Ligeti's Etude pour piano No. 11 ‘En Suspens’: Multidimensional
Analysis and Insight into the Composition Process
Hugues Bedouelle
4 rue Auguste Bartholdi, 75015 Paris, France–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Abstract
This study aimed at deciphering the inner logics of György Ligeti's Piano Etude
No. 11 and its relations with form. Two parameters were derived from pitch-class
set theory and used for statistical analyses: the sum ∑ of the six components (ic1 to
ic6) of the interval vector, which measures the complexity of a chord, and the ratio(ic1 + ic2 + ic6)/∑, which measures its dissonance. The results have shown that themusic is polyphonic and constructed from two complementary guidonianhexachords. The melodies generated from two short motifs, phrase structures,rhythms, average dissonance and complexity of harmony all vary between the partsand contribute internally to the tripartite form and progression. Polyrhythms result
from the superposition of cyclic rhythms whose periods are coprime integers. The
relations between hidden cyclic patterns of sound and pitch are reminiscent of themedieval motets. The outcome is complex but coherent.
Keywords: music analysis; musical form; piano etude; pitch-class set theory; polyrhythm
Citation: Bedouelle, H. (2020). Ligeti's Etude pour piano No. 11 ‘En Suspens’:
Multidimensional Analysis and Insight into the Composition Process. Retrieved fromhttp://hugues.bedouelle.free.fr/msc/bedouelle2020-ligeti-etude11.pdf
Uploaded: 6 May 2020
Copyright: © 2020 Hugues Bedouelle
Email: hugues.bedouelle@pasteur.fr
Web: http://hugues.bedouelle.free.fr/music.html–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

1. Introduction
György Ligeti (1923–2006) is one of the most original and creative composers of
the second half of the 20th century. Excellent monographs on his life and work
have been published in several languages, in particular English (Steinitz, 2003;
Floros, 2014; Boukobza, 2019, pp. 257–258). The last years of Ligeti's life werepartly occupied by the composition of his Etudes pour piano, in the tradition of
those by Scarlatti, Schumann, Chopin and Debussy. The first (1985), second© 2020 Hugues Bedouelle1

(1988–1994) and third (1995–2001) books contain respectively six, eight and four
Etudes (Ligeti, 1988; Ligeti, 1994; Ligeti, 2001a). They are considered as a major
set of 20th century compositions for the piano. A recent monograph summarizes
much of what is known so far about them (Boukobza, 2019).
Ligeti's writings inform us of his interests, influences and main researches
during the composition of his Etudes. He wrote them as studies of both pianistic
virtuosity and music composition, starting from simple central ideas and goingfrom simplicity to extreme complexity (Ligeti, 2013, p. 290). His influences haveincluded jazz, the music of sub-Saharan Africa, the Ars subtilior movement of the
14th century and the Studies for Player Piano by Conlon Nancarrow (1948–1992).
They inspired him to develop complex melodic and polyrhythmic structures, andexperiment the possibility of creating illusory melodic and rhythmic patterns by the
combination of several voices (Ligeti, 2013, pp. 127–129 and 287–290). In
parallel, Ligeti searched for new ways of thinking harmonically that would depend
neither on total chromaticism nor diatonicism (Ligeti, 2013, pp. 123–125). He
found that systems of complementary scales enabled him to construct new
harmonic configurations that were neither tonal nor atonal but in-between (Ligeti,
2013, p. 298). Ligeti was deeply interested in the theory of musical form. He
strongly advocated the view that form goes beyond the simple relations of the parts
between them and with the whole. The way in which the parts function internallyand act within the whole, has an important role (Ligeti, 2001b, pp. 147–162).
Ligeti's piano Etudes have been recorded by such exceptional artists as PierreLaurent Aimard (1997), Toros Can (2002), Idil Biret (2003), Fredrik Ullen (2006)
and several others since then. Aimard has worked directly with Ligeti, two of theEtudes are dedicated to him (Nos. 10 and 12), he has premiered many of them, andhis recordings of the Etudes are considered as an excellent rendition and areference for their interpretation (Bleek, 2019; Boukobza, 2019, pp. 239–249).
Etude No. 11, entitled En Suspens, occupies the middle position in the second
book of Etudes and it functions as a transitory point of rest in this cycle of pieces.
It was written in 1994 after the other Etudes of the second book and the Concerto
pour piano et orchestre (1988). Therefore, one can assume that many of themusical concepts and ideas that Ligeti had developed in his previous compositions
for piano would be included in Etude No. 11. This Etude has been the subject ofpartial analyses (Chang, 2007, pp. 140–145; Baik, 2009; Can, 2011; Pace, 2012;
Beffa, 2016; Marin, 2017; Boukobza, pp. 150–152) and proposed as a case study inpost-tonal theory (Straus, 2016, pp. 154–155). However, the results of theseanalyses are generally reported at a macroscopic level and do not explain thedetailed inner mechanisms of the piece. They privilege specific aspects, forexample the spatio-temporal structure of the piece (Chang, 2007) or the separate
voices of its polyphonic structure (Marin, 2017). The goal of the present work was
to decipher the internal logics of Etude No. 11 and its relations to form, and thus
gain insight into the composition process. Towards this goal, I performed a© 2020 Hugues Bedouelle

2

thorough multidimensional analysis of Etude No. 11, in particular of its scalarmaterial, melodic material, harmony and rhythm. I explored how the interactions
between these different musical elements contribute to the character, form andcomplexity of the piece while keeping coherence. The results and conclusions of
this analysis could interest pianists, composers, music analysts and historians.

2. Materials and methods
The Schott editions of Ligeti's Etudes pour piano and Concerto pour piano etorchestre were used in this work (Ligeti, 1985; Ligeti, 1988; Ligeti, 1994; Ligeti,
2001a). Etude No. 11 can be purchased separately. Its performances guided theanalysis in several instances. In the text below, the pitches of a chord are listedfrom the lowest to the highest without separation, e.g. E4B4A♭5D♭6. The pitches of a
melody are separated with hyphens, e.g. D♭6-E♭5-D♭6-B♭5-A♭5-G♭5. In the tables,
however, the pitches of chords are also separated with hyphens for simplicity.
The analysis of Etude No. 11 is complicated by the two different time
signatures of the bar: 6/4 for the upper staff and 12/8 for the lower staff or staves.
The position of notes in the score was indicated by a couple of integers (x, y)
where x is the bar number and y is the beat number, each beat with the time valueof a crotchet (quarter note), e.g. beat (9, 4) for beat 4 of bar 9. The interval of time
between beats y and z of bar x was noted (x, y–z).
Chords written on different staves are rarely struck simultaneously because of
the polymetre of the bar. To enable a detailed harmonic analysis, each bar was
divided into twelve sub-beats, each with the time value of a quaver (8th note). The
chords whose notes are struck on the same sub-beat, were named in-phase chords;
those whose notes are struck on different sub-beats but whose time lengths overlap,
were named out-of-phase chords.
The in-phase and out-of-phase chords were analyzed according to Pitch Class(PC) Set method (Forte, 1973; Walter, 2001). Chords whose normal forms are
related by a transposition, have the same function. The consonant intervals
correspond to components ic3, ic4 and ic5 of the interval-content vector (icvector); and the dissonant intervals to ic1, ic2 and ic6 (Schoenberg, 1978, pp. 18–
22). For statistical analyses, the set of ic-vectors was considered as a subset of N6
with the corresponding law of addition (N, natural numbers). In general, the sum of
ic-vectors is not an ic-vector; however, an example where it is an ic-vector isdescribed in the Discussion section. In this work, the complexity ∑ of a sextuple isdefined as the sum of its six components; the harmonic complexity ∑H of a pc-set(or chord) is defined as the complexity of its ic-vector. ∑H depends only on thenumber n of pitch classes in the pc-set and is equal to n(n–1)/2 (Forte, 1973, p. 19).
The complexity of a sum of ic-vectors is equal to the sum of their individual
harmonic complexities.

