Networks in transformational music theory (1)

For a long time, I promised a series of post on networks in music, and especially on our recent work, together with Andrée Ehresmann and Moreno Andreatta, on so-called ‘PK-nets’. So here is the first post in the series, in which I’m going to introduce the notion of Klumpenhouwer networks.

In order to do so, it is useful to look back at two great figures in recent music theory, namely Allen Forte (1926-2014),

and David Lewin (1933-2003) (Note: I do not know if this image is free of rights. Please contact me in any case) .

Allen Forte is particularly well-known for having introduced set-theoretic notions in the analysis of post-tonal music (though his work was preceded by previous theorists such as Milton Babbitt). His reference work is the following book:

  • The Structure of Atonal Music“, Allen Forte, New Haven & London: Yale University Press, 1973 (224p) – available on Amazon

The idea is to be able to categorize musical objects and describe the relations between them. In particular, Forte focused on pitch-class sets, i.e. unordered collections of distinct pitch classes (equivalence classes of notes modulo the octave), like \{A,C,E\flat,D\} for example.

Having pitch-class sets, one is then interested in relating them to each other. Among the many possibilities, we have already seen some in this blog, namely transpositions and inversions. For example, the pitch-class set \{C\sharp,G,G\sharp,A\},

and the pitch-class set \{G\sharp,D,E\flat,E\}

are transpositionally related, since the second is the transpose of the first by seven semi-tones. In a similar way, the pitch-class set \{B,E\flat,E,F\},

is inversionally related to the pitch-class set \{C\sharp,G,G\sharp,A\} through the element-wise inversion by I_0. Note however that, contrary to the previous posts on this blog, we are not speaking about transformations or group actions, but merely about relations between pitch-class sets, which can be more general. For example, Forte also considered what is known in music theory as the Z-relation, which indicates that two pitch-class sets are homometric, i.e. their interval content is the same. You can check for example that in \mathbb{Z}_{12}, the two pitch-class sets \{C,C\sharp,E,F\sharp\} and \{C,C\sharp,E\flat,G\} have the same interval content: they are homometric, or Z-related. The Z-relation is a complex object of study, and deserves its own series of posts, so I won’t detail it here today. The question of whether the Z-relation can be obtained by a group action has been studied by Amiot in a recent paper:

  • Z-Relation and Homometry in Musical Distributions“, Journal of Mathematics and Music, Vol. 5-2, pp. 83-98

Once a relation is defined, one can study the equivalence classes of elements of a set for this relation. Forte is known for having classified the equivalence classes of pitch-class sets, named set classes, for the transposition and inversion relations. Forte gave these set classes a code, the Forte number, but they can also be referred to by their prime form, which is informally the ‘most compact’ form of a pitch-class set.

For example, the above-mentionned pitch-class sets \{C\sharp,G,G\sharp,A\}, \{G\sharp,D,E\flat,E\}, and \{B,E\flat,E,F\} all belong to the same set class with Forte number 4-5 (prime form: [0,1,2,6]) for the transposition and inversion relations. The ‘4’ indicates that it is a tetrachord, and the ‘5’ that it is the fifth such set class. If one wishes to consider pitch-class set equivalence up to transposition only, then \{C\sharp,G,G\sharp,A\} and \{G\sharp,D,E\flat,E\} belong to the set class with Forte number 4-5A (prime form:[0,1,2,6]), whereas the pitch-class set \{B,E\flat,E,F\} belong to the set class with Forte number 4-5B (prime form:[0,4,5,6]), A and B being used to differentiate between inverted forms. The complete list of set classes can be found here, for example.

In this way, one can recognize set classes in a piece (especially in the case of atonal music), even when the actual pitch-class sets are transposed and/or inverted. For example, Webern made great use of set classes 3-3 and 3-2 in his music.

David Lewin introduced in the 80s his famous notion of Generalized Interval System, which introduced a framework for studying transformations between musical objects, based on group actions. I will not detail the mathematics of this part here, as I already did in the previous posts of this blog. The key point is that, as compared to Forte’s approach, we focus now on transformations and actions, and not merely on relations between objects. For example, with the appropriate setting, one can write T_2(\{C\sharp,G,G\sharp,A\}) = \{G\sharp,D,E\flat,E\}, to indicate that the T_2 transposition operation transforms the first pitch-class set into the second.

We have already seen the T_n and I_n operations, and the structure of the group (named the T\text{/}I group in the literature) they form. One can build an action of this group on the set of individual pitch classes, or on the set of pitch-class sets through its element-wise action. Obviously in this case, the action of the T\text{/}I group transforms pitch-class sets in transpositionally or inversionally related elements.

