Hello.

This piece is inspired by the music of the composer Iannis Xenakis, since precisely in this year 2022 the centenary of his birth is celebrated.

You can listen to it on my SoundCloud page:

https://soundcloud.com/ramon-capsada-blanch/hyperbolic-paraboloid-and-xenakis

To make it easier for you to read, although it is also on the SoundCloud page, I include below the presentation of the work in which I explain the details:

**Hyperbolic Paraboloid and Xenakis.**

For 47 acoustic timbre instruments interpretable only electronically.

When I knew that in this year 2022 the centenary of the birth of the composer Iannis Xenakis is being celebrated, I got the impulse of composing a piece inspired by his music. My motivation is given by the fact that the work of this composer can be considered among the most innovative and influential of those emerged in the second half of the 20th century and more particularly for being considered the most important pioneer in the use of mathematics and also of the computer in the musical composition, and that these are precisely my favourite fields of study and application in the creation of my musical pieces.

The characteristics of Iannis Xenakis music are very wide, very original and cover very diverse fields. Then I highlight those that I have taken into account in the composition of my musical piece and also explain how I have applied them.

**1. Orchestral music and electroacoustic music.** Xenakis was interested in the two types of music and has great works in the two fields. He is considered a precursor in various areas of creation with electronic means.

My idea has been to try to present a mixture of these two types of music, but not proposing a simple addition of instruments of the two types in a single set but choosing to use instruments with acoustic timbre but electronically emulated and, what is more Important, doing an interpretation that is only possible electronically: speeds, ranges, timbres unification. . ., which are impossible in real acoustic instruments and executions. Thus the audition of the music of this piece is born and dies in its electronic production.

Specifically I used a total of 47 instruments: 29 string instruments with unified range and 18 piano lines. The score that I have written (in the end attached its link) is not to be used in an orchestral interpretation, but as a basis for electronic production and also for its subsequent analysis. In the elaboration of this score I have applied a slightly artificial graphic distribution and a impression large size to enhance its visual character and thus help understand the geometric content of the work.

**2. Great and dense musical masses.** Although Xenaquis, as he advanced in his compositional production, he became increasingly interested in the small ensembles and soloists writing for them true masterpieces, in the works of his early stages he frequently uses large sound masses, what he calls *large clouds of point-notes* inspired by the musicality of natural mass phenomena: "*The sing of cicadas in summer, the hail against hard surfaces*"; or social: "*a political crowd of dozens or hundreds of thousands of people.*"

In my case I have tried to obtain a continuous gradation between passages of low sound density and among others of high density where they would simultaneously sound the 47 instruments, but never in unison but using a total *divisi*, in such a way each of the 47 instruments have a melodic line different, but, nevertheless trying to achieve, sometimes, the effect of a unique melody with a lot of power, in reality two melodies, the string and the piano. Sometimes, due to the intersection of a large number of different string melodies, an auditory synthesis is achieved and appearing new timbres.

**3. Musical space as a sound architecture.** The fact that Xenakis was an architect greatly influenced his music. The *Philips Pavilion*, designed for Expo 1958 in Brussels by the office of Le Corbusier, and also, much later in 1977, the transportable structure called *Le Diatope*, to houses the multimedia show based on his work *La Légende d'Eer*, they have a clear relationship with some of their musical works. He had a special interest in the geometric figure in three dimensions called **hyperbolic paraboloid** that used it both in these architectural constructions as well as in the composition of passages of his works, for example in the generation the sections of *glissandos* of his work *Metastaseis*.

This hyperbolic paraboloid (which shows a drawing in the image that accompanies my piece on the Soundcloud page) has been the base where I have built the bulk of the melodic and harmonic materials of my composition. It has been a different use than Xenakis in the aforementioned glissandos since he did it using the property that the paraboloid is a *ruled surface*, that is, it can be generated by a straight line when moving; The different straight lines that compose it were precisely violin *glissados*. In my case I have used its most internal structure when considering the different sections that originate by cutting it by parallel planes, generating two sets of parabolas, some that begin by growing and the others that begin by decreasing. In this way I can transform the continuum of this figure in a discrete collection of parabolic lines.

**4. Using mathematics**. As I have already commented, due to his technical formation he had good knowledge of mathematics and used them in the musical composition. The composer Olivier Messiaen, refers to him as follows: «… *You are lucky to be Greek, to be an architect and have studied special mathematics. Take advantage of them! Use them in your music* ”. And he obeyed him.

Two have been the main mathematical tools that I have used in my composition. The first, the definition of the mathematical functions that allow to generate the parabolic lines in a three -dimensional space and that make up the hyperbolic paraboloid. Each of the three dimensions (*x, y, z*) of the points that form the parabolas have a musical meaning. The *x* corresponds to time, the *y* corresponds to the pitch of the note and the *z* to each of the different voices or musical lines. As the paraboloid is rotated on an axis, different *views* of the disposition of the parables corresponding to the different delays of the musical voices are obtained in the corresponding counterpoints.

The second mathematical tool has been statistical analysis. Xenakis created the concept of *Stochastic Music*. He used different probability distributions to control some of the parameters of his musical masses. One of the most prominent is the parameter of the sound density that generally imposed that it followed the Poisson's probability distribution. In my case I have also wanted to control the sound density of my piece, analyzing the distribution of frequencies of the different densities and retouching the amount of musical lines that appear in the different musical units so that it will adjust more to the theoretical probability distribution.