© 2020 Hugues Bedouelle

33. Results
3.1. Overall organization
The score is 35 bars long. Most of the score is written on two staves. It is written
on three staves in bars 16–17 and 28–33 and the two lower staves are then linked
together. The upper and lower staves have different key and time signatures. The
upper staff corresponds to the right hand and the lower staff or staves to the left
hand, without ambiguity. The overlaps or crossings of hands are indicated bychanges of clefs in the corresponding staves (the words hand and staff are used
indifferently in the following).
The score can be divided into parts and phrases. Part 1 comprises bars 1–17;
Part 2, bars 18–26; and Part 3, bars 27–35. Part 1 can be further divided into fourphrases. Each of them ends with a minim and a rest in the upper staff and is exactly
25 crotchets long. The right hand plays the same notes in the third and fourthphrases as in the first and second ones, but transposed downward by an octave.
Therefore the four phrases of Part 1 form two phrase-groups. Part 2 comprises twophrases covering bars 18–20 and 21–26, respectively. Each phrase ends with adotted minim in both staves. Part 3 also comprises two phrases, covering bars 27–
30 and 31–35, but without clear separation between them. In Parts 2 and 3, the key
signatures are exchanged between the left and right hands at the junction between
the first and second phrases. Parts 1 and 2 are polyphonies of two voices, eachvoice comprising mainly dyadic chords and more rarely monads or triads. Part 3 isconstructed similarly, except that each main voice is accompanied by anornamental voice, for a total of four voices.
3.2 Scalar material
The precise nature of the scalar material in Etude No. 11 has taken a long time toemerge. The upper and lower staves have different key signatures, one with noalteration and the other one with five flats. For some authors, the absence ofalteration in one of the key signatures implies a C-major mode whereas the fiveflats in the other one imply a D♭-major mode, since the absence of accidentals
eliminates the minor modes (Baik, 2009, p. 51; Gottlieb, Neveux, Rignol & Thinat,
2001, p. 101). Other authors mention that: (i) the C pitch of the D♭-major scale andthe F pitch of the C-major scale are missing in the whole piece, (ii) the two scales
are hexatonic and (iii) they are related by a tritone transposition (Marin, 2017;
Boukobza, 2019, p. 150). In the present work, I made the alternative assumptionthat the F♯ pitch and not the F pitch is missing in one of the scales. In thisassumption, both scales are hexatonic, one derived from D♭ major and the otherone from G major with their leading tones omitted. They are devoid of tritone
intervals, deduced from one another by transposition of a tritone andcomplementary (Fig. 1A). This assumption opened new perspectives in theanalysis of Etude No. 11.
© 2020 Hugues Bedouelle

4

Figure 1. Melodic motifs in the upper staff of Etude No. 11. The music examples may
have been transposed by one or several octaves for clarity. A) The two hexatonic scales
used in the score. B–F) Melodic lines present in bars 1–4 (B), 5–6 (C), 18–19 (D), 23–24(E), 27-29 (F). G) Comparison of motifs present in bars 22 (top voice) and 29–30 (bottom
voice). H) Upper staff of bars 30–31. I) Comparison of motifs present in bars 5 (top voice)and 31 (bottom voice). J–L) Comparison of motifs present in the upper staff of bars 1 and 9(J), the lower staff of bar 9 (K) and the upper staff of bar 31 (L).

3.3. Melodic material
The main melodic line is played by the right hand in Etude No. 11. This assertion
is supported by several properties of the score. In Part 1, the melody of the righthand in bars 1–9 is exactly transposed by one octave in bars 9–17 while theaccompaniment by the left hand is modified. The first phrase ends with a halfcadence and the second phrase with a deceptive cadence in the right hand. The
dynamics and rhythmic density of the right hand are higher than those of the left
© 2020 Hugues Bedouelle

5hand and more elaborate. In Parts 1 and 2, the slurs delimitate longer musicalsegments and the harmony is more complex in the right hand than in the left hand.
The melodic lines of Etude No. 11 are constructed by combinations of two
motifs. Motif 1 comprises: i) a down skip, ii) an up skip, and iii) one or several
down steps. The intervals of the skips are variable. Motif 2 is a descending
hexatonic scale, often containing the characteristic step of minor third, either D♭B♭ or G-E.
The melody of Part 1 is constructed mainly from Motif 1. This motif andvariations with one additional anterior or posterior upskip are played at the
beginning of the first and second phrases by the right hand (Figs. 1B and 1C). The
accompaniment by the left hand appears erratic at first sight. However, by allowing
an anacrusis of two notes at the beginning of Part 1, the musical line of the left
hand rearranges in three-and-a-half meaningful phrases. Each phrase spans exactly
three bars and can be described as the succession of two segments of five and fournotes (Fig. 2A). Moreover, the melodic line that corresponds to the top notes of thedyads, has three noticeable features: (i) skips are predominant, (ii) successive steps
are avoided, (iii) the five-note segment of each phrase has its highest or lowest
pitch on the central note. The melody of the left hand also includes threesymmetrical patterns. The pentachord D4-B4-C5-G4-A4 in bars 1–2 and thepentachord A2-G2-C3-B2-D2 in bars 10–12 are in retrograde relation and bracket thethree-and-a-half phrases just described (Fig. 2A). A motif in dotted quavers in bar
12 is present in a quasi-inverted form in bars 13–14; the centre of inversion is
located between C and C# (Figs. 2B and 2C). A descending glissando in bar 16
mirrors an ascending glissando in bar 15. These symmetries give meaning to thecorresponding patterns.
The melody of Part 2 is constructed mainly from Motif 2 but with an important
contribution by Motif 1. The right hand plays Motif 2 twice at the beginning of thefirst phrase (bars 18–19, Fig. 1D) and again at the beginning of the second phrase(bar 21). It plays three combinations of Motifs 1 and 2 in the remainder of Part 2(bars 21–26; see Fig. 1E). The melodic line of the left hand has the same structure
in Part 2 as in Part 1 (compare Figs. 2A and 2D). It is organized in three phrases ofexactly three bars each. In each phrase, the five-note segment is a descending scale,in sharp contrast with the corresponding segment of Part 1, which was mainlycomposed of skips. The four-note segment is made of skips, except in the thirdphrase where it is a descending scale. The right and left hands play descending
scales in imitation in several regions of Part 2. The same descending scale of fournotes, played either by the right hand or by the left hand, brackets Part 2 (Figs 1D
and 2D).
The melody of the top voice at the beginning of Part 3 is nearly identical tothat at the beginning of Part 1 until it skips from E♭6 to G♭6 at the beginning of thethird bar (Figs. 1B and 1F). It then continues with a descending scale and ends witha motif that is a tritone transposition of a motif in Part 2 (Fig. 1G). The descending© 2020 Hugues Bedouelle6

scale is played as 7th dyads, except where the hexatonic scale does not allow suchan interval (bars 29–30). The motifs of the top voice at the beginning of the second
phrases of Part 3 and Part 1 are approximately related by a tritone transposition(Fig. 1I). In the upper staff of bars 30 and 31, two illusory melodic patterns inquavers, G♭5-F5-D♭5-F5-E♭5 and C6-B5-G5-B5-A5, are composed of notes from bothmain voice and accompaniment; they are related by transposition of a tritone andclearly recognizable on audition (Fig. 1H). In bars 33–34, the right hand plays A5C6-E6-G6, which is an arpeggiated minor 7th chord, and then G6-A6-B6-D7, which isan arpeggiated Mu chord (a major triad with added 2nd). These ascending arpeggi
are played mostly as 9th dyads and mirror the descending scale of bars 29–30.
Additional melodic similarities are present in bars 1, 9 and 31 (compare Figs. 1J,
1K and 1L).

