An Algorithmic Model Of György Ligeti's Étude No 1, Désordre (1985)

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Home Publications An Algorithmic Model of György Ligeti's Étude No. 1, Désordre (1985) :algorithmic-composition :common-music :ligeti :stanford-university Version: Mon 03/08/1999 See Also: Common Music source code (HTML, 18KB) Common Music source code (Common Music Source, 9KB) MIDI file of an example run (Standard MIDI File, 23KB)

ENGLISH

DEUTSCH

Revision: Wed 03/19/2003

Background: This short paper was written in 1999 in fulfillment of the 24-hour part of Stanford's Ph.D. University Special Area Exams and is hence somewhat cursory. It was presented here with minor corrections in its original version in an attempt to make it accessible to the interested outside the departmental archives. Thus, the code fragments given here pertain to the Common Music version current at that time, 1.3. Please see the links above for updated versions of the code. Also note that Rick Taube, the author of Common Music, has meanwhile reworked this paper into chapter 22 of his book Notes from the Meta-Level: An Introduction to Computer Composition, Swets & Zeitlinger (in print).

An Algorithmic Model of György Ligeti's Étude No. 1, Désordre (1985) Probably inspired by its title, the first of György Ligeti's Etudes for Piano has been analyzed in great depth with respect to a perceived antagonism between chaos and order in the composition (Kinzler, 1991). However, while the study does discuss possible algorithms that Ligeti might have used during the compositional process, it exhibits a fervent musicological passion for infinitesimally minute details. As a result, it largely fails to address matters related to the composition process and thus to explain Ligetis decisions from a perspective of algorithmic control. The purpose of this paper is to arrive at some idea of what specific knobs the presumedly employed algorithms offered Ligeti and, hence, what compositorial potential they represent within the structural framework of the near-finished etude. To accomplish this, the composition is first analyzed with what is believed to constitute reasonable neglect of accidental details and then modeled in the MIDI domain using Common Music as a score description language. As a model, it does make an honest attempt to be realistic, but also trades off accuracy against flexibility. After all, there is no point in implementing an

file://C:\DOCUME~1\BNT\LOCALS~1\Temp\N77Q22SC.htm

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algorithm that cannot be tossed around in parameter-space! The presumed algorithms are then discussed briefly with regard to their historical background. Following that, hints are given for taking the model for a spin and generating general désordre. A note regarding score references: since Désordre does largely away with synchronized barlines, references to the score need to be fully specified with page, system, and measure numbers—and, where necessary, an indication of the part. In the following, this information will be given in shorthand notation as P:S[u|l].M where P, S, and M denote page, system, and measure number, and—if necessary—u or l upper or lower part, respectively. Where systems begin with partial measures, the first complete measure is counted as measure 1. All numbers follow the (only) facsimile edition (Ligeti, 1985).

Structure Description The textural make-up of the etude is comparably straightforward in that each hand is assigned its own part, and that both parts share a common, layered substructure and a continuously pulsating eighth-note raster. Furthermore, each part is restricted to a mode—the right hand to white keys and the left hand to black keys. As crudely keyboard-oriented and commonplace as this use of modality may seem, it actually presents a rather practicable choice that allows Ligeti to let both technically relatively difficult parts share a register where so desired. Aside from these common characteristics, the parts are temporally quite independent and follow their own processual logic. That is, although stretches with near-synchronized phrase structures and prominent formal junctures exist, they are metrically and rhythmically independent when viewed on sub-formal levels. This temporal independence is central to the composition in that, on a large scale, most of its form-defining features emerge from it. Internally, each part is a combination of continuously rising eighth note runs in the background and a rhythmically pronounced and slower melodic line in the foreground. These foreground melodies are played forte throughout the piece and gain further prominence by progressing in increasingly dense chords—thus in effect establishing themselves as an autonomous layer


