This presentation is called "Transformational Music Theory: from group theory to film music analysis, and beyond..." I'd like first to thank Moreno Andreatta for inviting me to Strasbourg to give this presentation. This presentation is a bit of a strange mix : when Moreno invited me, he asked me to present both our collaborative work (together with Andrée Ehresmann) on the algebraic formalization of musical structures, and my work on film music analysis. At first glance, it might seem that there
is little overlap between the two... ...but we will see that there is an algebraic link between them that allows us to use our work on formalized musical structures, which uses group theory as well as category theory and apply it to film music analysis, on which it is particularly adapted. Before we delve into the subject, I'd like to propose a small musical quizz : I will play two musical extracts from two different film music scores and I'd like to ask you if you feel that these two audio ext
racts are similar either partially, totally, or if you feel that they have nothing in common at all. You could also try to guess from which movies these two audio cues are extracted. If you are seeing this video on YouTube, don't hesitate to take your time and pause, and even to play these audio files again. Here are the two musical files... So, do these two musical extracts feel similar in some way for way ? We will not give the answer right away, we will see it later in the presentation after
developing some necessary concepts... Before we begin the presentation itself, I'd like to introduce some notation and some basic concepts that will be used throughout. We will use equal temperament, meaning that we divide the octave in twelve equal parts, so we will not discuss here other tunings and temperaments. We will also assume octave equivalence, and thus we will speak about pitch classes: C, C#, etc. (12 of them) We will identify these pitch classes with elements of Z12, by arbitrarily
fixing C=0, C#=1, and so on. We could take any other pitch class as a reference, i.e. to be identified to 0, but usually C=0 in the literature. And thus, all mathematical operations in this presentation are assumed to be mod 12 by default. As hinted by the title of this presentation, we will use group theory and in particular group actions on sets and thus I will indicate the action of a group element g on an element x by either g(x) or g.x and by default, we assume a left action. It's not unusu
al to see right actions in the literature, I'm just used to left actions. Next, we will show chords graphically in Z12 as represented here. We have here a C major chord, with its constituent pitch classes C, E, and G, and we show it as a colored triangle, and similarly, we will represent a C minor chord as such. We will notate a major or minor chord as n_M or n_m n being the pitch class of the tonic of the chord, and M/m indicating whether the chord is major or minor. Thus, 0_M stands for C majo
r, 5_m stands for F minor, and so on. From the algebraic nature of Z12, we immediately have operations on pitch classes, which are of two types. The first ones are transpositions, notated T_k, for "transposition by k semitones" On a pitch class x, the action of T_k is T_k(x)=x+k. We can see here graphically the action of T_1, which is such that T_1(C)=C#, T_1(F)=F#, and so on... The second ones are inversions, notated I_k And on a pitch class x, the action of I_k is I_k(x)=k-x. We have represent
ed here graphically the action of I_3, which sends C# to D We can verify this quickly: C#=1, thus I_3(1)=3-1=2, and 2 corresponds to D. So we have I_3(C#)=D, and similarly I_3(C)=Eb, and so on... If you read papers on transformational music theory, you'll find many mentions of the "T/I group" This group is generated by the operations T_1 and I_0, and is isomorphic to the dihedral group D24 of order 24. Its presentation is as follows: it is quite obvious that T_1^12 is the identity, and all inver
sions are involutions, so in particular I_0^2 and we have this relation which is characteristic of the dihedral group. We have the following relations between transpositions and inversions transformations (and we will see them again later). We can notice that the T/I group does not act simply transitively on pitch classes, which is quite obvious since D24 is of order 24 and acts on a set of 12 elements, so it's impossible for the action to be simply transitive. We can extend the action on pitch
classes of the T/I group to the set of the 24 major and minor chords. You just need to apply an operation of the T/I group element-wise on the constituent pitch classes of the chord. For example, if we take a C major chord, i.e. the set {C, E, G}, we have an action of T_1 on C major by applying it on each pitch class, C is sent to C#, E to F, and G to G#, and we thus obtain a C# major chord. So we can write T_1(C major) = C# major. Similarly, we can apply inversions. Let's consider for example I
_0 applied on C major : we obtain the pitch classes F, G#, and C, i.e. an F minor chord. More generally, when applied on a chord n_M/m, transpositions T_k do not change the type of the chord, and return a chord of tonic n+k (mod 12). And when applied on a chord n_M/m, inversions I_k switch the type of the chord, and return a chord of tonic 5+k-n (mod 12). As before we can take T_1 and I_0 (acting on the set of major/minor chords, this time) as generators of a group... and the generated group, wh
ich we also call T/I (but it's now acting on major/minor chords) is also isomorphic to the dihedral group D24. We have the same presentation, and the same relations between transformations T_k and I_k. However, we have now a very interesting result: the action of this group on major/minor chords is simply transitive. This means that for each pair (x,y) of major or minor chords, there exists a unique element g of the T/I group such that g(x)=y. For example, we have I_2(Eb major) = E minor, which
