OF MATOLOGY · tion. Architect and theorist Peter Eisenman argues that the diagram is an expression...
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IN MUSIC AND ARCHITECTURE 14 APRIL 2014 ALEX PORTER COLUMBIA UNIVERSITY, COLUMBIA COLLEGE DEPARTMENT OF MUSIC DEPARTMENT OF ARCHITECTURE OF MATOLOGY
Transcript of OF MATOLOGY · tion. Architect and theorist Peter Eisenman argues that the diagram is an expression...
IN MUSIC AND
ARCHITECTURE
14 APR IL 20 14
ALEX PORTER
C O L U M B I A U N I V E R S I T Y,C O L U M B I A
C O L L E G EDEPARTMENT OF MUSIC
DEPARTMENT OF ARCHITECTURE
OF MATOLOGY
With much thanks to Joseph Dubiel, David Smiley,
Peter Susser, Irina Verona, and other Columbia
University faculty members whose discussion,
questions, and support made this project possible.
4
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the 20th century, is a good example of this because the pointed relationship
ABSTRACT
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Dialogue occurs between points. If points are
the disparate logical systems at play, then the
first diagram in architecture would be a dialogue
between “positive” or “built” things, and the
spaces they define. If we take points to be solids,
surfaces, roofs, and other such architectural
beings, a dialogical movement forms between
spaces and their containers. It is not a discus-
sion of figure-ground as much as one of presence
and its opposite, because one purpose of dialogue
is precisely to avoid the type of foreground-back-
ground which figure-ground establishes. The op-
position between Umberto Boccioni’s Develop-
ment of a Bottle in Space, and Donald Judd’s 15 Untitled Works
in Concrete illustrates the stakes of this dialogue, the ways in
which the two matters affect each other. We are thus given
occasion to title Judd’s Untitled as Development of Space in a
Bottle from its dialogue with Boccioni’s. The question provoked
is how an architect is to accomplish this dialogue by manipu-
lating only one of the participants directly. How does one turn
the Development of a Bottle into the Development of Space?
Music is a fitting reference for an architect, because a composer
also shapes the ephemeral, spatial, and sonic components of a
piece indirectly (the score). This pursuit returns to the dialogue
between the present and the un-present, in other words, the
dialogue of the abstract machine.
Dialogue occurs between points, in the lines which connect nodes in
Stravinsky’s representation of his ‘recent music’ in the 1950’s, the longtime
logo for Perspectives of New Music. Like Deleuze and Guattari’s Rhizome,
Stravinsky’s graphical system networks tones in Stravinsky’s rotational
array, instead presenting hexachords as autonomous pieces. The logical
systems which govern each node submit themselves to the movement
which occurs through the one to the other. A dialogue in music subordinates
musical material to musical movement, fabricating the material in dialogue
with a subversive contrary. Dialogue, then, is a question of analysis which
contains a question of ontology; what is the musical material, what is its
contrary, and in what way is it in dialogue with its contrary? Contrariety is
an ontological practice because it is embedded in a difference between two
points that construes such points for the composer or the analyst. Plato’s
dialogues are a deceptive practice, because they advance a
singular logos from two speakers. This is why Plato must banish
poets and musicians from his ideal city, to establish contrariety
and therefore dialogue between the philosopher and the artist.1
Analysis of a musical dialogue discusses contrariety of musical
material in terms of its reification in a composition. Locating
and exposing these dialogues in a piece’s mechanical procedure
leads backwards into the piece’s abstract ontology. This agenda
relates to Schenker’s division of foreground and background
in his analytical method, except for the fact that the abstract
machine does not rely on a structural background. Structure
remains a chicanery of the foreground’s performance; concept
is possible in the background.
Dialogue occurs between points. It is a singular which contains a plural. Despite his best
efforts, Hamlet cannot stage a dialogue without Polonius, and would be left with only logos,
logos, logos. Dialogue requires multiple points in order to move through, from one, to another.
This performative dia illustrates the limits of logos alone, an inability to conceptualize the pro-
cessional. Logos is static, d ialogos is a process. It is from this critique of the hegemony of
logos that we might consider the dialogue at hand: that between the abstract and the mechanical
which is here called diagrammatic.
From Deleuze and Guattari’s formulation in A Thousand Plateaus (1980), the abstract machine
is an anti-genealogy which is “not physical or corporeal… it is diagrammatic… It operates by
matter, not by substance; by function, not by form.”1 The system operates beyond substance
and thereby before presence, without correspondence to a form (matter) within the boundar-
ies of presence. Deleuze and Guattari replace genealogical, binary linguistics with a dialogic
formulation in which the lines between points are important; the points themselves are not.
They continue, “Substances and forms are of expression ‘or’ of
content. But functions are not yet ‘semiotically’ formed, and
matters are not yet ‘physically formed.’” This abstract machine
functions; it does not present fixed information. It avoids the
static, signification model by suggesting a performance. Unlike a
signifying linguistic in which something is, a performative model
is one in which something happens, a condition by which an
abstract machine must refer to an operation occurring outside of
the boundaries of physical form. A performance is a function, as
described above, in opposition to a form. It operates between
physically formed points; it is an implication, communicated
by suggestion. By its constitution as the movement between
points, performance is never properly present.
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Boccioni, Umberto. Development of a Bottle in Space . 1913. aStravinsky, Igor. Drawing of his ‘Recent Music.’ c.1950’s. bJudd, Donald. 15 Untitled Works in Concrete . 1980-1984 c, d
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architectural only by masking the diagram with a fixed layer of semiotic information which posits a
univocal, signification system, precisely the system to which Deleuze’s diagram is opposed.
This is not a conflict seeking to deny architecture’s compliance with the metaphysics of presence,
but one which questions the relationship between three diagrammatic performances (dialogue,
diaphora, diachronism) and the way in which architecture uses them to comply with presence (or
not?). The architectural theorist Anthony Vidler approaches this dilemma in his essay, “Diagrams of
Diagrams,” in which he argues that digital diagram practices upset the traditional, iconic statement of
a diagram by blurring the line between virtual and actual space.4 The study undertaken here removes
the traditional assumptions of material, form, program, etc. undermining the diagram’s autonomy.
Removing the architectural diagram from a substance-based approach to architectural problems
exposes the conditions of matter under which the diagram performs.
Because the conditions of the virtual are distinct from the conditions of presence, the essence
of a performative diagram cannot be simply, as architect Stan Allen says, “an abstract means of
thinking about organization.”5 This assumes a fixed order of things that denies the dialogue between
substance and matter, proposing local possibilities and relationships confined to concepts of
substance. Nor are these the conditions of “the social-discursive aspect of architectural practice,” 6
as the architectural team of Ben van Berkel and Caroline Bos have said, for this supplants the meta-
physical discussion of concept (the discussion that facilitates our comparison to music) with purely
social concerns—an architecture which does not address the conceptual dilemma under investiga-
tion. Architect and theorist Peter Eisenman argues that the diagram is an expression of Derridean
writing, which reveals a sub-conscious, already-written interiority. While this questions the truth of
substances themselves, it also posits a fixed nature to the sub-conscious, a singular interiority, which
is rooted in the relationship between “an authorial subject, an architecture object, and a receiving
subject.” 7 In other words, Eisenman reconfigures Derrida and Freud into a teleological practice, and
does not actively exploit the possibilities outside this telos. While each of these hypotheses has
venerable application in addressing certain architectural challenges, the question of performance at
hand demands an alternate condition: the condition of difference.
Diagram work is separate from “practical” drawings of plans, sections,
and perspectives because it is performative, not representational.1 Ar-
chitectural theorist R.E. Somol points out the difference between perfor-
mance and representation as the difference between the virtual and the
real. Diagram discourse in architecture has considered the diagram in
many sub-categories that tend to reduce to two polarized interpretations:
either the diagram generates something, or it explains something, both in-
terpretations placing heavy boots in the swamp of concept.2 Performance
creates the force behind a diagram-drawing, but the diagram becomes
an image of presence when it produces an architectural analogue. This
incompatibility between a machine reliant on a non-physical performance
and an image demanding a physical product is the ignored paradox of the
abstract machine in architecture.3 The diagrammatic operation becomes
The issue of dialogue and ontology is at stake in the associative figures of Terry Riley’s
landmark piece, In C (1964), as well as Stravinsky’s early piece, Three Pieces for String Quartet
(1914, rev. 1918). Following Dora Hanninen’s provocation of associative landscapes in A Theory
of Music Analysis, material in dialogue begin with the segmentation and association of figures dis-
cretely distributed in the space of either piece. According to Hanninen, segments, or groupings
of notes come in two varieties, the genosegment, the theoretical archetype which supports
exactly one criterion of sonic or contextual material, and the phenosegment, which is an audible
representation and thereby a repetition of one or more genosegments.2 For the discussion of
dialogue, the specific characteristic in question for an analysis constitutes the logos of that given
segment. The geno/pheno-segment distinction relates to the ontological nature (identity) of
musical beings, phenosegments, and their dialogue with musical Beings, genosegments.3 Just
as the architect, Greg Lynn, argues for myopic multiplicity of form, Hanninen’s biological imagery
tethers Being, as genetic identity, to being, as representational fact.
which substances are constituted. Performance asks the question: “What is ___?” By underpinning
formal procedures and operations with this ontological inquiry, the diagram advances “into a field of
knowledge which did not yet ‘exist’ prior to the moment of this formal conquest,”2 as the Austrian art
historian Hans Sedlmayr puts it. The internal duality of Deleuze and Guattari’s formulation, abstract
machine, suggests a challenging divide between performance and presence. Contrary to operative
thinking, the diagram posits abstraction before reality, not the other way around.
The correspondence between the presented diagram and the artwork’s becoming in presence
is the bridge which Nelson Goodman’s diagram categories begin to clarify. In Languages of Art,
Goodman distinguishes between the digital diagram and the analog diagram, saying that analog
diagrams correspond directly (syntactically and semantically) to the physical artworks they produce.
Digital diagrams, however, are “discontinuous throughout,” meaning that the notational scheme is
“one-one correlated with compliance-classes of a similarly discontinuous set.”3 The digital category
addresses the abstract machine in the break between discontinuous sets, even if this is as far as
Deleuze and Guattari further: “The abstract
machine is a pure Matter-Function—a diagram inde-
pendent of the forms and substances, expressions and
contents it will distribute.” Deleuze and Guattari cleave
the Matter-Function operation from the substance-
form reality, an autonomy under which the nature of
the dialogic performance might be realized. We must
separate the performative from the operative, the
abstract from the mechanical, in order to consider the
conditions of a passage between the two. Operative
thinking concerns forms, problems, and solutions, ad-
dressing the question: “How to ___?” Performative
thinking precedes this question, concerning matter by
Eisenman, Peter. Diagram of House VI . 1972-1975
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The architect Greg Lynn proposes the dialogue of difference in his conception of formal
species in architecture. Like Hanninen and Deleuze, Lynn works associatively to propose “the
practice of originating families of form,”8 for which one model is the art historian Rudolf Wittkow-
er’s diagrams of Palladio’s villas. In Wittkower’s reading of Palladio, the association of a family
of schematic plans suggests a generalized plan for Palladio’s villas, presented as the lower right
of Wittkower’s set. For Lynn, this is both an instance of morphological type, and a turn towards
Bataille’s informe, the morphological vagueness of geometric categorization. This categorical
vagueness is demonstrated in a comparison between the 18th century physiognomist Johann
Caspar Lavater’s drawing From Frog to Apollo and Hanninen’s associative reading of In C. The
boundaries between sonic categories of In C are blurred by the gradual changes of patterns,
and by the players following Riley’s instruction to “make all exits and entries as inconspicuous
as possible.”9 The transitions between plots of disparate field characteristics in In C become
movement between different families of sounds, much like Lynn’s project Embryological House.
Hanninen’s notion of an associative landscape is an abstract machine
in so doing, thus relating to Lynn’s Embryological House, Deleuze’s Rhizome
(also biological) and Schenker’s levels (organic). And yet, Hanninen’s
attention to the multivalent relationships between genosegments and phe-
nosegments departs from Schenker’s totalizing view of musical knowledge,
and the surface is seen as a product of genetic complexity. In other words,
Hanninen’s framework supports many possible genosegment backgrounds,
while Schenker attempts to point towards a singular deep background.
The division of an audible phenosegment from the tabula rasa genoseg-
ment with which it identifies could be considered a musical equivalent to
Derrida’s division of signifier from signified, and the possibility of associa-
tive freeplay thus instigated.4 There is therefore of analytical importance
to consider both the free association of genosegments and the contextual
digital diagram, plunging even analytic philosophy into the paradox of
discontinuity and presence. This is of importance for the diagram work
attempted here, in which analysis and genesis are parallel questions.
