Formalized Music. THOUGHT AND MATHEMATICS IN COMPOSITION. Revised Edition. Iannis Xenakis. Additional material compiled and edited by Sharon. Xenakis_Iannis_Formalized_Music_Thought_and_Mathematics_in_Composition .pdf (file size: MB, MIME type: application/pdf). Formalized Music THOUGHT AND MATHEMATICS IN COMPOSITION Revised Edition Iannis Xenakis Additional material compiled and edited by Sharon.

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Xenakis Formalized Music - Download as PDF File .pdf), Text File .txt) or read online. Formalized Music: Thought and Mathematics in Composition is a book by Greek composer, architect, and engineer Iannis Xenakis in which he explains his motivation, philosophy Create a book · Download as PDF · Printable version. PDF | On Jan 1, , Rob Wannamaker and others published 52–59 (from Xenakis, Formalized Music, 18–21; the gap in the figure.

Thus I believe that on this occasion music and archi- tecture found an intimate connection. We shall examine one by one the independent components of an instrumental sound.

Xenakis_ Iannis - Formalized Music. Thought and Mathematics in Composition

The interval between two points is identical with thc duration. Among all the possible sequences of points, which shall we choose? Put thus, the question makes no sense.

If a mean number of points is designated on a given length the question becomes: Given this mean, what is the number of segments eq ual to a length fixed in advance? The following formula, which derives from the principles of continuous probability, gives the probabilities for all possible lengths when one knows the mean number of points placed at random on a straight line.

See Appendix I.

Five key links

If we now choose some points and compare them to a theoretical distribution obeying the above law or any other distribution, we can deduce the amount of chance included in our choice, or the more or less rigorous adaptation of our choice to the law of distribution, which can even be absolutely functional.

The comparison can be made with the aid of tests, of which the most widely used is the X2 criterion of Pearson. In our case, where all the components of sound can be measured to a first approxima- tion, we shall use in addition the correlation coefficient. It is known that if two populations are in a linear functional relationship, the correlation coefficient is one.

If the two populations are independent, the coefficient is zero. All intermediate degrees of relationship are possible. Clouds of Sounds Assume a given duration and a set of sound-points defined in the intensity-pitch space realized during this duration.

Given the mean super- ficial density of this tone cluster, what is the probability of a particular density occurring in a given region of the intensity-pitch space? Poisson's Law answers this question: where! Lo is the mean density and fL is any particular density. As with durations, comparisons with other distributions of sound-points can fashion the law which we wish our cluster to obey.

We have been speaking of sound-points, or granular sounds, which are in reality a particular case of sounds of continuous variation.

Among these let us consider glissandi. Of all the possible forms that a glissando sound can take, we shall choose the simplest-the uniformly continuous glissando.

This glissando ean be assimilated sensorially and physically into the mathematical concept of speed. In a one-dimensional vectorial representation, the scalar size of the vector can be given by the hypotenuse of the right triangle in which the duration and the melodic interval covered form the other two sides.

Certain mathematical operations on the continuously variable sounds thus defined are then permitted. The traditional sounds of wind instruments are, for example, particular cases where the speed is zero.

A glissando towards higher frequencies can be defined as positive, towards lower fre- quencies as negative.

We shall demonstrate the simplest logical hypotheses which lead us to a mathematical formula for the distribution of speeds. The arguments which follow are in reality one of those "logical poems" which the human intelligence creates in order to trap the superficial incoherencies of physical phenomena, and which can serve, on the rebound, as a point of departure for building abstract entities, and then incarnations of these entities in sound or light.

Curiously, the notes that fall outside the computed set occur almost exclusively within the first exposition of G C while the notes heard in the four occurrences of this set marked rappel agree almost perfectly with the prediction. It is almost as if the first exposition is not of G C at all, but is of some other set which is mislabeled in the score and in Formalized Music.