we show here. (I_2 is the inversion along this axis, i.e. the C#-G axis) Or we could say that I_2 is the "interval", and I'm using quotes here because the notion of interval is usually defined between pitch classes (fifth, thirds, fourths, etc.), so I_2 could be said to be the "interval" between Eb major and E minor. We note in passing that we have here a curious object, a sort of labelled graph in which nodes are chords, and in which we also have labels on arrows, using elements of the T/I grou
p. We will come back later in the presentation on this kind of graph. We just introduced a notion of "interval" between chords, which is more general than the usual notion of interval between pitch classes, and that was the idea of David Lewin in the 80s, who initiated transformational music theory. David Lewin asked the following question: if we have a set X of musical elements which can be pitch classes, or chords, or time-spans, etc. how do you define a notion of interval between two elements
x and y of this set X ? To answer this question, David Lewin introduced "Generalized Interval Systems" (GIS) which he defines as a triple (X,G,int) where X is a set of musical elements, G is a mathematical group, the group of intervals for the GIS, and int is a function which takes two elements of X and returns an element of G, i.e. the interval between the two elements. We need some conditions to make it work: - for all x, y, z in X we must have int(x,y)*int(y,z)=int(x,z) (these are group elem
ents so we can multiply them). - and for all x in X, and g in G, there is a unique y in X such that int(x,y)=g. This definition of a GIS is a pragmatic one: the intervallic function int returns the interval between two musical elements, and the conditions make good sense: the first one says that intervals can be composed, and that it respects the group operation, and the second one says that for a given musical element x and a given interval g, there is only one other musical element which is at
the interval g from x. in the same way that given C and an interval of a fifth, only G is such that C-G is a fifth. It's a definition which agrees with the usual notion of interval between pitch classes, but is generalized to any set of musical elements. We have a direct link between Generalized Interval Systems and group actions, by the following proposition by Kolman in 2004 (also noticed by Lewin in his book) which says that the data of a GIS is equivalent to a simply transitive right action
of G on X. We have here the definition of a right group action, but it could be a left group action, we would just have to modify the multiplication order in this condition. What it also means is that for any two musical elements x and y, there is a unique interval g of G between them (i.e. g(x)=y) Before we continue, we take a peek into category theory A group action of G on X can also be viewed as a functor S from G to Sets. G in this case is considered to be a one-object category with morphi
sms being the elements of the group. The functor S from G to Sets sends the only object of G to the set X of musical elements. At first glance, it is not exactly clear what is the advantage of this categorical version; we will come back to it later in the presentation, and see what additional results this can bring us. I'm now introducing a new group, which we call the neo-Riemannian group PLR (sometimes also PRL) It is defined using three transformations. The first one is called P: if you apply
P to C major (C, E, G) you get the C minor chord, and vice-versa (P applied to C minor gives C major). This transformation preserves the fifth C-G, and switches the third (E to Eb, Eb to E) Then we define the L transformation: if we apply L to C major, we get E minor. In this case, we preserve the minor third (E-G). And inversely, if we apply L on E minor, we get C major. Finally, we define R such that R applied to C major gives A minor (we preserve the major third C-E), and inversely, R applie
d to A minor gives C major. More generally, we have the following relations: P is the transformation which switches the type of the chord, L applied on a major chord n_M gives a minor chord (n+4)_m and L applied on a minor chord n_m gives a major chord (n+8)_M R applied on a major chord n_M gives a minor chord (n+9)_m and R applied on a minor chord n_m gives a major chord (n+3)_M The transformations P, L, and R are involutions: when applied two times, you're back where you started. And these tra
nsformations are contextual : contrary to, say, transpositions T_k (which always give a chord of tonic n+k independently of the major/minor type of the chord), the tonic change with P, L, or R depends on the type of the chord. For example, with L, we have +4 if the chord is major, +8 if it is minor. The PLR group is generated by L and R, and is isomorphic to the dihedral group D24 (again !). Its presentation is shown here : LR is a transformation of order 12, and both L and R are involutions. P
is not included here, it can be omitted as a generator, since we have P=R(LR)^3, but we will often encounter it in musical analyses, as it has the same musical importance as L and R : as we have seen previously, just like L and R, it is a transformation which preserves two pitch classes and moves only the remaining one. And we have the following important result: the action of the PLR group on the set of major/minor chords is simply transitive. So it means we have a Generalized Interval System h
