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.
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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.
This is presumably the consequence of using a random drawing procedure to determine each sounding pitch in succession, with pitches drawn remaining eligible to be drawn again. The exposition of R in Herma comprises only notes, but, using such a random selection scheme, a much larger number of draws will usually need to be made before all 88 different possible pitches have been obtained. Consider, for instance, a situation in which 87 different pitches have already been drawn. The final pitch remains one among 88, so on average a further 88 draws will be required before it is drawn. Indeed, a computer simulation employing ten million trials of the pitch lottery has shown that an average of draws are required to obtain all 88 possible pitches with a standard deviation of draws , and that the probability of obtaining all 88 pitches in or fewer draws is extremely small about 3 in
However, the Denon recording of Herma by Takahashi was made in This suggests that the anomalies in the published score may have survived from the original manuscript. Not all contradictions are so easily resolved, however. For instance, the pitch B1 occurs twice in A and twice in A. The only subsequent sets in which B1 is repeatedly observed are G and G C.
Neither observation definitively places B1 in either A or A and, worse, the sets in question are themselves disjoint!
In the end, we surmise that B1 belongs in A on the weak evidence that other pitches in A sound several times during the exposition of that set i. The question of whether B1 belongs in B or B presents similar difficulties. We place it in B based on the fact that it appears in neither the exposition of A B C nor in that of F.
These can be used to algebraically compute the contents of pitch sets appearing later in the work. These are to be compared with the numeric entries representing the observed frequency of occurrence of each pitch in the score. Occurrences in the sections marked rappel have been included. 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.
Indeed, G is expected to contain 27 different pitches, but its exposition comprises only 12 notes. In the case of G C fewer than half of the scored notes fall within the predicted set.
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.
Section 3. The heights of the three peaks have been established. The third peak and the small arc connecting the straight directrixes of conoid. However, in electromagnetic music, problems of construction and of morphology were not faced conscientiously. Multiserial music, a fusion of the multimodality of Messiaen and the Viennese school, remained, nevertheless, at the heart of the fundamental problem of music. But by it was already in the process of deflation, for the completely deterministic complexity of the operations of composition and of the works themselves produced an auditory and ideological nonsense.
I described the inevitable conclusion in "The Crisis of Serial Music": Linear polyphony destroys itselfby its very complexity; what one hears is in reality nothing but a mass of notes in various registers. The enormous complexity prevents the audience from following the inter- twining of the lines and has as its macroscopic effect an irrational and fortuitous dispersion of sounds over the whole extent of the sonic spectrum.
There is consequently a contradiction between the poly- phonic linear system and the heard result, which is surface or mass. This contradiction inherent in polyphony will disappear when the independence of sounds is total. In fact, when linear combinations and their polyphonic superpositions no longer operate, what will count will be the statistical mean of isolated states and of transforma- tions of sonic components at a given moment.
The macroscopic effect can then be controlled by the mean of the movements of e1ement. The result is the introduction ofthe notion ofproba- bility, which implies, in this particular case, combinatory calculus. Here, in a few words, is the possible escape route from the "linear category" in musical thought. For if, thanks to complexity, the strict, deterministic causality which the neo-serialists postulated was lost, then it was necessary to replace it by a more general causality, by a probabilistic logic which would contain strict serial causality as a particular case.
This is the function of stochastic science. As a result of the impasse in serial music, as well as other causes, I originated in a music constructed from the principle of indeterminism; two years later I named it "Stochastic Music. But other paths also led to the same stochastic crossroads-first of all, Free Stochastic Music 9 natural evcnts such as the collision of hail or rain with hard surfaces, or the song of cicadas in a summer field.
These sonic events arc made out of thou- sands of isolated sounds; this multitude of sounds, seen as a totality, is a new sonic event. This mass event is articulated and forms a plastic mold of time, which itselffollows aleatory and stochastic laws.
If one then wishes to form a large mass of point-notes, such as string pizzicati, one must know these mathematical laws, which, in any case, are no more than a tight and concise expression of chain of logical reasoning. Everyone has observed the sonic phenomena of a political crowd of dozens or hundreds of thousands of people. The human river shouts a slogan. Then another slogan springs from the head of the demonstration; it spreads towards the tail, replacing thc first. A wave of transition thus passes from the head to the tail.
The clamor fills the city, and the inhibiting force of voice and rhythm reaches a climax. It is an event of great power and beauty in its ferocity. Then the impact between the demonstrators and the enemy occurs. The perfect rhythm of the last slogan breaks up in a huge cluster of chaotic shouts, which also spreads to the tail.
Imagine, in addition, the reports of dozens of machine guns and the whistle of bullets addipg their punctuations to this total disorder. The crowd is then rapidly dispersed, and after sonic and visual hell follows a detonating calm, full of despair, dust, and death. The statistical laws of these events, separated from their political or moral context, are the same as those ofthc cicadas or the rain.
They are the laws of the passage from complete order to total disorder in a continuous or explo- sive manner. They are stochastic laws.
Here we touch on one of the great problems that have haunted human intelligence since antiquity: continuous or discontinuous transformation. The sophisms of movement e.
One may produce continuity with either continuous or discontinuous clements.
A multitude of short glissandi on strings can give the impression of continuity, and so can a multitude of pizzicati. Passages from a discontinuous state to a continuous state are controllable with thc aid of probability theory. For some time now I have been conducting these fascinating experiments in instrumental works; but the mathematical char- acter of this music has frightcned musicians and has made the approach especially difficul t.
Here is another direction that converges on indeterminism. The study of the variation of rhythm poses the problem of knowing what the limit of total asymmetry is, and of the consequent complete disruption of causality among durations. The sounds of a Geiger counter in the proximity of a lIiIII radioactive source give an impressive idea of this.
Stochastics provides the necessary laws.