadic topos under the Grothendieck topology $j_P$ gave us the sheaf $\{ 0,3,4,7 \}$, and these are the tones of the major $0$-chord together with those of the minor $0$-chord, that is the two chords in the $\langle P \rangle$-orbit of the major $0$-chord. The group $\langle P \rangle$ is the cyclic group $C_2$.

For the sheafication with respect to $j_L$ we found the $T$-set $\{ 0,3,4,7,8,11 \}$ which are the tones of the major and minor $0$-,$4$-, and $8$-chords. Again, these are exactly the six chords in the $\langle P,L \rangle$-orbit of the major $0$-chord. The group $\langle P,L \rangle$ is isomorphic to $Sym(3)$.

The $j_R$-topology gave us the $T$-set $\{ 0,1,3,4,6,7,9,10 \}$ which are the tones of the major and minor $0$-,$3$-, $6$-, and $9$-chords, and lo and behold, these are the eight chords in the $\langle P,R \rangle$-orbit of the major $0$-chord. The group $\langle P,R \rangle$ is the dihedral group $D_4$.

More on this can be found in the paper Commuting Groups and the Topos of Triads by Thomas Fiore and Thomas Noll.

The operations $P$, $L$ and $R$ on major and minor chords are reflexions in one side of the chord-triangle, so they preserve two of the three tones. There’s a distinction between the $P$ and $L$ operations and $R$ when it comes to how the third tone changes.

Under $P$ and $L$ the third tone changes by one halftone (because the corresponding sides skip an even number of vertices), whereas under $R$ the third tone changes by two halftones (a full tone), see the pictures above.

The $\langle P,L \rangle = Sym(3)$ subgroup divides the $24$ chords in four orbits of six chords each, three major chords and their corresponding minor chords. These orbits consist of the

• $0$-, $4$-, and $8$-chords (see before)
• $1$-, $5$-, and $9$-chords
• $2$-, $6$-, and $10$-chords
• $3$-, $7$-, and $11$-chords

and we can view each of these orbits as a cycle tracing six of the eight vertices of a cube with one pair of antipodal points removed.

These four ‘almost’ cubes are the NE-, SE-, SW-, and NW-regions of the Cube Dance Graph, from the paper Parsimonious Graphs by Jack Douthett and Peter Steinbach.

To translate the funny names to our numbers, use this dictionary (major chords are given by a capital letter):

The four extra chords (at the N, E, S, and P places) are augmented triads. They correspond to the triads $(0,4,8),~(1,5,9),~(2,6,10)$ and $(3,7,11)$.

That is, two triads are connected by an edge in the Cube Dance graph if they share two tones and differ by an halftone in the third tone.

This graph screams for a group or monoid acting on it. Some of the edges we’ve already identified as the action of $P$ and $L$ on the $24$ major and minor triads. Because the triangle of an augmented triad is equilateral, we see that they are preserved under $P$ and $L$.

But what about the edges connecting the regular triads to the augmented ones? If we view each edge as two directed arrows assigned to the same operation, we cannot do this with a transformation because the operation sends each augmented triad to six regular triads.

Alexandre Popoff, Moreno Andreatta and Andree Ehresmann suggest in their paper Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis that one might use a monoid generated by relations, and they show that there is such a monoid with $40$ elements acting on the Cube Dance graph.

Popoff claims that usual presheaf toposes, that is contravariant functors to $\mathbf{Sets}$ are not enough to study transformational music theory. He suggest to use instead functors to $\mathbf{Rel}$, that is Sets with as the morphisms binary relations, and their compositions.

Another Elephant enters the room…

(to be continued)

Here, MaMuTh stands for Mathematical Music Theory which analyses the pitch, timing, and structure of works of music.

The Elephant is the nickname for the ‘bible’ of topos theory, Sketches of an Elephant: A Topos Theory Compendium, a two (three?) volume book, written by Peter Johnstone.

How can we get as quickly as possible from the MaMuth to the Elephant, musical illiterates such as myself?

What Mamuth-ers call a pitch class (sounds that are a whole number of octaves apart), is for us a residue modulo $12$, as an octave is usually divided into twelve (half)tones.

We’ll just denote them by numbers from $0$ to $11$, or view them as the vertices of a regular $12$-gon, and forget the funny names given to them, as there are several such encodings, and we don’t know a $G$ from a $D\#$.

Our regular $12$-gon has exactly $24$ symmetries. Twelve rotations, which they call transpositions, given by the affine transformations
$T_k~:~x \mapsto x+k~\text{mod}~12$
and twelve reflexions, which they call involutions, given by
$I_k~:~x \mapsto -x+k~\text{mod}~12$
What for us is the dihedral group $D_{12}$ (all symmetries of the $12$-gon), is for them the $T/I$-group (for transpositions/involutions).