ere and we can use elements of the PLR group as intervals between chords so if we take two chords, whether they are major or minor, there is a unique element of the PLR group which transforms the first chord into the second. That element is the interval between the two triads. So far, I've presented two groups and their action on the set of the 24 major/minor chords, the T/I group and the PLR group. And these actions share a very important property, called Lewin's duality. Lewin's duality says t
hat the action of the T/I group and that of the PLR group commute. We thus say that these two groups are dual in the sense of Lewin. A word of caution : we are speaking about dual groups, but this is not the same as in pure mathematics, for example when speaking about Pontryagin's dual. Here we always mention "dual in the sense of Lewin" We have an example shown here : if we start with Eb major, and we apply I_2 we get E minor and by then applying R, we get G major. Now if we do this the other
way around, starting with Eb major and applying R first we get C minor, and by then applying I_2, we get G major again. This diagram is commutative, you can swap I_2 and R, the result will be the same. Lewin's duality applies to simply transitive group actions, and is directly implied by the fact that the left and right regular representations commute. If we have a left action of a group G on a set X, we can construct the dual group (in fact, the dual left action) in the sense of Lewin. In order
to do so, one needs to bijectively identify elements of X with elements of G, by choosing one representative x0 of X as the identity e of G. Then, any element of X can be uniquely written as h.x0, with h in G. You can then define a new left action of G on X as shown here, and it will commute with the original left action. This result is given by Thomas Fiore and Thomas Noll in the following paper. We come back a little bit on the categorical version that we saw previously... as Lewin's duality
can also be described using category theory. If we have a simply transitive left action, then the associated functor S: G->Sets is representable, i.e. a Hom functor Hom(.,-) If we now look at the natural transformations from S to S, we can apply Yoneda's lemma, and thus a natural transformation will be uniquely determined by an element of X, i.e. by an element of G (since X and G are in bijection) The set of all such natural transformations form a group which is isomorphic to G. It is precisely
the dual of G in the sense of Lewin, indeed, if we detail the definition of the natural transformation we have here the components of the natural transformation, i.e. the direct action of an element g of G and we get here the right action of G, which commutes with the left action, hence Lewin's duality. As I've mentionned before, Lewin's duality is crucial for transformational music analysis (especially film music), and now that we have all the necessary concepts, I'll give the answer to our mus
ical quizz. I will show the movie extracts from which the audio cues were taken. The first extract was from Tim Burton's 'Batman' (1989) It's the main theme of Batman. And the second extract was taken from Russell Mulcahy's 'The Shadow' (1994). So we have Batman's main theme, written by Danny Elfman, and The Shadow's main theme, written by Jerry Goldsmith five years later. And if you felt earlier that these two themes were similar, or very similar, you had the right intuition, at least at the ha
rmonic level. We can analyze them using the action of the PLR group : Batman's theme begins on B minor, then oscillates to G major via the transformation L, then from B minor to C# major via the composite transformation LT6, T6 being the tritone transposition transformation so we go from B minor to G major by applying T6 first, then L. And in the main theme of The Shadow, we start on C minor, then G# major and back to C minor via the L transformation then to D major via the LT6 transformation ag
ain, the theme then continues by semitone movement to go back to C minor (with a chord type change). We have here a direct example of Lewin's duality, as the harmonic progression of The Shadow is transposed by one semitone from that of Batman but since the action of T/I and PLR commute, we have exactly the same labels, i.e the same transformations in both cases. Had we used operations of the T/I group instead of the PLR group to find the intervals between chords, we would have had very different
labels between the two harmonic progressions. Rather than seeing in the main theme of The Shadow some sort of plagiarism of the Batman's theme, it is my opinion that it is instead an homage. The Shadow is a character which was created in 1931, and we know that it inspired the creation of the character of the Batman (in 1939, if memory serves) There are many similarities between the two characters: they fight crime during the night, The Shadow uses guns, which Batman also used at the beginning I
f you have any doubts about it, you just have to look at the crossovers in comics books, and this is why I consider this as a musical homage, given the similarities between the two themes. I think there are also homages in other music scores from other Batman movies, For the first two Tim Burton Batman movies, the music was composed by Danny Elfman, and for the next two movies ('Batman Forever' and 'Batman & Robin') the music was composed by Elliot Goldenthal. We have a new main theme, which is
different. Here is it : Instead of having L, L, and LT6, we have PL, then its inverse LP, taking us from C minor to G minor and back, and then RT6 instead of LT6. It thus gives a modified chain of neo-Riemannian operations, which recalls the original theme by Danny Elfman. And then we have the 2022 movie 'The Batman' with the music and main theme composed by Michael Giacchino. Here is it : What happens is quite interesting : the theme oscillates between F# major and Bb minor and the interval bet