Apparently this is not the case, however, for the following reason. A 2 and A3 are definitely members of the atomic set A B C. In this case, the contents of the initial presentation of the set and the single presentation marked rappel agree with each other, although it would also appear that the set observed cannot be expressed in terms of A, B, and C. Particularly troublesome is the pitch D 3, which occurs three times in A, five times in B, and ten times in C.

This would imply that it is a member of A B C and therefore ought not to occur in F see Figures 8 and 9 in the Appendix.

Nonetheless, it is observed to occur nine times there. One possible explanation is that the composer deliberately introduced this pitch so that F would contain at least one representative of each pitch class.

Indeed, D would not be represented in F if the offending D 3 were excised. This explanation is partial at best since D 3 is also present in pitch sets occurring earlier in the work where it ought not to be found, such as in A B C and G C. Alternatively, the fact that D 3 is observed twice in C might prompt us to include it in C in spite of its repeated observation in C which would then account for its presence in A B C, G C and F. First, the sets of interest are not completely represented in the score.

It has already been remarked that this deficiency is manifest in R.

The problem recurs, however, with respect to various pitch sets throughout the score. In each case there exist pitches that are represented in the score neither within the set in question nor within its complement. The appearance of a pitch both within a given set and within its complement baldly contradicts the definition of complementarity. Inconsistencies appear to persist throughout the score based on the set-algebraic computations discussed above, in which significant disagreement between computed sets and their representations in the score is repeatedly observed.

Psychological studies investigating the ability of subjects to name musical tones with different pitches have found that they can reliably do so only when the number of tones is less than 5—6 although subjects with absolute pitch perform better. In addition to the fact that set macro-structure is only one of several factors influencing stream formation cf. Section 3 , other impediments to such identifications would exist.

This would again overtax the usual limits of short-term memory. Would like to have technical explanations. Let me know about Symbolic Music. In my opinion, they detract little from the effectiveness of the piece. This observation in itself, however, highlights certain deeper paradoxes associated with the compositional process.

Xenakis is clearly serious about his conception of the work as a temporal blackboard upon which is inscribed a set-theoretic argument demonstrating the equivalence of two different expressions for the target set F.

It is unrealistic to suppose that even musically astute listeners can apprehend and retain in memory large, harmonically amorphous pitch sets which are subjected to purely stochastic expositions. From a perceptual standpoint, the music is ergodic in pitch at the highest holarchical gestalt levels.

Certainly no sense is ever produced of an argument cast in pitch proceeding towards a conclusion. If it were otherwise, one supposes that the set-theoretic inconsistencies discussed above would not be performed and recorded by musicians of the highest competence.

Rather, the momentum of the music is entirely a product of its rhythmic and dynamic intensity and the tremendous overt technical virtuosity which must be displayed by its performer. Those are some clues to the elemental concerns of his music. But what happens when you hear his music goes beyond even the sensation of teeming natural phenomena or landscapes transmuted into music.

The Sieves of Iannis Xenakis

You'll hear a piano part of mind-bending complexity, which has the unique distinction, as far as I'm aware, of having a separate stave for each finger. You did read that right: Xenakis uses 10 staves in this piece. You'll hear clouds of minutely detailed orchestral sonority wrap around the solo part, like flocks of small birds mobbing an avaricious raptor; and you'll hear a near-continuous rhythmic intensity and textural violence that takes your breath away. Hearing this piece is as awesome an experience as watching some life-changing natural spectacle.

But there's something else as well.However, in electromagnetic music, problems of construction and of morphology were not faced conscientiously. Let n be the number of glissandi per unit of the pitch range density of mobile sounds , and r any portion taken from the range. A pair of disjoint sets click to enlarge Figure 8. Intersections between three sets, A, B, C, and their complements click to enlarge [A. Some expositions of sets such as those of B C and G do not exhibit all of the expected pitches, perhaps only because they are too brief.

There's a huge amount to discover in Xenakis's music, and much of his vast output is out there on YouTube.

Stochastics provides the necessary laws.