SYMMETRY, DISRUPTION, AND MISMATCH IN DALLAPICCOLA…alexaitken.co.uk/assets/Pitch Class Analysis on...

SYMMETRY, DISRUPTION, AND MISMATCH IN

DALLAPICCOLA’S TRE EPISODI:

A PITCH-CLASS ANALYSIS

ABSTRACT

This essay focusses on Dallapiccola’s own piano transcription of Tre Episodi from his

ballet Marsia. Using pitch-class-set theory, this essay will assess to what extent

Dallapiccola’s later preoccupation with symmetry in his twelve-tone works, a major

element in his mature style, was already present in this earlier work.

Transcribed by Dallapiccola in 1949, the Tre Episodi are excerpts from his 1942 ballet Marsia.

Symmetry became important in Dallapiccola’s mature style, as did the serialist techniques he

developed from the 1930s onwards.1 This essay explores whether symmetry was already

present in pitch-related elements of these earlier Episodes, with a specific focus on the

disruption of and adherence to symmetrical patterns, as well as links to Dallapiccola’s later

preoccupation with the number 5.2 A further discussion into the often-overlooked influence of

Messiaen’s modes will be followed by a close examination of elements of mismatch within the

Episodes (i.e. the extent to which levels of consonance, dissonance, and pitch-class circulation

do not align with the structural divisions).3

The analysis begins by assuming each bar is a set (to facilitate reference and comparison),

with the prime forms and interval vectors following Forte’s system in Appendix 1 of The

Structure of Atonal Music.4 This mechanistic, semiotic approach of dividing the music by bar

often yields little insight, but here serves as an effective way to begin this pitch-class (pc)

analysis, and gives significant results, as well as being a useful indication of the musical

density at any given point. This approach will then be critiqued and widened to show that the

observations from a more musically-intuitive segmentation often coincide with those from the

semiotic analysis.

It will be shown that the number 5 plays an important organisational role within this earlier

work, that the Episodes are shaped by elements of Messiaen’s mode III, and that ten-note sets

act as one of the main organisational forces. The essay will also demonstrate that symmetry

1 See Alegant, B. (2010) The Twelve Tone Music of Luigi Dallapiccola: 9-28. 2 The number 5 becomes important in later works, notably Parole di San Paolo (1964), whose row is divided into 5 + 5 + 2,

and in Ritmi (1952), where the rhythmic patterns are based on groups of five attacks. 3 Pitch-class circulation – the net result of pitch-classes subtracted and/or added from the previous bar. 5 pitch-classes

removed and 2 added gives a pitch circulation of -3 (three fewer pitch-classes). 4 Forte, A. (1973) The Structure of Atonal Music. Yale: Yale University Press.

SYMMETRY, DISRUPTION, AND MISMATCH IN DALLAPICCOLA’S TRE EPISODI: A PITCH-CLASS ANALYSIS

2

and disruption interact at multiple levels and, through schematic representations, will outline

the gradual evolution from disruption to symmetry on a larger scale.

The variety of processes in operation within these Episodes necessitates a mechanism by

which they can be hierarchically organised. Although Forte explicitly uses the Schenkerian

terms ‘foreground’, ‘middleground’, and ‘background’ in non-Schenkerian contexts, this

analysis will employ the more neutral terms ‘surface’, ‘underlying’, and ‘controlling

processes’.5

At the surface level the pattern of pitch-class totals per bar (here called the linear progression,

LP) contains a number of palindromic units (LP cells). The term linear progression is here

used in the mathematical sense, rather than either denoting a line, or a Schenkerian reduction

proceeding by step.

Episode 1 Episode 2 Episode 3

Bar LP Bar LP Bar LP Bar LP Bar LP Bar LP

7 5 1 3 97 6 124 5 138 6 49 9

8 6 2 3 98 5 125 6 139 8 50 8

9 6 3 5 99 5 126 6 140 6 51 10

10 5 4 3 100 5 127 5 141 8 52 8

5 3 101 6 142 6 53 9

14 8

15 10 12 5 109 5 134 6 201 6

16 10 13 4 110 5 135 5 202 6

17 8 14 4 111 3 136 5 203 5

15 4 112 3 137 5 204 6

38 7 16 4 113 3 138 6 205 6

39 8 17 4 114 5

40 8 18 5 115 5

41 7

FIGURE 1: PALINDROMIC LP CELLS IN EACH EPISODE

Each LP cell is either bordered by, or centres on, a bar with five pitch-classes, and/or itself

consists of five elements. The number 5 therefore assumes a two-fold function: it provides

two boundaries within which a symmetrical LP cell can operate, differentiating adjacent

symmetrical and non-symmetrical LP cells; and it acts as a point from, or towards, which the

momentum of LP can be generated or orientated (i.e. the LP cells head towards, or away

5 See Forte, A. (1986) Liszt’s Experimental Idiom in 19th-Century Music 10/3: 209-28


3

from, bars with five pitch-classes).6 These functions will be referred to as demarcative and

generative functions respectively. More noticeable are LP cells where symmetry is disrupted

by the final or penultimate elements (shaded red in Figure 2).


Bar LP Bar LP Bar LP Bar LP Bar LP

20 8 5 3 46 7 137 5 1 6

21 7 6 5 47 3 138 6 2 8

22 8 7 3 48 4 139 8 3 6

23 7 8 5 49 3 140 6 4 8

24 9 9 4 50 4 141 8 5 8

25 7 51 3 142 6

16 4 52 3 143 7 23 7

30 4 17 4 24 6

31 7 18 5 74 3 199 5 25 5

32 6 19 5 75 5 200 5 26 6

33 7 20 5 76 3 201 6 27 6

34 7 21 1 77 5 202 6

22 4 78 4 203 5 29 5

42 7 204 6 30 6

43 8 22 4 82 6 31 4

44 8 23 5 83 4 214 5 32 6

45 8 24 5 84 3 215 7 33 4

46 9 25 5 85 3 216 6

47 8 26 5 86 4 217 6 53 9

48 8 27 1 87 4 218 7 54 8

49 8 28 4 219 4 55 7

50 5 123 6 56 8

32 5 124 5 235 5 57 6

59 5 33 5 125 6 236 6

60 5 34 1 126 6 237 6 57 6

61 0 35 5 127 5 238 6 58 4

62 2 36 4 128 5 239 6 59 6

63 5 240 4 60 6

64 5 61 6

62 7

63 6

FIGURE 2: DISRUPTIVE LP CELLS IN EACH EPISODE

The number 5’s demarcative and generative role is significantly weakened within disruptive

LP cells (only eleven out of twenty-one such cells contain five elements, or are bordered by

or centred on the number 5). The demarcative and generative functions therefore act mainly

within symmetric LP cells, operating underneath the main interaction of symmetry and

6 LP momentum – the strength of motion towards or away from a particular number within the LP (e.g. the LP 213245 has

increasing LP momentum towards 5).


