A Theory for the Analysis of Spatial MusicDerived from Stockhausen’s Lichter-Wasser ()

Proposal for (Los Angeles)

by Paul Miller Prince St., No. Rochester,

paul@theoryofpaul.net( ) -

Graduate Student and Instructor of Music Theory,Eastman School of Music (University of Rochester)

A Theory for the Analysis of Spatial MusicDerived from Stockhausen’s Lichter-Wasser ()

Proposal for (Los Angeles)

Equipment required:Overhead Projector

A Theory for the Analysis of Spatial MusicDerived from Stockhausen’s Lichter-Wasser ()

Proposal for (Los Angeles)

Abstract. This paper proposes a new technique for the analysis of spatialized music by

using a recent work by Stockhausen, Lichter-Wasser, as a test bed. A tool called the motion

profile segment (MPSeg) is advanced to aid in the detection of spatial motives throughout the

work. Then, by generalizing the mathematical group properties of transformations in two

dimensions, we can relate spatial motives. These techniques can further our understanding of

the composer’s compositional strategies in the spatial domain, and may also be useful for other

spatialized works, especially those of Xenakis.

- 1 -

A Theory for the Analysis of Spatial MusicDerived from Stockhausen’s Lichter-Wasser ()

Proposal for (Los Angeles)

Published analyses of Stockhausen’s music often focus on issues of formal design and

pitch. Analysts such as David Lewin, Jerome Kohl, Hermann Conen, Richard Toop, Imke

Misch, and the composer himself have all offered detailed analyses of Stockhausen’s methods

of compositional design and coherence. While these contributions offer compelling views of

Stockhausen’s compositional craft, only rarely do they expound upon the spatial aspects of the

music – and even then, not in a way that sheds much light on their internal structure. Yet,

since his early electronic works of the s, Stockhausen has consistently devoted a significant

amount of attention in his compositional process to spatialization.

On the other hand, Maria Harley’s extensive work suggests many fruitful ways of

analyzing spatialized music; but despite her valuable observations, she does not offer any

techniques for analysis. This investigation will demonstrate that transformation theory and

statistical methods can be used to analyze the spatial movements in a recent work by

Stockhausen, Lichter-Wasser (). These methods can be generalized so that they can apply

to other spatialized music.

Envisioned as the first part in the Sunday opera from the gigantic Licht cycle, Lichter-

Wasser employs a core ensemble of musicians. The musicians are divided into two

orchestras based on register, and are arranged in a geometric pattern throughout the audience

(Example 1). Over the course of the -minute work, two melodic lines (one assigned to each

orchestra) weave their way through the performance hall. Each musician plays a note or a

group of notes before handing their melody off to the next player. There are nearly ,

such movements in Lichter-Wasser.

We can analyze the spatial motion in Lichter-Wasser by detecting spatial motives, or

sequences of movements that recur (Examples 2a and 2b). These motives are related by

various transformations within the physical space that the piece occupies. For example, some

cardinality 3 motives that recur within orchestra 1 and orchestra 2 are related by rotation,

retrograde, transposition and flip operations (Examples 2c-2e). While the two counterpointed

melodies that wind through the piece exhibit individualized spatial motion (due to the different

spatial configuration of each orchestra), the methodology I employ finds motivic releationships

across orchestras as well (Example 2f). In my presentation I will show how the operations of

transposition, rotation, flip, and multiplication operate in a two-dimensional space, and how

motives related by these operations create spatial coherence in Lichter-Wasser.

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How can we decide which spatial motives are important, even when there are many

good candidates to choose from? By analyzing the speed of motion in Lichter-Wasser, we

can discern motivic activity by detecting a recurring contour in the rate of change of motion

that I call an “MPSeg” (motion profile segment). The speed of motion in meters per second

can be calculated throughout most of the work’s twelve main sections, since the positions

of the players in the hall are known, and Stockhausen’s painstaking method of notating

rhythmic durations allows us to determine the timing of each note with the utmost precision.

In Examples 3a and 3b, the rate of motion is charted for sections 1 and 11. (In the score,

Stockhausen refers to these sections as “waves”.) If we calculate the distance that the two

melodies traverse in the first section, we find that each moves about one kilometer in roughly

. minutes, for an average speed of . m/sec and . m/sec, respectively. While the motion

profile (MP) of the first section is highly irregular, the slow rate of motion allows us more easily

to perceive motives that do not recur frequently.

The eleventh section (lasting only about seconds,) moves far more distance in much

less time; its average rate of motion is approximately m/sec and m/sec for each orchestra

respectively (Example 3c). Although a higher rate of motion should make it more difficult

to discern motivic activity, an MPSeg in the graph of the eleventh section recurs throughout

(indicated by horizontal brackets in Example 3b). While the eleventh section exhibits a far

higher density of spatial motives than the earlier section (Examples 4a, 4b), our hearing can

be guided by motives that create recurring MPSegs. Thus, even though the rate of motion

in the eleventh section is roughly eight times faster than the first section and there is greater

motivic saturation, there is still spatial coherence. Using this method to explore the other

sections of Lichter-Wasser will allow us to see how various sections of the work employ different

compositional strategies in the spatial domain.