When seen from particular perspectives, the dialogue moves in
multiple directions, none of which encompass it. Exposing the paradox
of the abstract machine prompts this examination of the conditions of
possibility a priori of diagrammatic thinking in a discussion of archi-
tecture and music. Though both disciplines are distinct, the parallel
discussions undertaken here hope to introduce a dialogue to describe
a diagrammatic conceptualization and its provocation of performance
as an ontological question. Removing the diagram from the content in
which it deals (architectural usage, musical pitch content, etc.) exposes
the condition under which it performs: the condition of difference.
Goodman goes. In application, the diagrams of architecture and music (plans
and scores, for Gsoodman) relate to each other in their digital components, the
musical score leading the architectural plan towards the digital. Just as the digital
breaks from the analog, so too does music help to open these diagrammatic
systems in presence. To his dismay, Goodman finds that “Architectural plans,
like musical scores, may sometimes define works as broader than we usually take
them.”4 This is problematic for Goodman’s theory, which leans on an obsession
with authorship and authenticity which is confounded by musical performance.
Perhaps we might step over the corpses of authorship to see this invocation
of the digital as the abstract possibility of the diagram made mechanical in one
instance. The particular instance, a plan or score, reveals its particularity by sug-
gesting a family of possibilities to which it refers. Goodman thus confirms the
pertinence to any diagram theory of the rift between abstract and concrete in his 1 D
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associations of phenosegments in the context of a composition in order to identify the multilevel
dialogue between associative segments.
In her discussion of five recordings of In C, Hanninen comments that performative possibilities
exist in the association of figures, in “the fit or misfit of associative and temporal proximity among
figures notated in the score and the realignment of figures in individual performances.”5 By describ-
ing the associative landscape in terms of fit or misfit, Hanninen rightly presupposes the figures to be
in dialogue with each other. A dialogue of fit and misfit among any of Riley’s 53 figures goes further
than simple consonance or dissonance, for it is shaped by the interaction of disparate strengths,
volumes, and contexts. This qualification must color any associations of phenosegments and subse-
quent deduction of their genosegments for a given presentation, for the resemblance of two sounding
figures to each other assumes varying degrees of sonic importance depending on their context in
a given performance. For this reason, Haninnen’s analysis of In C focuses on phenosegments as
“context-sensitive, shaped by the activity and relative strengths of individual criteria in a particular
NO
TE
S
a
b c
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Hanninen, Dora A. Associative Figures in Terry Riley’s “In C.” 2012. a, b, cLavatar, Johann Caspar. From Frog to Apollo . 1803 d
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variety. For Lynn’s project,
the dialogue between formal
geometry and Lynn’s distor-
tions groups the Embryologi-
cal House designs into one
family, each member of which
performs this dialogue in a
different but related way. The
dialogue between matter and
substance is thus performed
by challenging the hegemony
of substance over the identity
of the project. The matter—
the family of geometric distor-
tions—is not represented by a
singular substance.
The architectural theorist Jeffrey Kipnis
compares the idea of “families” in Lynn’s Embryologi-
cal House to different performances of Beethoven’s
sonatas. A single performance does not define a
sonata’s identity, but rather exemplifies possibilities
of the sonata as a diagram instructing performance.10
This returns to Hanninen’s analysis of disparate per-
formances of In C, showing the varied iterations of the
same instruction. What seems central to identifying
these families, Lynn’s Embryological House units as
well as performances of In C, with the instructions that
generate them, is the dialogic performance common
to all varied possibilities. For In C, the latitude of
performance within the performance instructions is
obvious, but we also expect some latitude from per-
formances of Beethoven, albeit latitude of a different
Given this eighth-note reference, the associative figures tend to gather
themselves about one pole or another, as a magnet might distribute positive and
negative charges in its field, or the poles of the Earth contract these into an aurora
borealis. The bilateral reference defines and separates Riley’s 53 figures by the
logical systems in which they seem to participate. Figures 2, 3, 4, and 5 share
a duration of 4 eighth-notes and define a field of simple, duple meter. These
durations can be doubled and quadrupled to increase the scope of the duple
meter field (6, 7, 9, 12, 13, 14, 15, 18, 28). This contrasts to both patterns heard
in simple, triple meter (1, 20, 21, 27, 43, 44, 45), and all those with dotted quarter
rhythms suggesting compound meters (11, 17, 19, 22, 23, 24, 25, 26, 29, 30, 31,
32, 36, 49). While it would seem that Riley’s scene has three players, the asso-
ciative field shows more contrariety between simple meters (2 eighth-notes) and
compound meters (3 eighth-notes), because these alter the conceptual stress
of the reference grid. As Hanninen observes, the figures are presented in a field
in which context and site have great influence over a figure’s reception. In the
opening five figures, figure 1 is written in triple meter while figures 2-5 use duple
meter. However, figure 1 repeats the same motif on each beat. The fact that the
meter differs graphically from that of the other members of the first associative
set (figures 1-5) carries little sonic relevance, because figure 1 seems equally
musical context.”6 Hanninen’s project maps the disparate associative landscapes of five per-
formances of In C over a given “plot,” drawing conclusions about the differences between the
performances. The plot is a given temporal field, which can be isolated based on (significant)
disjunctions in the musical fabric. For In C, a plot is the time span that a given associative
set of Riley’s figures is played in a certain performance. The indeterminacy of In C makes this
a valuable analysis for the musical dialogue, because it proposes a second order dialogue
between performances. Not only do figures form a network with each other, but performances
create highly dynamic topographies, employing divergent logical systems through which the
analyst can move.
Hanninen also proposes a first-order dialogue between associative figures (Riley’s 53
musical fragments) in In C, which will warrant further discussion here. Hanninen rightly says
that the sequence of figures causes small changes over time, and that figures can associate
both in sequence and out of sequence. This compares to many of the architectural diagrams
presented here, in which repetitions and small differences constitute development. On the
page, the figures in In C tend to align to metric structures of either 2 or 3 beats, and pitch
content in the orbit of C, E, or G. These conceptual frameworks are reinforced by the eighth-
note pulse on a high C, which serves as a reference for both meter and pitch. The fact that
Reich suggested the pulse after the initial rehearsal corroborates its importance as a two-part
reference within the expression of the piece’s concept.
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c
d e
Wittkower, Rudolf. Diagrams of Palladio’s Villas .1949 aHanninen, Dora A. Associative Landscapes for Figures 19-21 in Three Perforamnces of “In C.”2012. bLynn, Greg. Embryological House . 1999 c, dLynn, Greg. Study Diagram for the Deformation of Form. e
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and diminution (up a 5th, twice as fast). Thus, the figures of In C begin to connect out of sequence and
out of plot, meaning that figures which will never be heard together have melodic association. These
diachronic associations go further to suggest an associative dialogue in which Riley’s 53 figures are
themselves a selection from additional possibilities not in the score. The dialogue between figures in
In C demonstrates a far-reaching consequence of Hanninen’s theory of genosegment: that music can
be thought of as associative and not teleological, the very same qualities that define the rhizome for
Deleuze and Guattari.
plausible as the repetitions of a single quarter-note. This remains true for other figures in
simple, triple meter (20, 21, 27, 43, 44, 45), whose repetitions can be dissolved into the greater
fabric of simple meter.
The metric dialogue, then, is performed by the flickering contrariety between simple and
compound meter as the two ontological limits of metric repetition. This is particularly evident
in the plot of figures 19-31, the first part of which Hanninen surveys in several performances.
The prevalence of dotted quarter-notes in figures 19, 21-26, and 29-31 groups 3 eighth-notes,
but the full patterns of 22-26 do not align in their entirety to any compound meter. Moreover,
this plot is intermixed with figures that reference the quarter-note, and not the dotted-quarter-
note (20, 21, 27-30). Figures 21, 29, and 30 appear as phenosegments in both associative
sets, because these consist of dotted-half-notes, which reference groupings of 3 eighth-notes
(2 of these), and 2 eighth-notes (3 of these). The piece’s dialogue oscillates between con-
trarieties in these multivalent segments, as figure 21 is first heard in simple meter, and then
in compound based on the surrounding contexts. Due to this effect, Hanninen’s analysis of
performance variations is all the more salient, exemplifying the selection of possibilities from
a range of possibilities.
This suggests a second associative property of In C that connects melodic contours and
pitch classes. As Hanninen observes, figures 1-5 associate in a linear sequence; that is, the E
attacked by a grace note C in figure 1 is elaborated into an E-F-E pattern in figure 2, and the
grace note is left behind in figure 3, and so on. This process can take place in order (figs. 1-5,
9-13, 15-17, 22-26, 45-47), or out of order, as figure 2 connects to figure 16 by transposition
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SYLVANO BUSSOTI
2 4 8 10.5 12
2 4 8 16 21 24
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9.3
*
* Figures 14 and 42 is twice the metric value of this column, but is placed here in order to show its relationship with these figures.
*
*
* Figure 46 has 5 beats, but is placed here because of its strong quarter-note meter.
a
b
Porter. Associative Diagram for “In C.” aBussoti, Sylvano. XIV Piano Piece for David Tudor. 1969 b
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Performance, discussed as a dialogue between sub-
stances and their contraries, is an opportunity for architec-
ture to redeem itself from its reliance on fixed substance.
As Allen says, performative practices “are less concerned
with what things look like and more concerned with what
they do.”14 For Allen, this aims to suggest architecture as
a stimulus for social performances, but we can also think
of the performative faculty in more formal or metaphysi-
cal terms. Studying Allen’s diagram of field conditions,
the formal indeterminacy of spaces activates the dialogue
between shapes and spaces. Figure-ground is no longer
an accurate binary, because each field condition is paired
with a second possibility that shares characteristics with
the first. In Hanninen’s (and possibly Lynn’s) terms, the
specific presentation of phenosegments amongst alter-
natives helps the viewer to determine possible genoseg-
ments which confirm the association between pairs.
The dialogue between substance and space, a performance of Deleuze’s matter in ar-
chitecture, is at issue in Stan Allen’s idea of “field conditions.” Allen defines these by saying,
“Form matters, but not so much the forms of things as the forms between things.”11 The field
is a conception of space that motivates the relationship formed substances and the spaces
they contain. Taking inspiration from minimalist art, “contain” is an imperfect word for the
dialogue, but one which stands in for “forms between things”—meta-forms. Like additive con-
ceptions of symmetry that will later be discussed in the work of Morton Feldman, Allen’s field
is an activated indeterminacy that is formed, as Donald Judd puts it, by putting “one thing after
another.”12 In an image by artist Barry Le Va, Bearings Rolled (six specific instants; no particular
order; 1966-67), specific fields give way to a remarkable indeterminacy. Le Va is careful to note
that the iterations are specific instants that exist for a moment, but to undermine their absolute
presence by showing each instant alongside at least five alternatives (of which more could
certainly be imagined). He thus puts the substances, ball bearings (points), into dialogue with
space in such a way that space becomes more indeterminate through the diagram. Note that,
should any single instant of Le Va’s six iterations be constructed as architecture, the diagram
would become the only vestige of the additional possibilities that activate the field in this way.
As Allen puts it, in fixed representation, “architecture has surrendered its capacity to imagine, to
propose, or to construct alternative realities.”13
Violin every 23, and the eighth-note groups in the Violin II every 21 beats with a little internal variation.
These fixed proportions, as Kramer calls them, create disjunction but also call for the logical systems
to converge. In m. 43, the phenosegment of 4 eighth-notes aligns with what would be the 5th presen-
tation of the 23-beat Violin melody, creating an opportunity for Stravinsky to end the circus experi-
ment before all parts have converged.9 With these repeated, metrically irregular patterns, one con-
structive conception is to think about the alignment of parts as the convergence of these patterned
blocks of odd metric values. Admittedly, this largely ignores the musical material contained in the
patterns, and points of convergence have less sonic importance than conceptual importance. A
conceptual importance of convergence helps to relate the logos of each pattern (their beat durations,
pitch contents) to their reification in relation to each other.