4

disruption at the surface level of linear progression. Symmetric LP cells are subordinated in

favour of disruptive LP cells (twelve cases compared to twenty-one), since the introduction

of a new set towards the end of the cell, and therefore different pitch-classes, increases

pitch-class circulation and LP momentum, preventing stasis (note that 12 is the mirrored

counterpart of 21).7 In addition to the LP, which only relies on cardinality, there are also a

number of specific palindromic set progressions (here called set cells) in Episode 2, each

consisting of five bars.

Bar Set Bar Set Bar Set

1 3-4 56 5-21 201 6-32

2 3-4 57 5-21 202 6-32

3 5-21 58 5-27 203 5-25

4 3-4 59 5-21 204 6-32

5 3-4 60 5-21 205 6-32

FIGURE 3: PALINDROMIC SET CELLS IN EPISODE 2

The number 5 maintains its demarcative and generative function at the level of the pitch-

class set (the middle set cell is bordered by five-note sets, and the LP momentum of the two

outer cells is orientated towards five). In all three cells the number assumes an anchoring

function, since it lies at the centre of the block. There is also an overall symmetry to the set

cells, since the LP momentum is directed towards the middle in the two outer cells, but is

static in the central cell. The outer cells are also mirrored in the sense that LP momentum

strengthens towards the centre of the first cell, and weakens towards the centre of the third

cell.

As with LP cells, disruptive rather than symmetric processes are prioritised (thirteen cases

compared to three) at the level of the pitch-class set, as well as within the pitch-class totals.

The disruption occurs at the end of the cell, as shown in Figure 4.8

7 Pitch-class circulation – the net result of pitch-classes subtracted and/or added from the previous bar. 8 The musical context for each of these set cells may be gained through consideration of the first copy of the score.


5


Bar Set Bar Set Bar Set Bar Set Bar Set Bar Set

20 8-26 7 3-7 40 5-Z18 205 6-32 1 6-35 53 9-8

21 7-35 8 5-27 41 5-4 206 7-24 2 8-21 54 8-21

22 8-26 9 4-14 42 5-29 207 7-9 3 6-35 55 7-33

23 7-35 10 4-14 43 5-16 208 7-24 4 8-21 56 8-21

24 8-22 11 3-7 44 5-Z18 209 7-9 5 8-24 57 6-34

210 8-Z15

42 7-3 12 5-21 138 6-35 211 6-32 21 5-33

43 8-26 13 4-14 139 8-27 22 6-33

44 8-26 14 4-14 140 6-35 23 7-35

45 8-26 15 4-14 141 8-27 24 6-33

46 9-7 16 4-14 142 6-22 25 5-24

17 4-14

46 9-7 18 5-29 192 5-27 49 9-12

47 8-21 193 6-31 50 8-24

48 8-21 194 3-7 51 10-2

49 8-21 195 6-31 52 8-24

50 5-33 196 4-26 53 9-8

FIGURE 4: DISRUPTIVE SET CELLS IN EACH EPISODE

The first disruptive set cell in Figure 4, bb.20-24 of Episode 1, is shown below.9 9-11 in bar

24 disrupts the expected symmetry of the appearance of 8-26. (These sets will henceforth be

called the substituted and expected sets respectively.)

Bar Set

Pitch-Classes Used

(Numerical Order)

20 8-26(t7) 0245789E

21 7-35(t11) 024579E

22 8-26(t7) 0245789E

23 7-35(t11) 024579E

24 9-11(t2) 02345789E

FIGURE 5: EPISODE 1, BB.20-24

Note that 7-35 ⊂ 8-26 ⊂ 9-11.10 The substituted set increases LP momentum (the set cell

now has direction towards a nine-note set, instead of an eight-note set). Totalling the

number of appearances of each pitch-class produces primary cumulatives (CP), consisting of

two overlapping symmetrical blocks of five elements.

9 The notation t followed by an integer represents set transpositions by semitones (t0 is untransposed). 10 PC set 9-11 is RP to 9-12 (mode III).


6

PC CP

0 5

2 5

3 1

4 5

5 5

7 5

8 3

9 5

11 5

FIGURE 6: CP FOR EPISODE 1, BB.20-24

The number 5 continues its demarcative function, bordering each block. As the set cell

consists of five bars, seven invariant pitches can be seen in Figure 6. Totalling the number of

occurrences of each CP produces secondary cumulatives (CS) which, as shown below,

demonstrate the mode CP is 5.

CP CS

1 1

3 1

5 7

FIGURE 7: CS FOR EPISODE 1, BB.20-24

Figure 8 summarises the features of each cell in Figure 4.

FEATURE OCCURRENCES %

ELEMENTS OF SYMMETRY PRESENT AT CP LEVEL 13/13 100

SYMMETRICAL BLOCK(S) OF FIVE ELEMENTS PRESENT AT CP LEVEL 12/13 92

SYMMETRICAL BLOCK(S) OF 5 PRESENT AT CP LEVEL BROKEN BY FIFTH ELEMENT 9/13 69

LP MOMENTUM INCREASES ACROSS THE SET CELL 5/13 38

LP IS MAINTAINED ACROSS THE SET CELL 4/13 31

LP MOMENTUM DECREASES ACROSS THE SET CELL 4/13 31

MODE CP IS 5 3/13 23

BLOCKS OF FIVE ELEMENTS ARE ANCHORED BY THE NUMBER 5 AT CP LEVEL11 3/13 23

THE DISRUPTING ELEMENT IN THE BLOCK OF FIVE IS A MULTIPLE OF 5 3/13 23

THE EXPECTED/SUBSTITUTED SETS ARE K-RELATED 3/13 23

THE EXPECTED/SUBSTITUTED SETS ARE Kh-RELATED 2/13 15

ELEMENTS OF SYMMETRY PRESENT AT CS LEVEL 2/13 15

5 MAINTAINS DEMARCATIVE FUNCTION IN BLOCK(S) OF FIVE ELEMENTS AT CP LEVEL 2/13 15

5 DEMARCATES AND ANCHORS WITHIN BLOCK(S) OF FIVE AT CP LEVEL 1/13 8

OVERLAPPING SYMMETRICAL BLOCKS OF FIVE AT CP LEVEL 1/13 8

FIGURE 8: SUMMARY OF FEATURES WITHIN DISRUPTIVE SET CELLS IN FIGURE 4

11 The number 5 anchors when it is at the centre of the block.


7

The three most common features of the disruptive set cells involve symmetry. Figure 8

shows that, even within the disruptive set cells, symmetric rather than disruptive processes

are prioritised. Two of the three most prevalent features involve the number 5, which again

operates underneath, but supports, the interaction between symmetrical and disruptive

processes. Breaking the symmetry with the fifth element prevents stasis at the end of the

block, so the number maintains its generative function at the level of the disruptive set cell.

The anchorage and demarcation functions are not significant at this level.

Summarising the surface processes:

- The sets used create a pattern of pitch-class totals per bar (the linear progression). This,

and the sets used, create LP cells and set cells respectively, in which disruption is

prioritised over symmetry.

- Both set cells and LP cells are linked by the concept of linear momentum across the cell

(the strength of movement towards or away from a particular number within the LP).

- The number 5 has a three-fold function, and acts upon the LP and LP cells:

1. It demarcates LP and set cells;

2. It anchors (i.e. appears at the centre of a set or LP cell);

3. It generates LP momentum by being a point from, or towards, which the LP is

orientated.

