The Hexagon of Opposition in Music


François NICOLAS (Ens-Cirphles)


(Translation by Liam Flenady)





In what ways can the hexagon of opposition formalise what the musician calls “musical logic” and by doing so aid him in better understanding what “negation” and “deduction” mean for the music at work?

To respond to this question, this presentation will propose successively three ways of appropriating this hexagon for musical realities: the first – briefly outlined – will depart from the diatonic/chromatic opposition; the next – sketched with more detail – will touch upon the rhythm/timbre opposition. Finally, the third – largely developed – will undertake to formalise properly musical discursivity by orienting it according to three principles. These in turn will be counterposed term by term to the great logical principles of Aristotle: a principle of ambiguity (as opposed to the principle of identity), a principle of constrained negation (as opposed to the principle of non-contradiction) and a principle of the required middle (as opposed to the excluded middle).

These three principles configure the musical composition as an interaction between three “entities” (an initial entity, its alteration and an entity of another type), an interaction that this presentation will undertake to formalise in the form of a hexagon of opposition. The detailed interpretation of this will suggest in return that the development of musical discourse (that which takes the place of musical “deduction”) operates as a Borromean knotting of three forms of alteration (musically taking the place of “negation”).




I would like to reflect with you upon different ways of appropriating the hexagon of opposition for musical questions.

What are the musical oppositions that this hexagonal formalisation is capable of theorising and what in way can this kind of formalisation clarify a musical model in return?

I will make use here of the terms model and theory in the mathematical understanding that they have in the “theory of models”: the model here is the original that it is a matter of copying, the “canonical” model; the theory is that which formalises this model and in return gives rise to interpretations in the initial model.

The words are thus employed here in their original sense, which a neo-positivist trend has unfortunately reversed (where “model” then designates “the reduced model,” or “the scale model”).


In itself, music does not thematise what musical opposition means. Music presents itself in notes and in sounds, at a distance from all speech. The words “opposition” or “negation” do not take part in its repertoire, and the score does not know the signs “→” and “=/≠”.

The score simply mobilises the identity and difference of the signs of solfege: a musical sign will differ from another, either by its form (# and ♪), or by its place.

Thus music mobilises its own principle of differentiation in order to play with resemblances and dissemblances, similitudes and differences.

In the vocabulary of Alain Badiou (Logics of Worlds), we will say that music has its own transcendental that constitutes it as an autonomous world. This transcendental is solfege and the nominal categories of opposition do not belong to solfege…

In no way is all difference a matter of “opposition”: two musical objects (apprehended by writing or by the ear) could be different without for all that being opposed. Opposition is a particular relation, whereas difference remains a static relation that emerges as an exterior rapprochement rather than an endogenous process. Opposition is a dynamic and endogenous relation. Difference is a static and exogenous relation.


In other words, in order to identify the play of musical oppositions, it is necessary to have something other than the score alone; it is necessary to bring to the scene the musician, the one who executes and interprets the score: it is he in fact who will be able to declare and above all to effectuate (by playing!) the reality that such objective musical difference is a matter of opposition or not, and possibly negation.

It is the musicianly interpretation that gives a logic to the score, that renders it effective, that is to say sensible.


Thus I will here outline, as a musician, how to my eyes – and above all to my ears! – music puts to work oppositions, and how these oppositions can be formalised according to a logical hexagon that gathers us together here.


In order to do this, I will propose to you three attempts to appropriate our hexagon of musical oppositions in play.

·   The first is an attempt that will prove to be abortive since is will not be possible to stabilise for it a genuinely musical sense.

·   The second will bear upon the local organisation of material: it will be structured by the opposition of rhythm and timbre.

·   The third will bear upon the global development of musical discourse: it will pivot around the great opposition between variation and invariance.

1. A hexagon opposing tonal/atonal, major/minor, diatonic/chromatic

Let is depart from two contradictories tonal/atonal (a musical opus partakes of one or the other, if we understand by “atonal” that which is not tonal and not uniquely that which was inscribed under this name between 1909 and 1923: let us say that “tonal” operates here as a common name, not as a proper name).

The tonal could be seen as the sum of major and minor, which constitute its contraries. The atonal, for its part, is the product of the subcontraries diatonicism and chromaticism, which thus complete the hexagon:




The hexagon thus produced remains however unsatisfactory for at least three reasons:

·   the contradictions major-chromatic and minor-diatonic are not musically equivalent to the first contradiction tonal/atonal: the latter is strong, the formers are weak…

·   the subcontraries of the triangle diatonicism-chromaticism-tonality are forced: if one can maintain that a musical proposition can not be at the same time neither diatonic, nor chromatic, it is more difficult to maintain that it would not be able to be at the same time neither diatonic, nor tonal.