That said, if one waits for other dialogues to converge, the limiting factor will be the prime
number, 23, aligning to the other patterns. The 7-beat figure in the Cello begins at with the 23-beat
Violin melody in m. 4, and these two prime numbers would only repeat this relationship after the Violin
has repeated 7 times and the Cello repeated 23 times. The piece is curtailed on both ends, then by
convergences between the principal melody in the Violin I and the Cello (to begin) and Violin II (to
end). The loose part, the 21 beat phenosegment of 8 eighth-notes in Violin II, converges with the 23
beat Violin I melody at neither end of the piece, and this convergence would either require 5 additional
presentations of the Violin I melody, or 12 presentations of the melody before the piece begins. Like
Malevich’s Suprematist canvases from the 0.10 exhibition in 1915 Moscow, Stravinsky tenses these
Stravinsky’s concurrent patterns in the first
movement of Three Pieces for String Quartet (later
titled Danse) holds reveals a similar process of
selection in dialogue. The piece opens on a detuned
octave, C#3 against D4 in the Viola, such that the low
C3 string on the instrument is physically detuned in
order to sound the double stop. Like In C, this serves
as a harmonic reference, but one which reveals the
contrarieties at stake. The first sonority thus divides
the melodic patterns into two broad types: those in
the orbit of D (G, A, B, C in Violin I; D pizz. in Viola;
D, C in Cello), and those in the orbit of C# (F#, E, D#,
C# in Violin II; D#, C# in Cello).7 The pitch content is
entirely static throughout the piece, and so it would
seem that all oscillation between segments in dialogue
is effected by rhythm. Following Jonathan Kramer’s
rhythmic analysis in The Time of Music, these parts
are set in motion in non-converging beat patterns.8
After they begin, the Cello repeats every 7 beats, the
VLN I
VLN II VLN II
Cello23* 23 23 23 ...
21 21 21 21 ...21 21 21 21 ...
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
m. 4, start of played patterns
m. 43, end of played patterns*Numbers indicate the number of quarter notes in each block. The two basic numbers are 23 (VLN I), and 21 (VLN II, Cello [7])
a
b
c
Le Va, Barry. Bearings Rolled (six specific instants; no particular order) . 1966-67. aAllen, Stan. Field Condition Diagram . 1999 bPorter, Alex. Metric Tension Diagram in Stravinsky’s “Three Pieces for String Quartet.” c
it
ec
tu
re
arc
hite
ctu
re
ar
ic
mu
si
cm
us
ic
mu
si
cm
us
i
18
19
phenosegments in a deductive space outside the figures’ scope. Seen another way around, a
larger space is implied by the possibilities of convergence suggested by the system of logic,
only one piece of which is presented. Convergence as a dialogic device uses disequilibrium
to construct a dynamic process of static parts, a strategy which Lévi-Strauss calls dislocation,
“and it is only at the end that the pattern achieves a stability which both confirms and belies the
dynamic process according to which it has been carried out.”10 The understanding of conver-
gence advanced thus far moves from stable points of disparate logics to explain the disequilib-
rium in Three Pieces.
The particularity of this arrangement in the Violin II exposes another way to think about the
alignment between parts other than convergence. The set of 4 eighth-notes (F#, E, D#, C#) is
the only one to elude chronometric regularity, hovering around a 21-beat spacing, but presenting
itself at 22, 19, 21, and 22 beat intervals between repetitions. As far as the outer limits of this
figure are concerned, nothing changes about their placement if the figure is homogenized into
four, 21-beat sets, as is the case with the other phenosegment of 8 eighth-notes. When looking
at all segments defined by the pitch-set F#, E, D#, and C#, 4 and 8 eighth-notes, the shifted sets
of 4 eighth-notes are the only ones in which alignment between the D# and C# in
the Violin II and the E-flat and D-flat in the 7 beat Cello figure are at risk. The only
collisions between D-flat in the Cello, and D#-C# in the Violin II occur as a result
of Stravinsky placing the figure out of sequence (m. 25, 34). Had Stravinsky not
adjusted these, then the D# and C# of the 4 eighth-note figure would always align to
the E-flat and D-flat on the upbeat of the Cello, as happens in the first alignment (m.
7). In other words, there are three metric possibilities for the alignment of the D#
and C# to the Cello part: on the upbeat (m. 7), on the downbeat of the 2/4 measure
(m. 17), and mid-measure (m. 25, 34).11 Stravinsky’s inclusion of such alterations
as they produce all three possibilities questions the fixedness of the external con-
vergence points previously discussed. In other words, each alignment contains
different, absolute sonic properties of meter and pitch, and any one of them could
invariably be the alignment from which the pattern is determined.
This returns to the conclusions of In C, in which the dialogue between segments
of disparate logical fabrics correlates with additional possibilities for interactions,
Violin 1
Violin 2
Cello
Viola
2
o
7 16
Vln. 1
Vln. 2
Vlc.
Vla
25
a
b
Malevich, Kazimir. 0.10 exhibition in Petrograd, Moscow. 1915. See Erlich, Victor. “Russian Formalism.” The deductive structure of Malevich’s Suprematism tenses figures from spatial boundaries outside the figures themselves, as if the geometries were deduced from the space they occupy. aPorter, Alex. An Alternate Diagram for Alignment of Figures in Stravinsky’s “Three Pieces for String Quartet.” b
it
ec
tu
re
arc
hite
ctu
re
ar
ic
mu
si
cm
us
ic
mu
si
cm
us
i
20
21
other than those presented. In short, tension and
contrariety in the first movement of Three Pieces
for String Quartet establishes areas of contraction
within and outside of the sampling of possibilities
Stravinsky presents. These contractions, in both
pieces, define the adversarial nature of contrary
genosegments, as Deleuze describes: “There is
cruelty, even monstrosity, on both sides of this
struggle against an elusive adversary, in which
the distinguished opposes something which
cannot distinguish itself from it but continues to
espouse that which divorces it.”12 The tension
is entirely composed, impossible to represent in
terms of genosegment characteristics, and deter-
mined only by a dialogic analysis of the interac-
tion between parts.
1 S
ee
So
mo
l, R
.E.
“Du
mm
y T
ex
t, o
r T
he
Dia
gra
mm
ati
c B
asi
s o
f C
on
tem
po
rary
Arc
hit
ec
ture
.” p
. 82
S
ee
Eis
en
ma
n,
Pe
ter.
“Dia
gra
m:
An
Ori
gin
al
Sc
en
e o
f W
rit-
ing
.” p
. 2
7, a
nd
Ga
rcia
, M
ark
. “H
isto
rie
s a
nd
Th
eo
rie
s o
f th
e D
iag
ram
s o
f A
rch
ite
ctu
re.”
p. 2
43
P
ete
r E
ise
nm
an
ac
kn
ow
led
ge
s th
e a
ssu
mp
tio
n o
f th
e m
eta
-p
hy
sic
s o
f p
rese
nc
e b
eh
ind
dia
gra
m w
ork
. S
ee
, fo
r e
xam
ple
, E
ise
nm
an
, Pe
ter.
“Dia
gra
m: A
n O
rig
ina
l Sc
en
e o
f W
riti
ng
.” p
. 27
4
Se
e V
idle
r, A
nth
on
y. “
Dia
gra
ms
of
Dia
gra
ms.
” p
. 17
5
Alle
n, S
tan
. “D
iag
ram
s M
att
er”
p. 1
66
B
os,
Ca
rolin
e,
an
d B
en
va
n B
erk
el.
“Dia
gra
ms-
Inte
rac
tiv
e In
stru
me
nts
in
Op
era
tio
n”
p. 1
97
E
ise
nm
an
, P
ete
r. “D
iag
ram
: A
n O
rig
ina
l S
ce
ne
of
Wri
tin
g”
p. 2
98
L
yn
n, G
reg
. “M
ult
iplic
ito
us
an
d I
no
rga
nic
Bo
die
s.”
p. 3
49
H
an
nin
en
, D
ora
. “A
Th
eo
ry o
f M
usi
c A
na
lysi
s.”
p.
310
-311
. H
an
nin
en
cla
ims
this
to
be
Rile
y’s
dir
ec
tiv
e,
wh
ich
we
ca
n
ho
pe
fully
ac
ce
pt
giv
en
th
at
it u
sua
lly o
cc
urs
in
pe
rfo
rma
nc
es
of
In C
. H
ow
ev
er,
Ha
nn
ine
n g
ive
s n
o c
ita
tio
n f
or
Rile
y’s
co
m-
me
nt,
an
d i
t d
oe
s n
ot
ap
pe
ar
in t
he
in
stru
cti
on
s la
ter
ad
de
d
the
In
C’s
on
e p
ag
e s
co
re.
10
Se
e K
ipn
is, J
eff
rey.
“A
Fa
mily
Aff
air
.” p
. 19
611
A
llen
, Sta
n. “
Po
ints
+ L
ine
s.”
p. 9
2
Alle
n m
ake
s a
pu
n h
ere
, re
pe
ate
d f
rom
his
ess
ay
in
AN
Y m
ag
azi
ne
fro
m t
he
pre
vio
us
ye
ar
(19
98
) ti
tle
d “
Dia
gra
ms
Ma
tte
r.”
Th
is p
lay
s o
f D
ele
uze
’s
sep
ara
tio
n o
f m
att
er
an
d s
ub
sta
nce
, o
r m
ate
ria
l. “
Fo
rm m
at-
ters
” is
a f
orm
ula
tio
n r
ev
ers
ed
fro
m “
ab
stra
ct
ma
ch
ine
,” si
nc
e fo
rm i
s th
e s
ub
sta
nc
e,
pro
du
ct
of
the
ma
ch
ine
, a
nd
ma
tte
r is
th
e a
bst
rac
t c
on
ce
pt.
“M
att
er
form
s” m
igh
t b
e t
he
re
form
ula
-ti
on
to
co
rre
spo
nd
wit
h D
ele
uze
. A
llen
, w
ho
se
em
s a
co
m-
mit
ted
De
leu
zia
n,
like
ly s
ee
s th
is,
sin
ce
his
sta
tem
en
t b
eg
ins
wit
h t
he
mo
re e
asi
ly u
nd
ers
too
d,
ph
en
om
en
olo
gic
al
po
siti
on
(i
n f
orm
, th
ere
ex
ists
ma
tte
r),
an
d t
he
n d
isc
red
its
the
fo
rmu
-la
tio
n b
y a
dv
oc
ati
ng
fo
r a
n a
lte
rna
tiv
e o
rde
r—th
e f
orm
s b
e-
twe
en
th
ing
s.12
I
bid
. p. 9
113
I
bid
. p. 5
014
I
bid
. p. 5
3
This is why a “field condition” cannot be understood except in a
spatial dialogue possible in a diagram-drawing; association of particu-
lars allows for conceptions of general—the genetic material implicit in
the association. The forms between pairs do not always correspond in
Allen’s diagram, but possibilities of the spatial dialogues in any given
pair do. In this way, “field conditions” relate to both In C, and Three
Pieces for String Quartet. For In C, the contextual similarities between
successive figures create fields with similar sonic characteristics (for
example, e-minor pitch collection, and a feeling of compound meter in
figures 22-26). These fields have a degree of latitude to their presenta-
tion, which is the temporal indeterminacy of In C. The field is always the
space between figures, meaning the intervals, rhythmic relations, and
articulations common to or aggregated by multiple figures, endowing
a field with its sonic qualities. Turning to the score as a map of as-
sociative topography helps to define a field condition, like e-minor,
compound meter in the plot mentioned above, amongst its alterna-
tives. In Three Pieces, similar conclusions are possible by the fact
that the piece is varied by the spaces between patterns; the patterns
matter, but not as much as the space between patterns. NO
TE
S1
Se
e P
lato
. “T
he
Re
pu
blic
” B
oo
k X
. P
lato
’s s
up
-p
ress
ion
of
art
ists
sp
ac
es
the
art
ist
an
d t
he
ph
iloso
-p
he
r a
pa
rt f
rom
on
e a
no
the
r o
n id
eo
log
ica
l gro
un
ds.
T
his
en
tert
ain
s th
e p
oss
ibili
ty t
ha
t th
e a
rtis
t o
pe
r-a
tes
on
a d
iffe
ren
t sy
ste
m o
f lo
go
s th
an
do
es
Pla
to.
Dif
fere
nc
es
in l
og
os
allo
w d
ialo
gu
e t
o o
cc
ur.
2
Se
e H
an
nin
en
, Do
ra A
. A T
he
ory
of
Mu
sic
An
aly
-si
s. p
p. 6
2-7
33
Th
e
dis
tin
cti
on
b
etw
ee
n
[Be
ing
] a
nd
[b
ein
g/
pre
sen
t-b
ein
g]
loo
sely
fo
llow
s H
eid
eg
ge
r’s
inq
uir
y in
to w
ha
t h
e c
alls
Se
in a
nd
Sie
ind
em
, bu
t p
erh
ap
s in
a
n a
da
pte
d w
ay
wh
ich
do
es
no
t sh
are
He
ide
gg
er’
s c
om
mit
me
nt
to t
he
ge
ne
ral
be
co
min
g t
he
sp
ec
ific
. S
ee
He
ide
gg
er,
Ma
rtin
. “B
ein
g a
nd
Tim
e.”