Figure 2. Melodic motifs in the lower staff of Etude No. 11. The music examples may
have been transposed by one or several octaves for clarity. A) Motifs present in bars 1–12;
an anacrusis of three dotted crotchets has been introduced and the slurs have been modified
to emphasize two repeated motifs of five and four notes. B–C) Motifs present in bars 12 (B)and 13–14 (C). D) Motifs present in bars 18–26; the slurs emphasize two repeated motifs offive and four notes.

In conclusion, the melodic materials of the three parts derive from both Motifs
1 and 2 and are strongly interrelated, with emphasis on Motif 1 in Parts 1 and 3,© 2020 Hugues Bedouelle

7and emphasis on Motif 2 in Part 2. The melodies of the left hand have similar
constructions in Parts 1 and 2 but with a contrast between skips in Part 1 and steps
in Part 2. Additional melodic similarities create a network of relations andcoherence between the different phrases and parts.
3.4. Registers
In most of Part 1, the two voices play in different registers. The two voices overlap
briefly in bars 5, 6, 8 and 16. The two hands cross widely in bars 12–14 and thiscrossing gives the illusion of a third voice. In Part 2, the two voices share rather
similar registers. They overlap in bars 18–20, 22 and 24, and this overlap briefly
creates the illusion of a compound melody in bars 18 and 24. In most of Part 3, thetwo voices have different registers. However, they extensively overlap in bars 33
and 34 and it becomes difficult to differentiate them. They separate again at the endof bar 34 and this separation of register contributes to signal the end of the piece.
The pitches of the piece go from A0 to A7, i.e. nearly the full span of the modern
piano. There is a gradient of pitches towards the lower end of the register in Part 1.
By contrast, there are gradients of pitches towards the higher end of the register inboth Parts 2 and 3. The extreme pitches at the end of the parts function as
delimiters since all three parts start in the medium register. Most of Ligeti's Etudes
end in extreme registers and dynamics like En Suspens, with consequences on
timbre and harmony since the overtones of a pitch are audible or not according to
the register.
3.5. Segmental analysis
Melodic segments have often an intrinsic harmony that can be used to create
relations between them or harmonize them. The melody of the right hand in Parts 1
and 2 of Etude No. 11 is segmented by slurs. The melody of the left hand is
structured by alternating segments of five and four notes, as described above (Figs.
2A and 2D). Two melodic lines exist for each hand, corresponding to the top andbottom notes of their constitutive dyads. The pc-set classes of fifty five segments inPart 1, Part 2 and the beginning of Part 3 are reported in Table 1.
The whole hexachord (Forte number 6-32) is used in four segments of theright hand. It corresponds to a minor 11th chord (Persichetti, 1961, 82–85). The
pentachord subsets of the hexachord belong to three set classes. Set classes 5-35
and 5-23 are widely used, in eight and seven segments, respectively. Set class 5-35
corresponds to the maximally-even pentatonic scale. Set class 5-27, whichcorresponds to a major or minor 9th chord, is used only once and in its minor form.
The tetrachord subsets belong to seven set classes. The set classes 4-26, 4-23 and4-22 are widely used, four, seven and nine times respectively. Set class 4-26
corresponds to a major added-6th or minor 7th chord; 4-23, to a 7th suspension 4
chord; and 4-22 to a major added-2nd or -4th (Mu) chord. Set class 4-20, whichcorrespond to a major 7th chord, is not used at all. Set classes 4-14, 4-11 and 4-10,© 2020 Hugues Bedouelle

8which correspond to clusters, are used at most twice. The trichord subsets are usedin segments at the end of phrases. Thus, the tetrachords that have known functionsin jazz, are favored whereas those that correspond to clusters, are avoided. The
major 7th tetrachord and major 9th pentachord are totally avoided (see Paragraph
3.9).
Table 1. Segmental properties of Etude No. 11
Hand
RHt
RHb
LHt
LHb
RHt
RHb
LHt
LHb
RHt
RHb

Bars
1–9
1–9
1–12
1–12
18–26
18–26
18–26
18–26
27–28
27–28

Forte number
5-35, 5-35, 3-2 / 4-11, 6-32, 4-14/
5-23, 4-11, 3-9 / 4-26, 6-32, 4-23/
5-23, 4-23 / 4-26, 3-7 / 4-23, 4-23 / 5-23
4-23, 3-11 / 4-22, 4-22 / 4-14, 3-9 / none
4-22, 5-35, 5-23 / 4-26, 5-35, 6-32, 5-35, 3-2/
none, 6-32, 5-35 / 4-22, 4-22, 6-32, 5-23, 3-6/
5-23, 4-22 / 5-27, 4-23 / 5-35, 4-22/
none, 4-22 / 5-23, 4-26 / 5-35, 3-2/
4-23, 3-7
4-22, 3-7

RHt and LHt, top melodic lines of the right hand and left hand, respectively; RHb and LHb,
bottom lines of the right and left hands; (/), end of phrases. The segments of the right handare delimited by the slurs of the score, except for measures 27-28. The segments of the left
hand are alternately five and four notes long and presented in Fig. 2A and 2D.

3.5. Harmony of the separate right and left hands
The harmonic analysis of Etude No. 11 is complicated by two factors: the twocomplementary hexatonic scales and the different time signatures (6/4 and 12/8)
for the upper and lower staves. It may be performed at two different levels, theintra-harmony within each staff and the inter-harmony created by combining thetwo staves and therefore scales. Analysis of the intra-harmonies allows one toindependently characterize each of the two main voices of the polyphony. Analysisof the inter-harmony should consider both the in-phase chords, which are struck on
the same sub-beat (i.e. quaver), and the out-of-phase (or latent) chords, whose
notes are struck on different sub-beats but whose time lengths overlap. Here, astatistical approach was used to understand how consonance and dissonancecontribute to shape the overall form of Etude No. 11 (see Methods).
In a first step, I analyzed separately the chords played by each hand andcomputed their interval content vectors (ic-vectors). I then computed the sum ofthese ic-vectors for the different parts or phrase-groups (Table 2). There is no
tritone (ic6 = 0), which is consistent with the hexatonic scales used. In the firstphrase-group of Part 1 (bars 1–9) and upper staff, there are no minor 2nd or major
7th (ic1 = 0), and only few major 2nd or minor 7th (ic2 = 4). The dissonantintervals (ic1 + ic2 + ic6) account for 12% of the 33 intervals. In the lower staff,
the dissonant intervals account for 40 % of all intervals. Therefore, the right hand
© 2020 Hugues Bedouelle

