Thus we are, at this point in the exposition, still immersed in the lines of force introduced twenty five centuries ago and which continue to regulate the basis of human activity with the greatest efficacy, or so it seems. It is the source of those problems about which we, in the darkness of our ignorance, concern ourselves : determinism of chance, unity of style or eclecticism, calculated or not, intuition or constructivism, a priori or not, a metaphysics of music or music simply as a means of entertainment.
Actually, these are the questions that we should ask ourselves : 1. What consequence does the awareness of the Pythagorean-Parmenidean field have for musical composition ? 2. In what ways ? To which the answers are : 1. Reflection on that which is leads us directly to the reconstruction, as much as possible ex nihilo, of the ideas basic to musical composition, and above all to the rejection of every idea that does not undergo the inquiry [..]. 2. This reconstruction will be prompted by modern axiomatic methods.
Starting from certain premises we should be able to construct the most general musical edifice in which the utterances of Bach, Beethoven, or Schönberg, for example, would be unique realizations of a gigantic virtuality, rendered possible by this axiomatic removal and reconstruction.
it is necessary to divide musical construction into two parts [..] :1. that which pertains to time, a mapping of entities or structures onto the ordered structure of time ; and 2. that which is independent of temporal becomingness. There are, therefore, two categories : in-time and outside-time Included in the category outside-time are the duration and constructions (relations and operations) that refer to elements (points, distances, functions) that belong to and that can be expressed on the time axis. The temporal is then reserved to the instantaneous creation.
In Chapter VII I made a survey of the structure of monophonic music,
with its rich outside-time combinatory capability, based on the original texts of Aristoxenos of Tarentum and the manuals of actual Byzantine music. This structure illustrates in a remarkable way that which I understand by the category outside-time.
Polyphony has driven this category back into the subconscious of musicians of the European occident, but has not completely removed it ; that would have been impossible. For about three centuries after Monteverdi, in-time architectures, expressed chiefly by the tonal (or modal) functions, dominated everywhere in central and occidental Europe. However, it is in France that the rebirth of outside-time preoccupations occurred, with Debussy and his invention of the whole-tone scale. Contact with three of the more conservative traditions of the Orientals was the cause of it : the plainchant, which had vanished, but which had been rediscovered by the abbots at Salesmen ; one of the Byzantine traditions, experienced through Moussorgsky ; and the Far East.
This rebirth continues magnificently through Messiaen, with his �modes of limited transpositions� and �non-retrogradable rhythms,� but it never imposes itself as a general necessity and never goes beyond the framework of the scales. However, Messiaen himself abandoned this vein, yielding to the pressure of serial music.
In order to put things in their proper historical perspective, it is necessary to prevail upon more powerful tools such as mathematics and logic and go to the bottom of things, to the structure of musical thought and composition. This is what I have tried to do in Chapters VI and VII and what I am going to develop in the analysis of Nomos alpha.