4
Se
e D
err
ida
, Ja
cq
ue
s. “
Th
e E
nd
of
the
Bo
ok
an
d
the
B
eg
inn
ing
o
f W
riti
ng
.” (t
ran
s.
1976
) p
p.
6-2
6
De
rrid
a b
eg
ins
by
arg
uin
g t
ha
t th
e p
rob
lem
of
lan
-g
ua
ge
is
tha
t th
e t
rad
itio
na
l si
gn
is
pu
she
d t
o i
ts
limit
wh
en
it
ha
s n
o c
on
tex
t, a
t w
hic
h a
gu
ara
nte
ed
si
gn
ifie
d i
s lo
st,
an
d a
ne
w c
on
stru
cti
on
ou
gh
t to
su
pp
lem
en
t th
e t
rad
itio
na
l n
oti
on
of
a f
ixe
d,
refe
r-e
nti
al
sig
n.
5
Ib
id. p
. 310
6
Ha
nn
ine
n, D
ora
A. p
. 73
7
Th
e D
# a
nd
C#
in
th
e C
ello
is
wri
tte
n a
s E
-fla
t a
nd
D-f
lat
in t
he
sc
ore
.8
S
ee
Kra
me
r, J
on
ath
an
. P
rop
ort
ion
s in
Str
avin
-sk
y’s
Ea
rly
Co
mp
osi
tio
ns
in “
Th
e T
ime
of
Mu
sic
.” p
. 2
91
9
Str
av
insk
y is
sa
id t
o h
av
e w
ritt
en
th
is m
ov
em
en
t b
ase
d o
n a
clo
wn
ca
lled
“L
ittl
e T
ich
” in
Ap
ril
of
1914
. S
ee
Cra
ft,
Ro
be
rt a
nd
Ve
ra S
tra
vin
sky.
“S
tra
vin
sky,
In
Pic
ture
s a
nd
Do
cu
me
nts
.” p
. 12
610
L
év
i-S
tra
uss
, Cla
ud
e. �
Tris
tes
Tro
piq
ue
s.�
p. 1
91
11
A f
ou
rth
po
ssib
ility
ex
ists
, in
wh
ich
D#
an
d C
#
alig
n w
ith
th
e d
ow
nb
ea
t o
f a
3/4
me
asu
re,
wh
ich
o
cc
urs
in
ev
ery
re
pe
titi
on
of
the
8 e
igh
th-n
ote
fig
-u
re.
As
far
as
co
nc
urr
en
t p
itc
h c
lass
es
are
co
n-
ce
rne
d,
this
re
lati
on
ship
is
ide
nti
ca
l to
th
e c
ase
of
alig
nm
en
t o
n t
he
do
wn
be
at
of
a 2
/4 m
ea
sure
, b
oth
o
cc
urr
en
ce
s so
un
din
g D
#/E
-fla
t, C
#/D
-fla
t, a
nd
C
in t
he
sa
me
qu
art
er
no
te.
12
De
leu
ze, G
ille
s. “
Dif
fere
nc
e a
nd
Re
pe
titi
on
.” p
. 28
NO
TE
S
or
yth
eo
ry
th
eo
rr
ea
rc
hite
ctu
re
ic
mu
si
cm
us
ic
mu
22 23
Diaphora is a repeti-
tion of points, by which I mean
shapes, forms, and matters
that a diagram repeats. What
occurs between iterative rep-
etitions? Le Corbusier raises
the stakes of iteration by its
parallel in culture: “Civilizations
advance. They leave behind the
age of the peasant, the warrior,
and the priest to attain what is
rightly called culture. Culture
is the outcome of an effort of
Diaphora is a repetition of points, by which
I mean motives, ostinati, sets, and the like. The
agenda of repetition in music seems obvious
enough, invoked by Schenker as the originally
musical principle in his early writings. In an article
from 1895, Schenker discusses repetition as the
vehicle by which music escapes its parallel to
language (a proposition with which we certainly
disagree, but due to Schenker’s linguistic as-
sumptions, not his musical assumptions). In the
course of this argument, Schenker contends that
a repeated motive surpasses its semiotic compre-
hensibility, saying, “the musical motive is only a
sign for itself; or, to put it more accurate-
ly, it is nothing more and nothing less than
itself.”1 While this lays the groundwork for
Schenker’s concept of musical autonomy, it
also parallels Deleuze’s mask of repetition,
the diaphora which dissolves signification.
Through repetition, the motive becomes a
simulacrum indivisible from its own repro-
duction. The general is articulated with
distinct particulars, and these particulars
divide the univocal nature of the general.
The musical motive is nothing more or less
than itself.
Diaphora is a repetition of points; it is a performance between these
points, without these points. The more fervently Hamlet repeats his
father’s murder, the more fictitious becomes each repetition. These itera-
tions become masks which dissolve the face of the wearer, as Hamlet’s
sanity is destabilized. Hamlet’s performance is a diaphora which proves
to be the only trace of his father’s presence. Hamlet himself becomes
the repetition of his father, repeating dual roles as avenger and husband.
Each specific repetition of the murder (the ghost’s recollection, Hamlet’s
play, Polonius, Ophelia, etc.) reveals the instability of its general narrative.
Phora, in Greek, means “time” in terms of “instance,”1 and this is the crux
of Hamlet’s burden: to repeat actions through instantiations of the idea of
his father. By repeating, the general concept is construed by its distinct
particulars, and these particulars divide the univocal nature of the general.
Deleuze calls this category “internal repetition within the singular.”2 The
diaphora therefore questions the authority of a singular presentation of any
given repetition.
Difference breaches the surface as an operative condition in a diagram’s
performance; it is given a plane of immanence.3 The projection of diaphorat-
ic space prepares the diagram to reveal the condition of difference on which
it stands, for it must only resemble a figure, standing in for the impossible
presentation of difference. A “figure” of difference is a second paradox with
which to combat the conflict between the diagram and the metaphysics of
presence. This condition allows the “points” to fall behind the performance.
This recalls the architect Stan Allen’s apt formulation of a “stealth diagram,”
defined by an IBM advertising campaign during the 1998 Winter Olympics:
“You won’t see us but you will see what we do.”4
selection. Selection means discarding,
pruning, cleansing; making the Essential
stand out anew stripped and clear.”1 In a
word, Le Corbusier puts progress between
the repetitions. The Temple of Hera at
Paestum repeats in the Parthenon; the
horse-drawn carriage repeats in the au-
tomobile; architecture repeats as “plastic
invention.”2 According to Le Corbusier, the
difference is selection, and for architecture,
it is geometric selection. By repeating a
shape, an obligated is created to perform
diaphora—difference through repetition.
b
a
Temple of Hera at Paestum. 600-550 BC. Periclean Parthenon. 447-432 BC.
a
b
or
yth
eo
ry
th
eo
rr
ea
rc
hite
ctu
re
ic
mu
si
cm
us
ic
mu
24
25
Diaphora is predicated on the reading of diagram as shape. As for Le Corbusier, shape is
literal here, prisms, spheres, cylinders, etc. This idea is clearly shown in Peter Eisenman’s serial
diagram for House II (1969). The diaphora of the House II diagram shows a square prism that
divides into iterative and redundant spatial and structural divisions. Basing these divisions on the
structural linguistics of Noam Chomsky, the geometric process language of Eisenman’s simulacra
can be read as a set of shapes for the implicit difference between figures. The difference happens
between, allowing the diagram to imply a series of performative “forces” which effect these differ-
ences. Forces is not exactly the correct expression, for it suggests that the architect manipulates
some substance to create an architecture object. The diagram for House II avoids the opera-
tional performance of Lynn’s Embryological House by performing divisions and not manipulations.
Diaphora emerges clearly in these divisions that do not represent an object over time, but the
possibilities of this object in the first instance. The procession suggests the movement of differ-
ence that happens, that must happen in an axonometric space outside of presence.
makes a comparison between Rameau’s analytical model and the explicit/implicit possibili-
ties of genealogical linguistic structure espoused by Noam Chomsky (a figure against whom
Deleuze and Guattari are reacting). By comparing fundamental bass to an implicit statement,
like To please John is easy (versus John is easy to please), Keiler sees root bass as metalin-
guistic. This means that Rameau’s bass represents “potential attributes of chords” which
are implied, but not necessarily replicated by the score.3 The tension between divergent
attitudes toward Rameau’s bass results from a use of “[m]usic… [as] the language of analysis
for music,”4 a proposition which carries truth even beyond Rameau. This internal division
of musical syntax is a diaphora in that it operates in multiple ways by its repetition. An ana-
lytical line, Rameau’s, Schenker’s, or our own, repeats the musical language of the piece,
but to different effect. The following serial analysis of the Prelude to Stravinsky’s Requiem
Canticles compares to the parallel discussion of Eisenman’s diagrams for House II through
Keiler’s discussion of implicit divisions and metalinguistics, what I am calling diaphora.
Allan Keiler’s article on Rameau’s fundamental bass inspires thinking
about an internal division in music’s analytical language. Keiler describes
Rameau’s fundamental bass as a “fictitious (or analytic) bass line that
consists of the roots of the chords of a succession of harmonies…used
to represent the root movement of chords abstracted from the particular
inversions that actually occur.”2 This idea itself is the diaphora of diatoni-
cism; harmonic progressions are at once themselves and the simplified,
root motions for which they stand. Parallel attention to harmony and coun-
terpoint validate both entities, but, as root movement repeat in inversion,
an internal difference which relates the two is exposed. Keiler surveys two
historical attitudes toward Rameau’s bass line: that it either pre-exists the
composition which internalizes it, or that it distorts an analyst’s understand-
ing of the musical fabric by reducing out the true bass line. Keiler then
shapes are produced and reproduced. To continue Deleuze’s discourse, it is a dangerous lack in
matter, and not in substance, that employs this shape. Deleuze hypothesizes the diaphora as a
response to this shape’s inadequacy, saying, “Repetition is the pure fact of a concept with finite
comprehension being forced to pass as such into existence.”6 The shape is not meant as the object
of the diagram, but rather a trace of that which one cannot see.
Diaphora presents the shape as a sheaf, that is to say an assemblage, of possible shapes in a
given category, the category itself being subject to the internal difference of its multiple presenta-
tions. If we insist on this chicanerous word, sheaf, then the performance of difference is a supple-
ment for the différance (with an “A”) hypothesized by Derrida.7 One can think of the différance as the
meta-diagram revealed through the performance of difference, which challenges the authority of the
metaphysics of presence.8 Derrida explains that the différance cannot be exposed, for “[o]ne can
expose only that which at a certain moment can become present,”9 which the différance is not. If we
are to follow Merleau-Ponty, and to consider a shape as “pregnant with its form,”10 then a diagram-
The diaphora, a performance of difference,
construes a diagram’s presentation on a page, using
white space, lines, and that which one does see, as
a shape responding to difference. This formulation
comes from William Faulkner, in his novella As I Lay
Dying (1930, rev. 1957),{Faulkner, 1964 #5} in which
Hamlet’s logos is described as “just a shape to fill a
lack.”5 From Addie Bundren’s nihilism, the passage
first discredits logos altogether. However, one quickly
realizes the dialectic between shape and lack which
logos affords. The shape is imperfect, and makes a
subjective fit to cover a lack, but it is generated out
of this need. There must first exist a lack for which
Eisenman, Peter. Diagram for House II . 1969
or
yth
eo
ry
th
eo
rr
ea
rc
hite
ctu
re
ic
mu
si
cm
us
ic
mu
26 27
principle (columns). The same thing happens in plates 5 and 6, but the nine-square plan is now
shown as two sets of longitudinal arrangements, walls in 5, spaces in 6. The diagram thus uses
diaphora to perform the implicit possibilities of a single organizational principle: the nine-square
grid (Palladio). The comparison between plates, both successive and non-successive, constitutes
the legibility of their difference. In fact, it is the similarity between figures that so readily manifests
their difference.3 The diaphora does not follow a telos, nor is it overtly operative (as the processions
in House IV or VI are), but it simply emerges in an objective comparison. Just as Alan Keiler shows
the linguistic difference between explicit and implicit bass lines in Rameau’s root bass, Eisenman’s
House II diagram considers the implicit possibilities of the first two moves—the fission of a square
prism and the division of the plan into nine-squares. Eisenman’s reveals geometry’s internal differ-
ence as the possibility of division. The abstract machine is “layed out on a plane of consistency, …
from which the One is always subtracted (n-1).”4 The plates of House II can be considered as shapes
to demonstrate the movement of difference, a movement like subtraction.