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against the eight-note background rather than relating to it as melodies to their accompaniment. Yet, both foreground lines feature a phrase structure that is reminiscent of traditional melodic A A' B forms, where two closely related shorter phrases are followed by a longer and developing third phrase. In addition, the phrasing hints, despite the distortion that is induced by an asymmetrical meter in each part, at an underlying simple and song-like two-beat meter. A similarly asymmetric meter that is suggested in both background layers by eighth-note groups of differing lengths is relatively negligible compared to the prominent metric structure in the foreground layers. On a formal level, finally, the piece is structured by both, the temporal and pitch space-related behaviour of its two parts. A first section, 404 eighth notes long, from the beginning of the piece to 3:4.4, is mostly static in tempo, in spite of a small speed-up towards the end. The following section, which extends for 231 eigth notes until 4:4.7, accelerates the foreground voices until their beat equals an eighth note. At this point a typical ligetiesque cut interrupts the lower part, which had by then hit the bottom of the piano range, and resumes with both foreground lines in their original tempo and the lower part in the discant region, producing an aural effect of sudden slow-motion. This third section extends for the remaining 429 eigth notes to the end of the piece and is characterized by a static tempo in the upper part and a slowing down of the lower part. The Data Both foreground lines consist of cyclic repetitions of a constant step pattern (see Figure 1). With each cycle, this pattern is transposed diatonically by a constant interval. The pattern for the upper part consists of 26 steps, 7 for each of the A phrases and 12 for phrase B. The pattern for the lower part has 33 steps, again 7 for each of the A phrases but this time 19 in the extended phrase B.


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Figure 1: Upper and lower part foreground step patterns. Although these patterns bear a strong resemblance to material that has been generated by a rewrite system—a sequence of generations combined with an overall expanding gesture—they satisfy too many traditional musical criteria in order not to call for a standard musical description. The contours of both patterns are highly similar, and especially the upper part, which could have easily been generated by a context-sensitive grammar, exhibits strong motivic features. In contrast to the lower part's pattern, whose B phrase extends by 2 bars beyond the expected 8 bars and ends in a whole note, however, the upper part's B phrase falls short by 2 bars and loops instantly back to its beginning. There are overall 14 cycles in the upper part, each of which is transposed diatonically one step upwards and, due to its greater length, only 11 cycles in the lower part. The smaller number of cycles—which translates into fewer transpositions—is


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overcompensated for by a two-step diatonic transposition downwards and the fact that the lower part's pentatonic mode has fewer steps per octave than the heptatonic mode in the upper part. The cycles align with the score as follows: Cycles: Lower Part: Upper Part: # Transp. Start # Transp. Start =================== =================== ---------------------- Section I: ----------------------1: ds3 2:1.0 1: b4 2:1.0 2: c5 2:2.7 2: as2 2:3.5 3: d5 2:4.7 3: fs2 3:2.2 4: e5 3:2.6 --------------------- Section II: ----------------------4: cs2 3:4.4 5: f5 3:4.4 6: g5 3:4.11 5: gs1 4:1.3 7: a5 4:1.7 6: ds1 4:2.3 8: b5 4:2.5 7: as0 4:3.2 9: c6 4:3.3 10: d6 4:3.10 8: fs0 4:4.1 (1) --------------------- Section III: ----------------------4:4.7 11: e6 4:4.7 9: cs5 5:1.2 12: f6 5:2.4 10: gs4 5:3.7 13: g6 5:4.4 14: a6 6:2.4 (2) 11: ds4 6:2.4 (3)


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(1) Cycle 8 in the lower part reaches the and discontinues after m. 4 in phrase B treble clef (4:4.7). The first note of but for obvious reasons is given as fs5 (2) Cycle is cut off after m. 3 in Phrase (3) Cycle is cut off after m. 2 in Phrase

bottom of the piano range only to resume immediately in the this continuation should read fs to the right hand. B. A'.

