# Transformational Music Theory (17): wreath products

In this post, I will go back to traditional group-based transformational approaches and introduce the work of Julian Hook on so-called Uniform Triadic Transformations (UTT) and the corresponding group they generate. Julian Hook was, I think, the first to introduce in math/music theory the notion of wreath products, a well known construction in group theory.

I will be giving a short introduction to these notions in this blog post. For those interested, the classic references are given below.

• Hook, Julian. “Uniform Triadic Transformations.” Journal of Music Theory, vol. 46, no. 1/2, 2002, pp. 57–126. JSTOR, available here.
• Hook, Julian, and Jack Douthett. “Uniform Triadic Transformations and the Twelve-Tone Music of Webern.” Perspectives of New Music, vol. 46, no. 1, 2008, pp. 91–151. JSTOR, available here.
• Peck, Robert W. “Wreath Products in Transformational Music Theory.” Perspectives of New Music, vol. 47, no. 1, 2009, pp. 193–210. JSTOR, available here.

Before examining Uniform Triadic Transformations, let’s recall the classic $P$, $R$, and $L$ operations in neo-Riemannian theory. As we have done previously in this blog, let’s notate a major triad with root $n \in \mathbb{Z}_{12}$ as $n_+$, and a minor triad with root $n$ as $n_-$. The table below gives the three operations, along with the corresponding mapping of major and minor triads.

 P R L $n_+ \to n_-$ $n_- \to n_+$ $n_+ \to (n+9)_-$ $n_- \to (n+3)_+$ $n_+ \to (n+4)_-$ $n_- \to (n+8)_+$

Two things can be noticed from this Table :

• These three operations switch the nature of the chords, from major to minor, and vice versa. But these operations can be composed (it’s a group), and some products will leave the nature of the chords invariant and merely transpose them. For example, the operation $LP$ maps a major triad $n_+$ to a major triad $(n+8)_+$, and a minor triad $n_-$ to a minor triad $(n+4)_-$.
• The mapping of any of these three operations can be written in the general form $n_+ \to (n+p)_-$, and $n_- \to (n+q)_+$, with specific values of $p$ and $q$, with the additional condition that $p+q=12$ since these operations are involutions.

In fact, any operation in the $PRL$ group will be characterized by a couple of values $(p,q)$ such that $p+q=12$, and by a permutation $\sigma$ of the set chord types $\{+,-\}$, such that the corresponding mapping will be expressed by $n_+ \to (n+p)_{\sigma(+)}$ and $n_- \to (n+q)_{\sigma(-)}$.

We can now wonder if we could release the constraints on $p$ and $q$. This is precisely what Hook studied : he considered general transformations (UTTs) with mappings $n_+ \to (n+p_+)_{\sigma(+)}$ and $n_- \to (n+p_-)_{\sigma(-)}$ for any values $p_+$ and $p_-$ in $\mathbb{Z}_{12}$. Hook calls them triadic because, well, obviously they act on triads, and uniform because if a chord of root $n$ is mapped to a chord (possibly of a different type) of root $n+p_+$, then any transpose of the original chord (therefore of root $n+t$ for some $t$ in $\mathbb{Z}_{12}$) will be mapped to the same transposition of the chord of root $n+p_+$ (therefore of root $n+t+p_+$). In other words, any UTT commute with transposition. This restricts the possible permutations of the set of major and minor triads : it is possible to consider any permutation of this set of 24 elements as a potential musical transformation, but the group they would generate is enormous (of size 24! to be exact).

Hook introduced the following notation for UTT : a UTT is written $\langle p_+,p_-,\sigma \rangle$, with $\sigma$ being a permutation of the set chord types $\{+,-\}$, with the action therefore given as

$\langle p_+,p_-,\sigma \rangle (n_+) = (n+p_+)_{\sigma(+)}$, and

$\langle p_+,p_-,\sigma \rangle (n_-) = (n+p_-)_{\sigma(-)}$.

When dealing with major and minor triads, the permutation $\sigma$ is often written $+$ (the identity permutation), or $-$ (the other permutation of the set $\{+,-\}$). Thus, our usual neo-Riemannian $P$ operation corresponds to the UTT $\langle 0,0,- \rangle$, the $L$ operation corresponds to the UTT $\langle 4,8,- \rangle$, and the $P$ operation corresponds to the UTT $\langle 9,3,- \rangle$.

The UTT form a group under composition. It is a good exercise to verify that the product of two UTTs $\langle p_+,p_-,\sigma \rangle$ and $\langle q_+,q_-,\rho \rangle$ is given by the following formula.

