A musical research and its mathematical echoes: Heterophony
(Instruments of Algebraic Geometry, Bucharest,
20 September 2017 [1])
Franois
Nicolas, composer
(translation by Matthew Lorenzon)
You kindly offer me the possibility of speaking about music before a public of researchers in mathematics. I would like to take this opportunity to speak to you about a musical research.
I would like to take this opportunity
to clarify two things:
- The
specifics of musical research, in particular in relation to mathematical
research.
- How
a specific musical research - which I lead on the notion of heterophony (you can, next Sunday, come
and hear my Ricercare htrophonique [2],
which implements this concept) - is likely to resonate on the mathematical
side.
Researches
This research begins with the hypothesis
that, actually, there is such a thing as musical research.
Here, I understand the word research in a strong sense: it does not
mean the basic research for a musical theme by a composer (one finds a good
example of this type of research in BeethovenĠs sketchbooks), the basic
research on a phrase by a performer (think of the instrumental exploration of
Glenn Gould or of Harnoncourt), or even the research
for a libretto likely to carry an opera (think of Wagner or Schoenberg). In
short, it is not this type of musical research that limits itself to a given
work.
I would like to speak to you about a
more global research, which is likely to embrace multiple musical works, a
research configuring a compositional project of vast dimensions, a research
necessitating an interlacing between compositional practice and what I call musical intellectuality (that is to say
a formulation, in ordinary language - the same one in which I speak to you - of
musical issues). Or, to put it simply: a research that articulates musical practice
and theory on a grand scale.
The
difference, in music, between narrow research (on a theme, a phrase or a
libretto) and global research on a new compositional orientation can be
compared to the difference, in mathematics, between narrow research into a particular
demonstrative sequence and global research into a new theory - I will return to
this.
Certainly,
you know better than I that these two dimensions of mathematical research are
not without connections (demonstrating a given sequence can require a new
theory, and no new global theory is without its local demonstrative
inventions), but they set the two poles between which each real research must
learn to circulate.
In this way, in music as in
mathematics, a global research
associates theories and models, hypotheses and experiments, intuitions and
verifications. More specifically, as I have indicated, a significant musical
research associates composition and intellectuality, works and texts, writing
and listening.
I am
talking about compositional research
and I therefore exclude here musicological
research as well as technological research
(such as, while very interesting, is led at Ircam in
Paris, as in other equivalent centres of research around the world).
After this little comparison between
musical and mathematical research, a short note on their differences.
As you might expect, the opposition
between compositional and mathematical research rests on the criteria of validation
for the results of the research in question: music does not dispose of an
equivalent of mathematical demonstration, music does not produce its own
theorems, and one could also not say that it produces its own conjectures (if
it is true that in mathematics statements of conjectures and of theorems are
always equivalent [3], the
first being distinguished from the second only by the fact that they are awaiting
demonstration or refutation). In truth, music [4] knows
nothing of these demonstrations.
The criteria of validation of musical
research are thus rather to search elsewhere:
- On
the one hand, on the side of a certain practical success: do the works produced ÒworkÓ musically? But what does it mean exactly to
ÒworkÓ? Musicians do not agree on this term (like mathematicians can agree on
the term Òto demonstrateÓ).
- On
the other hand, on the side of a certain productivity:
are the ideas implemented musically stimulating? But, even there, the
stimulation will not be the same for all composers and the productivity criteria
will then no be massively shared by the musicians.
In truth, behind this question of
properly musical criteria runs the great question of beauty: does musical beauty always constitute a criterion of
validation of a compositional experiment, leaving it to share what musical beauty really means?
Now the point today is that there is no
more agreement between musicians on the notion of beauty as a criterion of
compositional success. One can even uphold that the contemporary is detached
from the modern around this precise point: where the modern (we would say, in
music, starting with Schoenberg) aims to recapture from classicism the flame of
beauty to found a specifically modern beauty, the contemporary (we would say,
in music, starting from 1968) disqualifies the opposition between the beautiful
and the ugly to promote the notions of innovation
and of creativity: a musical
experience will then be judged as novel and thus interesting, or as academic
and thus uninteresting, but the criterion of beauty will no longer be
considered pertinent.