·   it lacks, in this formalisation, a usual term that would form a contrary or subcontrary (depending on the way in which one understands this musically) with tonal, and that is the term modal.

In sum, one gets the sense of forcing the musical proposition in order to arrive at inscribing it in an inadequate formal framework.


Then, a better hexagon will be this one, which leaves the contrary of major and minor in order to introduce the contradiction of modal and non-modal. We can see that the opposition diatonic-chromatic is now, more logically, a contradiction (and no more a sub-contrary!), that the tonal is now the product of diatonic and non-modal (and no longer the sum of major and minor) and – last but not least- that our hexagonal figure is now well-structured by the triangle of contraries {modal, tonal, chromatic}.



But let us leave this first attempt and examine another way of musically approaching the hexagonal formation.

2. The Hexagon of Rhythm and Timbre

Let us depart this time from the classical contradiction between rhythm and timbre, a contradiction that we could configure equally as that of the melody and the chord: the melody is horizontal and materialises a rhythm (understood as a diachronic series of durations); the chord is vertical and materialises a timbre (understood as synchronic series of intervals).


Each of these terms – melody as much as chord – will give rise to two complementary forms: on one side, the rhythm of the melody gives rise either to a harmonic rhythm of successive chords, or to a polyphonic rhythm of superposed voices; on the other side, the timbre of the chord gives rise either to a harmony understood as a suite of chords, or to a counterpoint understood as a superposition (as “formants”) of voices.

If one divides this out amongst the six summits of the hexagon (disposed here horizontally rather than, as is customary, vertically),[1] we obtain the following hexagonal formalisation:



This formalisation sets in relief two new classical contradictions, on the one side, that of the chorale and counterpoint, and on the other side that of harmony and polyphony.

This hexagon can be divided into two trinities: that of the contraries (the chorale, polyphony and the timbre-chord) and that of the subcontraries (harmony, counterpoint and the rhythm-melody).

We see how the six summits “converge” upon a central object that figures here as an integral musical meshwork.


In sum, this formalisation is internally stable: the figuration proposed for each of the terms by way of points and dashes attests to its formal pertinence. We will consider it thus as musically acceptable.

It has, however, one defect that is musically unacceptable: this formalisation is essentially descriptive of the musical oppositions that are inscribed here. It remains static insofar as it teaches us nothing more than what we already knew (precisely what we needed to know in order to construct the hexagon).

A remark

However, we will remark that this formalisation suggests a refolding since certain objects, which were presented above without them belonging stricto sensu to our hexagon, prove to be reliable by connections (drawn above in thin lines) that we summarise here in the following schema:



Thus this would suggest another hexagonal disposition that would better take account of the relations of implication (those of the periphery), but less well of the transversal relations (those of opposition):



Here we come to a point that, in this type of formalisation, proves to be difficult for us musicians: it conjoins a formalisation of oppositions (by unoriented lines: contradictions in red, contraries in blue, subcontraries in green) and a formalisation of implications (by lines oriented in black).

We find in our example the properly logical difficulty that is our own: it does not go without saying that, musically, a musical regime of consequences (arrows in black) and a musical regime of oppositions (lines in colour) are strictly correlated.

·   The musical regime of consequences mobilises in general the figure of the chronological (the before and after, that which precedes and that which follows) but in an extremely open manner: it is not because B musically follows A that A→B (that is to say, that B develops A); and inversely, it is not because B develops A that B necessarily precedes A.

·   For its part, the musical regime of oppositions mobilises the very difficult question of the negation in music: what is it that could give a musical value to negation, knowing that the logical sign of negation does not exist in music, that all music, by definition, poses an existence, affirms a presence (in music, even silence is an affirmation: musical silence is not the negation of all sound; it is the affirmation of a musical duration like any other, written and thus counted as itself, not as zero 0 or the void set ø). Thus no music is able to negate without affirming a negative. We could say that in this sense music ignores the double negation since it cannot negate, as part of its very constitution, a negative: music can affirm a negative (a contrasting object, a different or altered one) or not affirm a positive (not develop, not repeat, not alter an object presented) but it is not able to effectuate a double negation.


Hence the interest for us in turning finally to a third hexagonal formalisation more forward looking and in the end more instructive for us musicians, a formalisation that will depart this time from the question: what is the negation in music? Or better: what does it mean to musically negate?