9plays more consonant dyads than the left hand in the first phrase-group. In thesecond phrase-group (bars 9–17), the right hand plays the same chords as in thefirst phrase-group, transposed downward by an octave, whereas the left hand plays
different chords. Therefore, the second phrase-group appears as a harmonic
variation of the first one. In the lower staff of the second phrase group, thedissonant intervals account for 17 % of all intervals. Overall, harsh dissonances are
avoided in the chords played by either the right or the left hand in Part 1.
In Part 2 (bars 18–26), the dissonant intervals account for 60 % of all intervals
for the right hand, and 79 % for the left hand. Thus harsh dissonances are favouredin the chords played by either the right or the left hand. In Part 3, I limited myself
to the accented chords. In the first phrase (bars 27–30), the dissonant intervals
account for 44 % of all intervals for the right hand and 33 % for the left hand. In
the second phrase (bars 31–35), they account for 47 % of all intervals for the righthand and 31 % for the left hand. Therefore, there is a minority of dissonantintervals in Part 3 and no significant difference in the dissonance contents of its
first and second phrases. In conclusion, when the right and left hands are
considered separately, they play consonant chords in Part 1, highly dissonant
chords in Part 2, and chords of intermediate consonance in Part 3.
Table 2. Statistics of the chords played by the right and left hands
Bars

Hand

1–8
1–8
9–17
18–26
18–26
27–30
27–30
31–35
31–35

RH
LH
LH
RH
LH
RH
LH
RH
LH

Chord
#
33
15
6
32
12
18
18
20
20

Sum of ic-vectors∑(0,4,13,8,8,0)(0,6,2,2,5,0)(0,1,0,1,4,0)(7,17,4,2,10,0)(4,7,0,1,2,0)(1,7,5,0,5,0)(0,9,6,5,6,0)(2,15,5,2,12,0)(2,9,7,5,13,0)

33
15
6
40
14
18
26
36
36

ic1+ic2+ic6
(%)
12
40
17
60
79
44
33
47
31

For bars 27–35, only the accented chords have been considered. Chord #, number of chords
analyzed, excluding the monads; Sum of ic-vectors, vector obtained by adding the icvectors of the analyzed chords; ∑, total number of intervals for the analyzed chords,
obtained by adding the six components of the vector in column 4 (see Methods).

3.6. Harmony of the in-phase chords
In a second step, I analyzed the chords that are played in-phase by the right and left
hands and therefore correspond to inter-harmony (Tables 3–5). For Part 1, the sumof the ic-vectors is equal to (5,10,17,16,9,8) for the 19 in-phase chords. The
consonant intervals (ic3 + ic4 + ic5) are in majority (65 % for 65 intervals). For
Part 2, the vectorial sum is equal to (24,14,8,8,7,6) for the 15 in-phase chords. The
dissonant intervals are in majority (66 % for 67 intervals). For Part 3, the vectorial© 2020 Hugues Bedouelle

10sum is equal to (13,24,33,34,26,16) for the 18 in-phase and accented chords. The
consonant intervals are again in majority (64 % for 146 intervals). Therefore, thetrend that I found by separately analyzing the right and left hands, is confirmed byan analysis of the in-phase chords. There is a harmonic contrast between Parts 1
and 3, which are rather consonant, and Part 2, which is highly dissonant, at least for
the in-phase chords. Tables 3–5 show that the majority of the in-phase chords(13/19) in Part 1 and accented in-phase chords (17/18) in Part 3 belong to a tertian
harmony (Persichetti, 1961, pp. 66–92). In contrast, only one of the 15 in-phasechords of Part 2 belongs to this type of harmony.
Table 3. Properties of the in-phase chords for bars 1–17
Position(1, 1)(2, 4)(3, 4)(5, 4)(8, 1)(8, 4)(8, 6)*(9, 3)*(9, 4)(11, 1)(11, 4)(12, 4)(14, 1)(15, 2)(15, 3)(15, 4)(16, 3)(16, 5)(16, 6)(17, 1)(17, 2)*(17, 4)Chord
E4-B4/A♭5-D♭6
G4/D♭5-B♭5
B3-E4/D♭5-A♭5
G4/D♭5-E♭5
B3-A4/G♭4 -E♭5
C4/E♭4-G♭4
C3-D4/B♭3-F4
A2-B2/A♭4-D♭5
E3-A3/D♭4-E♭4
G2/F4-D♭5
B2/E♭4-D♭5
B4/D♭4-F4
D7/B♭3-G♭4
E1/E♭4-G♭4
D2/D♭4-F4
C3/A♭3-E♭4
A5/G♭3-E♭4
C4/F3-D♭4
D3/E♭3-G♭3
E2/A♭2
D2-A2/B♭2-F3
A1-E2

IC-Vector
012120002001012120010101012111002001021120121110111111010101020100010101000300111000101100001110002001
100110101100000100101220000010Forte #
4-26
3-10
4-26
3-8
4-27
3-10
4-22
4-11
4-z29
3-8
3-6
3-8
3-12
3-2
3-3
3-11
3-10
3-4
3-3
na
4-20
na

Function
Minor 7th
Diminished
Minor 7th
Augmented 6th
Dominant 7th
Diminished
Major 9th (-7th)
Major 9th (-5th)none
Augmented 6thnone
Augmented 6th
Augmented
none
none
Major
Diminished
none
none
Major 3rd
Major 7th
Perfect 5th

The positions of the chords are given by the couple (bar, beat) and their functions according
to (Schoenberg 1983). (/) Separates the notes that are played by the left and right hands; (*)
Chords that are not in phase but were included in the table for the discussion on thetransitions between phrases and parts; na, not applicable.

3.7 Harmony of the out-of-phase chords
The above harmonic analysis was only partially satisfactory because it involved
only the in-phase chords. Are the out-of-phase chords a way of hiding or© 2020 Hugues Bedouelle11smoothing highly dissonant chords or intervals? I therefore analyzed the totality of
the chords, either in-phase or out-of-phase in two representative portions of thepiece, the first phrase-group of Part 1 and the entire Part 2. Ninety-eight chords
were analyzed for bars 1–9. The sum of the ic-vectors was equal to(53,67,119,106,55,59), with a large majority of consonant intervals (61 % of 459
intervals). One hundred and eight chords were analyzed for bars 18–26. The
vectorial sum was equal to (142,111,52,75,62,50), with a large majority ofdissonant intervals (62 % of 492 intervals). Therefore, the analysis of the combinedin- and out-of-phase chords was consistent with the analyses of the separate hands
or in-phase chords. The out-of-phase chords, which function through resonance, do
not appreciably blur the consonant or dissonant character of harmony in Etude No.
11.
Table 4. Properties of the in-phase chords for bars 18–26
Position(18, 1)(18, 4)(19, 1)(19, 4)(20, 1)(20, 4)(21, 1)(21, 4)(22, 1)(22, 4)(23, 4)(24, 1)(24, 4)(25, 4)(26, 4)

Chord
E5/F5
C5/B♭4
A4/E♭5
B4-C5/A♭4
D5/D♭5-A♭5
A4-E5/B♭4-A♭5A♭4-B♭4/B5-C6
F4/E5
D♭4/E5-C6
D♭5-E♭5/D5-C6
E♭4-D♭5/C6-B6
F6-G♭6/G5-E6
D♭6/C5-B5G♭5-F6/B6-A7
F5-G♭5-B♭5/C6-G6-B6IC-Vector
100000010000000001101100100011210111221100100000101100321000221100321000210000111111421242Forte #
na
na
na
3-3
3-5
4-5
4-2
na
3-3
4-1
4-2
4-1
3-1
4-z15
6-z6

Function
Minor 2ndMajor 2nd
Tritonenone
Viennese trichord
none
none
Major 7th
none
none
none
none
none
none
none

See legend to Table 3.