There are a number of underlying processes:

- Primary cumulatives often contain blocks of five elements.

- For a number of set cells the mode secondary cumulative is 5.

- The disruption prioritised within the set cells gradually yields to the emergence of

symmetry at a deeper level (within the primary cumulatives).

Figure 9 schematically represents the above. The left hand side shows whether symmetry or

disruption is prioritised. The vertical dimension represents the depth at which the process

can be thought of as operating.


8

PRIORITISES COMPONENTS FEATURE FUNCTION CONTROLLING FORCE

/LEVEL

SETS USED

DEMARCATION

LP

ANCHORAGE 5

DISRUPTION SYMMETRIC / ASYMMETRIC SURFACE

SET CELLS LP CELLS

GENERATION

LP MOMENTUM

SYMMETRY PRIMARY CUMULATIVES BLOCKS OF

5 UNDERLYING

SECONDARY CUMULATIVES MODE CS IS

FIGURE 9: SCHEMATIC REPRESENTATION OF SURFACE AND UNDERLYING PROCESSES OF PITCH-CLASS ELEMENTS

There is an overall trend to increase the LP momentum through the disruptive set cell (N.B.

in five cases in Figure 4). These instances are shown below. A < represents increasing LP

momentum; e.g. 7<9 means the expected seven-note set has been substituted with a nine-

note set. The number of invariant pitch-classes between the substituted and expected sets is

also shown.

BAR LP MOMENTUM INCREASE INVARIANCE %

E1, bb.20-24 8<9 7 88

E1, bb.42-46 7<9 4 44

E2, bb.205-211 7<8 6 75

E3, bb.1-5 6<8 2 25

E3, bb.21-25 5<6 4 67

FIGURE 10: LP MOMENTUM INCREASE AND INVARIANCE

The five instances are distributed symmetrically across the Episodes: two in Episode 1, one

in Episode 2, and two in Episode 3. The most prevalent set in Episode 1 is 5-33; the only five-

note whole-tone subset (and therefore RP to 6-35), and a sonority with twelve-fold

symmetry.12 But in Episode 2 the three most frequently used sets (4-14, 5-27, and 3-4)

display no aspects of symmetry; neither does 6-33, which appears nine times in Episode 3.

Figure 11 shows the sets used for each Episode. Sets displaying symmetrical properties in

their permutations are shaded grey.

12 Note that 5-33 contains five out of the six notes in a whole-tone scale, continuing Dallapiccola’s focus on the number 5.

Symmetry here is used in Forte’s terms (i.e. describing sets whose properties remain the same under repeated

permutations). Twelve-fold symmetry indicates that for twelve permutations the properties of the set remain unchanged.

INC

RE

AS

ING

DE

PT

H F

RO

M S

UR

FA

CE

OF

MU

SIC


9


Prime Form Set

Prime Form Set

01469 5-32

Prime Form Set

02 2-2

015 3-4

02479 5-35

016 3-5

04 2-4

016 3-5

01258 5-Z38

0135 4-11

048 3-12

025 3-7

013457 6-Z10

0157 4-16

0246 4-21

027 3-9 013467 6-Z13

0246 4-21

0248 4-24

0236 4-12

012458 6-15

0248 4-24

01348 5-Z17

0136 4-13

013478 6-Z19

01346 5-10

01458 5-21

0237 4-14

014589 6-20

01347 5-16

02357 5-23

0157 4-16

023468 6-21

02357 5-23

02468 5-33

0147 4-18

012468 6-22

01357 5-24

02469 5-34

0148 4-19

013568 6-Z25

02368 5-28

013589 6-31

0158 4-20 013589 6-31

01368 5-29

023579 6-33

0248 4-24 024579 6-32

02468 5-33

02468T 6-35

0358 4-26 02468T 6-35

012468 6-22

012358 6-Z40

0258 4-27

012358 6-Z40

023568 6-Z23

012469 6-Z46

01245 5-3

012568 6-Z43

013468 6-Z24

0123458 7-3

01236 5-4

013479 6-Z49

013469 6-27

0123469 7-10

01256 5-6

0123567 7-5

024579 6-32

0124569 7-Z17

02346 5-8

0123678 7-7

023579 6-33

0134579 7-26

01346 5-10

0123468 7-9

013579 6-34

013468T 7-34

01356 5-Z12

0123469 7-10

02468T 6-35

013568T 7-35

01268 5-15 0134568 7-11

0134578 6-Z37

01234579 8-11

01347 5-16

0234579 7-23

0123468 7-9

01345679 8-12

01457 5-Z18

0123579 7-24

012468T 7-33

0123468T 8-21

01367 5-19

0124679 7-29

013468T 7-34

0123568T 8-22

01378 5-20

0134679 7-31

013568T 7-35

0124579T 8-26

01458 5-21

012468T 7-33

01345679 8-12

01234578T 9-7

01478 5-22 013568T 7-35

0123468T 8-21

01234678T 9-8

01357 5-24

0123589 7-Z18

0124568T 8-24

01235679T 9-11

02358 5-25

0123568 7-Z36

01234578T 9-7

012345679T 10-3

02458 5-26

0124578 7-Z38

01234678T 9-8

012345689T 10-4

01358 5-27

01234678 8-5

01245689T 9-12

012345689T 10-4

02368 5-28

01245679 8-14

012345678T 10-2

0123456789T 11-1

01368 5-29

01234689 8-Z15

012345689T 10-4

01468 5-30

0123568T 8-22

0124578T 8-27

012345679 9-2

012345789 9-4

01234578T 9-7

0123456789T 11-1

FIGURE 11: SETS USED IN EACH EPISODE13

13 This analysis extends Forte’s system to encompass the six ten-note sets. They are labelled as follows:

10-1 0123456789 10-2 012345678T

10-3 012345679T 10-4 012345689T

10-5 012345789T 10-6 012346789T

Since there is only one eleven-note set, this is labelled 11-1.


10

The symmetrical sets do not consistently appear at moments of tension or structural

significance. Only 46 of the 107 sets used (43%) have some symmetrical properties,

suggesting that there is no agenda to specifically use sets with symmetry. But if the pitch-

classes of each set are placed within a pitch square and the resulting shape examined,

symmetry or near-symmetry can be seen in each set used.

The pitch-classes used within a set are placed within the inner square, beginning in the top

left corner with the lowest pitch-class and proceeding clockwise. The pitch-classes not used

are placed in the outermost square, again beginning in the top left corner with the lowest.

The level of symmetry in the inner square is then examined. Figure 12 shows the pitch

square for 6-35 (02468T):

FIGURE 12: PITCH SQUARE FOR 6-35

Two axes of symmetry can be seen, around zero and three.14 In each of the sets across the

Episodes symmetry is present in the pitch square, or thwarted by one pitch-class, which can

either be removed or repositioned in order to regain symmetry within the set. The near-

symmetrical 6-33’s pitch square is below. Symmetry is prevented by the pitch-class 2.

Symmetry or near-symmetry is inherent in each of the 107 sets used.