For example, between plate
1 and plate 2, fission of the original
square prism produces a movement
from one to two square prisms.
Between plates 3 and 4, these two
squares demonstrate distinct and
redundant structural and spatial
properties, made possible because
of the original fissure; the overlaid
prisms allow a nine-square division
(like that of Wittkower’s Palladio) to
manifest in two ways: as a spatial
principle (walls), and a structural
is a question of “How to ___?”, while the possibilities of technique is a
question of “What is ___?”
As in Three Pieces, the Prelude to Requiem Canticles distin-
guishes two groups of sound, the melodic, solo-brigade of single in-
struments above, and the accompanying ostinati below. Repetition
is the piece’s developmental paradigm, occurring at the scale of the
ostinato, the hexachord, and the motivic melody. The melodic repeti-
tions in both pieces are developmentally unmotivated, signs only of
themselves, because they return in four exact reproductions.5 In other
words, the melody announces its own repetition, and does not undergo
development by variation. The context of the accompaniment and the
agglutination of solo lines distinguish each repetition. Indeed, the lines
operate by distinction and not by development.
To consider repetition and difference in music, Stravinsky’s
compositional project lends many examples. The longevity of
Stravinsky’s interest in diaphora, static repetition, and its exposure
of difference, resonates in the movement from the previous dis-
cussion of Three Pieces for String Quartet: I (later titled Danse;
1914, rev. 1918), to one of his last pieces, Requiem Canticles: I
Prelude (1966). The comparison of these pieces under our param-
eters is useful because it demands that the analyst see through
stylistic or idiomatic procedure, and to see the performative, dia-
grammatic context. The fact that Three Pieces employs modal
folk melodies as a contextual device while Requiem Canticles uses
serial technique is less relevant to this discussion than the concep-
tual possibilities in the frameworks of either technique. Technique
framing the movement of différance, the architectural
compliance with the metaphysics of presence is not
assumed, but is a detour, containing a trace of the dif-
férance. This is why it is impossible to read a diagram
without the armature of diaphora, and equally impos-
sible to present a diagram in a literal, analog form.
Even Eisenman’s house loses the serial performance
of difference as a built form. So too does the Prelude
of Stravinsky’s Requiem Canticles require multiple it-
erations of the melody to communicate its ontology
separate from its presentation. The lack covered by
shapes is the erasure left in presence by the différance
that abandons presence.
matic sheaf, as a shape, can also be pregnant with différance; this is not to say that it
contains différance, but to suggest that it occupies the space, or the lack, in presence
that remains where an originary différance abandons presence.{Merleau-Ponty, 1974
#4} Through this lens, the mapping of semiotic information that is required to con-
cretize an architectural or musical diagram is only another shape “put in the place of
the thing itself,” representing “the present in its absence.”11 Like Faulkner’s shape,
which first requires a lack, the diagrammatic procedure begins with an absence for the
diagram to fill in presence. It is the same reason that Saussure proposes a semiotic
system of language predicated on the differences between terms and not positive
meanings.12 For Derrida, the lack is abandoned meaning, but it relates to architec-
ture and music as a lack posed by the conceptual challenges germane to a particular
work or set of works. This may be why R.E. Somol describes diagram-architecture “as
the framing and posing of problems rather than as the definition of solutions.”13 By
or
yth
eo
ry
th
eo
ri
cm
us
ic
mu
sic
mu
28
29
re
arc
hite
ctu
re
is only completed by inclusion of the first tone of the violin solo, A# (m.
4). The hexachord is thus established in a peculiar way, by momentarily
passing into the solo voice and returning to the lower voices to articu-
late the final tone. The A# in m. 4 has no univocal identity, because it
participates in the series of both the lower and upper voices. This mul-
tivalence is repeated towards the end of the piece, in which the missing
“A” in the lower voices’ hexachord (mm. 35-54; beta Retrograde, first
position) necessarily relates to the “A” which hinges two hexachords
in the cello/bass solo line (mm. 39-46; alpha and beta Inversion, sixth
position). In other words, an accurate reading of the piece’s pitch
content does not separate between formal components, but requires
their integration with each other. To recall the opening discussion, the
serial interlacing between parts sets the formal pieces in a dialogue.
Knowing that the piece is serially composed, the textural construction with a treble
melody and ostinato bass falsely suggests a hexachordal division of one element from
the other, the top voices from the bottom. The prelude has four formal units, each pre-
senting an iteration of the melodic simulacrum (I: mm. 1-7; II: mm. 9-19; III: mm. 20-33;
IV: mm. 35-46), and each separated by a brief silence (between 1 and 6 sixteenth-
rests). Stravinsky visually emphasizes these four units by cutting out empty bars, such
that the visual shape of each unit is paralleled by the others in a visual continuum. This
would seem to reinforce the separation between the upper and lower parts, because
the melody operates in a formal vacuum, closed to the ostinati. However, closer study
of pitch content proves the formal separation not to extend to a hexachordal segre-
gation. The lower voices outline the first four tones of the alpha hexachord from the
bottom registers up (mm. 1-3; F C B A), and pause on the fourth tone (mm. 4-6) before
completing the first formal unit with a fifth tone (m. 7; D). The serial unit, the hexachord,
1 I
nst
an
ce
, as
in: “
I h
av
e o
nly
se
en
Ha
m-
let
on
ce
[o
ne
tim
e].
” D
ia-i
nst
an
t im
ply
ing
re
pe
titi
on
s o
f th
is i
nst
an
t th
rou
gh
wh
ich
o
ne
mo
ve
s.
Ari
sto
tle
use
s D
iap
ho
ra a
s o
ne
of
the
fiv
e p
red
ica
ble
s, r
esp
on
sib
le
for
dif
fere
nti
ati
ng
be
twe
en
th
ing
s.
Se
e A
rist
otl
e. “
Top
ics.
” a
iv.
10
1 b
17-
25
.2
D
ele
uze
, G
ille
s. “
Dif
fere
nc
e a
nd
Re
p-
eti
tio
n.”
p. 1
3
Se
e D
ele
uze
, Gill
es
an
d F
elix
Gu
att
ari
. W
hat
is
Ph
ilo
sop
hy
. Ch
2. p
p. 3
5-6
04
A
llen
, Sta
n. “
Dia
gra
ms
Ma
tte
r” p
. 16
5
Fa
ulk
ne
r, W
illia
m.
As
I L
ay D
yin
g.
p.
172
.
Th
e
ex
ten
de
d
pa
ssa
ge
re
ad
s:
“He
ha
d a
wo
rd,
too
. L
ov
e,
he
ca
lled
it.
B
ut
I h
ad
be
en
use
d t
o w
ord
s fo
r a
lo
ng
tim
e.
I k
ne
w t
ha
t w
ord
wa
s lik
e t
he
oth
ers
: ju
st
a s
ha
pe
to
fill
a l
ac
k;
tha
t w
he
n t
he
rig
ht
tim
e c
am
e, y
ou
wo
uld
n’t
ne
ed
a w
ord
fo
r th
at
an
ym
ore
th
an
fo
r p
rid
e o
r fe
ar.”
6
De
leu
ze,
Gill
es.
“D
iffe
ren
ce
an
d R
ep
-e
titi
on
.” P
. 13
7
De
rrid
a, J
ac
qu
es.
“D
iffé
ran
ce
.” p
. 38
I
bid
. p. 1
09
I
bid
. p. 5
-610
M
erl
ea
u-P
on
ty, M
au
ric
e.
“Th
e P
rim
ac
y o
f P
erc
ep
tio
n a
nd
Its
Ph
iloso
ph
ica
l C
on
-se
qu
en
ce
s.”
p. 1
96
11
De
rrid
a, J
ac
qu
es.
“D
iffé
ran
ce
.” p
. 912
de
S
au
ssu
re,
Fe
rdin
an
d.
Co
urs
e
in
Ge
ne
ral
Lin
gu
isti
cs. p
. 117
13
So
mo
l, R
.E.
“Th
e D
iag
ram
s o
f M
att
er.”
p
. 26
NO
TE
S
Comparsion of Eisenman’s Diagram for House II and a unified view of Stravinsky’s score for the Prelude of Requiem Canticles
it
ec
tu
re
arc
hite
ctu
re
ar
ic
mu
si
cm
us
ic
mu
si
cm
us
i
30
31
1 L
e C
orb
usi
er.
“To
wa
rds
an
Arc
hit
ec
ture
.” p
. 18
3-1
84
.2
I
bid
. p. 1
89
3
Th
is i
s o
the
rwis
e d
esc
rib
ed
by
th
e a
da
ge
: “O
nly
th
at
wh
ich
is
alik
e d
iffe
rs,”
wh
ich
is
som
eti
me
s e
rro
ne
ou
sly
att
rib
ute
d t
o
De
leu
ze.
4
De
leu
ze, G
ille
s a
nd
Fe
lix G
ua
tta
ri. A
Th
ou
san
d P
late
au
s. p
. 2
15
S
ee
Be
nja
min
, W
alt
er.
“Th
e W
ork
of
Art
in
th
e A
ge
of
Its
Tec
hn
olo
gic
al
Re
pro
du
cib
ility
.” p
. 23
The impetus to divide the prism begs the question of its metaphysical
stability, especially in the context of architecture as the becoming-pres-
ent of a drawing. Considering Faulkner’s shape along with Eisenman’s
divisions, the prism which performs diaphora only stands in for the implicit
difference which the diaphora reveals. It is necessarily an imperfect shape,
insofar as it supplements that which cannot present itself—difference.
This acknowledges Le Corbusier’s shapes as only one particular of many
possible envelopes—a single shape amongst many in order to address a
given difference. Following Eisenman’s diagram through to its synthesis as
a “buildable” architecture object is problematic from the virtual perspec-
tive of a diaphora of (n-1). Eisenman immortalizes a single shape by syn-
thesizing many shapes, thus privileging the built shape among the others.
Implicit spatial divisions are disregarded for the sake of explicit divisions.
The built form gains an aura, to use Benjamin’s term, which was necessar-
ily absent in the reading of its diagrammatic repetitions.5 Diaphora in the
diagram prompts us to question the diagram’s movement into presence,
and the incompatibility of conceptual possibilities with singular ends. This
rift is not Eisenman’s failure, but the issue of presence and shape that his
process introduces.
they accomplish. In other words, the delicate interrelation
of hexachords to their neighbors in this piece prevents the
serial procedure from achieving the autonomy implied by a
rotational array.
Stravinsky encourages this interactive thinking by
including a ‘mystery’ hexachord in the “continuation” of
the Violin I and II solo lines (ie. m. 15-19). The second
parts of both lines play the same hexachord in retrograde,
relative to one another (mm. 15-19; F# D C# D# G# F).
Vexingly, this new hexachord is nowhere to be found in
Stravinsky’s rotational array for either row used in Requiem
Canticles, prompting more creative explanations. A first
possibility examines the Violin II line, and finds the alpha
Retrograde in first position (G# D# C# [G E] F#), if one
includes the G and E sounding in the lower voices (again
finding an intimate link between the texturally distinct top
and bottom, a link which does not occur in latter repeti-
tions of this Violin II solo mm. 30-33; 43-46). While this is
likely one source of the Violin II line at mm. 16-19, it fails to
This is further corroborated in the pitch content by the fact that the added Violin II solo
(ie. mm. 12-19), begins with the same hexachord as did the lower parts at the beginning of
the piece (alpha Prime, first position). This line stands in contrast to the repeated line of the
Violin I discussed above, beginning on A# (mm. 12-19). This juxtaposition recalls the separa-
tion between the two lines in the first formal unit, whereby the Violin I solo escaped the original
hexachord. In the second formal unit, it now plays against this same hexachord in the Violin
II. This is an example of the piece’s diaphora, in which the same pitch succession recurs with
a new “meaning” produced by its repetition in a new context. The original presentation (mm.