One
can however claim that beauty constitutes the properly artistic form of truth:
so in art, the name of truth is beauty.
To recuse the criterion of beauty from the contemporary musical experience
would be equivalent to recusing the criterion of truth from the scientific
experience - you have a sense that the degree of ideological upheaval in this
way operates from the modern towards the contemporaryÉ
This ideological upheaval has the consequence
that all compositional research becomes today ipso facto a research on the very criteria of what musical composition still means (the
ideology of the contemporary, here, promises - contrary to the modern - performance
against interpretation, the openness of new inscriptions against the closedness of ancient scores, mobility between artistic
practices against the fixity of delimitations between different artistic
worldsÉ).
My own research on the notion of
heterophony is part of such an ideological context: it is an attempt to revive
musical modernity, to take up the compositional questions left fallow at the
end of serialism (roughly from the turn of the 60s) against a nihilistic figure
of the contemporary that would like to impose itself as the only inventive
current.
Let us now turn to the more concrete
figure of this compositional research which has been
mine for some years.
A compositional research
Let us start with a quotation from
Pierre Boulez, taken from the last pages (written in 1960) of Penser la musique aujourdĠhui:
In my opinion, a certain lack of homogeneity of voices has
been confused with the abandonment of one of the richest principles of Western
music: two or more "phenomena" evolving independently of one another,
without ceasing to observe between them a responsibility of each instant. It is
thus necessary to conceive of the transformation of the notion of voice, not to
envisage its abolition, which annihilates one of the most important domains of
the dialectic of composition.
Boulez,
in saying this, orients musical composition according to three propositions:
1. Composition
must continue (rather than abolish) the ancestral principal of musical responsibility between independent
evolutions.
2. To
do this, composition must extend this principle of responsibility to non-homogeneous evolutions.
3. This
extension passes through a transformation (rather than an abolition) of the old
notion of voice.
Let us
summarize this in the following program: the dialectic of composition must take
up the old musical responsibility between independent voices and extend it to
heterogeneous voices, which then passes through a transformation of the ancient
notion of voice.
This
program is then explicitly opposed to a ÒcontemporaryÓ program that prefers to purely
and simply abandon the notion of co-responsibility and to abolish the notion of
the musical voice.
Incidentally
- we can distinguish three ways of revolutionizing a given domain:
1. A
revolution by destruction then reconstruction - this is, in politics, the paradigm
of insurrectional revolutions (from the French Revolution to the Russian
Revolution of 1917), destroying a given State to reconstruct in its place a new
type of State.
2. A
revolution by abandonment and then displacement - this is, in politics, the paradigm
of "liberated zones" (from anti-slavery uprisings to a long sequence
of the Chinese Revolution).
3. A
revolution by extension-adjunction - this paradigm is found this time in
mathematics (see for example the extension of the rational numbers to the real
numbers by adjunction of Dedekind cuts).
It can
be said that musical modernity separately experimented with these three types
of revolution: respectively with Boulez, Pierre Schaeffer and Schoenberg.
It can
also be argued that a new type of revolution should aim at combining these
three types rather than holding them incompatible or practicing them
separately.
Let us then reformulate the orientation
suggested by Boulez in order to better sketch out our research program: it is a
matter of extending polyphony (which practices collective responsibility
collectively) by adjoining new types of vocal configurations.
Just
another note: the musical ÒvoiceÓ is here a discursive notion and not at all
physiological.
Vocal configurations
What kinds of vocal configurations do
we have in music? Essentially five.
1. There
is monophony, or a configuration of
one voice - that's our singleton.
2. There
is then homophony, or a configuration
of several identical voices, that is to say, all playing the same thing - exemplarily
the two crucial moments in Wozzeck where the whole orchestra converges on the same B - that
is our One.
Two
examples of homophony
Verdi : Nabucco (Chorus of Slaves) [5]
Berg : Wozzeck (unison on a B) [6]
3. There
is then antiphony or the configuration
opposing two voices to one another, either simultaneously or alternating - see
the old Responses, the contrast refrain/couplets, etc. - this is our 1 + 1 = 2
4. There
is also polyphony or the
configuration made of multiple different voices (they donĠt play the same
thing) but cooperating in the same global realisation - this is exemplarily the
case of the different voices of a choral or of a fugue - this is our plural.