Let’s move from individual notes (or pitch classes) to chords (or triads), that is, three notes played together.

Not all triples of notes sound nice when played together, that’s why the most commonly played chords are among the major and minor triads.

A major triad is an ordered triple of elements from $\mathbb{Z}_{12}$ of the form
$(n,n+4~\text{mod}~12,n+7~\text{mod}~12)$
and a minor triad is an ordered triple of the form
$(n,n+3~\text{mod}~12,n+7~\text{mod}~12)$
where the first entry $n$ is called the root of the triad (or chord) and its funny name is then also the name of that chord.

For us, it is best to view a triad as an inscribed triangle in our regular $12$-gon. The triangles of major and minor triads have edges of different lengths, a small one, a middle, and a large one.

Starting from the root, and moving clockwise, we encounter in a major chord-triangle first the middle edge, then the small edge, and finally the large edge. For a minor chord-triangle, we have first the small edge, then the middle one, and finally the large edge.

On the left, two major triads, one with root $0$, the other with root $6$. On the right, two minor triads, also with roots $0$ and $6$.

(Btw. if you are interested in the full musical story, I strongly recommend the alpof blog by Alexandre Popoff, from which the above picture is taken.)

Clearly, there are $12$ major triads (one for each root), and $12$ minor triads.

From the shape of the triad-triangles it is also clear that rotations (transpositions) send major triads to major triads (and minors to minors), and that reflexions (involutions) interchange major with minor triads.

That is, the dihedral group $D_{12}$ (or if you prefer the $T/I$-group) acts on the set of $24$ major and minor triads, and this action is transitive (an element stabilising a triad-triangle must preserve its type (so is a rotation) and its root (so must be the identity)).

Can we hear the action of the very special group element $T_6$ (the unique non-trivial central element of $D_{12}$) on the chords?

This action is not only the transposition by three full tones, but also a point-reflexion with respect to the center of the $12$-gon (see two examples in the picture above). This point reflexion can be compositionally meaningful to refer to two very different upside-down worlds.

In It’s $T_6$-day, Alexandre Popoff gives several examples. Here’s one of them, the Ark theme in Indiana Jones – Raiders of the Lost Ark.

“The $T_6$ transformation is heard throughout the map room scene (in particular at 2:47 in the video): that the ark is a dreadful object from a very different world is well rendered by the $T_6$ transposition, with its inherent tritone and point reflection.”

Let’s move on in the direction of the Elephant.

We saw that the only affine map of the form $x \mapsto \pm x + k$ fixing say the major $0$-triad $(0,4,7)$ is the identity map.

But, we can ask for the collection of all affine maps $x \mapsto a x + b$ fixing this major $0$-triad set-wise, that is, such that
$\{ b, 4a+b~\text{mod}~12, 7a+b~\text{mod}~2 \} \subseteq \{ 0,4,7 \}$

A quick case-by-case analysis shows that there are just eight such maps: the identity and the constant maps
$x \mapsto x,~x \mapsto 0,~x \mapsto 4, ~x \mapsto 7$
and the four maps
$\underbrace{x \mapsto 3x+7}_a,~\underbrace{x \mapsto 8x+4}_b,~x \mapsto 9x+4,~x \mapsto 4x$

Compositions of such maps again preserve the set $\{ 0,4,7 \}$ so they form a monoid, and a quick inspection with GAP learns that $a$ and $b$ generate this monoid.

 gap> a:=Transformation([10,1,4,7,10,1,4,7,10,1,4,7]);; gap> b:=Transformation([12,8,4,12,8,4,12,8,4,12,8,4]);; gap> gens:=[a,b];; gap> T:=Monoid(gens); gap> Size(T); 8 

The monoid $T$ is the triadic monoid of Thomas Noll’s paper The topos of triads.

The monoid $T$ can be seen as a one-object category (with endomorphisms the elements of $T$). The corresponding presheaf topos is then the category of all sets equipped with a right $T$-action.

Actually, Noll considers just one such presheaf (and its sub-presheaves) namely $\mathcal{F}=\mathbb{Z}_{12}$ with the action of $T$ by affine maps described before.

He is interested in the sheafifications of these presheaves with respect to Grothendieck topologies, so we have to describe those.

For any monoid category, the subobject classifier $\Omega$ is the set of all right ideals in the monoid.

Using the GAP sgpviz package we can draw its Cayley graph (red coloured vertices are idempotents in the monoid, the blue vertex is the identity map).

 gap> DrawCayleyGraph(T); 

The elements of $T$ (vertices) which can be connected by oriented paths (in both ways) in the Cayley graph, such as here $\{ 2,4 \}$, $\{ 3,7 \}$ and $\{ 5,6,8 \}$, will generate the same right ideal in $T$, so distinct right ideals are determined by unidirectional arrows, such as from $1$ to $2$ and $3$ or from $\{ 2,4 \}$ to $5$, or from $\{ 3,7 \}$ to $6$.