3. The Hexagon of musical discourse

For this we will take up again what the question of what musical discourse means.

Preliminary remark

Logic is realised in music in three ways – or rather: there are three understandings of what “musical logic” means.

·   First of all there is the logic of writing. This touches upon this dimension of logic that Alain Badiou calls “transcendental”: music is endowed with a solfege – that is to say, a proper mode of writing – which is what takes a properly musical measure of sonic existences. I will not attempt today to formalise this fundamental dimension of musical logic (in my jargon I call it “the scriptural logic of the Music-world”) according to our principle, the hexagon of opposition.

·   Then there is “the discursive logic of the opus of music.” It is this that we will now explore in more detail in order to attempt to formalise it according to our hexagon.

·   There is finally “the strategic logic of the musical work,” which concerns the compositional project proper to a given work (a project taking up the projects of other works already in existence, responding to them, or contradicting them, etc.). Each musical work has for its real interlocutors other musical works (much more so than musicians or auditors). This strategic (or subjective) dimension musically at work will not be approached here.


Unsurprisingly: each of these three musical “logics” – more precisely each of these three understandings of what “logic” wants to mean in music – requires difficult operations of untangling with the question (itself already quite tangled) of the negation: if negation can be said in general in at least three ways (classical, intuitionist, paraconsistent) and if musical logic takes form specifically in three ways (solfege, discursivity and intension at work), then it would be necessary for us to examine at least nine possibilities![2]

Here I will content myself with clarifying the properly discursive aspect of musical logic by way of the hexagonal formation.

The three logical principles of musical discourse

We could introduce the logic of musical discourse by taking up the three great logical principles of Aristotle, in order to counterpose to them three musical principles according to what we could call a global antisymmetry between musical logic and classical Aristotelian logic.

Identity versus differentiation

Whereas “classical” logic prescribes the principle of identity (A, stated twice, is identical to itself in its different occurrences: A = A), the principle of musical logic, that we could call the principle of differentiation, states that each musical term stated twice bears, by that very fact, an alterity A≠A(). Or: no term, stated twice, is identical to itself. Or further still: in music, to repeat is, ipso facto, to alter.

Non-contradiction versus constrained negation

Where “classical” logic prescribes the principle of non-contradiction (I cannot state at one and the same time A and non-A except to spill over into inconsistency Non(A and non-A), music counterposes a principle of constrained negation: each musical object posed must compose itself with itself contrary, that is to say must compose itself by becoming, A (A and non-A) [3]. This is the way in which musical discourse will appear for us as a paraconsistent logic.

Excluded middle versus required middle

Where “classical” logic prescribes the principle of the excluded middle (between A and non-A a choice is necessitated since there is no third position: A or non-A), the musical composition poses a principle of the required middle: each musical term posed should be composed with another term that is other than the negation proceeding from the first, a neutral term (neutrum) since it is “neither one nor the other,” A  (A and B). This is the way by which musical discourse will appear for us as an intuitionist logic.


In each of these senses to compose a musical discourse, this is to pose together (com-pose) three terms: an initial musical term, its alteration and another term, {A, A’, B}.

Example of the fugue

We can remark that this principle of composition delivers the initial matrix of all fugues: each fugue commences by posing a theme A (called subject), in order to immediately set in train a varied repetition A’ (called response), which is associated contrapuntally to a new object B (called counter-subject) that prolongs and contrasts with the subject A.

See here for example the beginning of the Art of the Fugue (Johann-Sebastien Bach), played on the organ by Wolfgang Rübsam (Naxos).



The response (A’) varies the theme-subject (A) in four ways simultaneously:

· by repeating it (thus by displacing it in time: on hearing it again, the thing is by this very fact heard differently: no longer as initial presentation but as reprise),

· by displacing it to another voice in the polyphony (here from alto to soprano),

· by transposing it to the dominant (for reasons of harmonic differentiation),

· and finally by subjecting it to some slight modifications (that we could call “mutations”) so as to adjust the initial ambitus (a fifth I-V) to a new ambitus (a fourth V-I).




Formalising the relations rather than the objects…

In order to formalise such a musical development in the framework of our hexagon, it is necessary for us to operate at the level of the musical relations between objects rather than directly between the objects concerned: contradiction, contrary and subcontrary will need to be situated not between the musical objects (our themes, motifs, harmonies, voices, rhythms…) but between the musical relations that they sustain.

Thus one could not speak properly of the subject and the counter-subject as contradictions, contraries and subcontraries, but one could by contrast maintain that the relation of varied repetition between subject and response opposes itself to the contrast between response and counter-subject, etc.