ween these two chords in the PLR group is the transformation L. We thus have an L-mediated oscillation between these two chords, and it doesn't go further [SPOILER WARNING] and I'm going to spoil the movie a little bit: the character of Batman in that movie is not yet the Batman we know it is still a "character in construction", and this is why I think we never see the LT6 transformation, endlessly looping on the L transformation. Coming back to Lewin's duality and why it is important for film m
usic analysis, I'm showing here two harmonic progressions, the first one being C minor to G# minor, mediated through the neo-Riemannian transformation PL, i.e. applying L first then P. We can listen to this progression here : And the second progression is Eb minor to B minor, mediated through the same neo-Riemannian transformation PL, which is logical since the second progression is transposed from the first by a minor third and thus by Lewin's duality, we get the same interval in the neo-Rieman
nian group PLR. I will replay C minor to G# minor, and then Eb minor to B minor. Well, these harmonic progressions feel a bit like a cliché, and I chose to play it on the organ on purpose... we can imagine something evil or malefic going on. The neo-Riemannian operation PL has been associated through use to this feeling of evil/malefic. The PL transformation is also called the 'Tarnhelm' transformation, from Wagner's tetralogy and in particular the first opera 'Das Rheingold'. I'm going to show
the specific moment to which it corresponds. In 'Das Rheingold', the Tarnhelm is a magic helmet forged from the cursed Rhinegold, and which allows one to shapeshift. We will see how the 'Tarnhelm' transformation is used in this extract, and you will see a countdown before it happens. And another typical example of 'Tarnhelm' transformation can be found in the Imperial March in Star Wars. In Rogue One, we have something reminiscent of the Imperial March, with another use of the 'Tarnhelm' transfo
rmation. There are countless other examples : for example in the TV show 'Buffy the Vampire Slayer' you have a 'Tarnhelm' transformation in almost each episode. There are also many occurences in the trilogy 'The Lord of the Rings'. You can watch the rest of this YouTube video, in which I compiled many examples of 'Tarnhelm' transformations in film music. And so, because of Lewin's duality, the same neo-Riemannian operation occurs independently of transpositions thus giving the same feeling. And
this is why the neo-Riemannian group PLR can successfully be used for film music analysis, or at least a specific style of film music called 'pantriadic chromaticism' and I urge you to check out Frank Lehman's book 'Hollywood Harmony', which is simply fantastic. In his book, Frank Lehman defines 'pantriadic chromaticism' as an harmonic organization dominated by major and minor triads and characterized, among other things, by 1/ non-diatonicity meaning that pitch classes do not belong to a diaton
ic or quasi-diatonic set, like a usual major scale, 2/ non-centricity, meaning that the harmonic progression does not refer univocally to a single key beyond a few chords and 3/ non-functionality, meaning that it doesn't use the known diatonic functional process such as fifth motions, cadences, etc. and instead new functional processes will be used. Among these functional processes, we find what Lehman calls 'absolute progressions', harmonic relations between chords that should be considered by
themselves independently of any tonal context. The 'Tarnhelm' transformation is such an example, associated with a specific feeling. Often these absolute progressions are highlighted by harmonic oscillations. We have seen such an example in Batman's main theme, mediated by the L transformation. These oscillations are often repeated. There are other functional processes that I will not detail here, but you can read about it in Lehman's book. I will only focus here on absolute progressions. I will
give a few examples, and I'd like to begin with the T6 transformation, i.e. the tritone transposition. The tritone may evoke scary stuff (it has this 'diabolus in musicae' tag attached)... but we really have to demystify it, and I urge you to watch the fantastic YouTube video about it by Adam Neely. If I had to give only one example, it would be the opening credits of The Simpsons. A superb example of a tritone. I didn't take the intro of The Simpsons at random, by the way... ... the main theme
has been composed by Danny Elfman, hence the link to the earlier Batman theme. This is a melodic tritone, though, and I'd like to speak instead about the tritone transposition applied on chords. There are two versions: the major T6 transposition, for example applied on D major (D, F#, and A) gives the G# major chord and we can see that the T6 transposition has many interesting properties : first we can see graphically that this is a central point inversion (D sent to G#, Eb to A, etc.), we have
then the notion of tritone itself, the largest interval which can be found between pitch classes, but at the chord level, we end up with a chord for which two pitch classes have moved by one semitone only, and a pitch class which goes the full way, from C to F#. We also have a minor T6 version, since the T6 transformation does not change the type of the chord, so if we apply it for example on F minor (F, G#, C) we get the B minor chord. We can see the same properties as before : a central inver