$\langle p_+,p_-,\sigma \rangle \langle q_+,q_-,\rho \rangle = \langle p_{\rho(+)} + q_+,p_{\rho(-)} + q_-,\sigma \rho \rangle$

It is a group of size 288, and the reader versed in group theory will have recognized here the typical signature of a wreath product of $\mathbb{Z}_{12}$ by $\mathbb{Z}_2$, usually notated $\mathbb{Z}_{12} \wr \mathbb{Z}_2$. A wreath product is a particular case of a semidirect product $(\mathbb{Z}_{12} \times \mathbb{Z}_{12}) \rtimes \mathbb{Z}_{12}$. It is generated by the two UTTs $\langle 1,0,+ \rangle$ and $\langle 0,0,- \rangle$.

Now, it’s quite obvious that the neo-Riemannian $PRL$ group is a subgroup of Hook’s UTT group, but there are many other subgroups, and in particular many other subgroups with a simply transitive action on the set of major and minor triads. In this old blog post, I discussed about the possible group extensions of $\mathbb{Z}_{12}$ by $\mathbb{Z}_2$. As an alternative to the usual dihedral group $D_{24}$ behind the $\text{T}/\text{I}$ group and the neo-Riemannian $PRL$ group, we could consider the cyclic group $\mathbb{Z}_{24}$ which is generated by the usual transposition $z=T_1$ and by an inversion $h$, such that a presentation of this group is as follows.

$\langle z, h \mid z^{12}=1, h^2=z, h^{-1}.z.h = z \rangle$

These two generators corresponds to the two UTTs $\langle 1,1,+ \rangle$ and $\langle 0,1,- \rangle$ respectively, and thus $\mathbb{Z}_{24}$ is also a subgroup of the UTT group. As we will see later, musical transformations from this group show up often in transformational analyses of various pieces.

Notice that if we decide to use a group of musical transformations which has a simply transitive action on the set of major and minor triads, then for any pair of chords the group element which describes the transformation between them is by definition unique. On the opposite, when using Hook’s UTT group, there may be multiple possibilities since we now have 288 group elements for a set of 24 musical elements.

For example, let’s consider a very classic example that we already talked about, namely Beethoven’s 9th symphony, and in particular that passage at 2’29” in the second movement, which you can listen to in the video below.

We have already seen that this succession of chords (C major – A minor – F major – D minor – etc.) can be described with an alternating sequence of neo-Riemannian $R$ and $L$ operations. However, it can also be described by the repeated application of just one UTT, namely $\langle 9,8,- \rangle$, which is called the “mediant transformation”.

Another example, from Beethoven’s string quartet op. 18 n°6, in the fourth movement, in the passage which begins at 19’56” in the video below.

We have here a succession of triads which is elegantly described by the repeated application of the UTT $\langle 5,2,- \rangle$.

But perhaps one of the most striking example of the use of Uniform Triadic Transformation is given by Roeder’s analyses of Arvo Pärt’s music. Here’s the reference article :

• John Roeder, “Transformational Aspects of Arvo Pärt’s Tintinnabuli Music”, Journal of Music Theory, vol. 55/1, 2011, pp. 1-41, available here.

I will focus in particular on his analysis of Pärt’s Beatitudes, which you can listen to below.

Roeder divides the chord progression in group of three triads as follows :

• F minor – C# major – Bb minor
• F major – D minor – Bb major
• G minor – Eb major – C minor
• G major – e minor – C major
• etc.

Each group can be described by the repeated application of the $\langle 9,8,- \rangle$ UTT. But there’s more ! For each successive pair of groups, the triads of the second group are related to the triads of the first one by the UTT $\langle 1,0,- \rangle$. But there’s even more ! For each pair of successive pairs, the triads are related by the UTT $\langle 2,2,+ \rangle$. And thus we get this superb transformational diagram below.

How general is Hook’s framework of UTTs ? Well, if you look at the way we constructed the UTT and the corresponding group, you’ll notice that we encoded major and minor triads as pairs $n_s$ where $n$ is a root, and $s$ is a sign/type of chord. Thus, major and minor triads are considered as atomic, indecomposable entities, and their pitch-class content is not included in this consideration. In fact, you could replace the set of major and minor triads by, say, the set of all forms of the set classes [0,1,5] and [0,4,5] and the construction would remain identical. In the most general form, you just have to consider musical elements which you can bijectively identify to a product $(n,s)$ where $n$ is a group element from a group $Z$ and $s$ is a group element from a group $H$, and you can then consider the wreath product $Z \wr H$ as a group of UTTs acting on the set of these musical elements (even if we are not considering triads anymore, it has been the usage in the literature to speak about Uniform Triadic Transformations for such group elements). In his Ph.D. dissertation, and in the article presented above, Hook considers tetrachords from twelve-tone rows as atomic musical elements, whose transformations in Webern’s music can be described using UTTs.