Yet, as we will see very soon, this passage from musical objects to morphisms that link them forces us to take leave of the strictly musical terrain of the score for that of the musician who reads and interprets this score. In fact it is indeed necessary to see that musical relations between objects are not presented as such in the score: they are certainly present here but they are in no way presented, and less still represented.

If I take for example our fugue, the soprano voice presents me indeed with an alteration of the theme initially presented by the alto but nothing in the score tells me that it is a matter of an altered repetition. Of course, no musician will ignore this status as altered repetition, and as a consequence all will phrase the two entries according to the same principle of articulation. But while the two musical objects are indeed presented by the score, their relation itself is not.

If I want to formalise the logic of musical discourse, in other words to treat musical relations, present but not presented, between musical objects (which, themselves, are at the same time present and presented by the score), it is necessary for me to undertake an initial musicianly “formalisation” of musical discourse – I will call it musicianly categorisation – capable of being then formalised in the framework of our hexagon of oppositions.

Musicianly categorisation

This categorisation will result from the three principles affirmed above.

It goes without saying that it is the musician – not music – that affirms them… in matters of music.

The musical variation could take two forms: that of alteration of the same object (via its repetition: A→A’) or that of a contrast (A/B).



Alteration, inherent in the repetition of a musical object, has for its contradiction the fixity that is attached to the unicity of a non-repeated object, of an object only announced a single time, or in other words an hapax H.



In the same way, variation has for its contradiction an invariance, which sums up two possible modalities: the fixity of the object presented only once (the unicity of the hapax H) or the fixity of the simple repetition (A repeated remains indeed A – beyond its variant A’ [4]):



Repetition is distinguished here from variation: variation transforms an object internally (as in our fugue where the response transforms the subject by a transposition and a mutation) whereas repetition contents itself with taking up the identical again (but by displacing it: between two voices, chronologically). The figure of alteration (varied repetition or variation associated to a repetition) sums or collects together these two possibilities:




Let us inscribe all this in a “philosophico-logical” hexagon:




Let us complete our formalisation by adding a centre that will suggest its implicitly Borromean logic:[6] the notion of properly musical “discursivity” is naturally situated at the heart of this formalisation.

And we can detail this hexagon thus:



On the side of the triangle of contraries that link together (in blue) the three figures of the conjunction or the product, we have:

·   the principle of universal variation (the musical discourse as a train of variations: Y, which follows X, always differs from X, in one way or another) projects itself on the one hand towards a general alteration (when A is redeployed, it is ipso facto varied: A→A’) and on the other hand towards a singular contrast (when what follows A is a completely different object B: A→B);

·   regular repetition (in music, the discursive regularity consists in repeating) projects itself on the one hand towards a general alteration (repetition alters the repeated object A) and on the other hand towards a specific invariance (which allows for the recognition of the object A repeated in its series of alterations Ai and by doing so allows for the differentiation of it from B);

·   the particular unicity of the hapax (H is only announced once) projects itself on the one hand towards a particular form of invariance (that which is attached precisely to the unique presentation) and on the other hand towards a particular aptitude for creating discursive contrast (in the train of A).

On the side now of the triangle of subcontraries that link together (in green) the three figures of summation or of gathering together:

·   general alteration (proper to our first principle of differentiation: A repeated is varied in A’) is the sum of repetition (reprise) and variation;

·   the singular contrast (proper to our principle of the required middle) is the sum of the variation of the objects set in train by the discourse along with the unique presentation proper to the object H;

·   specific invariance (that allows for the regrouping of the objects into related families) is the sum of the particular type of hapax H and the aptitude of the regularly repeated objects. This generates a family of related objects.

The stakes of this third hexagon?

How does this formalisation allow us to better understand what “musical discourse” means?

To develop?

Our three initial principles have indicated to us that the composition of a musical discourse involves posing together (com-posing) three terms: an initial musical term A, its alteration A’ and another term B: {A, A’, B}.


But of course, to compose is not simply to pose these objects one up against the other, it is above all to “develop” them; it is by this that one can recognise a composer: not so much by his capacity to invent beautiful and immutable melodies but more so by his capacity to develop the more banal motifs and to make real musical use of them.


Our hexagon indicates then that to compose is to develop the initial triple disposition by forming a simultaneous play of variations (or internal deformations of objects), repetitions (or simple displacements of objects) and of unique interventions attached to this or that object-hapax.