sion, but also a parsimonious motion for some pitch classes These characteristic properties may evoke many things: something different/distant because of the tritone interval but also something close, because of the parsimonious motion of some of the pitch classes. And I encourage you to check out these references, Lehman's book of course, in which he discusses T6 but also this paper by Scott Murphy, in which he analyzes the use of major T6 in sci-fi movies This paper can be freely accessed on t
he Internet. In his paper, he links the specific properties of the T6 transformation (distant/close) with the context of sci-fi movies. I'd like to show you a few examples, from a compilation I made of T6 uses either in classical/opera music or film music Here is it. These examples are available on my YouTube channel, and there are many more. Same as before, you will see countdowns to occurences of T6 transformations. As we have seen, there are two version of T6 (major/minor), major T6 often bei
ng used in grandiose settings as in the sci-fi movies, whereas minor T6 is more associated with evil/malefic, for example with the Ark of the Covenant, or with the emperor in Star Wars (hem... a small technical glitch with videos here) so we can have two different feelings associated with the T6 transformations. As another example of neo-Riemannian transformation being used in absolute progressions, we have the SLIDE/S operation it can be considered as a composite S=RPL of the basic neo-Riemanni
an transformations or as an atomic/indecomposable transformation, and its action on major/minor chords is such that it sends a minor chord to the major chord whose tonic is one semitone below, and vice-versa, since S is an involution. Its use in film music is relatively recent, say, from the 90s onward, Frank Lehman has studied its use in the representation of genius in film music, and I'd like to show you some other examples. It is often associated with mystical/supernatural events. (only a few
examples, otherwise it would be too long) You'll find other compilation videos on my YouTube channel, in particular, one with S and T6 examples. Yet another example of neo-Riemannian transformations: the F and N transformations. They can both be viewed as composite transformations, F being equal to PRL, and N being equal to PLR, or as atomic transformations, especially when used in absolute progressions. F sends a major chord to a minor chord whose tonic is 7 semitones above, and vice-versa sin
ce F is an involution N sends a major chord to a minor chord whose tonic is 5 semitones above, and similarly, N is also an involution. And I refer you to the recent Ph.D. dissertation of Daniel Obluda, in which he analyzes the use of F and N in which he shows that they can be associated with feelings of grandiose, or wonder, even heroism/adventure. I'd like to show you a few examples of F transformations. Again, everything is accessible on the YouTube channel. We have seen uses of F; as for N, y
ou can watch them in the same video. We can summarize all these transformations in what I call the "Transformation Tonnetz" The Tonnetz itself is a geometrical space built from major and minor chords, each node is a pitch class, and thus triangles are chords, for example here a C major chord (C, E, G) and you stitch triangles (chords) by their common pitch classes. For example here, C major, and E minor, which shares with C major the two pitch classes E and G. And thus we build a geometrical spa
ce, which has the same topology as a torus, you can verify that you can stitch the upper-lower, and left-right edges, and in which the neo-Riemannian transformations P, L, and R play a role, since they each preserve two pitch classes. Since the PLR group acts simply transitively on the set of major/minor chords, instead of labelling triangles with chord names, we can label them, after choosing one chord as a reference (i.e. corresponding to the identity of the PLR group), we can label them with
elements of the PLR group. This is what I've depicted here. only for involutions (transformations which change the type of chord [NOTE: not exactly, see later]) We have the three basic neo-Riemannian operations P, R, and L. You have here the transformations that we have seen previously, S, F, and N. Here, the T6 transformation which *is* an involution, but doesn't change the type of the chord. And around it, other transformations which I labelled PT6, LT6, and RT6. And two involutions which I th
ink don't have a name in the literature, which I call X and Y. I will not detail them here. So we have a summary of neo-Riemannian transformations in this "Transformation Tonnetz". I would like now to come back to the mention I made earlier in the presentation, when I said that we have this mathematical object with chords on nodes, labels on arrows, and indeed, so far I've presented musical analyses using labelled graphs. For example here, the main theme of the Shadow, with chords on nodes, and
PLR operations on arrows. This is a good time to speak about the research work done in collaboration with Moreno Andreatta and Andrée Ehresmann on the mathematical formalization of transformational networks. In the 90s, Henri Klumpenhouwer and David Lewin developed the so-called "Klumpenhouwer networks" mainly to study pitch classes and their relations. Informally, these are directed graphs, with pitch classes on the nodes and with arrows labelled with elements of the T/I group (acting on pitch
classes). such that the labels are compatible with the node contents, and that the composition of labels is respected. We have an example here of such a Klumpenhouwer network. I_2 is indeed the transformation such that I_2(E)=Bb, T_9 is also the transformation such that T_9(Bb)=G, and the composite T_9*I_2 is indeed I_11, such that I_11(E)=G. It seems there is no universal definition of Klumpenhouwer networks, or transformational networks in general David Lewin was one of the first to propose a
