X: There you have pinpointed, and I congratulate you on it, a subject which has never been penetrated, that of the very essence of music; what is its role, its aim in present day society? One might say "it's to pass the time", or "it's for pleasure", or "it's a spiritual diversion" when evaluating the nobility of classical works. For contemporary ones "free intellectualism, exploration of the unknown etc." The Pop song brings large rewards because it is consumed by the masses, it enters their souls, their lives, and stops there; one can very well do witout the musical questings of the avant-garde, one already has the musical past of Europe, which is not so bad; people keep discovering old bits of nonsense, mummies that they've dug up, and to hear all that could last a lifetime so what's the point of taking the prickly risks of a sortie into the contemporary world? Musical exploration today is in the same state that mathematics was 80 years ago, when they were considered a crazy fantasy. When Riemann started non-Euclidean geometry, people said it was a freak. Now the result of these scientific researches is tangible enough today, and spectacular in the cosmic field.
MB: What consequences could musical research such as yours have in the distant future?
X: There are two answers to that. First of all, if one approaches music in the same way that I approach mine, it marches with and intermingles with mathematics. Mathematicians are beginning to appreciate this interaction and to react to it, making new propositions most beneficial on the purely material plane. It is the widening of the horizons of the pioneers which gives birth to applications which are profitable to all. The second answer concerns the importance that music can play in the achievement of man through his creative faculties. If one allows these faculties the opportunity to develop, the whole of society is affected, and this will give to humanity an even richer knowledge, and therefore an ever greater mastery.
MB: Teilhard de Chardin expressed the hope that "the age would come when man will be more pre-occupied with knowing than having."
X: Music is certainly a basic tool for helping to fulfil this hope. Pythagorism was born of music. Pythagoras built arithmetic, the cult of numbers on musical foundations. This is splendid; it is Orphic. In Orphism music fulfills the function of the redeemer of souls in their escape from the infernal cycle of reincarnations. If one wishes to be reborn on a higher plane one must look after one's soul. This is to be found also in Homer; it is the Orphic thesis. It is for religious reasons therefore that Pythagoras discovers the processes by which music is made, and then the relation between length of string and a note, and following that the association between sounds and numbers; moreover, as geometry was being born at the same period, Pythagoras interested himself in it. By adding arithmetic to it he laid the foundations of modern mathematics; thus he was able to invade the realm of astronomy, invent the theory of the spheres, the theory of the music of the spheres, of the harmony of the spheres, which survived right up to Kepler; the Keplerian discoveries could never have been made but for the contribution of Pythagoras. In the old days music therefore became, quite simply a branch of mathematics. Euclid wrote an entire book called "Harmonics," in which he treated music on a theoretical level. This was the position in the West right up to the middle ages: up to the end of the 9th century, when Hucbald, in his Musica Enchiriadis was analyzing plainchant and speaking of music in ancient classical terminology. With the appearance of polyphony, there occured a divorce.. Today there are fewer reasons for upholding this divorce then for suppressing it.
MB: Isn't this to go against the stream in a reactionary way, which would be rather odd for a composer of the avant-garde, to hold that the mathematical theories of Pythagoras were engendered from a music that preceded them, and then today, 25 centuries later, to maintain that music should go back to mathematics?
X: Not at all. Music is by definition an art of montage, a combinatory art, a "harmonic" art, and there is plenty to discover and to formulate in this domain. I think I have defined two basic structures, one which belongs to a temporal category in musical thought, the other is independent of time, and its power of abstraction is enormous: ancient music was based on the extra-temporal, which allowed to be conjoined to mathematics; present-day music, since it is polyphonic, has almost dispensed with the extra-temporal factor in music, to the advantage of the temporal...the combination of voices, of modulations, melody, all this is made in time. This music has lost, for example, all that it possessed of modal structure, which was based on tetrachords and "systems" and not on the octave scale; it has lost all that to the advantage of the temporal, that is to say of time-structures.
It is urgent now to forge new ways of thinking, so that the ancient structures (Greek and Byzantine) as well as the actual ones of the music of western countries, and also the musical traditions of other continents, such as Asia and Africa, should be included into an overall theoretic vision essentially based on extra-temporal structures.
MB: Since music gave birth to mathematics, ought it now to seek refuge in a return to mathematics?
X: It isn't a refuge that music asks of mathematics, it is an absorption that it can make of certain parts of mathematics. Music has to dominate mathematics, and without that it becomes either mathematics or nothing at all. One should remain in the realm of music, but music need mathematics for they are a part of its body. When Bach wrote the Art of the Fugue, he produced a combinatory technique which is mathematical. In this new kind of music which I am propounding, and which at present is no more than the first hesitant stammering, mathematical logic and the machine will give a formidable power in the context of an extremely wide generalization. Did you know that at the age of 16, in 1938, I tried to express Bach in a geometrical formulae?...but the war came...
MB: I want to come back to the public which is listening to you in the concert hall: there they all are, creasing up their foreheads over your programme-note where, in presenting your work you cite Poisson's Law which you follow up with several algebraic equations. Why do you bother to do all this? Aren't you being a bit of a tease?
X: If the listener doesn't understand any of it, it is first of all useful to show him to himself as ignorant. Because the laws which I cite are universal ones and treasures of humanity, real treasures of human thought. To be unwilling to know them is as uncivilized as to refuse to recognize Michelangelo or Baudelaire. Furthermore, these formulae, which 10 years ago were the property of only a specialist, are now the common property of the average student of elementary mathematics. in a few years time they will be in every schoolboy's satchel. The level of learning, this also evolves. There exists an inner beauty in mathematics, beyond the enormous enrichment it brings to those who possess it, even a part of it; pure mathematics approaches poetry.