The rhythmic picture is less clear due to rasterization effects in the acceleration and deceleration of the patterns. The sequence of rhythms in the upper and lower parts read as follows: Rhythms Upper Part: ; ------------------------------; Sect. I Little Disorder [404 e] 3 5 3 5 5 3 7 ; cycle 1 3 5 3 5 5 3 7 3 5 3 5 5 3 3 4 5 3 3 5 3 5 3 4 5 3 8 ; cycle 2 3 5 3 4 5 3 8 3 5 3 4 5 3 3 5 5 3 3 4 3 5 3 5 5 3 7 ; cycle 3 3 5 3 5 5 3 7 3 5 3 5 5 3 3 4 5 3 3 5 3 5 3 4 5 2 7 ; cycle 4 -- Speed-Up 2 4 2 4 4 2 5 2 3 2 3 3 1 1 3 3 1 1 3 ; ------------------------------; Sect. II Rapid Accelerando [231 e] 1 2 1 2 2 1 3 ; cycle 5 -- start is in sync with cycle 4l 1 2 1 2 2 1 3 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 3 ; cycle 6 1 2 1 2 2 1 3 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 3 ; cycle 7 1 2 1 2 2 1 2


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1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 2 ; cycle 8 1 2 1 2 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 2 ; cycle 9 1 2 1 2 1 1 2 1 2 1 2 2 1 1 1 2 1 1 1 1 2 1 1 1 1 2 ; cycle 10 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 ; ------------------------------; Sect. III Static 3 5 3 5 5 3 8 ; cycle 11 3 5 3 5 5 3 8 3 5 3 5 5 3 3 5 5 3 3 5 3 5 3 5 5 3 8 ; cycle 12 (same) 3 5 3 5 5 3 8 3 5 3 5 5 3 3 5 5 3 3 5 3 5 3 5 5 3 8 ; cycle 13 (same) 3 5 3 5 5 3 8 3 5 3 5 5 3 3 5 5 3 3 5 3 5 3 5 5 3 8 ; cycle 14 3 5 3 5 5 3 8 ; 3 5 3 5 5 8 ; cuts off here

Rhythms Lower Part: ; ------------------------------; Sect. I Static 3 5 3 5 5 3 8 ; cycle 1 3 5 3 5 5 3 8 3 5 3 5 5 3 3 5 5 3 3 5 3 5 3 5 5 3 8 3 5 3 5 5 3 8 ; cycle 2 3 5 3 5 5 3 8 3 5 3 5 5 3 3 5 5 3 3 5 3 5 3 5 5 3 8 3 5 3 5 5 3 8 ; cycle 3


[429 e]

[404 e]

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3 5 3 5 5 2 7 ; -- Speed-Up 3 4 3 4 4 2 2 4 4 2 2 3 2 3 1 3 3 1 4 ; ------------------------------; Sect. II Rapid Accelerando [231 e] 1 3 1 2 2 1 3 ; cycle 4 -- start is in sync with cycle 5u 1 2 1 2 2 1 3 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 3 1 3 1 2 2 1 3 ; cycle 5 1 2 1 2 2 1 3 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 ; cycle 6 1 2 1 2 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 ; cycle 7 1 2 1 2 2 1 2 1 2 1 2 2 1 1 2 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 ; cycle 8 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 ; ------------------------------; Sect. III Allargando [429 e] 5 3 3 5 3 5 3 5 5 3 8 3 5 3 5 5 3 8 ; cycle 9 3 5 3 5 5 3 8 3 5 3 5 5 3 3 5 6 3 3 5 3 5 3 6 5 3 8 3 6 3 5 5 3 9 ; cycle 10 3 5 3 5 6 3 8 3 5 3 6 5 3 3 5 6 3 3 5 3 5 3 6 5 3 9 3 7 3 8 9 3 13 ; cycle 11 3 11 3 21

The Model This analysis translates fairly straightforward into a model of the combined foreground structure. First, the tempo is set


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according to the specification in the score: ;;; ;;; Tempo ;;; ;;; Set as ration in order to work around floating point rounding problems ;;; in cm timing. (defparameter *eigth-note-pulse* (rationalize (rhythm 'e (tempo 76 w)))) This cosmetic hack allows calls to make-item-stream to be embedded in item stream constructor macros. ;;; ;;; Allow the use of make-item-stream in constructor macros (defprop make-item-stream :item-expand t)