I finish attaching some links:

Score of my piece:

https://www.dropbox.com/s/gc072hondneezat/Hyperbolic%20Paraboloid%20-%20Full%20ScoreHD.pdf?dl=0

Information about the life and work of Xenakis:

https://en.wikipedia.org/wiki/Iannis_Xenakis

To listen to Xenakis music:

https://soclassiq.com/en/classical_music_masterpieces/Iannis_Xenakis/ID/5778/

**Happy Centenary Xenakis !!**

## Replies

Hi Ramon, I have listened to and enjoyed your work here although it is challenging for a number of reasons. I am glad that you post here and that you make a good effort to explain your composition process and its connection to Xenakis.

I think I understand the basic principle here, the selection of musical notes is determined by the three dimensions of points on the curves of a hyperbolic paraboloid. So as the figure is drawn or traced the musical parameters of individual notes are generated at preselected intervals (points) on the curves of the figure. You then alter these raw forms to be more musical as necessary?

My question here is: Once you have your sequence of parameters do you then use those to create a midi sequence for each of 47 tracks in a DAW? So I assume you have a program that can convert those three number parameters into that midi sequence?

Can you comment on any of the software used here and which DAW you are using? If I'm totally confused here there's no need for you to spend a lot of time on this but thanks anyway!

Hi Ingo.

First, I want to thank you for the interest you have shown in my composition by having listened to it carefully and read the explanations with caution, and that you also want to go deeper into it.

You have perfectly understood the basic principle on which I have based the creation of music: each note of the musical composition corresponds to a point of the hyperbolic paraboloid. As the points have three dimensions (just like the point determined in a corner of a box) there will only be three musical parameters of the notes that are determined by this geometric model, which are pitch, duration and timbre. The remaining parameters (mainly dynamics and articulations) are left free to the composer's artistic criteria. However, a great difficulty appears: the surface of the hyperbolic paraboloid is continuous, that is, it is formed by infinite infinitesimally close points. But music (microtonal and glissandos excepted) does not work with infinite infinitesimally close notes but with discrete series of 12 notes. So, which points to choose, how to convert the continuity of the geometric surface into discrete sets (formed by separate values) of point-notes. I have achieved this "discretization" process in two steps: First I have reduced the geometric surface to two sets of 24 and 23 parabolas respectively, achieved by cutting the surface with two collections of planes equidistant between them and perpendicular to the two main axes of the figure. These two sets of parabolas, although they greatly simplify the geometric figure, represent it perfectly since it maintains its structure and shape. The second step in achieving discrete note sets has been to convert these parabolas into "stepped" lines; once a parabola is fixed (which corresponds to a certain timbre) each of its points will only have two dimensions that can vary, which correspond to time and to pitch; that are continuously changing. For this reason, each parabola is a mathematical function that relates two variables (time and pitch). Converting the parabola into a "stepped" line means that as time increases, the pitch remains constant until it reaches the value of one of the notes of the chromatic musical scale, which will again remain constant until reaching a new value of musical scale, and so on. With this it is achieved that only the 12 classes of pitch appear and also that notes of different durations appear, thus in the areas of the parabola close to its vertex where the variation rate of the pitch variable is small (becoming zero just at the point of the vertex) is where notes of longer duration appear and in the extreme areas of the parabola, the notes are shorter and faster since it is there where the variation rate of the pitch variable increases very quickly.

Therefore, and answering your question, the most important alterations that I have applied to the initial geometric shape to achieve more musicality have been those that I have described here: the selection of two sets of parabolas and the conversion of each parabola into a step function. In addition to these two, I have also applied alterations from the musical parameters that are totally detached from the geometric shape which, as I have already indicated, are the intensity of the notes (which I regulate with the corresponding dynamics) and also the different articulations. I have also used a last type of "alteration" that has been very important in the final result, which has been to be able to freely choose the different types of overlaps of the melodic lines obtained from each parabola, raning from the presence of a single melodic line, going through the successive entries of the different melodic lines and until arriving at the simultaneity of all the voices at the same time. This process has been fundamental in the creation of the different sound densities of the piece. Although its determination has been free, it also has a geometric interpretation such as the vision that you can obtain from the different parabolas depending on the point of view that the viewer has of the geometric figure.

Regarding the computer software used, basically there have been two:

And with this I am ending, that perhaps I have extended too much.

Greetings.

Ramon

Fascinating stuff Ramon, thank you for taking the time! I wondered where the number 47 came from, so each melody has its own parabola or segment of parabola. I guess you experimented with the placement of the planes slicing the original figure to get to that number?

You mention that the entry points and overlaps for each melodic line were freely chosen so I picture you sliding them around in Sibelius. Then you say that your Python app does "creation of the delays for each melodic line" which sounds like maybe the program was suggesting a pattern for the polyphony?

As I said before, I know you are busy so no need to spend time on these questions; you've been very helpful already.

Once again you are correct in your assumptions (you are very insightful ;-) ). The 47 melodic lines are the result of searching for the best configuration of distances between lines and, at the same time, audibly convincing tonal ranges. The entry points for each melodic line are chosen freely in that they are not determined by the properties of the geometric figure, but once the decision has been made according to my own criteria, I have used, as you suppose, two ways of applying it, for the entries of a more systematic nature I have used algorithms in the computer application that will facilitate its construction given the high number of lines. Instead, for the more unique overlays and later modifications, I've used direct editing in Sibelius.

Hi, Ramon! I'm enjoying your piece here. Somehow it has, when listening to some part of it, a penrose steps idea going on in my head. It's certainly new to me.

Regards,

Sam

Thank you very much Sam for listening to my music and I am very glad that you enjoyed it. I find the reference you make to the Penrose staircase interesting, since it has great geometric strength and a component of illusion and magic, which I would love for my work to transmit as well.

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