3.8. Harmonic complexity
The harmonic density of a chord, measured as the number of its pitch classes, is avery rough parameter. I have therefore introduced a new parameter, the harmoniccomplexity of a chord, which is defined as the sum of its six ic-vector components(Methods). I deduced the average harmonic complexities of the in-phase chords inthe three parts of the score from the vectorial sums that are given in Paragraph 3.6.
The average harmonic complexity is equal to 65/19 = 3.4 for the 19 in-phasechords of Part 1; it is equal to 67/15 = 4.5 for the 15 in-phase chords of Part 2;
finally, it is equal to 146/18 = 8.1 for the 18 accented and in-phase chords of Part
3. Thus, the average harmonic complexity increases from Part 1 to Part 2 (from 3.4© 2020 Hugues Bedouelle12

to 4.5), and sharply from Part 2 to Part 3 (from 4.5 to 8.1).
Table 5. Properties of the in-phase chords for bars 27–35
Position(27, 1)(27, 4)(28, 4)(28, 5)(29, 1)(29, 4)(30, 4)(30, 5)(31, 1)(31, 4)(32, 4)(32, 5)(33, 1)(33, 4)(34, 4)(34, 5)(35, 1)(35, 4)Chord
E4-B4/A♭5-D♭6
C4-G4-A4/E♭5-B♭5E3-G4/D♭5-B♭5
C4-A4/G♭5-E♭6
E4-C5/A♭5-G♭6
G4-A4/E♭5-D♭6
A2-E3/G♭4-D♭5
B2-G3/A♭4-E♭5B♭3-F4/D5-A5
F3-B♭3-A♭4/D5-C6
D♭3-A♭3-F4/B5-E6
G♭2-D♭3-B♭3/G5-A5-D6
F3-E♭4/A4-B5
B♭4-A♭5/D5-E6
E♭6-A♭6/C6-D7
C5-E5-B5/D♭6-A♭6-E♭7F♯5-C♯6/D6-A6-E7
B4-F♯5/D♯6-A♯6-C♯7

IC-Vector
012120123121004002004002020301020202012120101310101220032221113221313431020202020202111111323430122230122230Forte #
4-26
5-25
4-28
4-28
4-24
4-25
4-26
4-19
4-20
5-34
5-32
6-z19
4-25
4-25
4-z29
6-14
5-27
5-27

Function
Minor 7th
Seven-sixth chord
Diminished 7th
Diminished 7th
Augmented 7th
Dominant 7th flat 5
Minor 7th
Minor major 7th
Major 7th
Dominant 9th
Dominant 7th sharp 9
Major 5th+Suspended
Dominant 7th flat 5
Dominant 7th flat 5
none
Major 7th+Suspended
Major 9thMajor 9th

See legend to Table 3. Only the chords that are accented in both hands have been
considered.

3.9. Transitions
Since the scales of the upper and lower staves have no seventh degree, leading to a
tonic, I analyzed the initial and final chords of the different phrases, and the chords
that are present in the transitions where the two staves exchange their scales. I
found that the initial and final chords of the phrases in Parts 1 and 3 are obtained as
subsets of a major 9th chord (Fig. 3A).
In Part 1, the initial chord E4B4A♭5D♭6 [beat (1, 1)] of the first phrase-groupcan be considered either as a major added-6th chord or as a minor 7th chord. The
latter corresponds to a major 9th chord with the root omitted. The final chord
C3D4B♭3F4 [beat (8, 6)] of this phrase-group corresponds to a major 9th chord withthe 7th degree omitted. The initial chord A2B2A♭4D♭5 [beat (9, 3)] of the second
phrase-group corresponds to a major 9th chord with the 5th degree omitted. The
penultimate chord D2A2B♭2F3 [beat (17, 2)] of this phrase-group correspond to amajor 9th chord with its 9th degree omitted, and therefore to a major 7th chord.
The last chord of the first phrase C4A♭4E♭5 [beat (4, 5)] and the last chord of thethird phrase C1A♭3E♭4 [beat (13, 2)] are identical major triads and would bedominant chords if the piece was written in D♭ major. The last chord of the fourth© 2020 Hugues Bedouelle

13phrase A1E2 [beat (17, 4)] is a perfect 5th and ends Part 1.
In Part 3, the initial chord E4B4A♭5D♭6 [beat (27, 1)] of the first phrase isidentical to the initial chord of Part 1 [beat (1, 1)]. The transitional chord
B♭3F4D5A5 [beat (31, 1)] between the first and second phrases is identical to thepenultimate chord D2A2B♭2F3 [beat (17, 2)] of the fourth phrase in Part 1, exceptthat the two hands are permuted. The final chord of Part 3 is an authentic major 9th
chord [beat (35, 4)].

Figure 3. Initial and final chords of phrases. A) For each chord, the figure indicates thecoordinates of the chord in the score, its original disposition, its tertian equivalent and the
major 9th chord from which it may derive. B) Chords at the junction between the first andsecond phrases of Part 3. The tertian equivalents of the four chords are F♯AC♯E, A♭C♭E♭G,
CD, and B♭DFA, respectively. C) Final chords of the score. Only the relevant parts of thebars are represented.