As Forte pointed out in his work, one would also like to compare pitch-class sets which are not necessarily related by transposition or inversions. Take for example the following extract of Webern’s Three Little Pieces of Cello and Piano, op.11/2.

The three segments of three notes form pitch-class sets which are clearly not related by any inversion or transposition, as can be seen below.

However, they share common characteristics which are apparent when one uses Klumpenhouwer networks.

Klumpenhouwer networks were introduced by Henry Klumpenhouwer in its doctoral dissertation in 1990, and later developed by Klumpenhouwer and Lewin. The reference papers on this subject are:

  • “Klumpenhouwer Networks and Some Isographies That Involve Them.”, Lewin, D.,  Music Theory Spectrum, Vol. 12 (1990), pp. 83–120.
  • “A Tutorial on Klumpenhouwer Networks, Using the Chorale in Schoenberg’s Opus 11, No. 2.”, Lewin, D., Journal of Music Theory, Vol. 38 (1994), pp. 79–101.
  • “The Inner and Outer Automorphisms of Pitch-Class Inversion and Transposition: Some Implications for Analysis with Klumpenhouwer Networks.”, Klumpenhouwer, H., Intégral, Vol. 12 (1998), pp. 81–93.
  • “Thoughts on Klumpenhouwer Networks and Perle-Lansky Cycles.”, Lewin, D., Music Theory Spectrum, Vol. 24-2 (2002), pp. 196–230.

The list is not exhaustive: in fact, the notion sparked considerable interest, to the point that a large part of an issue of Music Theory Online was devoted to the subject.

Informally, a Klumpenhouwer network is a directed graph such that

  • The vertices are labelled by pitch classes, and
  • the edges are labelled by transformations of the T\text{/}I relating the pitch classes of the source and target vertices,
  • composition between transformations being respected in the network.

It is interesting to note that no definitive definition of Klumpenhouwer networks exists, and that it can vary from one paper to another, which motivated in part the formal definition in the categorical approach we took with Andrée Ehresmann.

Let’s go back to the Webern’s example above, to see an example of a Klumpenhouwer network. The first three notes can be arranged in the following network.

Similarly, the other three-note segments can be arranged in the following networks.


We note that, even though these pitch-class sets are not transpositionally or inversionally related, they share the same organization of their pitch classes: namely, two of them are related by an I_8 inversion, whereas two others are related by an I_9 inversion, the T_1 transformation being obtained by composition. By showing the internal transformations between the constituents of the pitch-class sets, Klumpenhouwer networks allow us to see the similarity between these pitch-classes.

As said above, we often would like to be able to relate musical objects. In the case of Klumpenhouwer networks, we thus need a way to relate networks to each other. The above example is a particular case of Klumpenhouwer network isography, an important notion that we will refer to often in the following posts. First of all, recall that the automorphisms of the T\text{/}I group are given by the pairs (k,p) with k \in \{1,5,7,11\} and p \in \mathbb{Z}_{12}. The action of an element (k,p) \in \text{Aut}(T\text{/}I) on the elements of T\text{/}I is given by

  • (k,p)(T_n) = T_{kn}, and
  • (k,p)(I_n) = I_{kn+p}.

In the literature on transformational music theory, the usual notation for automorphisms of the form (1,p) is \langle T_p \rangle, whereas the notation for automorphisms of the form (11,p) is \langle I_p \rangle. Automorphisms of the form (5,p) or (7,p) are rarely used. With this mind, we define the following notions of Klumpenhouwer network isography.

  • Two Klumpenhouwer networks are said to be in positive isography, if the edge labels of the second are obtained through the transformation of the edge labels of the first by an automorphism of the type \langle T_p \rangle.
  • Two Klumpenhouwer networks are said to be in negative isography, if the edge labels of the second are obtained through the transformation of the edge labels of the first by an automorphism of the type \langle I_p \rangle.
  • The strong isography relationship is a particular case of positive isography, in which the applied automorphism is \langle T_0 \rangle., i.e. the identity on transformations. This is the case for the above Webern example.

Since automorphisms are themselves transformations, this means that ‘networks of networks’ can potentially be formed, wherein the vertices would be labelled by Klumpenhouwer networks, and edges would be labelled by isography transformations. Indeed, in its 1990 article, Lewin saw the potential of this recursive approach and described such networks of networks in his analyses. The end of the article (“Afterword”) is quite interesting in this regard, as Lewin tells us how he gradually developed and expanded on Klumpenhouwer’s ideas, first identifying the automorphisms of the T\text{/}I group as the key relations between networks (Klumpenhouwer had only considered positive isographies at first), and then realizing their recursive potentialities.

In the next posts, I will detail the approach we have taken to formalize Klumpenhouwer networks using category theory and expand them to more general settings.

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