4-7) used a shared nodule (A#) between the Prime hexachord and the Retrograde in Violin I. The
second presentation suggests the two hexachords as contrary to one another. Deleuze works
with this type of contrariety to posit extremes outside the metaphysics of “the large” and “the
small.” He says, “Each contrary must further expel its other, therefore expel itself, and become
the other it expels. Such is the movement of contradiction as it constitutes the true pulsation of
the infinite.”6 In other words, diaphora operates to expose an infinite difference between the one
and the other, the first and the second. Just as the A# becomes the first solo line for the Violin
I as it completes the prime hexachord for the lower voices, so too does the opposition between
the Violin I line and the original, prime hexachord necessitate the becoming-other of each. Thus,
diaphora shows a falsity to pure serial analysis in this piece, which identifies all repetitions of the
same hexachord equally in the rotational array, but these repetitions can be very different in what
NO
TE
S
Two photos of Eisenman’s House II as it was built. Hardwick, Vermont. 1970.
32
33
ic
mu
si
cm
us
ic
mu
si
cm
us
i
explain the E# in m. 15, and the D in m. 19, and seems rather distant from the clearly retrograded
hexachord common to both Violin I and II in this passage. A second possibility cuts and splices
pieces of relevant hexachordal information from Stravinsky’s rotational array to construct the
hexachord in question (F# D C# D# G# F). I propose this option in order to contradict the idea
that the rotational array separates all possible hexachords from each other, something already
shown to be false. The rotational array in this piece can be thought of as a first-order repetition,
a repetition which stands in place of the intervallic information (the difference between pitch
classes) common to each rotation. From Deleuze, “Repetition is truly that which disguises itself
in constituting itself, that which constitutes itself by disguising itself. It is not underneath the
masks, but formed from one mask to another, as though from one distinctive point to another,
from one privileged instant to another, with and within the variations.”7 By privileging intervallic
information of the row rather than pitch information, Stravinsky establishes the various hexa-
chords as masks, shapes, or figures articulating the space between. This is already apparent
in Stravinsky’s rotational array, which does not include all possible permutations of pitch, but
only those with intervallic relevance. Indeed, reading an alpha and beta hexachord horizontally
across the rotational array counts as reading “the row,” even though it is only forms of the row’s
intervals, and not all 12 pitches, that appear in these lines. The hexachord itself is a shape to
articulate this set of intervals, not to articulate pitches successively, which is to say a shape
filling a difference.
1 S
ch
en
ker,
He
inri
ch
. “T
he
Sp
irit
of
Mu
sic
al
Tec
hn
iqu
e.”
p. 3
21
2
Ke
iler,
Alla
n.
Mu
sic
as
Me
tala
ng
ua
ge
: R
am
ea
u’s
Fu
nd
a-
me
nta
l B
ass
in
“M
usi
c T
he
ory
: S
pe
cia
l To
pic
s.”
ed
. R
ich
mo
nd
B
row
n. p
. 84
3
Ib
id. p
. 98
4
Ib
id. p
. 10
05
T
he
ori
gin
al
me
lod
ic p
rese
nta
tio
n i
n t
he
Vio
lin I
(m
m.
4-7
) is
ob
vio
usl
y t
run
ca
ted
as
co
mp
are
d t
o t
he
su
cc
ess
ive
re
pe
ti-
tio
ns.
T
his
dif
fere
nc
e a
ffir
ms
the
pri
nc
iple
of
ag
glu
tin
ati
on
in
th
e m
elo
dy
(th
at
ea
ch
re
pe
titi
on
is
giv
en
an
ad
dit
ion
al
solo
v
oic
e),
an
d c
orr
ob
ora
tes
the
im
po
rta
nt
co
nn
ec
tio
n b
etw
ee
n
the
se
co
nd
ha
lf o
f th
e V
iolin
I m
elo
dy
an
d t
ha
t o
f th
e V
iolin
II
me
lod
y (
ie.
mm
. 15
-19
) in
th
at
the
Vio
lin I
is
un
ab
le t
o s
en
-si
bly
pre
sen
t th
e s
ec
on
d h
alf
of
its
me
lod
y w
ith
ou
t p
lay
ing
it
ag
ain
st t
he
Vio
lin I
I.6
D
ele
uze
, Gill
es.
“D
iffe
ren
ce
an
d R
ep
eti
tio
n.”
p. 4
57
D
ele
uze
, Gill
es.
“D
iffe
ren
ce
an
d R
ep
eti
tio
n.”
p. 1
7
P
I F C B A A# D C# D# G# F# E G
II F E D D# G A# C# F# E D F B
III F D# E G# B F# C# B A C F# G#
IV F F# A# C# G# G C# B D G# A# D#
V F A C G F# E C# E A# C F D#
VI F G# D# D C C# C# G A D C A#
I
I F A# B C# C G# A G D E F# D#
II F F# G# G D# C A E F# G# F B
III F G F# D B E A B C# A# E D
IV F E C A D D# A B G# D C G
V F C# A# D# E F# A F# C A# F G
VI F D G G# A# A A D# C# G# A# C
α-hexachord β-hexachord
R
I G E F# G# D# C# D A# [A] B C F
II G A B F# E A# D C# D# E A F#
III G A E D G# F D E F A# G D#
IV G D C F# D# F D D# G# F C# C
V G F B G# A# C D G E C B D#
VI G C# A# C D A D B G F# A# A
IR
I G A# G# F# B C# C E F D# D A
II G F D# G# A# E C C# B A# F G#
III G F A# C F# A C A# A E G B
IV G C D G# B A C B F# A C# D
V G A D# F# E D C G A# D D# C#
VI G C# E D C F C D# G G# F# F
F A# B C# C G# A G D E F# D#
F F# G# G D# C A E F# G# F B
F G F# D B E A B C# A# E D
F E C A D D# A B G# D C G
IR
I G A# G# F# B C# C E F D# D A
II G F D# G# A# E C C# B A# F G#
III G F A# C F# A C A# A E G B
IV G C D G# B A C B F# A C# D
V G A D# F# E D C G A# D D# C#
VI G C# E D C F C D# G G# F# F
D C# D# G# F F#
pulse (mm. 1-14)
VLN 1 (mm. 4-7, etc.)
pulse (mm. 15-54)
Cello/Bass (mm. 39-46 etc.)
VLA (mm. 26-33, etc.)
VLN 2 (mm. 12-15, etc.) VLN 2 (mm. 16-19, etc.)
VLN 2 (mm. 15-19, etc.) VLN 1 (mm. 15-19, etc.)
F G# D# C# D F#
cm
us
ic
mu
si
cm
us
ic
mu
si
c
NO
TE
S
Porter, Alex. Diagram of Rotational Array and Its Usage in the Prelude of Stravinsky’s “Requiem Canticles”
or
yth
eo
ry
th
eo
rr
ea
rc
hite
ctu
re
ic
mu
si
cm
us
ic
mu
34
35
Diachronism temporizes points which function through simultaneous presentation.
For architecture, this puts the movement from paper to presence under review. The tradi-
tional separation between analytical and generative diagrams is a distinction based on the
diagram’s place in the creative process—before or after the building. Diachronism is a per-
formance which synthesizes the two. The generative diagram returns to end, and the analyt-
ical diagram moves to the beginning. In the experience of architecture, movement through a
building defines a successive sequence, a sequence which diachronism returns to the simul-
taneity of its original diagram. The analytical diagram moves to the beginning as the simul-
taneous origin of a concept that is necessarily experienced temporally. Benjamin discusses
this duality, the architectural drawing as beginning and end, by saying that images do not
“re-produce architecture. They produce it in the first place… [W]hat is crucial in the consid-
eration of architecture is not seeing but the apprehension of structures. The objective effect
of the buildings on the imaginative being [vorstellungsmässige Sein] of the viewer is more
important than their ‘being seen.’”1
The drawing that produces the
building is then reconstructed
by the imaginative being who
perceives the building temporal-
ly. This is the reconditioning of
movements to time; that one dis-
regards the temporization of expe-
rience in the diagrammatic return.
Diachronism synthesizes both
the production and the reception
of building into a diagram at the
beginning and the end.
Diachronism is the temporization of points while maintaining the
manifold lines between. This idea has long been at stake in the practice
of music analysis. Schenker opens an analysis of a Bach Sarabande (BWV
1009) by saying, “The eye can follow and encompass the lines of a painting
or architectural structure in all their directions, breadth and relationships;
if only the ear could hear the background of the Ursatz and the continu-
ous musical motion of the foreground as profoundly and as extensively!”1
Thus, Schenker’s system could be described as a synthesis of the visual
and temporal which is meant to set time out of joint through simultaneity.
Schenker’s Ursatz is always a movement, one which conditions time in the
foreground. This means that a generic movement in the background condi-
tions the possibilities available to the master for elaboration, prolongation,
and other temporizing techniques of the foreground. Thus, Schenker’s
concept of structure is one which determines the status of temporal events. This is
what could be considered conditioning time, for Deleuze, and might explain Schen-
ker’s tendency to prefer diachronic readings of extended pitches over specific,
rhythmic commitments. When Schenker presents his late graphic analyses, like that
of this Bach Sarabande, the visual presentations use time to condition movement,
because the background necessarily occurs simultaneously with the foreground.
In other words, the primary accomplishment of the background, its unfolding in
free composition, depends upon the simultaneity of its visual concurrence with the
foreground. Diachronism affords Schenker a temporal basis for the simultaneity
with which he extends pitches in the foreground to shape the background. This is
even more obvious in Schenker’s parenthetical discussions of “the non-geniuses,
who must therefore compose entirely in terms of the succession of surface events,
just as they hear and read in terms of successive events.”2
Diachronism temporizes points in space, but
maintains their performance outside of time. The
contraction of time can refer to Hamlet’s excla-
mation upon seeing his father’s ghost, “The time
is out of joint.”1 If time is jointed, Hamlet expects
to experience it in the same order as he lives it,
for its procession to follow his movements. He
does not expect to speak to his father after his
father’s death. Hamlet’s crisis is the recognition
of a break between movement and chronos, that
movements become strange because they are
subservient to chronos. Chronos becomes an
alternate conception of “time,” one in which the
strokes of time achieve greater autonomy from the procession of events which usually conditions
them. One can generally witness chronos in visualizations which plot time lines and movements, so
that the first and the last are literally one in the same.2 In Deleuze’s book on Kant, the unhinging of
time acknowledges the space of succession and the space of simultaneity. Deleuze writes, “Time
is no longer related to the movement which it measures, but movement is related to the time which
conditions it.”3 Diachronism measures the movements of points in space conditioned by time, and
stratified by simultaneity.
The disjoint in time is the disparity between the diachronic order that reveals différance through
simultaneity and the physical order in which true simultaneity is impossible. Music at first seems
to challenge to this proposition, for two tones, two melodies, and two intervals may be simultane-
ously heard. However, these counterexamples do not reflect the kind of simultaneity important to
the diagram. Simultaneity refers to the relationships, connections, and differences which can only
be represented at a visual level. In architecture too, built works are physical facts which behave like
Olivier, Laurence. Still from “Hamlet.” 1948.
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together yet apart.”2 In the diagram, difference also appears in the re-
lationship of programmatic elements. The three different classes
of program (living, working, and sleeping) project a sequence
to differentiate between, and to temporize sleeping, working,
and living. There is a dialogue which relates these three
activities to a Möbius strip (or at least to a sort of figure
8), and a diaphora which shows them to be repetitions
of equal conceptual importance, distinguished by their
sequential and spatial arrangements. These two perfor-
mances are intelligible as concepts because of their simul-
taneous presentation. Two activities can exist at once, that
each might be understood separately in order to subtract the
relationships between.
The analytical program diagram of UN Studio’s Möbius House
(1993-1998) sets the time out of joint by visually sequencing temporal
activities. The diagram for Möbius House is a programmatic arrangement
of activities based on a pattern of relationships resembling a Möbius
strip. It is here called analytical because it does not claim a rigorous
relationship to the generation of built form, as previous diagrams by
Eisenman, and Lynn have. The diagram explains an aim of the project’s
general arrangement, and perhaps assumes a less formal role in the
design process—a process of designing substances with matters. Ben
van Berkel and Caroline Bos, the principal architects at UN Studio,
present this project in their 2006 book under the heading, Living as Con-
tinuous Difference, suggesting the continuously separate movement “of
two intertwining pathways, which trace the progress of two people living
listening to a performance, because it is not held in the
same temporal tension as a performance would be.
If diachronism puts the time is out of joint, then the
temporal hierarchies which stratify music in the spatio-
temporal present are reversed. The visual score becomes
the thing conditioned by time, and the temporal tension of
the performance is slackened. Diachronism is the motor
for this subversion, because it synthesizes temporal points
into a new sort of temporal tension, the tension of simulta-
neity. Thus, the spatio-temporal present, the metaphysics
of presence, loses its authority over movement in a musical
performance, and a theoretical stratification of movements
in visual time receives privilege.