Another well-known problem in transformational music theory is the question of extending the neo-Riemannian framework to seventh chords. I won’t detail here all the work that has been done in the past 30 years, but I’ll concentrate on a recent paper by Sonia Cannas :

• Cannas S., Antonini S., Pernazza L. (2017), “On the Group of Transformations of Classical Types of Seventh Chords”. In: Agustín-Aquino O., Lluis-Puebla E., Montiel M. (eds) Mathematics and Computation in Music. MCM 2017. Lecture Notes in Computer Science, vol 10527. Springer, Cham, available here.

In this paper Cannas considers five types of seventh chords, namely

• the dominant seventh, i.e. any set of the form $\{n, n+4, n+7, n+10\}$,
• the minor seventh, i.e. $\{n, n+3, n+7, n+10\}$,
• the half-diminished seventh, $\{n, n+3, n+6, n+10\}$,
• the major seventh, i.e. $\{n, n+4, n+7, n+11\}$, and
• the diminished seventh, $\{n, n+3, n+6, n+9\}$.

Notice that Cannas has to consider distinct diminished sevenths for each root $n$ in order to use UTTs : in other words, a C# diminished seventh is considered as distinct from a E diminished seventh even though their pitch-class content is the same.

Cannas then considers analogues of the neo-Riemannian operations for seventh chords, based on individual pitch-class change by one semitone or tone. For example, she defines the $L_{13}$ operation as the involution which exchanges a dominant seventh chord with a half-diminished seventh chord four semitones higher, as represented below.

But now, we have more than 2 types of chords, so what does the $L_{13}$ operation do to the other ? Cannas takes the decision that this operation leaves all other chords unchanged. Thus, in UTT notation, it can be described as $\langle 4,0,8,0,0,(1,3) \rangle$. In this case, we consider the permutation $(1,3)$ on the set of the five different chord types. In this way, Cannas defines 17 operations and studies the structure of the resulting group. The whole group of UTTs for the five different types of chords would be $\mathbb{Z}_{12} \wr S_5$, but it so happens that the operations Cannas defines are such that the transpositions in the corresponding UTT is always sum to 0 (mod 12). Thus, the group Cannas considers is a subgroup of $\mathbb{Z}_{12} \wr S_5$ isomorphic to $\mathbb{Z}_{12}^4 \rtimes S_5$ ($\mathbb{Z}_{12}^4$ corresponds to $\mathbb{Z}_{12}^5$ quotiented by the the subgroup of elements which sum to 0 (mod 12)).

I hope this post has made the notion of Uniform Triadic Transformation clear. On a computational note, you can practice transformational music analysis using Hook’s UTT group by hand, or directly use it from Opycleid, the Python package I developed, in which Hook’s group is already implemented.

1. One immediately wonders about the other two sevenths, the minor major {n, n+3, n+7, n+11} and the augmented major {n, n+4, n+8, n+11} – each the inversion of the other. But I suppose that question ought to be directed to Cannas et al.

2. M. A. Coury-Hall says:

Is the following statement correct from above? « For example, the operation LP maps a major triad n_+ to a major triad (n+8)_+, and a minor triad n_- to a minor triad (n+4)_-. » Doesn’t LP map to ? [cf. Table 1, p. 80, Hook, 2002] So LP maps a major triad n_+ to a major triad (n+4)_+. etc.

3. It depends on whether you are considering multiplication on the left (as I always do), in which case the operation LP maps n_+ to (n+8)_+ (by applying P first, then L), or multiplication on the right, as some authors do, in which case the operation LP maps n_+ to (n+4)_+ (by applying L first, then P).

4. Hi, Could I check that the semi-direct wreath product is meant to be written as (Z12 x Z2) x| Z12 ? Wreaths are new to me but the Z2 must be there in place of one of the Z12s?

Secondly, for (p+,p-,s).(q+,q-,r), assuming we apply q first before p, then isn’t it the sign of p that depends on the permutation r, not the sign of q? E.g. if we start with a minor chord, the translation is first q- then p(r(-)) ? Seems like just a typo, but thanks for clarifying as I’m trying to check if I understood it correctly.

5. About the second part of your comment, you are absolutely right, it was a typo. This has been corrected in the post. Thank you for informing me. As you can see from the structure of the group law, Z2 acts on Z12xZ12 by permuting the components, so the wreath product really is a semidirect product (Z12 x Z12) x| Z2.