These techniques of analyzing Lichter-Wasser suggest broader application to works that

are spatialized in different ways, especially Xenakis’s Terretektorh and Nomos Gamma. The

paper proposed herein will use these methods to investigate the spatial design of Lichter-Wasser

more completely. Then I will set the stage for looking at other kinds of spatialized music

in a more analytically fruitful way by generalizing the mathematical group properties of the

four basic operations of transposition, rotation, flip and multiplication in a two-dimensional

space.

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References

Brant, Henry (). “Space as an Essential Aspect of Musical Composition.” In Contemporary Composers on Contemporary Music, ed. Elliott Schwartz, Barney Childs. New York: Da Capo Press.

Coenen, Alcedo (). “Stockhausen’s Paradigm: A Survey of His Theories.” Perspectives of New Music ⁄: -.

Conen, Hermann (). Formel-Komposition: Zu Karlheinz Stockhausens Musik der siebziger Jahre. Mainz: Schott.

Harley, Maria Anna (). Space and Spatialization in Contemporary Music History and Analysis, Ideas and Implementations. Ph. D. Dissertation, McGill University.

---------- (). “Musique, Espace et Spatialisation: Entretien de Iannis Xenakis avec Maria Harley.” Circuit. Revue Nord-Americaine de Musique du XXe Siecle ⁄: -.

Kohl, Jerome (). “Into the Middleground: Formal Syntax in Stockhausen’s Licht.” Perspectives of New Music ⁄: -.

Lewin, David (). “Making and Using a Pcset Network for Stockhausen’s Klavierstück III .” In Musical Form and Transformation: Four Analytic Essays. New Haven: Yale University Press.

---------- (). Generalized Musical Intervals and Transformations. New Haven: Yale University Press.

Misch, Imke (). “On the Serial Shaping of Stockhausen’s Gruppen für drei Orchester.” Perspectives of New Music ⁄: -.

Stockhausen, Karlheinz (). Composition Course on Lichter-Wasser. Kürten: Stockhausen Verlag.

---------- (). “Music in Space.” Trans. Ruth Koenig, in Die Reihe vol. . Pennsylvania: Presser.

Toop, Richard (). Six Lectures from the Stockhausen Courses Kürten . Kürten: Stockhausen Verlag.

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

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23

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25

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27

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29

30

V5 F2 P1 V3 Fa1

B Sax

Va5 Eh F1

T1 Va1

K Va3 Fa2 Va2 P2

H1

T V1

V2 T2

Va4 Eu

Ob V4 H2 Kb Tu

30 m

eter

s

27 meters

(1, 4) (26, 4)(7.25, 4) (13.5, 4) (19.75, 4)

(1, 29)(7.25, 29) (13.5, 29) (19.75, 29)

(26, 29)

(1, 10.25)

(1, 16.5)

(1, 22.75) (26, 22.75)

(26, 16.5)

(26, 10.25)(13.5, 10.25)

(13.5, 16.5)

(13.5, 22.75)

(7.25, 16.5) (19.75, 16.5)

(radius = 12.5 m

)

(9.08, 20.92) (17.92, 20.92)

(17.92, 12.08)(9.08, 12.08)

(4.66, 25.34)

(22.34, 25.34)

(22.34, 7.66)(4.66, 7.66)

conductormixing console synth.

V5F2P1V3Fa4BSaxVa5EhF1T1Va1KVa3Fa2Va2P2ThV1H1V2T2Va4EuObV4H2KbTu

1.00 29.007.25 29.00

13.50 29.0019.75 29.0026.00 29.00

4.66 25.3422.34 25.34

1.00 22.7513.50 22.7526.00 22.75

9.08 20.9217.92 20.92

1.00 16.507.25 16.50

13.50 16.5016.5016.50

19.7526.00

9.08 12.0817.92 12.08

1.00 10.2513.50 10.2526.00 10.25

4.66 7.6622.34 7.66

1.00 4.004.004.004.004.00

7.2513.5019.7526.00

Key

Coordinates of eachinstrument

violin 5flute 2

trombone 1violin 3

bassoon 4bass clarinet

saxophoneviola 5

english hornflute 1

trumpet 1viola 1

clarinetviola 3

bassoon 2viola 2

trombone 2tenor horn

violin 1horn 1

violin 2trumpet 2

viola 4euphonium

oboeviolin 4horn 2

e-flat clarinettuba

Example 1. Spatial Layout of Lichter-Wasser *

instrument abbreviation x-coord.

y-coord.