5. Finally,
there is cacophony, or the configuration of voices anarchically piled up and
rivalling each other - for example, the pode of Chronochromie (Messiaen) - this is
our pure multiple.
Example
of cacophony
pode
of Chronochromie
(Messiaen, 1959–60) [7]
In
this way we distinguish the plural - the
repetition of the same type of term (1 + 1 + 1 + 1 ...)
- and the multiple - the simple
overlapping of terms of disparate types. Thus, the plural mobilizes the similar
and the multiple the heterogeneous - for example, two tonal voices will be dissimilar if they do not share the same
tonality and two voices will be heterogeneous
if for example one is tonal and the other dodecaphonic.
Let us summarize: Music knows the
unicity of monophony, the unity of homophony, the duality of antiphony, the plurality of polyphony and the multiplicity of cacophony.
We can
summarize these five vocal configurations as follows:
In so doing, we formalize oppositions
and connections that structure our five vocal configurations, but in reality
this draws a well-known figure from logic: that of a hexagon of oppositions,
which indicates to us the existence (at the bottom) of an absent vertex making
a strong opposition with cacophony.
It is this vertex that we are going to
add to our plan: let us call it (by a neologism) juxtaphonie - which suggests a notion of collage in this type of vocal collective (see for example Charles
Ives or Bernd Alois Zimmermann).
We can thus schematise the following
hexagon (which will this time formalize different types of qualitative relationships
between voices rather than different quantities of voices):
It is the very game of this hexagon in its entirety that we now propose to
call heterophony.
Before justifying the name heterophony globally given to this
hexagon, let us examine the logical properties of this hexagon of oppositions.
- The
hexagon rests on three vertices formalizing three collective logics
incompatible two by two: polyphony carries
cooperation (between the different voices), antiphony
carries rivalry between them and juxtaphony
carries indifference.
- Each
pair from these three vertices configures a vertex of another type which formalizes the interaction between different voices (common interaction in
polyphonic cooperation and antiphonic rivalry), the non-cooperation (common to antiphonic rivalry and to juxtaphonic
indifference) and the non-rivalry
(common to juxtaphonic indifference and polyphonic cooperation).
- These
three new vertices are in a logical relation of subcontrariety
(globally incompatible, they remain compatible two by two). They include our
three remaining vocal configurations: cacophony
taken as an emblem of the interaction between different voices, homophony as the emblem of non-rivalry
and monophony as the emblem of non-cooperation.
- Each
of the six vertices is in strong contradiction with its symmetrical opposite:
one must choose between polyphonic cooperation or non-cooperation of
monophonies, between antiphonic rivalry and homophonic non-rivalry, between
juxtaphonic indifference and cacophonic interaction.
In total, our formalization classically
distinguishes:
- two types of vertices: the
"products" inscribed in blue in a white rectangle and the
"vertices" inscribed in green in a gray
oval;
- three types of opposition between
vertices: the contradictions (in red) between a "product" and a
"sum", the contraries (in blue) between the "products", and
subcontraries (in green) between the "sums";
- a type of implication between
vertices (arrows in black from a "product" to a "sum") - for
example: "if cooperation, then interaction and/or coexistence" ...
Heterophony
What then does ÒheterophonyÓ name?
Heterophony
here names a remarkable property of this logical hexagon of oppositions: its Borromean knot structure - it is the mathematician Ren Guitart who untied it [8].
This means that cooperation, rivalry and indifference (or polyphony, antiphony
and juxtaphony) can be tied in pairs by the play of the third.
It is precisely this global property of
Borromean interlacing that I propose to call heterophony.
Heterophony
will therefore be defined here as this new global type of vocal
collectivization which braids (Borromeanly)
polyphonic cooperation, antiphonic rivalry and indifferent juxtaposition
between voices of all types.
Thus heterophony is the interlacing of
collectives of different types. It designates a global Collective made of
sub-collectives - hence its reasonance with
the political idea of people
...