The landscape architect, Galia Hanoch-Roe, has compared the ephemerality of music to
the experience of spaces, describing their shared unfolding “in a linear manner over time.”3
Recalling Goethe’s adage that architecture is petrified music,4 the issue of movement as inher-
ently temporal in music and inherently fixed in architecture is a starting place for this discussion
of the way in which time conditions movement. Hanoch-Roe’s answer separates the spatio-
temporal, of which the sonic is a category, from the visual, of which score is a category. The
process of score-making in music petrifies the spatio-temporal, and spatial experience in archi-
tecture liquidates architecture’s representational images. This sets the visual and the spatial
against one another, and Hanoch-Roe encourages a becoming-visual of musical representation,
and a becoming-spatial of architectural representation. Simultaneity is thus addressed by the
untemporized score, “the silent reading of a score” in which “[t]he person chooses the tempo,
accentuation, and the linearity of the process, and may stop, turn back, return, and do as he
pleases.”6 Interpretive time clearly relates to a spatial experience, perhaps more directly than
“the ‘originary’ différance”5 and the metaphysics of presence might form a dialectic. This dialectic
between presence and deferral offers a space for architecture and music to comply with the meta-
physics of presence, the space with which most existing diagram theory begins. The fracture in time
between virtual space and actual space under the conditions of différance is bridged by Derrida’s
theory of deferral.
Recognition of the thing deferred is an acknowledgement of the non-successive concept of time
which accompanies diachronism. The totalizing aspect of time stratifies the present and future into
dimensions of the past, the present for Deleuze being conceived as a contraction of past instants.
The past is the synthesis of time because it remains fixed while pulling the present into itself and
calling the future into the present. For Deleuze, the movement of the present to the past is paradoxical
because the present becomes the past instantaneously, which is the same as saying that the present
cannot pass unless it is already past. Deleuze completes his paradox with that of pre-existence, that
“each past is contemporaneous with the present it was, the whole past coexists with the present in
signs for the simultaneous diagram, with a gap in time
between themselves and the subjects who perceive
them little by little. In other words, diagrammatic time
is out of joint because it is non-Cartesian, making a
rift between the all-at-once, and the once-at-once.
The time is out of joint because of this irremov-
able rupture in time, which Derrida attempts to explain
with the idea of deferral between the moment of the
shape’s appearance and the moment of its significa-
tion. The diagram relates to its analog in presence
by “defer[ing] the moment in which we can encounter
the thing itself.”4 Derrida properly calls this deferral
the temporization of différance, an effect by which
diachronism
module
iteration
repetition
memory
Porter, Alex. Diachronism Diagram . 2013. aUN Studio. Diagram for Mobius House . 1993-1998. b
a
b
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39
presence or not-presence. The concept is about things happening all at once, singular experiences
of the program disrupt time between a diagrammatic order of simultaneity and a present order of suc-
cession.
Diachronism transfers the diagram’s simultaneity into the built form. Form is an envelope for the
analytical concept—a stage on which the diagram can be acted out in time. This recalls Stan Allen’s
interpretation of Diagram Architecture as “an architecture that establishes a loose fit of program and
form, a directed field within which multiple activities unfold, channeled but not constrained by the ar-
chitectural envelope.”3 The simultaneous concept of the diagram is thus activated by the static form,
even if movements condition its use. In other words, the form is a datum, fixed in time, that registers
movements, such that form presents experience diachronically. Emphasizing its own immobility, the
Möbius House contains a memory of the movements it conditions, returning the experiential order
to the simultaneous diagrammatic concept which precedes it. Form makes its diachronic return to
concept by reconstructing the simultaneity with which it began.
As a built project, the simultaneity of
activity on which these performances rely is
impossible when the diagram is transposed
into entirely positive arrangements. Sleeping
and working cannot occur simultaneously, and
the movement between becomes circulation;
all movements condition the work in terms of
temporal experience in which performative si-
multaneity is lost. Indeed, the dilemma of
presence is that extant points cannot be sub-
tracted, impeding the performance of (n-1) that
reveals the virtual. Any subtraction leads to zero
amongst equal positives, following the binary of
Open scores, graphic scores, and other visualizations for sonic-temporal events
in music corroborate this argument. Ligeti’s piece of early electronic music for tape,
Artikulation (1958), is accompanied by an interpretive score by the editor, Rainer
Wehinger, in the edition printed in 1970. This system of modulating dots and lines
set against a Cartesian timeline works like one of Hanninen’s associative landscapes.
The graphical notations form genosegment sets which permit manifold variations at
the level of phenosegment. However, the time in this score is more garishly out of
joint, because despite the timeline, it is unclear what exactly these phenosegments
produce, affording more attention to their associative possibilities amongst the various
shapes. Wehinger does present a “key” with his notation, reproduced here, which
stipulates qualities of pitch and timbre which the shapes and colors represent. Time,
our present focus, is conspicuously absent, deferring to the timeline which remains
less salient than the associations.
1 S
ha
kesp
ea
re,
Will
iam
. A
ct
I, S
ce
ne
V,
L 1
90
2
M
att
he
w
20
:16
“S
o
the
la
st
will
b
e fi
rst,
a
nd
th
e
firs
t w
ill
be
la
st.”
Je
sus
co
nfo
un
ds
the
pro
ce
ssio
n a
nd
th
ere
by
ma
kes
a d
iac
hro
nic
pe
rfo
rma
nc
e.
On
e th
ing
aft
er
an
oth
er
is i
mp
oss
ible
. A
ll th
ing
s a
t o
nc
e i
s it
s a
lte
rna
tiv
e.s
3
D
ele
uze
, G
ille
s. “
Ka
nt’
s C
riti
ca
l P
hilo
s-o
ph
y.”
p. v
ii4
D
err
ida
, Ja
cq
ue
s. “
Dif
féra
nc
e.”
p. 9
5
Ib
id. p
. 10
6
De
leu
ze,
Gill
es.
“D
iffe
ren
ce
an
d R
ep
-e
titi
on
.” p
. 82
relation to which it is past, but the pure element of the
past in general pre-exists the passing present.”6 Time
is totalized by these ordinary movements, and reveals
its simultaneous ground of the past. Simultaneity is a
product of time conditioned by this repeated movement.
Thus, diachronism excavates the ground
of the past, and provokes Hamlet’s reversal, that the
operative quality of time conditions the performative
movements in a piece of architecture or music. “One thing after
another” becomes “all things at once,” and then returns to “one
thing after another” with the motion of return conditioned by a
newly stratified temporal space. This process is activated by the
diachronic disjunction exposed when past meets present. NO
TE
S
a
b
c
Wehinger, Rainer. Artikulation (Score) . 1970. aWehinger, Rainer. System of Symbols . 1970. bUN Studio. Representations of Mobius House . 1993-1998. c
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to 2 is called: “First revolution—Slicing of panels.” As plate 1 moves to plate 2, plate 2 also passes
as plate 3 becomes present, because plate 3, “Volumetric realization Panel Surfaces” includes and
builds upon plates 1 and 2. This procession creates a network of implicit, temporal relationships
enabled by the simultaneous presentation of plates in series. Each plate must simultaneously be past
and present, and these marks are made on the future. Subsequently, a single plate is incompatible
with its singular presentation, as it necessarily sits between other plates. As Deleuze and Guattari
contend, “It is composed not of units but of dimensions, or rather directions in motion. It has neither
beginning nor end, but always a middle (milieu) from which it grows and which it overspills.”5 The
simultaneity in time creates a middle between middles, all participating in a simultaneous movement
suggested by the space between.
Simultaneity must again be ruptured by the particular when the Max Reinhardt Haus is “crystal-
lized” as a building.6 While simultaneity enabled the present to pass by its already being past, the
building can only be an index of this process, the past now seen through the present. If architecture
is measured “by its performative effects as much as by its durable presence,”7 the rupture in time that
the generative performance necessitates is problematic because it potentially constrains the perfor-
mance of an architecture object to a remembrance of the past, and not a projection of the future.
Architecture must, thus, be redefined in a spatial continuum of images in order for the generative
diagram to return to the present. Diachronism is not a question of experience here, but a testament
to the possibility of a building with traces of the past that generates it.
The diagram for Eisenman Architects’
Max Reinhardt Haus (1992), a diagram
which generates form, also separates si-
multaneity from particularity in time. Using
a möbius strip with formal rigor, the diagram
performs the crystallization of form as
the product of a series of operations on a
square panel revolving around the axis of a
vertical möbius strip.4 The diaphora in this
diagram works similarly to that of House
II, subtraction from a series which follows
the paradigm of n-1. The Max Reinhardt
Haus diagram is distinct from that of House
II because it follows a clear set of articu-
lated moves, a matrix processing toward a
goal. This means that the diagram’s plates
are read in a certain order, and each has a
past and a future. Plate 2 is directly called
forward by plate 1, and the motion from 1
roughly even distribution of page types based on
staff systems per page, these can only demon-
strate true symmetry if we allow 3 to equal 4.
Feldman thus provokes the question of
the degree zero to constitute symmetry. In his
essay, Crippled Symmetry, from 1981, Feldman
addresses the “disproportionate symmetry” of
Near and Middle Eastern rugs, “in which a sym-
metrically staggered rhythmic series is used: 4:3,
6:5, 8:7, etc., as the point of departure.”7 The
symmetry of disproportionate halves is an additive
process, suggesting the possibility that Feldman
would have begun with 36 pages of three staff
systems each, and added a fourth system to the
pages of the second half. Feldman comments
on similar, additive asymmetries in Stravinsky’s
Requiem Canticles, which oscillates between “A”
and “B” segments in which “A” stays the same
while “B” changes duration, but still remains the
In its attention to both time and graphic, Morton Feldman’s late work, Crippled Symmetry
(1983), for Flute, Vibraphone/Glockenspiel, and Piano/Celeste uses graphic simultaneity to
unhinge time from a Cartesian understanding. As in his earlier piece, Why Patterns? (1978),
Feldman aligns all the measures of music on the page in an even grid, even though there is
no expectation for the three players’ measures to coexist in performed time. Feldman’s insis-
tence on traditional notation references the Cartesian practice internal to notation, in which even
bar lines uniformly translate into the procession of time. This traditional linkage between the
visual order of music and its spatio-temporal order is the hinge which Feldman unpins. Given
Feldman’s rejection of graphic score in all his late work, his critique more acutely sits within
the discipline, through issues like graphic and time which have existed all along. In Crippled
Symmetry, the 38-page score is graphically regulated by exclusively 9-bar systems. 18 pages
have three staff systems per page, 18 have four, and the remaining two pages (pp. 14-15) have
8 and 5 staff systems respectively. Pages with 3 staff systems tend to be toward the beginning
(1-12, 17, 19, 22-24, 33; mean = 12, median = 9.5), and pages with four staff systems tend to
be toward the end (13, 16, 18, 21, 25-32, 34-38; mean = 26.44, median = 28.5). The medians of
these two page sets match, with respect to the beginning and end of the piece, both 9.5 pages
from the outer limits. This illustrates a kind of symmetry which Feldman sought to demonstrate
in his graphic, in which a method to divide the score into halves of equal, symmetrical graphical
space reveals the impossibility of these two parts as symmetrical matches. In recognizing the
a
b
Eisenman Architects. Diagram of Max Reinhardt Haus. 1992. aFeldman, Morton. Crippled Symmetry, p. 1 . 1983. b
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43
1 B
en
jam
in,
Wa
lte
r. “T
he
Rig
oro
us
Stu
dy
of
Art
.” p
. 71
2
Se
e B
os,
Ca
rolin
e a
nd
Be
n v
an
Be
rke
l. U
N
Stu
dio
: d
esi
gn
mo
de
ls,
arc
hit
ect
ure
, u
rba
n-
ism
, in
fra
stru
ctu
re. p
. 15
03
A
llen
, Sta
n. “
Dia
gra
ms
Ma
tte
r” p
. 18
4
Th
is i
s a
dir
ec
t c
on
tra
st t
o t
he
fir
st u
se o
f th
e m
öb
ius
stri
p,
wh
ich
UN
Stu
dio
de
scri
be
s a
s �n
ot
use
d w
ith
ma
the
ma
tic
al
rig
ou
r.�
Se
e Ib
id. p
. 15
05
D
ele
uze
, Gill
es
an
d F
elix
Gu
att
ari
. A
Th
ou
-sa
nd
Pla
tea
us.
p. 2
16
D
av
idso
n,
Cy
nth
ia.