= instrument in orchestra 2

= instrument in orchestra 1

(radius = 6.25 m)

= location of a loudspeaker

* The hall in which the premiere took place measured 27 × 30 meters.

V5 P1 V3 Fa1

B Sax

Va5 Eh F1

T1 Va1

K Va3 Fa2 Va2 P2

H1

T V1

V2 T2

Va4 Eu

Ob V4 H2 Kb Tu

F2 V5 P1 V3 Fa1

B Sax

Va5 Eh F1

T1 Va1

K Va3 Fa2 Va2 P2

H1

T V1

V2 T2

Va4 Eu

Ob V4 H2 Kb Tu

F2

Example 2. Motivic Design of the 1st Section (”Wave”) in Lichter-Wasser

Example 2a. Motives that occur in Orchestra 1 Example 2b. Motives that occur in Orchestra 2

c

d

c

d

Example 2c. Relationship of motives a and b

Example 2d. Relationship of motives c and d

Example 2e.Relationship of motives e and f

Example 2f.Relationship of motives a and c

c

[transpose(6.25,-6.25)][rot(180)]a = b [retrograde]c = d or [flip(vertical)]c = d

or

[flip(horizontal)][rot(180)]c = d

[transpose(0, -12.5)][retrograde][flip(horizontal)]e = f

[mult(2)][rot(315)][transpose(4.4, 8.1)]a = c(The order of operations is always from right

to left.)

a

b

e

f

a

b

e

f

a

Example 3a. Rate of Motion or Motion Profile in the 1st Section (”Wave”) of Lichter-Wasser

0 50 100 150 200

0

5

10

15

20

25

30

35

25 75 125 175 213

rate

of m

otio

n (m

eter

s per

seco

nd)

elapsed time (seconds)

Orchestra 1

Orchestra 2

Total Distance Traversed

Number of Movements

Average Speed

Orchestra 1 Orchestra 2

801 m

105

3.8 m/sec

1061 m

100

5.0 m/sec

Example 3b. Rate of Motion or Motion Profile in the 11th Section (”Wave”) of Lichter-Wasser *

0 10 20 30 40 50

0

50

100

150

200

rate

of m

otio

n (m

eter

s per

seco

nd)

elapsed time (seconds)

5 15 25 35 45

Orchestra 1 Orchestra 2

1794 m

210

37 m/sec

1687 m

165

34 m/sec

Key

Example 3c. Comparisonof the Rate of Motionin Sections 1 and 11of Lichter-Wasser

Section 1 (213 seconds) Section 11 (49 seconds)

*brackets indicate a recurring motion profile segment

Approx. Rate of Movement .5 moves/sec .5 moves/sec 4 moves/sec 3 moves/sec

Example 4a. Frequency of Cardinality 3 Motives in the 1st Section (”Wave”) of Lichter-Wasser

Example 4b. Frequency of Cardinality 3 Motives in the 11th Section (”Wave”) of Lichter-Wasser

Orchestra 1 Orchestra 2

eh à t1 à va3v2 à v1 à va2

f1 à v2 à kv3 à eh à t1va5 à k à va3t1 à va3 à thk à va3 à bk à ob à v4va3 à th à v2va2 à va1 à ehth à v2 à v1v1 à va2 à va1t2 à f1 à v3ob à v4 à kbv4 à kb à t2kb à t2 à f1

33

22222222222222

p1 à sax à fa1fa1 à p2 à tub à p1 à saxb à va5 à h1sax à p1 à bsax à fa1 à p2p2 à th à eueu à fa2 à btu à eu à fa2

p1 à b à va5fa1 à sax à p1va5 à h1 à va4va3 à th à va4fa2 à b à va5h1 à va4 à h2va4 à h2 à fa2eu à tu à p2h2 à fa2 à eutu à p2 à eu

333333333

2222222222

Orchestra 1

v1 à va2 à va1

v3 à f2 à v5f1 à v3 à f2va3 à v2 à kbva2 à va1 à ehv2 à kb à t2t2 à f1 à v3v4 à v1 à va2kb à t2 à f1

v5 à k à obf2 à v5 à kk à ob à v4ob à v4 à v1

eh à t1 à va3t1 à va3 à v2va1 à eh à t1

eh à va3 à v2va1 à eh à va3

14

1313131313131313

12121212

111111

22

eu à h2 à p1h2 à p1 à btu à eu à h2

fa1 à p2 à tusax à fa1 à p2va5 à h1 à va4fa2 à sax à fa1p2 à tu à euth à fa2 à saxh1 à va4 à thva4 à th à fa2

p1 à b à va5b à va5 à h1

p1 à b à v5b à v5 à va5

v5 à va5 à h1fa1 à th à tuva5 à h1 à fa1th à tu à euh1 à fa1 à th

131313

1111111111111111

1010

33

22222

Orchestra 2

frequency motive frequency motive

frequency motive frequency motive