14 The axes in the pitch square are labelled according to the lower of the two numbers they pass between in

either the inner or outer square (e.g. the axis passing between zero and six is zero; that between one and seven

is one, etc.). Axes passing between numbers are labelled as the lowest number + 0.5 (e.g. that passing between

zero and one and between six and seven is labelled 0.5).

1 3

E 0 2

4

T

8 6 5

9 7


11

The appendix at the back of the score booklet lists the sets used in the Episodes, and

whether they possess symmetry within the pitch square. For near-symmetrical sets the

movement, or removal, of a pitch-class required to provide symmetry within the pitch circle

is stated, along with the axis around which the resulting symmetry acts.

Extracting the pitch-classes that prevent symmetry across the larger sequence of sets in each

Episode produces another set (here labelled the preventative set). For Episode 1 the set 9-11

is produced (01245789T – Figure 14 below).

FIGURE 14: PREVENTATIVE SET FOR EPISODE 1 FIGURE 15: REPLACEMENT OF PC7 WITH 6

This is also a near-symmetrical set (if pitch-class 7 is replaced by 6 – as shown in Figure 15 –

symmetry is achieved around 3). Note that 9-11 is RP to Messiaen’s mode III (01245689T).

Messiaen’s modes of limited transposition, and their pitch-class labels, are shown below.

1

E 0 2 3 4

5

T 9 7

8 6

3

E 0 1 2

4

T 5

9 8 7

6

3

E 0 1 2

4

T 5

9 8 6

7

FIGURE 13: 6-33 PITCH SQUARE FOR 6-33


12

Mode I 02468T 6-35 (the whole tone scale)

Mode II 0134679T 8-28 (the octatonic scale)

Mode III 01245689T 9-12

Mode IV 01236789 8-9

Mode V 012678 6-7

Mode VI 0124678T 8-25

Mode VII 012346789T 10-6

FIGURE 16: MESSIAEN’S MODES OF LIMITED TRANSPOSITION

For Episode 2 a symmetrical 11-note set (around 5) is formed.15

FIGURE 17: PREVENTATIVE SET FOR EPISODE 2

For Episode 3 another near-symmetrical set, 10-5, is produced. Replacing pitch-class 1 with

6 would achieve symmetry around 0.

FIGURE 18: PREVENTATIVE SET FOR EPISODE 3

This set is a superset of Messiaen’s mode V (012678).

15 Note too how the symmetry here occurs because five pitch-classes fall consecutively either side of the line of

symmetry, which itself passes through pitch-class 5.

E 0 1 2 3

4

T 5

9 8 7 6

E 0 1 2 3

4

T 5

9 8 7

6


13

Summarising the preventative sets:

- For Episode 1 it is 9-11; a near-symmetrical set maximally similar to mode III.

- For Episode 2 it is a symmetrical 11-note set (with symmetry around pc5).

- For Episode 3 it is 10-5; a near-symmetrical set, and a superset of Messiaen’s mode V.

The preventative sets therefore yield a looser symmetry:

EPISODE: 1 2 3

SYMMETRY IN PREVENTATIVE SET: NO YES NO

SYMMETRY

FIGURE 19: PREVENTATIVE SET SYMMETRY ACROSS THE EPISODES

Figure 20 adds these findings to the schematic diagram.


/LEVEL

SETS USED

DEMARCATION

LP

ANCHORAGE 5


SET CELLS LP CELLS

GENERATION

LP MOMENTUM


5


SYMMETRY / SET SYMMETRY

NEAR

SYMMETRY UNDERLYING

PREVENTATIVE

SET

NEAR SYMMETRICAL 9-11 IN E1 MAXIMAMALLY MODE III

SIMILAR TO

SYMMETRY SYMMETRICAL 11-1 THROUGH 5 IN E2

NEAR SYMMETRICAL 10-5 IN E3 SUPERSET OF MODE V

FIGURE 20: REVISED SCHEMATIC REPRESENTATION OF SURFACE AND UNDERLYING PROCESSES OF PITCH-CLASS ELEMENTS

Figure 20 adds mode III and mode V to the underlying processes. Dallapiccola was known to

be influenced by Messiaen’s modes of limited transposition, and so it would now be logical to

extend the investigation to further modes, having observed the presence of mode III and V.

INC

RE

AS

ING

DE

PT

H F

RO

M S

UR

FA

CE

OF

MU

SIC


14

Any mode I (whole-tone) influences on sets have already been pointed out where they occur.

The following section explores the strength of influence of the octatonic collection (mode II)

on the Episodes.16 Superimposing two Z-related sets often yields octatonic collections. The

diagram below takes the first two Z-related sets that appear in Episode 1 as examples, and

compares their combined normal form with the octatonic set 8-28 (0134679T).

SET APPEARS AS COMBINED SHARED PITCH-CLASSES NON-OCTATONIC

NORMAL FORM WITH 0134679T PITCH-CLASSES

5-Z17 2679T

(t6)

12345679T 7 out of 9 25

(9-3) (78% of the set 9-3)

6-Z40 13456T

t(i6)

FIGURE 21: EXAMPLE OF COMBINED Z-RELATED SETS IN EPISODE 1

The table below summarises the above for all combinations across the Episodes.

Episode 1

Set 1 Set 2 Combined Normal Form Forte Octatonic Pitch-Classes % of Set Not Octatonic

5-Z17 6-Z40 12345679T 9-3 7 78 25

5-Z17 7-Z17 679TE123 8-19 6 75 2E

6-Z40 7-Z17 9TE123456 9-4 6 67 25E

Episode 2


6-Z10 6-Z13 0123467 7-4 6 86 2

6-Z10 6-Z19 0123456789 10-1 7 70 258

6-Z10 6-Z49 234678TE0 9-12 6 67 28E

6-Z10 7-Z36 2345679TE0 10-5 7 70 25E

6-Z10 7-Z38 9TE023467 9-11 7 78 2E

6-Z10 8-Z15 TE012345678 11-1 8 73 146

6-Z13 6-Z19 TE0123467 9-3 7 78 2E

6-Z13 6-Z49 TE012345678 10-4 7 70 28E

6-Z13 7-Z36 9TE01234567 11-1 8 73 25E

6-Z13 7-Z38 9TE013467 9-10 8 89 E

6-Z13 8-Z15 TE012345678 11-1 8 73 36

6-Z19 6-Z49 678TE124 8-27 7 88 6

6-Z19 7-Z36 9TE0123567 10-4 7 70 25E

6-Z19 7-Z38 1234679TE 9-11 7 78 2E

6-Z19 8-Z15 5678TE012 9-5 7 78 06

6-Z43 6-Z49 23456789TE 10-1 7 70 369

6-Z49 7-Z36 789TE02345 10-5 7 70 039

6-Z49 7-Z38 2346789TE 9-4 6 67 28E

6-Z49 8-Z15 45678TE012 10-6 8 80 06

7-Z36 7-Z38 2345679TE0 10-5 7 70 25E

7-Z36 8-Z15 89TE012356 10-3 8 80 1T

7-Z38 8-Z15 0123456789TE 12-1 8 67 258E

Episode 3


6-Z23 6-Z24 E023579 7-34 6 86 7

6-Z37 6-Z23 789E0235 8-27 7 88 7

6-Z37 6-Z24 789TE0235 9-7 7 78 7T

FIGURE 22: COMBINATIONS OF Z-RELATED SETS ACROSS THE EPISODES

16 For further discussion of Dallapiccola and the octatonic, see Alegant, B. (2006) Octatonicism in Luigi Dallapiccola’s

Twelve-Note Music in Music Analysis 25/1: 39-87.