Then an algorithm is defined for the upper foreground. As suggested by the anaylsis above, the algorithm is driven by four sets of data: the part's mode, its step pattern, its transposition pattern, and its rhythmic pattern. ;;; ;;; Upper Foreground (defparameter *upper-mode* (transpose (mode white 2 2 1 2 2 2 1) 'c4)) (defparameter '( 0 0 1 -1 -1 2 2 2 4

*upper-fg-steps* 0 2 1 -1 1 3 2 -2 3 5 4 -1 0 3

2

6

; Phrase a ; Phrase a' 5)) ; Phrase b

(defun make-upper-fg-notes (start-note) (intervals (make-item-stream 'items 'cycle *upper-fg-steps*)


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from (steps 1 initially-from (modeify (degree start-note) *upper-mode*)))) (defun make-upper-fg-rhythms () (items 3 5 3 5 5 3 7 3 5 3 5 5 3 7 3 5 3 5 5 3 3 4 5 3 3 5 3 5 3 4 5 3 8 3 5 3 4 5 3 8 3 5 3 4 5 3 3 5 5 3 3 4 3 5 3 5 5 3 7 3 5 3 5 5 3 7 3 5 3 5 5 3 3 4 5 3 3 5 3 5 3 4 5 2 7 2 4 2 4 4 2 5 2 3 2 3 3 1 1 3 3 1 1 3 1 2 1 2 2 1 3 1 2 1 2 2 1 3 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 3 1 2 1 2 2 1 3 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 3 1 2 1 2 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 1 2 2 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1


; cycle 1

; cycle 2

; cycle 3

; cycle 4

; cycle 5

; cycle 6

; cycle 7

; cycle 8

; cycle 9

; cycle 10

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(items (items 3 5 3 5 5 3 8 ; 3 5 3 5 5 3 8 3 5 3 5 5 3 3 5 5 3 3 for 3) 3 5 3 5 5 3 8 ; 3 5 3 5 5 3 8 3 5 3 5 5 3 ; cuts ))

cycle 11-14 5) cycle 14 off here

(algorithm upper-fg-mono midi-note () (vars (ntes (make-upper-fg-notes 'b4)) (rhys (make-upper-fg-rhythms))) (setf note (unmodeify (item ntes) *upper-mode*) rhythm (* *eigth-note-pulse* (item rhys :kill t)))) The algorithm for the lower part foreground is driven by the same types of data. However, since it is eventually pitch-warped into the treble clef after it hit the bottom of the keyboard (4:4.7), its transposition pattern adds a warp function that shifts notes 5 octaves (i.e. 20 mode steps) up after 7 cycles, two 7-note phrases (A and A') and 4 measures of the B phrase (8 notes). (defparameter *lower-mode* (transpose (mode black 2 3 2 2 3) 'cs4)) (defparameter *lower-fg-steps* '(0 0 1 0 2 2 0 1 1 2 1 -2 -2 -1 1

1

2

2

0 -1 -4 -3

0 -1

3

2

; Phrase a ; Phrase a' ; Phrase b 1 -1 0 -3 -2 -3 -5))

(defun make-lower-fg-notes (start-note) (let* ((cycle-length (length *lower-fg-steps*)) (warp-point (+ (* cycle-length 7) 7 7 8)))


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(intervals (intervals (make-item-stream 'items 'cycle *lower-fg-steps* :for 1) on (items (items 0 for warp-point) (items 20 in sequence for most-positive-fixnum)) for cycle-length) from (steps -2 initially-from (modeify (degree start-note) *lower-mode*))))) (defun make-lower-fg-rhythms (items 3 5 3 5 5 3 8 3 5 3 5 5 3 8 3 5 3 5 5 3 3 5 5 3 3 5 3 5 5 3 8 3 5 3 5 5 3 8 3 5 3 5 5 3 3 5 5 3 3 5 3 5 5 3 8 3 5 3 5 5 2 7 3 4 3 4 4 2 2 4 4 2 1 3 1 2 2 1 3 1 2 1 2 2 1 3 1 2 1 2 2 1 1 2 2 1 1 3 1 2 2 1 3 1 2 1 2 2 1 3 1 2 1 2 2 1 1 2 2 1 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 1 2 2 1 1 2 1 1 1 1 1 1 1 1 2