To compose is thus to knot together these three ways of giving each object a future in the work (a future at work) in a logic that we could in fact conceive as Borromean since:

·   the quotient of discursivity by repetition is indeed contrast[7] (which is the sum of variation and the unique presentation);

·   the quotient of discursivity by variation is indeed invariance (which is the sum of the unique presentation and repetition);

·   the quotient of discursivity by the unique presentation of H is indeed alteration[8] (which is the sum of repetition and variation).

Two musicianly positions on musical discourse…

In sum, which musical stakes are hidden behind this little formal game? What does this hexagonal formalisation teach us about properly musical discursivity?


First of all, it reminds us that musical operations on objects are more important that the musical objects themselves: the composition of a musical discourse plays itself out less in the choice of the object mobilised (motifs, harmonies, “themes,” instruments, rhythms…) than in the compositional handling that confers a musical destiny on these objects. But this the musician already knows: the hexagon only formalises this conviction.


But above all this formalisation clarifies a musical duality that, in our hexagon, takes the form of the contradictory[9] duality between contrary products (here written within rectangles) and subcontrary sums (here written within ovals), a duality that could be seen as that which relates on the one hand what is produced by the score and on the other what is summed by the play of the performer: whereas the composer (the one who is found at the origin of the score) thinks in terms of variation, repetition, and unicity (what is it that I should repeat, what is it that I should vary, and what is it that I should only state once?), the interpreter (and necessarily then the auditor whose labour consists ultimately in living the work as rendered sensible by the performer) thinks in terms of alteration, contrast and invariance: what is it in this discourse that remains the same, what is it that forms a contrast and what is it that is altered?

Hence the idea of figuring our hexagon slightly differently, by regrouping, as we have done above for our hexagon of timbre and rhythm, to have on one side the products, and on the other the sums:



The triangle of the contraries (in blue) figures here as the labour of the score; the triangle of the subcontraries figures by contrast the labour of the interpretation (and subsequently the audition).

Thu we find again the intuition evoked before that the score, rooted in the writing of solfege, operates in the framework of an intuitionist logic and that interpretation, attached to sensible labour (proper to the body and the ear), labours in the framework of a paraconsistent logic.

We could then read this “sketch” from left to right (all the arrows are found here oriented in this sense)[10] as a path going from construction (proper to written material) towards expression (proper to sonic material).

In sum, this arrangement of our hexagon would figure, under the form of the opposition of left/right, the dialectic of the eye and the ear, of the read and the heard, the written and the sonic, the score and the audition.


It would then be necessary to complete this “sketch”[11] by directly linking together “specific invariance” and “universal variation”[12] in order to formalise a retroactive component of the labour (that of the performer or auditor) upon the comprehension of the score and its capacity to project different dimensions at work in the musical discursivity:



I will stop here at these considerations, which remain, you will have understood, open and only held in a provisional state of my own research into what logic and negation could mean in music.



[1] We will see, towards the end, the specific reason for this rearrangement.

[2] The current philosophical work of Alain Badiou would suggest to us that the properly subjective logic of the intension would rather be paraconsistent.

We could intuit that the logic of solfege combines an intuitionist logic (for entire sections of the solfegic transcendental, for example for the marking of intensities or even durations) and a classical logic (since, at the precise point of a sonic object, the score should “classically” stand out against its recognition: is it noted, inscribed, or abandoned to the vagaries of the musically contingent sonic realities?

[3] where non-A≡A’

[4] It is indeed for this reason that we write A’ and not B, C, D… or H.

[5] The singularity here is a particular manifestation of universality; generality here has two sides: that of universality or that of regularity (understood as the contradictory of singularity); the specific here is the contradictory of the universal in that it appears as a choice between the particular or the regular (or precisely what is not marked by a singular power of the universal)…

[6] One should refer here to the work of René Guitart. See for example “L’idée d’objet borroméen, à l’articulation entre les nœuds et la logique lacanienne,” ERES | Essaim, no. 28 ( 85.htm)

[7] What remains of the discourse quotiented by repetition is the contrast between types of objects. Thus, in the case of the fugue, if all this only operates upon variations on the one hand of the Subject A, and on the other of the Counter-subject B, we will say that its musical discourse is summarised by the contrast of {A} and {B}.

[8] What of discourse is not expunged by the sole unique presentation is the general principle of the repeated…

[9] The contradiction of a product is a sum, and vice versa.

[10]  It is precisely for this reason that we have privileged for some time this “horizontal” orientation of the hexagon of opposition…

[11] understood in the more precise sense of sketches that Charles Ehresman developed in the theory of categories…

[12] that is to say, in our diagram, a colimit (inductive) and a limit (projective)