The transition between the first and second phrase-groups in Part 1, and thetransition between the first and second phrases in Part 3 are worth analyzing. The
final chord [beat (8, 6)] of the first phrase-group in Part 1 and the initial chord [beat© 2020 Hugues Bedouelle14(9, 3)] of the second phrase-group are two derivatives of a major 9th chord, witheither the 7th or the 5th degree omitted (Fig. 3A). Therefore, they generate asmooth transition. There is no exchange of scales between the hands in Part 1.
The transition between the first and second phrases of Part 3 is trickier (Fig.
3B and legend). The last three in-phase and accented chords of the first phrase are
the following: A2E3G♭4D♭5 [beat (30, 4)] is a minor 7th chord; B2G3A♭4E♭5 [beat(30, 5)] is a minor-major 7th chord; B♭3F4D5A5 [beat (31, 1)] is a major 7th chord.
The right hand proceeds through a succession of three parallel perfect fifths:
G♭4D♭5-A♭4E♭5-D5A5. The left hand plays the following chords: A2E3 is a perfect
5th; B2G3 is an augmented 5th/minor 6th; D3C4 is an augmented 6th/minor 7th; andB♭3F4 is a perfect 5th. Thus, we observe a smooth transition from a minor 7th chord
to a major 7th chord and from one hexatonic scale to the other one. The arrival on
the final chord of the piece is somewhat similar (Fig. 3C). The main voice of theright hand plays a succession of four parallel major 9th, A5B6-C6D7-D♭6E♭7-D6E7[beats (34, 3) to (35, 1)], then a minor 7th, D♯6C♯7 [beat (35, 4)]. The main voice ofthe left hand plays a succession of four parallel perfect 5th, E5B5-F♯5C♯6-D5A5B4F♯5 [beats (34, 5) to (35, 4)]. The chords of both hands resolve in a major 9th
chord. Overall, the major 7th chord is only present at the end of part 1 and in thetransition between the two phrases of Part 3. The major 9th chord is only present at
the end of Part 3 (Tables 3-5). These two chords do not correspond to any of the
melodic segments analyzed (Paragraph 3.4). Thus, they seem reserved for specialtransition functions in the score.
3.10. Rhythm
Previous works had recognized the existence of cyclic rhythms in Etude No. 11 but
without analyzing the interactions between them and with the melody (Marin,
2017; Boukobza, 2019, pp. 151-152). Here, I have divided the rhythms into twoclasses for a precise analysis: short-range and long-range. The short-range rhythms
result from the different time signatures of the upper and lower staves, 6/4 and 12/8
respectively. Several long-range rhythms can be recognized in the different parts.
Part 1 comprises four phrases of exactly 25 crotchets each, as described above. On
the upper staff, each phrase comprises three uneven length segments, delimited byslurs. The rhythmic grouping is by crotchets (1) and minims (2): 111222;
11111122; 11121. The last crotchet is a rest separating the phrases (Figs. 4A and
4B). Since the numbers 25 and 6 (or 25 and 24 = 4x6) are coprime, each phrase andtherefore each cycle of the corresponding rhythm start on a different beat of thebar, with an increment of one crotchet per cycle.
On the lower staff, the rhythmic grouping is by dotted crotchets (1) and dotted
minims (2): (12)111121122, where the dotted crotchet and dotted minim inparentheses are only present in bar 1 at the beginning of the piece. This rhythm isdifferent from the one previously described (Marin, 2017; Boukobza, 2019, pp.
151-152). It is strictly repeated three and a half times between the two symmetrical© 2020 Hugues Bedouelle15

melodic patterns in bars 1–2 and 10–12 (Paragraph 3.3). The rhythm issubsequently varied with dotted quavers and a glissando but is still recognizable
until the end of its fifth cycle in bar 16. Since this cyclic rhythm is 12 dotted
crotchets (18 crotchets or three bars) long, each cycle starts on the same beat of thebar, i.e. on the fourth dotted crotchet. The new cyclic rhythm that I propose here,
has the advantage over the previously published ones that it rearranges the melodic
line of the lower staff in a meaningful, repeating structure (Paragraph 3.3).
The cyclic rhythms of the right and left hands have periods of 25 and 18
crotchets, respectively. Since these two numbers are coprime, the combined longrange rhythm of the left and right hands varies in the different phrases of Part 1. A
regular and preconceived pattern of accent marks contributes to the complexity of
the rhythm. For the right hand, they are on the first note of every group of threecrotchets and on every minim (underlined in the above pattern). For the left hand,
there is no accent mark, except in bars 12–14 (Figs. 2B and 2C). In most of Part 1,
there is a two versus three polyrhythm (left hand versus right hand). However, thedotted crotchets of the left hand are split into two dotted quavers in bars 12 to 14
and every second dotted quaver is accented. This split generates a four versus threepolyrhythm (left hand versus right hand).
The long-range cyclic rhythm of the right hand and phrases are 50 quavers
long in Part 1 whereas the short-range cyclic rhythm of the left hand is 3 quavers
long. Consequently, their combined rhythm varies in the different phrases since thenumbers 50 and 3 are coprime. The first dotted crotchet of the left hand occurs at
the beginning of phrase 1, one quaver after the beginning of phrase 2, two quavers
after the beginning of phrase 3 and again at the beginning of phrase 4. Overall, theshort-range and long-range cyclic rhythms at work in Part 1 have different
frequencies and the arithmetic relations between their periods, based on coprime
numbers, generate variations and diversity in the combined polyrhythm.
Part 2 comprises two phrases of unequal length, encompassing bars 18–20 and
21–26 respectively. The lower staff has the following rhythmic pattern of dotted
crotchets (1) and dotted minims (2): 111121122. This pattern occurs once in thefirst phrase and twice in the second phrase, and is exactly three bars long. I
observed that the left hand has fully identical long-range rhythms in Parts 1 and 2.
Accents are present in both right and left hands of Part 2. They modify the firstnote of descending scales or reinforce dissonant chords.
In Part 3 (bars 27–35), each hand plays two voices, the primary voice
consisting of chords in crotchets and the secondary voice consisting of anembellishment of one of the chord tones, mostly written in quavers. The left andright hands follow rhythmic patterns that extend exactly on two bars, are repeatedfour times from bar 27 to bar 34 (Figs. 4C and 4D), and result in a continuous flowof notes. In each couple of bars, the 24 quavers of the right hand are grouped by
two or four. The 24 quavers of the left hand are grouped by three except the last six
ones, which are grouped by two. In bars 33 and 34, some of the groups of three© 2020 Hugues Bedouelle

16

quavers are replaced by irregular divisions in four or five quavers (tuplets), or bysixth semiquavers. This creates four, five or six versus three polyrhythms (left handversus right hand) and an acceleration of notes. Every chord of the primary voices
is accented. In bars 27–30, these accents create a compound rhythmic pattern thatis similar to that in Part 1 (Fig. 4).

RH: •-•-•-o---o---o---•-•-•- (A)LH: •--o-----•--•--•--•--o-RH: •-•-•-o---o---•-•-•-o--- (B)
LH: ---•--•--o-----o-----•-RH: •-•-•-•-..•-..•-•-•-•-.. (C)LH: •-.•-.•-.•-.•-.•-.•-•-•RH: •-•-•-•-..•-..•-•-•-•-.. (D)LH: •-.•-.•-.•-.•-.•-.•-•-..
RH: •-...ø-----LH: •-•-.ø------(E)

Figure 4. Rhythmic patterns in Parts 1 and 3. RH, right hand; LH, left hand; (•-)
crotchet; (•--) dotted crotchet; (o---) minim; (o-----) dotted minim; (.) rest in themain voice; (ø-) final chord. (A) Pattern of bars 1–2; (B) bars 3–4; (C) pattern repeatedthree times in bars 27–28, 29–30 and 31–32; (D) slightly different pattern in the last beat ofbars 33–34; (E) pattern of bar 35.

3.11. Dynamics
In Part 1, the dynamics of the right hand are mostly piano and those of the left
hand are lower or at most equal to those of the right hand. Variations of dynamics(crescendo or decrescendo) are marked only for the right hand. These dynamic
marks support the following assumptions: (i) the right hand plays the melody andthe left hand, an accompaniment; and (ii) harsh dissonances are avoided. In Part 2,
the similar dynamics of the right and left hands, and the accents emphasize thedissonant character (see Paragraphs 3.5–3.7). In Parts 1 and 2, the crescendi anddecrescendi underline the highest pitch in the different fragments of phrase andgive a vocal character to the melody. Overall, the dynamic range of Parts 1 and 2
extends only between ppp and mp. In bars 27–30 of Part 3, the dynamic marks
indicate that the accented chords of both right and left hands should be playedmezzo forte and the intervening ornamental notes played pianissimo so as toemphasize the similarity between the first phrases of Parts 1 and 3 (see Paragraph© 2020 Hugues Bedouelle17

3.10). In bars 33–34, the right hand plays arpeggiated chords at higher and higher
pitches along a crescendo that leads to the only forte dynamics of the score, and isfollowed by a steep diminuendo towards the final chord, which is played
pianissimo and clearly indicates the end of the piece.
The transition between the first and second phrases of Part 3 is a very special
moment of the score since: (i) it leads to the last phrase and coda of the score, (ii) it
occurs with an exchange of the scales between the hands, and (iii) mostimportantly, it occurs without interruption of the musical flow. All the othertransitions between phrases include either a minim and a rest as in Part 1 or adotted minim as in Part 2 at the end of the upstream phrase. Moreover, they occur
without a change in dynamics. In Part 3, the transition is emphasized by the
dynamics: the last two beats of the first phrase undergo a decrescendo to
pianississimo whereas the second phrase starts piano (Fig. 1J).