() ; cycle 1 3 5 3 5 3 5 5 3 8 ; cycle 2 (same) 3 5 3 5 3 5 5 3 8 ; cycle 3 2 3 2 3 1 3 3 1 4 ; cycle 4 starts in synch w/ cycle 5u 1 2 1 2 1 2 2 1 3 ; cycle 5 1 2 1 2 1 2 2 1 2 ; cycle 6 1 2 1 2 1 2 2 1 2 ; cycle 7 1 2 1 1 1 1 1 1 2 ; cycle 8


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1 1 1 1 1 1 1 1 1 1 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 6 3 5 5 3 5 3 5 6 3 5 3 6 5 3 7 3 8 9 3 11 3 21 ))

1 1 3 3 3 3 3 3 3

2 1 1 8 8 3 5 9 8 3 5 13

5 3 3 5 3 5 3 5 5 3 8 ; cycle 9 6 3 3 5 3 5 3 6 5 3 8 ; cycle 10 6 3 3 5 3 5 3 6 5 3 9 ; cycle 11 ; cuts off here

(algorithm lower-fg-mono midi-note () (vars (ntes (make-lower-fg-notes 'ds4)) (rhys (make-lower-fg-rhythms))) (setf note (unmodeify (item ntes) *lower-mode*) rhythm (* *eigth-note-pulse* (item rhys :kill t))))

At this stage, both foreground processes are implemented and may be listened to by issuing a Stella [Top-Level]: mix upper-fg-mono,lower-fg-mono 0 Enhancements A further improvement adds octaves and simulate the background layer: ;;; ;;; Add Octaves and a fake background. ;;; ;;; Background notes are "muted" when the foreground is active. To ;;; accomplish this, the names of the background containers must follow the


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;;; foreground container names alphabetically. Local variables are set up ;;; to communicate information between containers. (defparameter *foreground-amplitude* .55) (defparameter *background-amplitude* .4) (merge desordre-v1.0 () (let ((upper-fg-time 0) (upper-fg-note 0) (upper-fg-eighths 0) (lower-fg-time 0) (lower-fg-note 0) (lower-fg-eighths 0)) (algorithm 1a_upper-fg_v1 midi-note (amplitude *foreground-amplitude* channel 0) (vars (ntes (make-upper-fg-notes 'b3)) (rhys (make-upper-fg-rhythms))) (if (evenp count) (setf upper-fg-time time upper-fg-eighths (item rhys :kill t) upper-fg-note (item ntes) note (unmodeify upper-fg-note *upper-mode*) duration (* *eigth-note-pulse* upper-fg-eighths) rhythm 0) (setf note (+ note 12) rhythm duration))) (algorithm 1b_upper-bg_v1 midi-note (length 1064 rhythm *eigth-note-pulse* channel 1) (setf note (unmodeify (item (steps 1 (2 weight .33) (3 weight .15) (4 weight .05) in random for (expr upper-fg-eighths) from (expr upper-fg-note))) *upper-mode*) amplitude (if (= time upper-fg-time) 0 *background-amplitude*)))


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(algorithm 2a_lower-fg_v1 midi-note (amplitude *foreground-amplitude* channel 2) (vars (ntes (make-lower-fg-notes 'ds3)) (rhys (make-lower-fg-rhythms))) (if (evenp count) (setf lower-fg-time time lower-fg-eighths (item rhys :kill t) lower-fg-note (item ntes) note (unmodeify lower-fg-note *lower-mode*) duration (* *eigth-note-pulse* lower-fg-eighths) rhythm 0) (setf note (+ note 12) rhythm duration))) (algorithm 2b_lower-bg_v1 midi-note (length 1064 rhythm *eigth-note-pulse* channel 3) (setf note (unmodeify (item (steps 1 (2 weight .2) (3 weight .05) in random for (expr lower-fg-eighths) from (expr lower-fg-note))) *lower-mode*) amplitude (if (= time lower-fg-time) 0 *background-amplitude*))))) Finally, this code adds realism by simulating the chords that occur in both foregrounds in section three: ;;; ;;; Finally, add more realistic foreground voicings. (merge desordre-v2.0 () (let ((upper-fg-time 0) (upper-fg-note 0) (upper-fg-eighths 0) (lower-fg-time 0) (lower-fg-note 0) (lower-fg-eighths 0))


