In his text "Vers une philosophie de la Musique" (1968) Iannis Xenakis assumes that there are two structures, within and outside of time. To take an example from music: A scale is an extra-temporal structure, a melody formed with it an inner-temporal one. In recent centuries, according to Xenakis, the focus has shifted more and more to the inner-temporal structures, and the important question of the extra-temporal structures has been pushed into the background. Mathematics is a universal tool at our disposal, which allows us to deal with the extra-temporal structures and gives us everything we need to choose from them.
His two crucial tools for this are sieve theory and group theory. So we’re going to make an excursion into mathematics:
The sieve theory selects certain from a predefined set of elements, for example the sieve 3.0 starting with element 0 chooses every third.
With the three functions of composition (∧), overlap (∨) and contrast ( ) we can also create our familiar scales from the 12 chromatic steps.
For example, the formula for a major scale is:
(3,2 ∧ 4,0) ∨ (3,1 ∧ 4,1) ∨ (3,2 ∧ 4,2) ∨ (3,0 ∧ 4,3) Figure 2
The decision to use a semitone as the step width serves only for clarity, but it's possible as well for quarter tones or even finer subdivisions. And in other respects, the method is of course flexible: Xenakis also uses pitches as an example, since they and the intervalls between them are so easily recognizable. Of course, such a method can be applied to any parameter.
This may seem like a mathematical gimmick at first, but it has consequences. Such a mathematical derivation facilitates a more sober look at “our” scale: it is one structure among many and if you change only one parameter, you end up with a completely different scale – but with the same permission.
Basically, the sieve theory serves a selection of individual elements from a predefined set – not only from pitches, of course.
Group theory: Elements mathematically form a group if they have three properties.
∘ represents any linkage. Examples for groups:
The set Z of integers at the conjunction "+" (neutral element is the 0, inverse is with the other sign).
The set Q of rational numbers at the conjunction "+" (neutral element is 0, inverse is in each case with the other sign) without zero at the conjunction "*" (neutral element 1, inverse in each case 1/n).
An equally infinite group is formed by the intervals with the "+" link. The neutral element is the unison. All the extensions that algebra makes from such simple groups can also be understood with any scale system. (To be exact, one must separate into the body of the resulting pitch and the vector space of the intervals with which one moves. And physically, pitch has a limit going down.)
These are all examples of infinite groups. But there are also finite ones. For example (if one is willing to abstract from octavations) the sequence of the 12 notes of our well-tempered tonal system, combined with the 12 ascending intervals from unison to majo seventh. Mathematical: M = {0, 1, 2, ..., 11} +mod12. The neutral element is the unison, the inverse element is the complementary interval.
You would have exactly the same thing if you imagine a piece of cake that is rotated around its top by 30°at a time. There are 12 possible angles for the rotation (0*30° to 11*30°), neutral element is 0*30°, the inverse to 1 is 11, to 2 is 10 etc. It becomes even more manageable if you take a square and its images on itself, each of which can be reached by rotating it by 90°.
Figure 3 shows the 4 positions that can be reached in this way. Again, these turns form a group. For example, just as I rotated from A to D, I can also rotate from D and end up at C. An overview of these possible combinations is provided by the grayed-out portion of Figure 4.
Just to repeat: The neutral element is A, associativity is given (A∘C)∘D=A∘(C∘D)=B, to each element there is an inverse element, which makes it back to A.
The white part of Figure 4 shows the extension of possibilities if I also allow the reflection around the vertical axis (A becomes A' thus).
If the corners are designated as 1, 2, 3 and 4 (from the top, then from left to right, then down) as shown in Figure 3 for the positions A-D, the sequences for the 8 positions result in the numbers shown in the right table in Figure 4.
What is gained by this? Unfortunately, I haven't found anything that applies group theory to a tangible everyday example. But what I find obvious is, that here eight of the 24 possible permutations of the numbers 1-4 are not simply taken out arbitrarily, but that a selection of eight permutations related to each other is created. And if I link B with C (i. e. originally the rotation by 90° with the one by 180°), I'm going from 3, 1, 4, 2 via 4, 3, 2, 1 to 2, 4, 1, 3.
The sieve theory, as said, selects elements from a set, the group theory, without making any further selection among the elements, creates a link between them.
With Xenakis, this happens less concrete - but also more interestingly. He leaves the two-dimensional space. A cube can be brought into 24 different positions (to the imagination: in each case 1-6 looking forward, then one can turn it from each of these positions still 3 times, thus 6*4=24 positions). The cube has 8 corners, which can be numbered again and read off as numbers, just as in the example square. 8 numbers can be arranged in 40320 sequences, which are now just reduced to 24.
This is exactly what Xenakis did in his preparation for the composition of nomos alpha. As in the example with the square, a table of the various links can be created, which can be found as Figure 5. The large group of all rotations still has two subgroups, once formed from I-C, then from I-L2. Finally, the individual positions can be grouped together to form I-L2 as a and Q1-Q12 as q. Figure 6 then results, again mathematically a group.
As an aside, this is certainly tongue-in-cheek fun: for a piece that will be the exact opposite of random music, Xenakis is using a cube, whose Latin name Alea has earned random music the name Aleatorik.