4. Discussion
4.1. Pianistic technique
Etude No. 11 is proposed to students in their eighth year of piano practice (Gottlieb
et al., 2001, p. 101). What are the technical difficulties that the score may present
for students at that level? The two staves have different key and time signatures.
The two hands overlap or cross widely in several parts of the score and inparticular they overlap in the middle of a glissando. In bars 15–16, the left handplays glissando and crescendo while the right hand strikes chords decrescendo,
which requires a good independency of the two hands. The score includes
polymetres and irregular divisions of the metre (tuplets), giving rise to three versus
two, four versus three and five versus three polyrhythms. Accented notes alternate
between the two hands. Some chords that are expressly noted as non-arpeggio,
exceed the span of an average hand. For example, the left hand plays E3G4 at beat
(28, 4); B♭2A♭3D♭4, D♭3A♭3F4 and G♭2D♭3B♭3 in bar 32. Ligeti indicates that thepiece should be played ‘avec l'élégance du swing’. In my opinion, the ‘swing’
character comes from the polymetre and both hands should play strictly in rhythm.
Parts 1 and 2 are of medium difficulty and worth studying for themselves. Theinterpreter needs large hands and a superior technique to play Part 3 and especiallybars 33–34 in the right tempo.
4.2. Character of the piece
The two complementary hexatonic scales that are used in Etude No. 11 for the rightand left hands, are transpositions of the hexachord C-D-E-F-G-A that was
described by Guido of Arezzo in the 11th century and the seed for the diatonic
scale (Reisenweaver, 2012). Ligeti used different scales for the right and left hands
in other piano pieces. In Etude No. 7 Galamb Borong and in the 3rd movement of
the Concerto pour piano et orchestre, Ligeti used two complementary whole tone
© 2020 Hugues Bedouelle18

scales (Ligeti, 2013, p. 132). In Etude No. 1 Désordre and in the 5th movement of
the piano concerto, Ligeti used complementary pentatonic anhemitonic anddiatonic scales (Svard, 2000; Ligeti, 2013, pp. 299–300). In this way, the composer
creates an illusory and shimmering harmony that comes from the equal
temperament but does not belong to it any more from a sound viewpoint (Ligeti,
2013, p. 298). In Etude No. 12 Entrelacs, the key signatures are identical to thoseof Etude No. 11. However, the C and F pitches are present in both staves.
Therefore, the two scales are not complementary, even if their union reconstitutes
the 12 tones of the equal temperament.
The analysis of the scalar material in Etude No. 11 could explain its title En
Suspens. The piece was first entitled Convexe-Concave, a title quasi-identical tothat of Escher's lithography No. 56, entitled Concave and Convexe. It was thensuccessively entitled Souplesse, En suspension, Engrenage (i.e. gears), and finally
En Suspens (Beffa, 2016, p. 353). I propose the following explanation: the staves
use scales that are derived from the D♭-major and G-major scales but are devoid of
their leading tone, C and F# respectively. Therefore, phrases of music, written
either on the upper or lower staff, cannot end with an authentic cadence. All thecadences are imperfect (or suspended), hence the title of the piece En Suspens.
The ic-vectors of the diatonic and Guido's scales differ by the ic-vector
111111 of the all-interval tetrachords (Forte, 1973, p. 18). As an aside, thesetetrachords are used at least once in each of the three parts of Etudes No. 11(Tables 3–5). The structure of Guido's scale and therefore its potential for melodies
are very close to those of the diatonic scale. However, the corresponding melodies
are neither tonal nor atonal but somewhere in-between for lack of leading tone. As
observed in the Results section, the slur and dynamics markings give a cantabile
character to the melodies in Parts 1 and 2 of Etude No. 11. In Part 1, this characteris reinforced by the overall consonant harmony. In Part 3, the key melodic notes
are accented in the upper staff and the combined rhythmic pattern of the accented
chords in the upper and lower staves strongly parallels the rhythmic pattern in bars
1–2 (Fig. 2), thus recalling the melodic gesture of Part 1.
Several of Ligeti's compositions have a specified polymetre of the bar,including Etude No. 4 Fanfare, No. 5 Arc en ciel, No. 11 En Suspens and the 3rd
and 5th movements of the Concerto pour piano et orchestre. Etudes Nos. 5 and 11
have similar polymetre of the bar, 3/4 (6/8) and 6/4 (12/8) respectively; they have
similar indications to the interpreter: ‘Andante con eleganza, with swing’ and‘Andante con moto, avec l'élégance du swing’. These indications suggest that
Ligeti used hemiola as a means of rigorously recreating the swing character of jazz.
The polymetre of the bar is refined in Part 3 of Etude No. 11, since the 12 quaversof a bar are subdivided either into 3+3+3+3 or 3+3+2+2+2, in a reminiscence of
the Ottoman Aksak rhythmic system (Cler, 1994). Part 3 of Etude No. 11 also
shares a high content in 7th and 9th chords with Etude No. 5, putting it on the edgeof tonal and jazz sounding.
© 2020 Hugues Bedouelle

19

The up-skip of Motif 1 induces a tension whereas the down steps of Motif 2
induce a relaxation (Persichetti, 1961, p. 235). In Part 1, where Motif 1 is
preponderant, the tension is counterbalanced by the tertian, rather consonant
harmony. In Part 2, the dissonant harmony is counterbalanced by the
preponderance of Motif 2. In the first phrase of Part 3, Motifs 1 and 2 are in
equilibrium. Thus, the tensions and relaxations brought about by melody andharmony are well balanced in most of the score. This balance, together with thesoft dynamics, gives the music its quiet and elegant character. In the second phraseof Part 3, a tension is created by the rising pitch, acceleration of notes andcrescendo until the only forte dynamics of the score and it is suddenly released by
the deceleration of notes, rapid diminuendo and final pianissimo chord. This lastphrase has the character of a coda.
4.3 A subtle organization
A first level of rhythmic organization in Etude No. 11 is provided by hemiola, athree versus two polyrhythm. It results from the two different time signatures forthe right and left hand, 6/4 and 12/8, is short range and runs through the wholescore, except for a few beats in Part 3 (Fig. 4). Hemiola has been widely used inWestern music and is a defining characteristic of African music (Temperley, 2000).
A second level of rhythmic organization is long range and reminiscent of the
distinction between cyclic rhythms of sound (talea) and pitch (color) in theEuropean music of the 14th century. Three levels of long-range organization can be
distinguished in Etude No. 11: those of sound, pitch and phrase. In Part 2, thesounds of the lower staff follow a cyclic rhythm whose period equals three bars.
The pitches of the lower staff have a cyclic organization in two segments of fiveand four notes that has also a period of three bars (Fig. 2D). Thus, sound and pitch
have different but isoperiodic cyclic organizations. The music of the upper staff
does not follow a cyclic organization but the lengths of the first and second phrases
are equal to one and two periods of three bars respectively. Thus, Part 2 is globallyorganized through a time-length of three bars. In Part 3, the sounds of both upperand lower staves follow cyclic rhythms. The two rhythms are different but
isoperiodic, with the same period of two bars. The pitches are not cyclicallyorganized. The length of the phrases is equal to two periods, if one excludes the
exceptional final bar. Thus, Part 3 is organized through a time length of two bars.
In Part 1, the sounds and pitches of the lower staff have the same cyclic rhythm andorganization as in Part 2, at least until bar 12, with a period equal to three bars (Fig.
2A). As regards the upper staff, the sounds follow a cyclic rhythm whose period isequal to 25 crochets and the phrases have the same length: the period of the soundrhythm and the length of the phrases are isometric. The pitches are not cyclicallyorganized but the music of periods 1 and 2 is repeated, transposed downwards by
one octave, in periods 3 and 4. Thus, the sound rhythms, pitch organizations andphrase lengths of the upper and lower staves are incompatible. Because the© 2020 Hugues Bedouelle