definition for transformational networks with a series of definition and conditions ensuring that the node and arrow labels were compatible with respect to the (semi)group action and that composition of arrow labels was respected. Other authors have sometimes added additional conditions, which are not always compatible between authors. In transformational networks, the necessary composition of arrow labels is strongly reminiscent of category theory, in which morphism composition is a crucial poi
nt of the definition of a category itself. Our work was thus to formalize transformational networks using category theory. We will generalize what we've seen previously, by replacing the group G with a more general category C together with a functor to the category Sets. If you look at the definition of a Generalized Interval System, it's very close to the definition of a category. Among the many definitions of a category, one can define it as a monoidal object in the category of quivers, on a f
ixed node set which is almost what we have with the definition and conditions on the intervallic function int in a GIS. In our work, we proposed a definition for "Poly-Klumpenhouwer networks" (PK-Nets) which generalize Klumpenhouwer networks The definition is given by the following diagram : we have a category C and a functor S: C->Sets we also have a (small) category Delta and a functor R: Delta->Sets and a PK-net is defined by a 4-uple (R,S,F,phi) with F being a functor from Delta to C, and p
hi is a natural transformation from R to SF. The category C and the associated functor S generalize group actions it will be the 'transformational context (or framework)' for the analysis. and the category Delta and the associated functor R represent the backbone of the transformational network. This allows us to have more general network structures as compared to Klumpenhouwer networks, we have an example here where each node is a set of pitch classes, and not just a single pitch class. We have
here C and Eb transformed by T6 into F# and A and we can even change the cardinality of the sets, as we inject these two pitch classes into that set so I_11 sends F# and A to F and D respectively, and we have an additional pitch class Bb and the composition I_11*T6 is indeed I_5. This representation is a simplified version of the diagram I presented on the previous slide, If we detail it, we have Delta, a category with 3 objects and 3 non-trivial morphisms between them, The functor F gives labe
ls to these morphisms in the T/I group, so that f is mapped to T6, g to I_11, and the composite gf to I_5. The functor *R* maps these objects to sets, two with cardinality 2, and one with cardinality 3, and by the definition itself of a natural transformation, and that's the beauty of it, we can be sure that the action of the elements of the T/I group is respected for example that C is indeed sent to F#, and that F# is sent to F, and that composition is respected. We see here the advantage of us
ing category theory as the definitions of functors and natural transformations directly ensure that composition and actions are respected. As we talked about graphs and quivers before, we can notice that a PK-net is almost a morphism of graphs, We have here the diagrammatic definition of a PK-Net and if you take the categories of elements of the functors R and S this one over Delta, and this one over C, then the definition of the functor F and the natural transformation phi gives us a functor be
tween the category of elements which makes this diagram commute. You can then take the forgetful functor to quivers, and you will get a morphism of quivers. Note that by doing so, you're losing the notion of composition of morphisms in a category. With this category-theoretical formalization of transformational networks, we can thus represent musical analyses with labelled graphs (their simplified form), as shown here. I'm coming back to film music, and I'd like to focus on what I call "near-T6"
transformations : these are the PT6, LT6, and RT6 transformations which are shown here, obtained by applying T6, then one of the basic neo-Riemannian operations P, L, or R. Since P, L, and R preserve two pitch classes out of three, the "near-T6" transformations applied on chords will move only two pitch classes out of three by a tritone, hence their name. We see them in the "Tonnetz of Transformations" which I presented earlier, on which I added some information... I've colored in red, the chor
ds which share at least one pitch class with the reference chord, so with our reference chord here, all these chords share at least one pitch class with it. And then I've colored in yellow the chords which belong to the diatonic scale defined by the reference major chord. So if our reference chord is C major, chords belonging to the C major scale are C major, D minor, E minor, F major, etc. in yellow here So we see here that the "near-T6" transformations, along with T6, and the H transformation,
I haven't talked about the H transformation, but you can check out the compilation videos on the YouTube channel, well these transformations bring us into a peculiar place in the Tonnetz, which share nothing in common with the reference chord, neither on pitch classes, nor in a diatonic way. We find examples of "near-T6" transformations in film music, and I've made a compilation of them, they are often found in magical/mystical/supernatural settings, although a more thorough examination of thei
r use is probably needed. Here is it. So far, we've seen examples of RT6, I'll now show examples of LT6 I won't have time to show all of them... I'd like to focus on some specific examples in which the "near-T6" nature of these transformations may have a close link with the context of the movie. The first example is the birth of Dr Octopus in Spider-Man 2 (2004). First, I'm going to show you the movie extract, we'll then see what happens in the harmonic progression. Watch how chord changes and s
