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ISMM Bucarest (Roumanie), 1994
Copyright © ISMM 1994
EXAMPLE 1. Melodien by Ligeti (pitch-time representation)
As you will see, there will be no mathematics in my presentation. Considering the title of the Symposium "Music and Mathematics", I must confess that my presentation is mostly music oriented. But I shall try to explain how it could be related to more theoretical and mathematical problems. I would like to explain too how the things I am working on could be related to the very interesting and brilliant lectures that we have already heard during the Symposium, concerning music theory. To that purpose, a classical distinction could help me clarify matters.
Xenakis defines three domains:
To explain what I mean by inside time domain, let us say first that a score, for instance, belongs to the inside time domain. The most basic thing that we can say about a score, from my point of view, is that two relations can defined between events:
By "other levels", I mean weaker representations of the musical content of a score; that is to say that you can forget some information. Let us take an example: suppose you are studying the basic rules of serialism; from this point of view, you need to know that a pitch occurs before another pitch, but you don't need to have information about durations of the pitches, nor about registers of those pitches. So you can forget durations and registers, but you need to preserve the fact that events are successive or simultaneous.
To formalize those two basic relations, I use a theoretical framework which is known in computer science as formal language theory. The corresponding algebraic structure is the free monoid, with an operation called concatenation. It is possible to extend the structure with a new operation which is called superimposition. The resulting algebraic structure is a lattice. We will not develop this point now, but we consider an example to make it more intuitive. Let us assume that two occurrences of an event a are successive; then you can distinguish these two occurrences. But if the two occurences of the same event a are simultaneous, you cannot distinguish them. For this reason, the superimposition is an idempotent operation, whereas the concatenation is not. The idempotent property is one of the basic properties of the lattice structure (for more details see Chemillier 1989).
In the traditional formal language theory, what is studied is sets of sequences which can be computed by different kinds of machines. Since the works of Turing, one has a theoretical and even mathematical classification of machines. We are interested in adapting this point of view to the study of music. That is to say, we try to considere music sequences (which belong to the inside time domain) as the result of abstract computations, and we try to study the process of such computations.
The example by Ligeti we shall study will be considered from this point of view. We shall try to discover the process which is involved in the passage we shall analyze. But there will be no theoretical considerations about this process (which is quite simple). We shall only try to represent this process by means of the programming language Lisp, which provides, as we shall see, powerfull functions which are convenient to describe the process. Gérard Assayag and I have made a computer reconstruction of the example in PatchWork which is a Lisp environment with graphical and musical interface.
In the pitch-time domain represented in Example 1, one can see two special types of areas. The first one is marked by cross-hatching. This corresponds to what could be called textures. A texture is a musical surface of the pitch-time domain resulting from the superimposition of different melodic lines, in which a short motive is repeated and progressively transformed. Many examples of such textures can be found in Ligeti's compositions. The other type of areas represented in black in Example 1 corresponds to sections of the piece in which a part of the pitch space is filled chromatically. The fragment of Melodien I shall try to analyze is the texture which begins the first part of the piece, from measure 14 to measure 30.
Textures of this type have been studied in a recent paper written by Jane Piper Clendinning (Clendinning 1993). This paper focuses on a few pieces composed by Ligeti in 1968-69, which are called "pattern-meccanico" compositions: Continuum for harpsichord, Coulée for organ, the fifth movement of the Second String Quartet, and the eighth of the Ten Pieces for Wind Quintet. They all share a specific feature of construction, which is analyzed in detail in the paper. As we shall see, the principles involved in our analysis of the texture from measure 14 to measure 30 of Melodien are not very different from those described by Jane Piper Clendinning for the pattern-meccanico concept.
From a rythmical point of view, the notes in each part of the texture have equal durations, but these durations are different from one part to the other, so that the texture consists in the superimposition of different speeds. At the end of the section, an accelerando is applied to each part of the texture.
In each part of the texture, rests are inserted to delimit motives. These motives are progressively transformed from the begining to the end of the section. At the begining, the transformations are simple: notes are progressively added to the motives; for instance, the first motive of the piccolo part is F6 A6, then it becomes Eb6 F6 A6, then Eb6 F6 F#6 A6, and so on. One can also mention that the notes of the motives are the same in the different parts of the texture: for instance in measure 17, the notes are Eb6 F6 F#6 A6 in the piccolo part, F6 A6 Eb6 Gb6 in the xylophone part, Eb6 F6 A6 F#6 in the celesta part, and A6 F6 Gb6 Eb6 in the violin B solo part. Let us consider this passage of Melodien.
The maximal number of notes in a motive is ten notes. This maximal number is reached measure 22 for the xylophone, celesta and violin B solo part. In the piccolo part, no motive has more than nine notes. After this point (measure 22), the notes of the motives in the different parts of the texture are not the same, and the evolution of the texture becomes more complicated. We notice that at the end of the process (measure 29), the motives are nearly in ascending order for the piccolo and xylophone parts, and in descending order for the celesta and violin B solo parts. We shall explain the full process in action during this section. As we shall see, this explanation relies on what we shall call virtual reference motives, that is to say motives which are not necessarily played in the different parts of the texture.
F6 A6 Eb6 ( ) ( ) ( ) ( ) ( ) ( ) ( )
and progressively notes are undeleted up to measure 22.