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(algorithm 1_upper-fg_v2 midi-note (amplitude *foreground-amplitude* channel 0) (vars (ntes (make-upper-fg-notes 'b3)) (rhys (make-upper-fg-rhythms)) voicing) (setf upper-fg-time time upper-fg-eighths (item rhys :kill t) rhythm (* *eigth-note-pulse* upper-fg-eighths) upper-fg-note (item ntes) note (unmodeify upper-fg-note *upper-mode*) ) ;; add: upper octave until 1 after cycle 11 ... (if (< count 261) (sprout (object midi-note start time note (+ note 12) rhythm rhythm channel 0)) ;; ... then add: 2 voices until 2 before cycle 12 ;; 3 voices until 11 after cycle 12 ;; 4 voices thereafter (loop with nvoices = (if (< count 284) 1 (if (< count 297) 2 3)) with voicing = (read-items (intervals 1 2 3 4 5 6 in heap for (expr nvoices) from (expr upper-fg-note))) for degree in voicing do (sprout (object midi-note start time note (unmodeify degree *upper-mode*) rhythm rhythm channel 0))))) (algorithm 2_upper-bg_v2 midi-note (length 1064 rhythm *eigth-note-pulse* channel 1) (setf note (unmodeify (item (steps 1 (2 weight .33) (3 weight .15) (4 weight .05)


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in random for (expr upper-fg-eighths) from (expr upper-fg-note))) *upper-mode*) amplitude (if (= time upper-fg-time) 0 *background-amplitude*))) (algorithm 3_lower-fg_v2 midi-note (amplitude *foreground-amplitude* channel 2) (vars (ntes (make-lower-fg-notes 'ds3)) (rhys (make-lower-fg-rhythms)) voicing) (setf lower-fg-time time lower-fg-eighths (item rhys :kill t) lower-fg-note (item ntes) note (unmodeify lower-fg-note *lower-mode*) rhythm (* *eigth-note-pulse* lower-fg-eighths)) ;; add: upper octave until 23 after cycle 8 ... (if (< count 254) (sprout (object midi-note start time note (+ note 12) rhythm rhythm channel 2)) ;; ... then add: 2 voices until 22 after cycle 9 ;; 3 voices thereafter (loop with nvoices = (if (< count 286) 1 2) with voicing = (read-items (intervals 1 2 3 4 in heap for (expr nvoices) from (expr lower-fg-note))) for degree in voicing do (sprout (object midi-note start time note (unmodeify degree *lower-mode*) rhythm rhythm channel 2)))))


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(algorithm 4_lower-bg_v2 midi-note (length 1064 rhythm *eigth-note-pulse* channel 3) (setf note (unmodeify (item (steps 1 (2 weight .2) (3 weight .05) in random for (expr lower-fg-eighths) from (expr lower-fg-note))) *lower-mode*) amplitude (if (= time lower-fg-time) 0 *background-amplitude*))))) The controls which this model offers over pitch structure and transposition factors are believed to be adequate to Ligeti's approach. The temporal processes, however, contain a fair amount of handcoding and are thus less easily transformed, although possibilities exist to replace them with more formal techniques.

References Kinzler, Hartmut. György Ligeti: decision and automatism in Désordre, 1er Étude, Premier Livre, in: Interface 20(2):89-124. Swets & Zeitlinger, 1991. Ligeti, György. Études pour piano: premier livre. B. Schott's Söhne, Mainz, 1985. http://www.fictive.com/t/pbl/1999_desordre/ligeti.html


©2003 Tobias Kunze. All Rights Reserved.

18. 2. 2006

An Algorithmic Model Of György Ligeti's Étude No 1, Désordre (1985)

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