20

anacrusis of two notes and the positions of the slurs hide the cyclic organizations of
the lower staff, Part 1 appears organized through a time-length of 25 crotchets, i.e.
just above four bars.
The organizations of the three parts of Etude No. 11 are clearly different. They
are reminiscent of the medieval motets. Part 2 of Etude No. 11 could be classified
as isorhythmic by reference to Ars nova, and Parts 1 and 3 as pan-isorhythmic byreference to Ars subtilior (Apel, 1972, p. 427). Given that Guido's hexachord was
described in the 11th century and that the isorhythmic and pan-isorhythmic motets
are inventions of the 14th century, one can assume that Ligeti was inspired by the
medieval music for the composition of Etude No. 11.
4.4. Form and progression
The main melodic line is played by the right hand and strongly structures form in
Etude No. 11. The musical idea of Part 1 is conveyed by two phrases of about fourbars each (25 crotchets), forming a phrase-group. The first phrase starts with aclear statement of Motif 1 and ends with a chord that would be a major dominanttriad in the scale of D♭ major. The second phrase appears as a repetition of the firstphrase with variations and it ends on a major 9th chord, with the 7th omitted (Table3). Both phrases share the same rhythmic pattern. The second phrase group of Part
1 has the same structure as the first one. Thus both phrase-groups have structuresthat are reminiscent of a musical period (Schoenberg, 1967, pp. 20–81).
The musical idea of Part 2 is conveyed by two phrases of unequal lengths,
comprising three and six bars. The first phrase starts with a clear statement of
Motif 2, which is immediately repeated. The second phrase repeats three times a
combination of Motif 2 and Motif 1 in sequence and appears as a development of
the first phrase. Thus, the phrase-group of Part 2 has a structure that is reminiscentof a sentence (Schoenberg, 1967, pp. 20–81).
The melodic structure of Part 3 is close to that of Part 1. It includes twophrases of four bars each and a concluding bar. Both phrases start with a statementof Motif 1 and share the same rhythmic pattern. A descending scale in the firstphrase is mirrored by ascending arpeggiated chords in the second phrase that givethe latter the character of a coda. The descending scale stops two crotchets before
the end of the first phrase and reverses in five ascending steps, including achromatic step, that lead to a major 7th chord, a swap of the key signatures
between the staves and the beginning of the second phrase (Fig. 1H). Theseelements, together with a jump in dynamics (Paragraph 3.11), provide a clear
articulation even though the second phrase is a continuation of the first one,
without interruption. In summary, the different phrase-groups of Etude No. 11 have
melodic structures that are reminiscent of the period and sentence although these
two forms are classically defined in a tonal context. The local forms of the melodic
lines participate in the contrast between the three different parts of Etude No. 11
and therefore to its overall form.
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Sound rhythm, pitch organization, phrase length, their cyclic patterns and theirsubtle interactions clearly differentiate the three parts as detailed in Paragraph 4.3.
Harmony participates in form. Dissonances are avoided in Parts 1 and 3, whichmostly include tertian harmonies. In contrast, harsh dissonances are favoured inPart 2, which mostly includes cluster harmonies. Thus, melody, rhythm andharmony contribute to the contrasts between the three parts of the score. The
harmonic complexity of the in-phase chords increases significantly from Part 1 toPart 2 and then strongly from Part 2 to Part 3. Therefore, there is a gradient of
harmonic complexity when going from the beginning to the end of the score, which
contributes to the musical progression. Likewise, the gradients of register in thethree parts and the increased frequency of notes towards the ends of Parts 1,
through a long glissando, and Part 3 contribute to form and progression. Overall
the piece has a ternary structure AA'BA".
4.5. Conclusion
From the analysis of Etude No. 11, one can deduce or confirm some traits ofLigeti's compositional method. At an early stage, the elementary materials of thepiece (scale system, melodic motifs, main rhythmic patterns, etc.), theirinteractions and the global structure of the piece are decided or even calculated.
Ligeti has qualified his Etudes as constructivist pieces (Ligeti, 2013, pp. 44 and52). However, the composition is not automatically generated from these early
decisions. The complex interactions between the different musical layers areplanned to generate diversity but the composer creates additional relations between
the elements of the nascent score to add meaning and coherence.
By using complementary scales, Ligeti could write a music that is neither tonal
nor atonal but in-between. In fact, this strategy enabled him to use compositional
techniques that belong to the pre-tonal, tonal and post-tonal musical worlds
altogether. In Etude No. 11 for example, he used varying relations between soundand pitch rhythms that are reminiscent of the 14th century motets. He used
contrasts between melodies, between tertian and cluster harmonies, between phrase
structures, and the possibility to create tension and relaxation by disjunct andconjunct intervals that are reminiscent of tonal structures and progressions.
However, he also used parametric delineators of form that act in a non-tonal andnon-thematic way, and are characteristic of post-tonal music, e.g. temporal density,
harmonic complexity, dynamics and register (Howland, 2015).
The multivalency of Ligeti's compositional approach gives his music acharacter of both familiarity and strangeness, and therefore attractiveness. In Etude
No. 11, Ligeti has employed and grouped many compositional techniques in ashort piano piece and shown how they can give a melodic, complex but
nevertheless coherent, very pleasant and beautiful piece of music. Etude No. 11 isperhaps less spectacular than other Ligeti's Etudes but it is nevertheless technicallydifficult. It is a masterpiece of composition in which Ligeti's reflections, art and© 2020 Hugues Bedouelle

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genius found ways of expression. I did not invoke Ligeti's life, the development ofmusic in the 20th century or the chaotic history of this period of time to analyze
and explain Etude No. 11. Other authors have done it in a convincing manner for
the whole set of Etudes (Steinitz, 2003; Floros, 2014; Beffa, 2016; Boukobza,2019). Conversely, I hope that the work reported here provides a glimpse in
Ligeti's mind as he was conceiving, composing and writing Etude No. 11.

Acknowledgments
A manuscript of this article was submitted to the journal ‘Music Analysis’: anoriginal version on 14 May 2019 and a revised version on 25 Sept. 2019. I thankone of the two anonymous reviewers for his encouragements, constructive
comments and useful advice.

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