EXAMPLE 2. Xylophone part
When the last note E5 arrives (state 10 measure 22), a new operator is activated. As we can see in Example 2, the B5 in state 9 is decreased a half step down to the Bb5 in state 10. In the next state (state 11), one can see that some other notes are decreased: D#6 and D6 are decreased one step down to respectively C#6 and C6, and C6, E5 and Ab5 are decreased a half step down to respectively B5, Eb5 and G5. We can also notice that the delete operator is once more activated, because in state 11, the G5 has been deleted and replaced by a nil symbol. In fact, this G5 is virtualy present in the motive, because in next state (state 12), it is undeleted again, and at the same time decreased one step down to F5. This situation leads us to introduce our second operator:
In state 15, one can find a new transformation. In order to explain the whole process, we need to introduce a new operator. Recall that at the end of the process, the motives in the xylophone part are in ascending order. This means that the highest notes must be pushed to the right of the motives. For this reason, we have to introduce a third operator which displaces notes to the left hand side of the motives (or to the right hand side in the case of the celesta and the violin B solo parts where the final states are in descending order):
The four parts of the texture can be analyzed in the same manner. They all rely on a ten note motive, and these four reference motives consist of different permutations of the same ten notes. These reference motives are virtual in the sense that they do not necessarily appear in the corresponding part. For instance, as we have seen in the xylophone part, when the last note E5 arrives (state 10), the B5 is decreased a half step down to Bb5, so that the reference motive is never played in this part:
EXAMPLE 3. Reference motive for the xylophone part
The same holds for the violin B solo part: when the last note of the motive arrives, other notes are decreased so that the reference motive is not played. The only part in which the reference motive is actually played is the celesta part: it appears as state 15 at the end of measure 21. In the piccolo part, something interesting should be mentionned. As we have already pointed out, the piccolo part contains no motive with more than nine notes. This means that at every stage of the process, at least one note is deleted. In fact, the reference motive for this part is represented in Example 4:
EXAMPLE 4. Reference motive for the piccolo part
When the E5 arrives (measure 22), the F6 is deleted (and other notes are decreased). Then the E5 is decreased down to D5 ; the F6 comes back as an Eb6, but another note is deleted. Then the E5 (which has become a D5) is one more time deleted, and will never be undeleted again until the end of the process, so that there will never be a ten note motive in this part.
EXAMPLE 5. Representation of the process with two levels
The Lisp environment used for the computer reconstruction of the process is PatchWork. PatchWork combines a Common Lisp interpreter, an elegant graphical interface in which functions are represented as boxes with input/output, and special boxes for music notation and Midi connexions. This system running on a Macintosh has been developped at Ircam. In Example 6 one can see the full process represented as a patch. The four virtual motives are stored in chord boxes. One can verify that the ten notes are the same in the four boxes. These boxes can be open, and an arpeggio representation of the chords would then appear, showing the different permutations of the ten notes used in each instrumental part of the texture. The process described in Example 5 is represented in this patch by the box named processus. It takes as input the virtual motive of the corresponding instrumental part, and the data for the action of the three operators, which are stored in files represented in the patch by the boxes piccolo, xylo, celesta, violon. The output value returned by the processus box is the sequence of notes played by the corresponding instrument. It is then processed by the rythme box, which computes durations for this instrumental part of the texture. The final accelerando is not computed: it is stored as data in the const boxes. The four instrumental parts of the texture are collected in the trame box. Some melodic elements have been added to the resulting polyphony: they are stored in the melodies box. Part of the string background is stored in the tenues box.
EXAMPLE 6. Computer reconstruction in PatchWork
More informations about this patch can be found in a paper written by Gérard Assayag (Assayag 1993). A simulation of the passage has been played on a Yamaha SY77 synthesizer connected to PatchWork running on a Macintosh.
As a second remark, I would like to mention the fact that the process described in this presentation does not pretend to be the exact description of what Ligeti did to compose this passage. It only gives a coherent explanation of the musical content of the passage. However, Ligeti has heard the computer reconstruction of the passage (at Ircam). He said "it is correct". But he also made some comments about the way he composed Melodien. The whole pitch content of the piece was defined as an harmonic skeleton. From this harmonic skeleton, he then chose the pitches actually played by the different parts of the full orchestral score. I think it would not be too unrealistic to assume that this informal operation (to chose pitches from an harmonic skeleton) can be related to the delete operator that we have described, and to the decomposition of the process into two levels we have introduced: the underground level, and the surface level.
EXAMPLE 7. Another derivation (xylophone part, state 15 to state 19)
It is possible to compute all the possible derivations describing the process involved in each part of the texture. This can be done by introducing the notion of subwords of a word, i.d. words obtained by deleting letters in a word. For instance, the word aba may be obtained as a subword of ababa in different ways: aba(b)(a), ab(a)(b)a, a(b)(a)ba, and (a)(b)aba. There exists a simple recursive formula which computes all the different ways a given word is obtained as a subword of another given word. The notion of subword may be adapted to formalize the relation between the motives wi occurring in the instrumental parts of the texture, and the corresponding ten note motive mi at the underground level. One can then write a program computing all the underground sequences which can be associated to the surface sequence corresponding to one part of the texture. Unfortunately, there are many thousand solutions!
Our analysis of the texture should then be improved. One of the problems could be the fact that we have treated separately the four parts of the texture. A more precise analysis can be obtained by considering the surface level as the sequence of all the motives occurring in the four instrumental parts. If we compute all the underground sequences which can be associated to this new surface sequence, the program gives a surprising result: there is only one possible solution. The uniqueness of the solution means that the delete operator does not lose too much information: it is possible to restore the underground level in a unique way. This underground level produced by the computation could be considered as the "harmonic skeleton" of the passage. A full description of this analysis would exceed the limits of the present paper.
Chemillier, Marc. 1989. Structure et méthode algébriques en informatique musicale. Thèse, Université de Paris VII.
Clendinning, Jane Piper. 1993. "The Pattern-Meccanico Compositions of György Ligeti." Perspectives of New Music 31, no. 1: 192-234.
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