15

The final column of non-octatonic elements contains all three transpositions of 4-28 (an

octatonic subset and the complement of the octatonic scale).

0 1 2 3 4 5 6 7 8 9 T E

0 3 6 9 1 4 7 T 2 5 8 E

4-28 TRANSPOSITION (t0) (t1) (t2)

FIGURE 23: DIVISION OF THE 12 PITCH-CLASSES INTO THREE OCTATONIC TRANSPOSITIONS

Whilst this is not particularly significant, since 4-28 does not play a prominent role in the

music, the distribution of the transpositions across the episode is, perhaps coincidentally,

symmetrical: only one transposition appears in the outer episodes (t2 in Episode 1 and t1 in

Episode 3), and all three transpositions appear in the Episode 2. Note too that symmetrical

distribution of 1 3 1 also adds to five.

The table below looks at the octatonic content of the five most prevalent sets.

Episode 1

Set Appearances Octatonic Pitch-Classes %

5-33 7 3 60

8-12 6 7 88

8-26 5 6 75

7-35 4 5 71

8-21 4 6 75

Episode 2


4-14 21 3 75

5-27 14 3 60

3-4 13 2 67

4-26 9 4 100

6-Z25 8 4 67

Episode 3


6-33 9 4 67

8-21 4 6 75

8-24 4 5 63

7-9 4 5 71

6-35 3 4 67

FIGURE 24: OCTATONIC PERCENTAGES FOR THE FIVE MOST PREVALENT SETS IN EACH EPISODE

Mode II therefore cannot be added to the underlying processes, since there is only one

subset (4-26). Seven sets (and the most prevalent set for the first two Episodes) are,

however, mode III subsets (3-4, 4-14, 4-26, 5-27, 5-33, 6-35, and 8-24), adding strength to

the conclusion that mode III elements act as underlying processes. The following section

therefore investigates the influence of mode III.


16

Figure 32 repeats the above process, but looking at the level of mode III content.

Episode 1

Set Appearances Mode III Pitch-Classes %

5-33 7 5 100

8-12 6 6 75

8-26 5 7 88

7-35 4 6 86

8-21 4 7 88

Episode 2


4-14 21 4 100

5-27 14 5 100

3-4 13 3 100

4-26 9 9 100

6-Z25 8 5 83

Episode 3


6-33 9 4 67

8-21 4 7 88

8-24 4 8 100

7-9 4 6 86

6-35 3 6 100

FIGURE 25: MODE III PERCENTAGES FOR THE FIVE MOST PREVALENT SETS IN EACH EPISODE

There is a greater number of mode III than octatonic subsets, but the subsets are too small to

be significant in Episodes 1 and 2 (since mode III is a large set it has many subsets). More

important is the presence of the larger subsets 6-35 and 8-24 in Episode 3. Figure 26

compares Figure 25’s sets to each of Messiaen’s modes of limited transposition. The set’s

inclusion in the mode, either as a subset or superset is marked with a Y. The double line

divides supersets from subsets for each mode (supersets being above the line); the absence

of a double line indicates the mode’s cardinality is greater than the largest prevalent set.

Set Mode I Mode II Mode III Mode IV Mode V Mode VI Mode VII

8-26

8-24 Y Y

8-21 Y

8-12 Y

7-35

7-9 Y

6-Z25 Y

6-35 Y Y Y

6-33 Y Y

5-33 Y Y Y Y

5-27 Y Y

4-26 Y Y Y

4-14 Y Y Y

3-4 Y Y Y Y Y

Totals 2 1 7 2 1 3 12

FIGURE 26: SUBSET/SUPERSET RELATIONSHIPS BETWEEN MOST PREVALENT SETS AND MESSIAEN’S MODES OF LIMITED TRANSPOSITION


17

Mode VII appears to be behind all but two of the most prevalent sets, but this in itself is not

surprising, since mode VII is a ten-note set, and therefore is more likely to be related to other

sets. This may, however, be a deliberate choice; mode VII consists of two symmetrical

pentads (i.e. each has the same interval content), orientated around the number 5.

0 1 2 3 4 6 7 8 9 T

5

FIGURE 27: MESSIAEN’S MODE VII PENTADS AROUND PC5

Figure 28 below adds the findings of this section onto the evolving schematic diagram.


/LEVEL

SETS USED

DEMARCATION

LP

ANCHORAGE 5


SET CELLS LP CELLS

GENERATION

LP MOMENTUM


5



NEAR

SYMMETRY

PREVENTATIVE

SET


SIMILAR TO

SYMMETRY SYMMETRICAL 11-1 THROUGH 5 IN E2 UNDERLYING



NEAR

SYMMETRY MOST PREVALENT MODE VII TWO PENTADS 5

PREVENTATIVE SETS SUBSETS AROUND

SET

NEAR SYMMETRICAL 9-11 IN E1 MAXIMAMALLY TWO TETRADS MODE VII

SIMILAR TO AROUND

SYMMETRY SYMMETRICAL 11-1 THROUGH IN E2 MODE III

NEAR SYMMETRICAL 10-5 IN E3 SUPERSET OF

MODE V

FIGURE 28: REVISED SCHEMATIC REPRESENTATION OF SURFACE AND UNDERLYING PROCESSES OF PITCH-CLASS ELEMENTS

INC

RE

AS

ING

DE

PT

H F

RO

M S

UR

FA

CE

OF

MU

SIC


18

Figure 28 shows that, although mode VII interaction is the most prevalent because of its

large size, modes III and V are also present as underlying processes. Mode III, like mode VII,

can be divided into two symmetrical blocks (in this case tetrads) either side of pc5.17

0 1 2 4 6 8 9 T

5

FIGURE 29: MODE III DIVISION INTO TWO SYMMETRICAL TETRADS EITHER SIDE OF PC5

Note too that modes III and VII fall symmetrically either side of mode V.

Examining the sets used in the Episodes, whilst useful, does not account for supersets. The

diagrams below reduce each set used of cardinality n to its superset of cardinality (n + 1) or,

where the superset of (n + 1) is not used in the Episode, to its superset of cardinality (n + 2)

or (n+3). Each set is linked to the superset/subset with the lowest Forte number. For visual

clarity, multiple superset/subset relations are not displayed.