An
interesting and unexpected musical example of such heterophony can be found in
the discourse of the burning Bush intervening at the very beginning of Moses und Aaron.
Three
examples of heterophony
Spoken/sung
heterophony: the burning Bush in Moses
und Aaron (Schoenberg) [9]
Heterophonic voice: Sequenza III (Berio) [10]
Heterophony in improvisation (jazz): Art Ensemble of Chicago [11]
To fix these ideas pedagogically, let
us advance these two minimal matrices of heterophony, which assemble in
different ways (simultaneously and successively) different types of vocal
collectives (or phonies):
Here is an example of such a
heterophony, extracted from my Ricercare
htrophonique:
(juxtaphony between strings & antiphony with the piano)
In doing so, I propose to hear heterophony in the strong sense of a new
type of musical discourse.
The issue of heterophony, as we have
seen, is to tie or intertwine different heterogeneous vocal configurations - one
could say: different heteronomous musical discursivities.
But in doing so, it is a question of producing a vocal ensemble
which affirms a new type of global unity between its different vocal
configurations and not one that is content with a juxtaphony or a cacophony of
different polyphonies. The formal
name of this new heterophonic type of unity is knot; its musical name is discourse.
It will therefore be said that the stake of this tying will lie in its capacity
to compose a new type of discourse: a heterophonic
discourse, which cannot be limited to a heterophony of different discourses.
In the
example given above, it is therefore necessary that the ensemble composed of
the two successive interventions of the strings and the simultaneous
intervention of the piano make, all together, ÒoneÓ musical discourse (in a
new, enlarged sense of the word discourse) and this, beyond the
disparity of the three remarks thus reported.
To
give two political equivalents of this question:
o Is
it legitimate to continue to speak of the "Chinese revolution"
(1927-1949) once the three types of revolution which are combined in it
(successively or by partial superposition) have been identified?: one of abandonment-displacement, the other of
destruction-reconstruction, the third of adjunction-extension;
o It
is also a question of being able to think of the collective existence of a
people not as a collective of individuals (the serialized mass of Sartre in the
Critique of dialectical reason) but
as a political unit of a new type between different mass collectives (workers,
peasants, women, young people...).
This is the formal challenge of the
compositional research in question.
The formalization of research, the
creation of compositional "models" for this formal theory of heterophony,
is at stake in musical compositions that concretely support this research.
Before returning to it, a few words about other uses of the term heterophony in the musical field.
The meaning I give to heterophony differs considerably from
the traditional musical use of this term. The term comes from ethnomusicology,
naming a sort of approximate homophony - when large groups of musicians play
vaguely in unison. Here, the hetero-
prefix no longer names alterity in so far as it
opposes the same homo- of
homogeneity, but denotes an alterity in a homogeneity, a localized alteration
of a global homogeneity. The two meanings of heterophony - the ethnomusicological
sense and the compositional meaning that I give it - are therefore radically
heterogeneous.
Two
examples of heterophony in the musicological sense
J.-S. Bach - Cantata BWV 80 "Ein' feste Burg ist unser Gott", Aria for soprano with oboe obligato
Mozart - Piano Concerto in C minor, K491, first movement, bars 211-214
I have
presented my musical research to you by privileging its formalized approach, in
particular by privileging a deductive form of mobilized formalization. But this
is of course a rhetorical presentation of the first results, in no way a
chronological account of the research in question.
In
fact, the idea of heterophony, as
clarified by the formalization here outlined, proves to have always been at
work in my compositions, such that the research in question here is a
revelation of intuitively pre-existing hypotheses.
In my
early compositions, the latent interest in this heterophony, which I was not
able to name until recently, was already there: for example, in Deutschland (a work from thirty years
ago). And I can now retrace a great part of my work as a composer by following
the red thread of this idea.
I will
not do it here, but ultimately this late formulation of an intuition that is
always-already at work would validate the well-known hypothesis that we would
have only one idea in all our life!