cene cuts are related. The harmonic progression is shown here. We can see that RT6 happens at a particular moment during the "creation" of Dr Octopus. It's the moment when interfacing with the robotized arms begins, and when they begin to shiver. At this moment, the failsafe chip is still there, so we are not facing yet the super-villain Dr Octopus, and thus, instead of using for example a full T6 transformation, hinting at Otto Ottavius evil alt identity, we have here a "near-T6" transformation
(RT6 in this case) which anticipates the super villain to come. Notice the final F transformation, usually associated with wonder/grandiose, which is the case in this scene. We'll now see another example of "near-T6" transformation, which we can associate with a near-transformation into a super-villain. And we'll show also how neo-Riemannian analysis and tonal analysis can complete each other, a point studied by F. Lehman in his book "Hollywood Harmony". We're going to look at the scene of the
birth of Sandman in Spider-Man 3 (2007). The scene begins in C# minor, and two neo-Riemannian operations are used at specific moments. The first one is the transformation L, taking us from C# minor to A major, and which corresponds to the first failed attempt of Sandman to take form, you can see him collapsing. The second one is LT6, taking us from C# minor to Eb major, with a final resolution on G# major. And we have to look at the role G# major plays in this progression. This chord happens at
the beginning, but in the context of C# minor, it doesn't really fit, one would expect a minor chord on the fifth scale degree, but we get a major one G# major doesn't really belong there. It's the transformation LT6 that allows us to go to Eb major, and to end this part on an authentic/perfect cadence V-I, for which G# major is a first scale degree major chord, and you can notice that Sandman expires at this moment. One could almost say that it is the transformation LT6 which gives birth to San
dman. Now I'd like to focus on another point, namely the composition of musical transformations same as morphism composition in category theory. If you take this harmonic progression, i.e. F major to F minor via the transformation P, then F minor to C# minor via PL, i.e. the 'Tarnhelm' transformation, then you have, between F major and C# minor, a PLP transformation which is the hexatonic pole transformation H. This latter operation perhaps won't be perceptible if you hear these chords in progre
ssion, but it's there (some would say hidden behind). As an example of such composition of transformations, I'd like to show a scene from Dr Strange 2 and I have to warn you about massive SPOILERS ahead : feel free to skip this part if you have not seen the movie. It's a scene in a parallel universe in which Dr Strange faces a memorial statue of himself. Here is this scene. This scene contains the two Dr Strange main themes. Danny Elfman composed the score of Dr Strange 2, while it was Michael G
iacchino who composed the score for the first Dr Strange movie. We hear Elfman's main theme, then Giacchino's, then Elfman's again. The harmonic progression goes from F minor to E major via the transformation SLIDE/S, then to B minor via the transformation F, then to G major and Eb minor, via L first and then H. The use of S is quite adapted : we have this strange statue in a completely different universe, hence the supernatural feeling, The use of F works well too given the 'wonder' feeling whe
n facing this statue. And the use of H... (I haven't talked much about it in this presentation) (again, check out the examples on YouTube, as well as Richard Cohn's paper) H is associated with feelings of "uncanny", which is the case here when we see Strange facing a statue of himself. And that gives us this initial harmonic progression. Now, in the neo-Riemannian group PRL, we have the following equation between S and F S equals F times T6 (S=FT6). Thus, hidden in this harmonic progression, we
have a T6 transformation between F minor and B minor, and similarly, hidden between L and H, we have LP (the inverse of the 'Tarnhelm' transformation) It's a minor T6 transposition, and we've seen it associated with evil/malefic events, and the 'Tarnhelm'-like transformation is also associated with such things. Warning, spoilers : The memorial statue evokes the actions of an heroic Dr Strange in this universe, but we later learn in the movie that it hides a lie, and that the Dr Strange of that u
niverse had been corrupted by the Darkhold. Hence the link we can make with these T6 and LP transformations. Let's go back to some more mathematical aspects of transformational networks as we formalized them. We've seen how we can give a category-theoretical definition of such networks, and we'll now see how we can transform networks, i.e. a notion of morphism of transformational network. The definition of such a morphism is given here, it may look a bit complicated, We call it a "PK-homography"
between two PK-Nets (R,S,F,phi) and (R,S',F',phi'), so we assume they share the same functor R. A more general definition exists, but let's focus on this one. We need two pieces of information to define it : 1/ a functor N from C to C', such that F'=NF. N will transform the labels on the arrows of the transformational network. 2/ and a natural transformation nu from SF to S'F' such that phi' equal nu times phi. This natural transformation will map the musical elements. We call it a "PK-isograph