17 Symmetrical in this case since both blocks have the same interval content.


19

10

-3

10

-4

9-7

9-8

9-1

1

8-2

6

8-2

2

8-2

1

8-1

2

8-1

1

7-3

7-1

07

-Z1

77

-26

7-3

47

-35

6-Z

46

6-Z

40

6-3

5

6-3

3

6-3

1

5-Z

17

5-2

15

-23

5-3

35

-34

4-2

1

4-2

4

3-1

2

FIG

UR

E 3

0:

SU

PE

RS

ET

RE

DU

CT

ION

FO

R S

ET

S U

SE

D I

N E

PIS

OD

E 1


20

10

-31

0-4

9-7

9-4

9-2

8-5

8-1

48

-Z1

58

-22

8-2

7

7-Z

38

7-Z

36

7-3

5

7-3

3

7-3

1

7-2

9

7-2

4

7-2

3

7-Z

18

7-1

1

7-1

0

7-9

7-7

7-5

6-Z

10

6-Z

13

6-1

56

-Z1

96

-20

6-2

16

-22

6-Z

25

6-3

16

-32

6-3

56

-Z4

06

Z4

36

Z4

9

5-Z

38

5-3

5

5-3

2

5-3

0

5-2

9

5-2

8

5-2

7

5-2

6

5-2

5

5-2

4

5-2

2

5-2

1

5-2

0

5-1

9

5-Z

18

5-1

6

5-1

5

5-Z

12

5-1

0

5-8

5-6

5-4

5-3

4-1

24

-13

4-1

44

-16

4-1

84

-19

4-2

04

-24

4-2

64

-27

3-9

3-7

3-5

3-4

FIG

UR

E 3

1:

SU

PE

RS

ET

RE

DU

CT

ION

FO

R S

ET

S U

SE

D I

N E

PIS

OD

E 2


21

10

-4

10

-2

9-7

9-8

9-1

2

8-2

4

8-2

1

8-1

2

7-9

7-3

37

-34

7-3

57

-Z3

7

6-3

5

6-3

4

6-3

3

6-3

2

6-2

7

6-Z

24

6-Z

23

6-2

2

5-1

05

-16

5-2

35

-24

5-2

85

-29

5-3

3

4-2

4

4-2

1

4-1

6

4-1

1

3-5

FIG

UR

E 3

2:

SU

PE

RS

ET

RE

DU

CT

ION

FO

R S

ET

S U

SE

D I

N E

PIS

OD

E 3


22

Each Episode can be reduced to two of three ten-note sets – 10-2, 10-3, and/or 10-4. (Ten

being a multiple of five continues Dallapiccola’s preoccupation, of course.) 10-2 is a superset

of the whole-tone scale; 10-3 of the octatonic scale; and 10-4 of mode III. So whilst the

whole-tone and octatonic sets (i.e. modes I and II) do not control the pitch-based musical

processes, the sets used reduce to three ‘organisational forces’: the whole-tone, octatonic,

and mode III sets.18 The presence of mode V in the preventative set of Episode 3, and all of

the most prevalent sets in the Episodes being linked to mode VII indicate that mode V and

VII infiltrate to a certain extent.

The Episodes therefore make use of modes I, II, III, V, and VII. This sequence (1,2,3,5,7),

apart from consisting of five elements, is a Fibonacci sequence (where any term is the sum of

the previous two) disrupted by the fifth element. Note too that the prime forms for modes I,

III, and VII can be arranged into two symmetrical blocks either side of pc5, whereas modes II

and V cannot. This gives further symmetry:

MODE

I II III V VII

SYMMETRICAL BLOCKS AROUND PC5 Y N Y N Y

FIGURE 33: PC5 SYMMETRY WITHIN MESSIAEN’S MODES OF LIMITED TRANSPOSTION USED

Figure 34 below, completes the schematic diagram with the findings above.

18 The looser term ‘organisational force’ is used here to recognise that modes I, II, III, V, and VII are, whilst not important

enough to be labelled controlling forces, involved in the compositional process to a significant extent.


23


/LEVEL

SETS USED

DEMARCATION

LP

ANCHORAGE 5


SET CELLS LP CELLS

GENERATION

LP MOMENTUM


5



NEAR

SYMMETRY

PREVENTATIVE

SET


SIMILAR TO

SYMMETRY SYMMETRICAL 11-1 THROUGH 5 IN E2 UNDERLYING



NEAR

SYMMETRY MOST PREVALENT MODE VII TWO PENTADS 5

PREVENTATIVE SETS SUBSETS AROUND

SET

NEAR SYMMETRICAL 9-11 IN E1 MAXIMAMALLY TWO TETRADS MODE VII

SIMILAR TO AROUND

SYMMETRY SYMMETRICAL 11-1 THROUGH IN E2 MODE III

NEAR SYMMETRICAL 10-5 IN E3 SUPERSET OF

MODE V

BY 5TH ELEMENT

DERIVATIVE SETS DISRUPTED (ALSO BLOCK OF 5)

FIBONACCI

10-2 SUPERSET OF MODE I (WHOLE TONE)

10-3 SUPERSET OF MODE II (OCTATONIC)

SYMMETRY 10-4 SUPERSET OF MODE III

E3 PREVENTATIVE SET SUPERSET OF MODE V

MOST PREVALENT SETS SUBSETS OF MODE VII

FIGURE 34: FINAL SCHEMATIC REPRESENTATION OF SURFACE AND UNDERLYING PROCESSES OF PITCH-CLASS ELEMENTS

Note how the path from surface to underlying processes maps neatly onto the emergence of

symmetry from disruption (shown on the left hand side). But mode III and VII’s large

cardinalities give them many relationships with other sets. In the absence of larger subsets

(of cardinality six or above), the significance of concluding that the Episodes have modes of

limited transposition as an organising force is substantially reduced.

INC

RE

AS

ING

DE

PT

H F

RO

M S

UR

FA

CE

OF

MU

SIC


24

Likewise, this semiotic analysis results in the idea that the Episodes are each controlled by

ten-note sets. Whilst a neat conclusion, ten-note sets, because of their size, also have many

relationships, so this is not an effective way to understand this music, and gets beyond the

point where any conclusions drawn through a semiotic approach can be considered

significant. Widening the approach to look at more musical factors (i.e. looking across bars,

rather than bar-by-bar), gives messier results, yet still provides some interesting

observations.

The second copy of the score in the separate booklet now highlights more musically intuitive

divisions. Episode 1 can be understood as consisting of WT1 sonorities, plus one of three

trichords (3-6, 3-7, or 3-8) from WT2, operating at different transpositions. This is

summarised below, and also labelled in the score.

FIGURE 35: WHOLE-TONE ORGANISATION IN EPISODE 1

Note that there are five whole-tone orientated sections. The organisation of sonorities can be

read as two symmetrical blocks either side of a 5/4 bar:

WHOLE-TONE DIATONIC WHOLE-TONE 5/4 WHOLE-TONE DIATONIC WHOLE-TONE

FIGURE 36: SYMMETRICAL READING OF PATTERN IN EPISODE 1

The final section (bb.62-65) can then be interpreted as a recommencement of this cycle. The

three trichords in operation (3-6, 3-7, and 3-8) collectively make 5-9 (a more significant

subset of Messiaen’s mode III because of its larger size).