Three
examples of heterophony in my precedent works
Cadence of Duelle (2001) [12]
Ten voices in four configurations : two heterophonic (4 pianos | 4 violins) and two monophonic-polyphonic-antiphonic (harpsichord | flute)
Instress (2007) [13]
Heterophony of three configurations: polyphonic (three voices: string trio), monophonic (flute) and variable (piano)
Htrophonie presto (2016) [14]
Heterophony by collage of five
configurations:
–
heterophonic (chorus of six
languages)
–
antiphonic (counterpoint of two
monophonic songs)
–
polyphonic (Magnificat - Bach)
–
antiphonic (two musical voices
coming from jazz)
–
homophonic and/or polyphonic (Presto, for percussions).
Reasonances
To
emphasize this point, I will even tell you that I have recently realized that
my whole personal and individual life has been guided by this principle of heterophony if it is true that for
decades now I have led a life interweaving six simultaneous voices, sometimes
polyphonically unified, sometimes cacophonically
divergent, sometimes indifferently juxtaposed, a life that can be called
"heterophonic" since it intertwines daily survival (waged work and daily
tasks), a life as a parent (to numerous offspring), a life as a lover, a life
as an activist, a life as a musician and a life as an intellectual (who is
interested in mathematics, for example).
I have
already suggested the reasonances
of this notion of heterophony in political matters.
I
would like to point out that I am organizing a fifty-year anniversary week of
the fifty years since May Ġ68 which will be called Htrophonies/68: its title highlights the reasonances of the notion of
heterophony with the exuberance of the collective thought freed during this
historical uprising.
But I
would like to conclude by saying how this same notion of heterophony could also
concern certain aspects of mathematical research.
For
this purpose, I would like to examine the heterophonic dimension of two
mathematical theories: in analysis, the theory of integration and in
arithmetic, the theory of extensions of numbers.
Theory of Integration
We can
classically distinguish three times in the theory of integration, three times
which have deposited three different conceptions of integration indexed by
three proper names: Riemann, Lebesgue and Kurzweil-Henstock.
As we
know, these three theories of integration are partly compatible, even
complementary (Kurzweil-Henstock in a sense "completes" Riemann by
the introduction of a gauge on the abcissaĠs axis),
as rival parties (Riemann and Lebesgue "rival"
with respect to the privileged axis of integration: abcissa
or ordinate?). Could one not then consider the interweaving of these three
mathematical "voices" as configuring, in total, a heterophonic
conception of integration?
This
could then be drawn as follows:
Theory of numbers
Let us
now take the arithmetical theory of numbers which
proceeds by successive extensions from whole numbers - let us say ℕ,
ℤ,
ℚ,
ℝ,
ℂ.
We can
diagrammatise this in a little more detail:
Do we
not see here, too, the way in which a vast theory is developed heterophonically by playing simultaneously with different
numerical configurations which maintain different
relations between them?
- relations of polyphonic cooperation (to
pass, for example, from rationals to the reals by
Dedekind cuts),
- relations of antiphonic rivalry or juxtaphonic
coexistence (depending on whether one extends ℝ
to the body of the complexes - losing then the relation of order - or to the
body of the surreals - losing algebraic completeness
this time).
So it
seems - and I will conclude on this - that compositional research aimed at
taking up the old notion of musical discourse and extending it into a
heterophonic discourse, intertwining polyphony, antiphony and juxtaphony, can -
beyond its own musical productivity (which is, of course, its main, immanent
criterion of success) - resonate with similar concerns in widely differing
areas of thought.
At
this point, the risk would of course be to constitute the musician - in this
case the composer - as a prophet of humanity (it is known that Jean-Philippe
Rameau began to drift in this direction from 1749, dĠAlembert
rightly immediately set himself against these new claims of musical
intellectuality to try to set the course of the sciences of the time).
Let
us suggest that the risk nowadays would be all the greater for musical intellectuality that musicians could be
tempted to substitute themselves for the contemporary poets deploring melancholically
the end of the "age of the poets".
The
musician must therefore drastically self-limit this musical research on the
notion of heterophony. But this does not prevent him from addressing to the
mathematicians some friendly gesture of the hand to signify to them that they
are not alone in trying to invent a new modernity of thought.
Thank
you.
***