y" if N is an isomorphism, and nu is an equivalence (an inversible natural transformation) so the PK-isography itself is inversible. Among PK-homographies, we'll look at specific types, the first one being "complete PK-homographies". These are PK-homographies in which nu: SF->S'F' can be written as nu=\tilde{nu}F, with \tilde{nu} being a natural transformation from S to S'N. These are homographies in which the whole transformational context, i.e. the functor S: C->Sets, is transformed all at onc
e. We have then the following result if C is a group G, and S is representable, the group of complete isographies, i.e. the automorphism group of the functor S, is isomorphic to the holomorph of G (the semidirect product of G and Aut(G)). In this group of complete isographies, we have a normal subgroup isomorphic to G, corresponding to PK-isographies of this type, and this is the dual group in the sense of Lewin of G, in a more general setting, since there are more automorphisms of the functor S
. Let's go back to 'The Shadow' for an application of these complete PK-isographies, in particular on the leitmotif of Shiwan Khan, the villain in 'The Shadow'. The first occurence is in the following scene : We have an harmonic progression Eb minor, E major, G minor, Eb minor and the transformations between them : LRL, RT6, and PL. Further occurences of the Shiwan Khan leitmotif are different, though they are all built on the same transformational chain. Here we have C# minor, A major, F minor,
C# minor. with the associated transformations L, H, and PL. One can show that we have two distinct PK-isographies between these transformational networks, i.e. we can have a morphism of networks between them, for example via the PK-isography sending L to R, and R to RLR, (and the mapping of the remaining neo-Riemannian operations can be deduced from this group automorphism) and which has a natural transformation sending major chords to major chords whose root is five semitones above, and sendin
g minor chords to minor chords whose root is two semitones below. This musical analysis is incomplete, because one would need to show how this PK-isography is related to the content of the movie. But at least, on the mathematical side, there is a morphism between the two networks. Another type of PK-homography / morphism between networks: the local PK-homographies. They are such that the natural transformation nu: SF->S'F' can be written as nu=S\hat{nu} with \hat{nu} a natural transformation fro
m F to F'=NF. It means that you are now transforming networks by a local transformation of their nodes. I won't give all the category-theoretical details here. If we take the two Shiwan Khan leitmotives from before, the top one can be transformed by a local PK-isography, on the first node from Eb minor to C# minor with a neo-Riemannian transformation RLRL, on the second node from E major to A major with the transformation LR, and similarly on the remaining nodes, in both case with the transforma
tion RLRL. One can notice that this local homography seems regular with respect to the transformation on the nodes, but again this analysis is incomplete as once again, we need to relate this to the content of the movie. To conclude this presentation, I'd like to offer some perspectives, based on the category-theoretical approach we saw, So far, we've employed group actions, and particularly the action of the PLR group on major/minor chords, and we've seen that we can recast this as a functor fr
om G to the category Sets. What can we do from here ? A first idea would be to change the group the PLR group is isomorphic to D24, which is an extension of Z12 by Z2, it's not the only possible extension of Z12 by Z2, there are others which I studied in a 2013 paper. So we could use them to analyze harmonic progressions. A second idea would be to use a category C instead of a group G, for example a groupoid, a subject we worked on together with Moreno Andreatta and Andrée Ehresmann in a 2019 pa
per. This can be useful if you wish to add other types of chords beyond major and minor, for example diminished chords, And finally, we have used the category Sets, but we could change this as well, we could use the category Rel of sets and binary relations between them, with a quantale Q we could use the category Rel of sets and Q-valued relations between them, or even the category Vect of vector spaces. The 'Cube Dance' is an example of the use of binary relations. It's the graph for the so-ca
lled parsimonious relation P_1,0 introduced by Douthett and Steinbach. The major and minor chords are shown here, as well as the augmented chords. Chords are related by P_1,0 if they differ by only one pitch class moving by one semitone. For example, between C major and C minor, only the pitch class E moves to Eb, and thus they are related by P_1,0. Same for C major and E minor, only the pitch class C moves by one semitone to B. We can see that G# augmented is related to six chords by P_1,0 Thus
, functions between sets are not enough, and we have to use binary relations Together with Moreno Andreatta and Andrée Ehresmann, we showed that we could generalize our categorical definition of networks to the case of binary relations, with some additional conditions. Another example is the work on metrical networks, together with Jason Yust, in which we are interested in meter and metrical relations (so we're not working with chords anymore, but with temporal relations) We have defined metrica
l relations as specific binary relations between timepoints/attack points and we've defined the notion of metrical networks, giving us information about how a piece is organized temporally. We have applied this to Brahms' music. Thank you for your attention ! For those interested, more examples of uses of neo-Riemannian transformations can be found on my YouTube channel.
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