AGGREGATE BARS W-T COLLECTION TRICHORD ADDED NO.OF BARS SONORITY FEATURE

12-1 1-4 WT1 + WT2 4

5-14 WT1 3-8 t.11 10

15-16 WT1 3-8 t.7 2 WHOLE-TONE

17 WT1 3-8 t.3 1

12-1 18-19 WT1 + WT2 2

20-27 WT1 3-7 t.4 8 DIATONIC GLISSANDI


10-4 37 1 5/4 BAR

38-39 WT1 3-8 t.5 2

40 WT1 3-8 t.1 1 WHOLE-TONE

41-42 WT1 3-8 t.9 2

43-46 WT1 3-7 t.4 4 DIATONIC GLISSANDI

47-55 WT1 3-8 t.3 8

56-58 WT1 3-8 t.11 3 WHOLE-TONE

59-60 WT1 3-8 t.9 2

61 1 SILENCE

62-65 WT1 3-8 t.11 4 WHOLE-TONE


25

The infiltration of mode III elements in this reading is strengthened upon an examination of

Episode 2, based again on the more musically intuitive divisions. Here the divisions either

contain more significant mode III subsets or build ten-, eleven- and twelve-note aggregates.

Figure 37 below summarises the Episode, with mode III subsets highlighted in bold. Note too

that there is one palindromic and one disruptive set cell, both consisting of five elements.

FIGURE 37: EPISODE 2 ORGANISATION OF AGGREGATES AND MODE III ELEMENTS

The presence of larger ten-note aggregates in this reading now adds strength to the earlier

conclusion that each Episode is to some extent controlled by a ten-note set (see Figures 30-

32). Both the semiotic and musically intuitive divisions demonstrate a process involving

these larger sets. The presence of six prominent mode III subsets now allows mode III to be

labelled a controlling collection in Episode 2. Further, as shown in Figure 38 below,

comparing the prime form of 9-12 with that of each non-mode III subset or superset shows

that each set differs by only three pitch-classes, which form the set 3-8 (one of the trichords

in circulation in Episode 1).

AGGREGATE BARS SET COMMENTS NO.OF BARS

1-8 5-21 8

9-16 7-Z37 8

12-1 17-47 31

48-55 6-Z19 8

56-67 8-17 8-17 ⊂ 9-11 (RP TO 9-12) 12

10-2 68-72 5

73-77 6-14 5

11-1 78-83 6

11-1 84-89 6

90-93 9-7 4

94-96 9-7 3

10-2 97-102 6

10-5 103-110 8

11-1 111-127 17

10-5 128-137 10

10-4 138-145 8

11-1 146-156 11

157-160 7-21 4

12-1 161-166 6

167-171 8-17 8-17 ⊂ 9-11 (RP TO 9-12) 5

12-1 172-185 14

186-195 7-21 10

196-205 7-35 7-35 ⊂ 9-11 (RP TO 9-12) 10

11-1 206-214 9

10-1 215-218 4

219-224 6-Z19 6

12-1 225-232 8

233-239 7-33 7

12-1 240-242/1 2.1

241/1-246 6-Z19 4.4


26

SET PRIME FORM DIFFERING PITCH-CLASSES FROM 9-12

9-12 PRIME

FORM 7-35 013568T 3

01245689T 8-17 01345689 3

9-7 01234578T 37

10-1 0123456789 37

10-2 012345678T 379

10-5 012345789T 37

11-1 0123456789T 37

379 = 026 IN PRIME FORM (3-8)

FIGURE 38: PRIME FORM COMPARISONS BETWEEN NON-MODE III SUBSETS/SUPERSETS AND MODE III

Episode 3 is again concerned with whole-tone collections with added trichords, and is

summarised in Figure 39 below.

FIGURE 39: WHOLE-TONE ORGANISATION IN EPISODE 3

There is a symmetry to the organisation in Episode 3 in terms of the sonorities employed.

The similar process involving trichords gives the set of Episodes a looser symmetry, with the

outer ones involving whole-tone processes, and the middle Episode being controlled by

mode III. Note too that the very last chord in Episode 3 in the left hand is 3-8 – the most

prominent circulating trichord.

The final section looks at the level of mismatch within the Episodes, and will still use

musically-intuitive, rather than bar-by-bar divisions. Since in some cases these differ from

the divisions created for the above pitch-class analysis, these revised divisions (determined

AGGREGATE BARS W-T COLLECTION TRICHORD ADDED NO.OF BARS SONORITY

1 WT1 1

2-3 WT2 2

4 WT1 1

12-1 5-8 WT1 + WT2 4

9-12 WT1 3-8 t.5 4

13-16 WT1 3-6 t.3 4


21-23 WT1 3-6 t.3 3

24-31 WT1 3-6 t.11 8

32-33 WT1 3

34-35 WT2 3-8 t.0 2

36-41 WT2 3-6 t.2 6

42-45 WT1 3-2 t.7 4 DIATONIC

46-48 WT2 3-8 t.10 3

49 WT1 3-12 t.3 1

50 WT2 3-6 t.10 1

51 WT1 3-8 t.5 1

52 WT2 3-6 t.6 1

53 WT1 3-8 t.5 1


56 WT1 3-6 t.5 1

57-60/2 WT1 3-8 t.9 3/2

61-63/1 WT2 3-6 t.4 2/1

63/2-64 WT1 3-8 t.9 1/2

65-69 WT1 3-8 t.7 5


27

more by motivic factors) are indicated by black lines on the first copy of the score. The level

to which various parameters comply with (or go against) these structural divisions will be

examined; features going against the divisions will be termed structurally misaligned,

though this is not intended as a pejorative term.

The graphs below plot the level of consonance and dissonance against each bar, with the aim

of determining whether any process is present, and whether levels of consonance or

dissonance interact with, or go against, structural divisions. Vertical black lines on the

graphs indicate the divisions outlined in the score. Each graph also has a linear trendline

indicating the overall tendency of increase, decrease, or stasis. The consonance values are

the sum of ICs 3, 4, and 5 (the consonant intervals) in the IV for the set/bar. Likewise,

dissonance values are the sum of ICs 1, 2, and 6 (the dissonant intervals).

Episode 1 has 5 structurally misaligned consonance peaks/troughs, and 7 misaligned

dissonance peaks/troughs. These are indicated with red arrows.

FIGURE 40: IV CONSONANCE GRAPH FOR EPISODE 1

0

5

10

15

20

25

30

35

40

0 10 20 30 40 50 60 70

Le

ve

l o

f C

on

son

an

ce

Bar


28

FIGURE 41: IV DISSONANCE GRAPH FOR EPISODE 1

In order to determine whether consonance and dissonance themselves interact, Figure 42

below superimposes Figures 40 and 41 (dissonance is shown by the red line), and shows

both factors increasing and decreasing in proportion, locked in phase with each other, with

the level of consonance being greater than dissonance. There is still a decreasing trend

through the Episode.