What we have so far is a structured field of possibilities, but the question remains how to deal with it. The first question that arises is the order. For this, Xenakis falls back on Fibonacci, whose series we know above all with the function "+" on the set of natural numbers (i. e. 0, 1, 1, 2, 3, 5, 8 etc. ). In general, it can be formulated as follows: Rn-2 ∘ Rn-1 = R. The table in Figure 7 shows the Fibonacci series starting at D in the field of cube rotations. In total there are one possibility with 2 steps until the repetition of D, one with 5, ten with 6, three with 10, three with 16 and finally 9 times 18 steps. Xenakis chose two of these longest chains, D-Q3 and D-Q12.
In the sequence of the 18 variants, there are six groups of three: one item from the I-L2 range is followed by two from the Q1-Q12 range. Likewise, nomos alpha has 6 parts. The determination of the six-part structure is thus only secondarily a decision, primarily a consequence of the mathematically worked out structure. After each of these parts there is a short further part that follows other structuring.
Figure 8 shows the two overviews of what kind of sounds Xenakis uses (this determines the first row of rotations) and the duration, volume and density of each part. There are 3 different readings for each of these (referred to as α-β-γ), with the right-hand table differing only in density and duration. The 6 parts of the piece have the sequence β-γ-α-β-γ-α. So if the combination Q7, Q4 appears again (as well as Q11, Q8 and Q1, Q11) the specification is still different.
In addition, of course, there is the question of pitches. Here, as one would expect, Xenakis uses the sieve theory, starting with the quarter tone as a step. The exact construction of the sieve he uses for this has not become clear to me. However, its structure is basically like that of the sieve for major scales, but with different variables:
L(m,n)=(n,i ∨ n,j ∨ n,k ∨ n,l) ∧ m,p ∨ (m,q ∨ m,r) ∧ n,s ∨ (n,t ∨ n,u ∨ n,w)
Each of the six sections has its own sieve, the parameters of the following sieve are derived from the one used before.
At the beginning of each of the three parts of a section, not only the reading is given (i.e. at the beginning β), but also (as well as in between) the cube rotations and the variables m and n of the sieve. (For the 2. Part 13, 11, for the 3. Part 11. 5 etc.)
Up to this point, the structure of the piece is not only kept quite clear, but also well documented. This results in a composing instruction like: "an action in this shape, with a certain duration, volume and event density". The selection of pitches is for a whole part. But how Xenakis then chose the pitches for the individual particles, how he arrived at the respective varieties and dynamic developments, unfortunately remains unclear.
So let us turn to the question of what prompted Xenakis to compose in this way.
In the already mentioned text "Vers une philosophie de la Musique" (1968) Iannis Xenakis begins with a long derivation from Greek philosophy: the real beginning of our civilization was the step towards a rational explanation of the world, which was taken in Ionia a good 2500 years ago. A development that, Xenakis wrote in 1968, trumps all mysteries and religions and has never been as universal as it is today: "the US, China, the Soviet Union, Europe, currently the greatest protagonists, renew it with a homogeneity and a uniformity that I would almost call disturbing."
Xenakis describes this development in more detail in terms of two philosophers.
First of all, to Pythagoras. Things are numbers or all things are provided with numbers or all things behave like numbers, depending on the stage of development of the Pythagorean doctrine.
Xenakis' other point of reference is Parmenides. Being is One, it is all and it is continued. There is no non-being. Whereby it is Xenakis' rationale that is of most interest: "What need should have made it (being) start a little earlier or a little later if it had come from nothing?"
The increasing digitalization of our world as well as the ever increasing valuation of all being according to its countable monetary value can also be seen as an effect of Phytagoras, just as the theorem of the conservation of energy is, so to speak, the physical formulation of Parmenides' consideration.
So, in a nutshell, we need a reason to understand things and we can use numbers to access things. We still live in a Pythagorean/Parmenid age.
Implicitly, this all boils down to determinism, to LaPlace's demon who, if he knew the locations and motions of all atoms, could predict the future. There is no room for the freedom of the human will in this view.
The opposite position is established by Epicurus, who takes up Democritus's atomic theory but assumes that it is tiny, random perturbations that join different atoms together to create objects.
Xenakis illustrates how we stand between these two poles with the calculus of probability: we will never be able to predict the next result of a coin toss, yet according to the law of large numbers we can say that among 1000 tosses there will be about 500 for each side.
Two questions arise from this for Xenakis:
1. What consequence must the reflection of the Pythagorean-Parmenidean field have for musical composition?
2. Which way.
His answers:
To 1. Reflecting on what is leads us directly to the reconstruction, as far as possible ex nihilo, of the foundations of music and, above all, to discarding all that has not been questioned.
This also includes the difference between structures within and outside of time mentioned at the beginning. Of course, a reconstruction also includes the structures outside of time (e.g. new scales by means of sieve theory ).
To 2. This reconstruction will be inspired by modern axiomatic methods.
Nomos alpha represents an extreme point in Xenakis' oeuvre. In the literature it is described as deterministic music. But up to which point this was carried out (and how) unfortunately remains open.
Matthias Lorenz, 2003/2021