This gives us that $\Omega$ consists of the following six elements:

• $0 = \emptyset$
• $C = \{ 5,6,8 \} = a.T \wedge b.T$
• $L = \{ 2,4,5,6,8 \}=a.T$
• $R = \{ 3,7,5,6,8 \}=b.T$
• $P = \{ 2,3,4,5,6,7,8 \}=a.T \vee b.T$
• $1 = T$

As a subobject classifier $\Omega$ is itself a presheaf, so wat is the action of the triad monoid $T$ on it? For all $A \in \Omega$, and $s \in T$ the action is given by $A.s = \{ t \in T | s.t \in A \}$ and it can be read off from the Cayley-graph.

$\Omega$ is a Heyting algebra of which the inclusions, and logical operations can be summarised in the picture below, using the Hexboards and Heytings-post.

In this case, Grothendieck topologies coincide with Lawvere-Tierney topologies, which come from closure operators $j~:~\Omega \rightarrow \Omega$ which are order-increasing, idempotent, and compatible with the $T$-action and with the $\wedge$, that is,

• if $A \leq B$, then $j(A) \leq j(B)$
• $j(j(A)) = j(A)$
• $j(A).t=j(A.t)$
• $j(A \wedge B) = j(A) \wedge j(B)$

Colouring all cells with the same $j$-value alike, and remaining cells $A$ with $j(A)=A$ coloured yellow, we have six such closure operations $j$, that is, Grothendieck topologies.

The triadic monoid $T$ acts via affine transformations on the set of pitch classes $\mathbb{Z}_{12}$ and we’ve defined it such that it preserves the notes $\{ 0,4,7 \}$ of the major $(0,4,7)$-chord, that is, $\{ 0,4,7 \}$ is a subobject of $\mathbb{Z}_{12}$ in the topos of $T$-sets.

The point of the subobject classifier $\Omega$ is that morphisms to it classify subobjects, so there must be a $T$-equivariant map $\chi$ making the diagram commute (vertical arrows are the natural inclusions)
$\xymatrix{\{ 0,4,7 \} \ar[r] \ar[d] & 1 \ar[d] \\ \mathbb{Z}_{12} \ar[r]^{\chi} & \Omega}$

What does the morphism $\chi$ do on the other pitch classes? Well, it send an element $k \in \mathbb{Z}_{12} = \{ 1,2,\dots,12=0 \}$ to

• $1$ iff $k \in \{ 0,4,7 \}$
• $P$ iff $a(k)$ and $b(k)$ are in $\{ 0,4,7 \}$
• $L$ iff $a(k) \in \{ 0,4,7 \}$ but $b(k)$ is not
• $R$ iff $b(k) \in \{ 0,4,7 \}$ but $a(k)$ is not
• $C$ iff neither $a(k)$ nor $b(k)$ is in $\{ 0,4,7 \}$

Remember that $a$ and $b$ are the transformations (images of $(1,2,\dots,12)$)
 a:=Transformation([10,1,4,7,10,1,4,7,10,1,4,7]);; b:=Transformation([12,8,4,12,8,4,12,8,4,12,8,4]);; 
so we see that

• $0,1,4$ are mapped to $1$
• $3$ is mapped to $P$
• $8,11$ are mapped to $L$
• $1,6,9,10$ are mapped to $R$
• $2,5$ are mapped to $C$

Finally, we can compute the sheafification of the sub-presheaf $\{ 0,4,7 \}$ of $\mathbb{Z}$ with respect to a Grothendieck topology $j$: it consists of the set of those $k \in \mathbb{Z}_{12}$ such that $j(\chi(k)) = 1$.

The musically interesting Grothendieck topologies are $j_P, j_L$ and $j_R$ with corresponding sheaves:

• For $j_P$ we get the sheaf $\{ 0,3,4,7 \}$ which Mamuth-ers call a Major-Minor Mixture as these are the notes of both the major and minor $0$-triads
• For $j_L$ we get $\{ 0,3,4,7,8,11 \}$ which is an example of an Hexatonic scale (six notes), here they are the notes of the major and minor $0,~4$ and $8$-triads
• For $j_R$ we get $\{ 0,1,3,4,6,7,9,10 \}$ which is an example of an Octatonic scale (eight notes), here they are the notes of the major and minor $0,~3,~6$ and $9$-triads

We could have played the same game starting with the three notes of any other major triad.

Those in the know will have noticed that so far I’ve avoided another incarnation of the dihedral $D_{12}$ group in music, namely the $PLR$-group, which explains the notation for the elements of the subobject classifier $\Omega$, but this post is already way too long.

(to be continued…)