FIGURE 42: SUPERIMPOSED CONSONANCE AND DISSONANCE GRAPHS FOR EPISODE 1

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70

Le

ve

l o

f D

isso

na

nc

e

Bar

0

5

10

15

20

25

30

35

40

0 10 20 30 40 50 60 70

Le

ve

l o

f C

on

son

an

ce

/D

isso

na

nc

e

Bar


29

Note that three misaligned peaks fall either side of the one aligned peak at bar 37, giving a

sense of symmetry to the consonance/dissonance alignment within the Episodes. Note too

that for each of the main peaks consonant and dissonant intervals often differ by five. Bar 37

is also in 5/4 and can be considered to be the central focus of Episode 1.19 The consonance

and dissonance graphs for Episode 2 are below, as well as the final superimposition.

19 This will be dealt with in the second essay.

FIG

UR

E 4

3:

CO

NS

ON

AN

CE

GR

AP

H E

PIS

OD

E 2


30

FIG

UR

E 4

4:

DIS

SO

NA

NC

E G

RA

PH

EP

ISO

DE

2


31

FIG

UR

E 4

5:

CO

NS

ON

AN

CE

/D

ISS

ON

AN

CE

GR

AP

H E

PIS

OD

E 2


32

The graphs for Episode 2 again show consonance and dissonance peaks/troughs locked in

phase with one another, but are here aligned to the structural divisions of the Episode. As

with Episode 1, the consonance values are greater.

The graphs for Episode 3 are below.

FIGURE 46: IV CONSONANCE GRAPH FOR EPISODE 3

FIGURE 47: IV DISSONANCE GRAPH FOR EPISODE 3


33

FIGURE 48: IV CONSONANCE/DISSONANCE GRAPH FOR EPISODE 3

Figures 46 and 47 show twelve misaligned consonance and eight misaligned dissonance

peaks/troughs.

The linear trendlines of the graphs show that the outer Episodes share a general decrease in

the level of consonance and dissonance, with Episode 2 showing a general increase

throughout. There is therefore a kind of symmetry to the overall fluctuation of dissonance

and consonance. The graphs also clearly show a point at which the dissonance and

consonance are at their maximum (shown to the nearest percentage):

Episode 1 = bar 19 (30% through)



Note how these percentages are all multiples of 5.

The graphs below look at pitch-class circulation for each episode.20

20 Pitch-class circulation describes the net result of pitch-classes subtracted and/or added from the previous bar. Five

pitch-classes removed and two added gives a pitch circulation of minus three (three fewer pitch-classes).


34

FIGURE 49: PITCH-CLASS CIRCULATION FOR EPISODE 1

Many of the peaks/troughs align structurally. The most noticeable anomalies (bb.2-3, 18-20,

26-28, and 35-36) initiate new aggregate cycles immediately after the previous one has been

completed.21 The restarting of an aggregate cycle (and therefore the subtraction of a large

number of pitch-classes from the previous bar) accounts for these areas of greater pitch-

class circulation. The aggregate cycles within Episode 1 do not therefore align structurally.

The pitch-class circulation graph for Episode 2 is shown below.


21 The term aggregate cycle is used to describe the gradual construction of the twelve-tone aggregate over a number of bars.


35

Again, the major peaks/troughs do not align structurally. The locations of maximum pitch

circulation again occur when an aggregate cycle is restarting but, as with Episode 1, these

points do not align structurally. The pitch-class circulation graph for Episode 3 is below.


Only one peak is misaligned – bb.18-19 – and does not coincide with the build-up of an

aggregate. Across the Episodes most consonance/dissonance and pitch-circulation

peaks/troughs align structurally; any which do not can be attributed to locations where a

new aggregate cycle is beginning. Introducing only a few pitch-classes at the start of a new

aggregate cycle necessarily reduces the consonance and dissonance values, and results in a

large pitch-circulation change. The misalignment of aggregates may be due to rhythmic,

dynamic, and/or motivic reasons, and will therefore be explored in the second essay.

This essay has shown that the number 5 plays an important organisational role within this

earlier work, and, through both semiotic and motivic divisions of the music, that the

Episodes utilise ten-, eleven-, and twelve-note aggregates. Several modes of limited

transposition are present as controlling elements, with mode III being a controlling

collection in Episode 2. The outer Episodes have been shown to have mode I (the whole-tone

scale) as a controlling collection, giving a looser symmetry to the episodes, with the outer

ones being controlled by the whole-tone collection. The use of modes I, III, and I of course

add to five.

It has been demonstrated that symmetry and disruption interact at multiple levels and,

through schematic representations, the gradual evolution from disruption to symmetry has


36

been shown to match neatly with the progression from the surface to the underlying forces.

The elements of mismatch between consonance, dissonance, and pitch-circulation have been

explored and found to have aligned in most cases with structural divisions (segmented

according to similar motivic and/or rhythmic features). The cases where there is a mismatch

with the structure have been attributed to the conclusion and initiation of aggregate cycles.

It is suspected that the location of these aggregate cycles and the reason for their structural

misalignment are both associated with rhythmic and motivic features; something which will

be investigated in the second essay.

BIBLIOGRAPHY

Alegant, B. (2010) The Twelve Tone Music of Luigi Dallapiccola. Rochester: University of Rochester Press.

Alegant, B. (2006) Octatonicism in Luigi Dallapiccola’s Twelve-Note Music, in Music Analysis 25/1: 39-87.

Cook, N. (1987) A Guide to Musical Analysis. Oxford: Oxford University Press.

Dunsby, J. and Whittall, A. (1988) Music Analysis in Theory and Practice. London: Faber.

Eckert, M. (1985) Octatonic Elements in the Music of Luigi Dallapiccola, Music Review 46: 35-48.

Fearn, R. (2003) The Music of Dallapiccola. Rochester: University of Rochester Press.

Forte, A. (1973) The Structure of Atonal Music. Yale: Yale University Press.

Forte, A. (1986) Liszt’s Experimental Idiom in 19th-Century Music 10/3: 209-28.

Forte, A. (1991) Debussy and the Octatonic, Music Analysis 10/1-2: 125-69.

Lester, J. (1989) Analytic Approaches to Twentieth-Century Music. New York: Norton.

Lewin, D. (1998) Some Ideas about Voice-Leading Between PC-Sets, in Journal of Music Theory 30/ii: 79-102.

Mancini, D. (1998) Twelve-tone Polarity in Late Works of Luigi Dallapiccola, in Journal of Music Theory 30/ii: 203-24.

Messiaen, O. (1944) Le Technique de mon language musical. Paris: Leduc, anonymous translation from

www.courses.unt.edu/jklein/files/Messiaen1_0.pdf

Schuijer, M. (2008) Analysing Atonal Music: Pitch-Class-Set Theory and its Contexts. Rochester: University of Rochester Press.

Online resources:

www.jaytomlin.com/music/settheory/help.html

www.solomonsmusic.net/setheory.htm

www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/

Set calculations done online from:

www.jaytomlin.com/music/settheory

www.mta.ca/faculty/arts-letters/music/pc-set_project/calculator/pc_calculate.html

SYMMETRY, DISRUPTION, AND MISMATCH IN DALLAPICCOLA…alexaitken.co.uk/assets/Pitch Class Analysis on...

Documents

Transcript of SYMMETRY, DISRUPTION, AND MISMATCH IN DALLAPICCOLA…alexaitken.co.uk/assets/Pitch Class Analysis on...