Tra
cin
g E
ise
nm
an
. p
. 2
22
7
Alle
n, S
tan
. “D
iag
ram
s M
att
er”
p. 1
68
L
ibe
skin
d,
Da
nie
l. “I
n S
ea
rch
of
Arc
hit
ec
-tu
re”
Ch
am
be
r W
ork
s
As such, Daniel Libeskind’s prefaces his collection of
drawings, Chamber Works (1983), with this statement:
Diachronism is the category of listening that
attempts to hear these connections, such that
memory reconfigures time. For example, Feldman’s
visual score shows all three parts coming together
on p. 14, playing on the downbeat of every other
measure. The parts all occupy the same system,
allowing the anomalous 8 systems on this page, and
seemingly collapsing the barriers of time signature
between the parts. He recognizes this as mislead-
ing, and notes, “It should be understood that this
page (like the others) is not a synchronized score”11
Feldman’s insistence against synchronization
stresses the score’s visual information to an even
greater degree, since it does not give much tem-
poral-acoustical information. This recalls Stravin-
sky’s practice of cutting out empty bars and in an
imagined visual unity between all parts, as is shown
earlier.
All of the measures with notes on p. 14 are
given 3/16 time. They are paired with measures of
other-half which “A” expels.8 Feldman gives this musical concept, imperfect symmetry, a visual
dimension in what he calls the “notational imagery.”9 In so doing, he describes his process as
“a visual rhythmic structure,”10 comparable to the rhythm canvases of Jackson Pollock. The
synthesis of two fundamentally musical concepts, visual rhythm and imperfect symmetry, results
in the notational grid in Crippled Symmetry.
Within this grid, Feldman liberates the visually regulated measures from their analog in time
by giving different, and constantly changing, time signature markings to every part. Goodman’s
category of score as digital diagram is unset by this, because the score shows a visual syn-
chronization which will never manifest in performance, if the score is read properly. In other
words, Feldman invites the silent listener (the analyst) to make immediate associations on the
page which may be less salient in performance, and for the listener to draw conclusions at a
distance from the score. This departs from the absolute, retrograde symmetry at work in the
fourth movement of Ruth Crawford Seeger’s famous String Quartet from 1931, symmetrical
by retrograde. It also departs from the sonic symmetry of “parallel periods” in 18th and 19th
century music. The rift between sonic symmetry and visual symmetry thus demonstrates two
qualifications for symmetry: retrograde symmetry, where pitches are reflected but not neces-
sarily durations, and sonic parallelisms, where durations must be equal to give the formal unit
symmetrical halves. For Feldman, symmetry has a repetitive, acoustical component, and a
retrograded, visual component, and the dialectic between these two elucidates symmetry as a
conceptual category.
Architecture is neither on the inside nor
the outside. It is not a given nor a physical
fact. It has no History and it does not follow
Fate. What emerges in differentiated expe-
rience is Architecture as an index of the re-
lationship between what was and what will
be. Architecture as non-existent reality is a
symbol which in the process of conscious-
ness leaves a trail of hieroglyphs in space
and time that touch equivalent depth of un-
originality.8 NO
TE
S
a
b
Pollock, Jackson. Autumn Rhythm. 1950. aFeldman, Morton. Crippled Symmetry, p. 14 . 1983. b
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45
1 S
ch
en
ker,
He
inri
ch
. “D
as
Me
iste
rwe
rk i
n d
er
Mu
sik
, Ja
hrb
uc
h 2
” p
. 55
2
S
ch
en
ker,
He
inri
ch
. Elu
cid
atio
ns
in “
Th
e M
as-
terw
ork
in
Mu
sic
: Vo
lum
e 1
” p
. 113
3
Ha
no
ch
-Ro
e,
Ga
lia. S
cori
ng
th
e P
ath
: L
ine
ar
Se
qu
en
ces
in M
usi
c a
nd
Sp
ace
in
“R
eso
na
nc
e:
Ess
ay
s o
n t
he
In
ters
ec
tio
n o
f M
usi
c a
nd
Arc
hi-
tec
ture
.” p
. 86
4
Th
is i
s c
ollo
qu
ially
re
pe
ate
d a
s “A
rch
ite
ctu
re
is f
roze
n m
usi
c,”
an
d a
pp
ea
rs i
n t
his
fo
rm i
n H
a-
no
ch
-Ro
e’s
ess
ay.
5
S
ee
Go
eth
e,
Jo
ha
nn
Wo
lfg
an
g v
on
. “C
on
ve
r-sa
tio
ns
wit
h E
cke
rma
nn
.” (M
arc
h 2
3, 1
82
9)
6
Ha
no
ch
-Ro
e, G
alia
. p. 8
6-8
77
Fe
ldm
an
, M
ort
on
. C
rip
ple
d
Sy
mm
etr
y,
in
“RE
S: A
nth
rop
olo
gy
an
d A
est
he
tic
s, N
o. 2
” p
. 91
8
Ib
id.
p.
91
“In
o
ne
o
f th
e
mo
ve
me
nts
o
f S
tra
vin
sky
’s R
eq
uie
m C
an
ticl
es
the
re i
s a
co
n-
tin
uo
us
pla
y b
etw
ee
n A
an
d B
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. 14
silence of variable length, 1/8, 1/4, 3/8, 1/2, 5/8 etc., always increasing duration by one
eighth-note. The pattern of silence over pages 14 and 15 gradually increases from 1 eighth-
note in the first measure, first system, to 12 eighth-notes in the sixth measure, third system,
down to 1 eighth-note in the first measure, sixth system, and up to 14 eighth-notes by the
last measure on the page. In this pattern, the symmetrical increase and decrease of rest
values (1 to 12 to 1) allows the viewer to perceive symmetrical parts, but also to recognize
their inconclusive presentation and imagine the possibilities that would, perhaps, create a
more perfect symmetry (adding a complementary ascent from 1 to 14 before the section
with 1-12-1; completing the set from 1 to 14 to 1; etc.). This same operation occurs on
page 30, in which visually synchronized measures containing a four-note figure in 3/16 are
interspersed with additive silences. Taking this into performance, the diachronic listener is
much more likely to hear a symmetrical expansion and contraction of the space between
notes of a particular part than the fact that this symmetry is imperfect. The procedure
becomes even less determined for the listener on p. 15, in which “synchronous” bars of 3/8
continue in the Piano and Vibraphone with oscillating rest values (4 up to 14 down to 5 up to
7), but the flute changes figure, a break in construction which is not audible when the parts
are heard discretely. Feldman thus uses a procedure, symmetric expansion, which would
be conceptually accepted from an acoustical perspective, and at least partially denied from
a visual perspective. Like the hand-stitched rugs or the Rothko paintings of which Feldman
speaks, Crippled Symmetry describes symmetry as a conceptual which pre-exists its visual
actuality.
Similar additive procedures work to define and subvert symmetrical forms in the
piece’s first section. Feldman emphasizes the piece’s formal break lines by changing in-
struments when a part has completed a symmetrical (and usually a motivic) unit. The vibra-
phone repeats the same motive (E-flat, D-flat, D, D-flat, C) for the first two systems, then
changes to Glockenspiel to play a different, but related pattern (D-flat, E-flat, C, D). In the
first two systems on page 1, the Vibraphone and Piano/Celeste use similar expansionary
and contractionary processes as happen later. The Vibraphone begins with its 5 note figure
in 5/16 in m. 1, and a sixteenth-rest is added to each successive measure, such that the
metric scheme is 5/16, 6/16, 7/16, etc. Over the first 13 measures, this pattern moves from
5/16 to 11/16 to 5/16, and closes the formal unit with an additional 5 measures. Three of
these measures are included in a repeated unit of the first 16 measures, creating another rift
between the visual and the acoustical. In performance, a repeat mark would seem to con-
stitute a level of symmetry between the two repeated halves, even if this does not play out
in the visual symmetry. In other words, the difference between repetitive, sonic symmetry
and visual, retrograde symmetry is at stake in a diachronic listening of Feldman’s piece.
Two possibilities of symmetry in the smae Rug presented by Feldman in his essay Crippled Symmetry . 1981. “Disproportionate symmetry” in the rug as made LeftPerfect, visual symmetry by reflection in the same rug Right
46
47
The question of diagram as a practice, then, is the question Or Not? It is a
question that must be asked if a diagram migrates into presence and hopes to
maintain the performative conditions of difference under which it operates. Or
Not? is not a challenge to presence, but is a challenge to its authority over works
of music and of architecture by calling attention to the diagrammatic, virtual, un-
presentable différance moving below the figural surface of design process. It is,
perhaps, the same question posed by Isozaki in his Fujimi Country Clubhouse, a
lingering question-mark on the authority of presence. It is a subversive question
which identifies the conditions of presence (conditions of site, context, instru-
mentation, form etc.), and challenges their hegemony using the conditions of
possibility here presented. The practice of Or Not? is reflected in some of my
recent work.
In an initial reading, the question presupposes a metaphysics of
presence by building a binary along the axis of presence: [To be / or /
not to be], in which or is the axis of presence. These two paths establish
ontological limits prefacing Hamlet’s question: that presence exists (as in
To be), and that un-presence does not exist (not to be); the conclusion of
the first discourse accepts the hegemony of the metaphysics of presence.
This closes the system of presence to the circle under examination here,
and defines the points by positive terms. In order for Descartes to be, he
must first think, a proposition which he would here have to do as a non-ex-
istent being, something rather unlikely to occur. Architecture could either
be built, or not exist; there is no between in which the diagram operates.
Music yields to sound, or to silence; the abstract machine is supplanted
by the recording with only two absolute modes of being. Both of these
Diagram is the constellation of points which
stretches over the chasm separating the virtual from
the present, and can be witnessed in the various
breaches of dialogue, diaphora, and diachronism in
presence. From these, a diagram is the hidden sum
made manifest by its constituent parts, but is also the
ontological motor which generates these parts. Just as
Descartes’s cogito, ergo sum returns to itself in a circle,
in which ergo sum is already implicit in the subject of
the verb cogito, a diagram follows an orbit to connect
its perceptual work to its generative work.1 As such,
a diagram is reified as a teleology which is also an
ontology. What mass occupies the center of this orbit?
As we conclude Hamlet’s diagram, it is a question of
being, Hamlet’s interrogative “To be, or not to be?”2
as the information given by it. Hamlet utters the second to be under different conditions than
the first, because it is a repetition of first. As in Schenker’s foundational work on harmony, the
passing tone motion from one scale step to the next always retains a vestige of the first in the
sounding of the second. The same is true of Eisenman’s diagram for the Max Reinhardt Haus,
in which the procession of plates always refer to past ground.
This veritable difference wedges a space between the two terms, and the passage through
this space demands a diagrammatic motion which performs the difference between terms. The
diagrammatic reading, then, is: [To be / or not / to be / ?], and the passage through the wedge
between two statements of /to be/ is written as an or not, spacing the two terms. This or not
is the movement of différance between two terms for being: To Be (both capitalized now, for
clarity) and to be. As Derrida says, “différance is not, does not exist, is not a present-being,”4
and this effects the separation of an /or not/ from the two terms for being-present in Hamlet’s
question. According to Derrida, the distinction between the To Be and to be “exceeds the
alternative of presence and absence”5 (an alternative seen in the initial reading of Hamlet’s
question), because it is an alternative to two presence-s.
or
yth
eo
ry
th
eo
r
conclusions are untrue, for we would be wrong to conclude that an architectural project does
not exist because it is only a drawing, so too does a composition necessarily exists before it
is played. The musical case has even higher stakes, for the piece still exists between perfor-
mances, and each performance relies on this abstract force which connects the performance to
the score.
To deconstruct this passage under the conditions of différance reverses this conclusion,
because it properly includes the non-present realm of which Libeskind speaks. Recalling that
Hamlet himself must leave the sensible realm in order to fulfill the imperative of différance,3 his
question must be deconstructed thusly, outside the scope of presence. If the conditions of dif-
férance allow a movement away from presence, then the question no longer reads as: [To be /
or / not to be], because we may recognize a new distinction between the repetitions To be and
not to be, wholly separate from the positive-negative construction. While each term includes
some form of the construction /to be/, the two terms are woven of the different fabrics, for one is
capitalized while the other is lower case, and this capitalized To be pre-exists for the lower case
to be in that it precedes it in time. From Stravinsky, understanding the repetition is as important
OR NOT?
yth
eo
ry
th
eo
ry
th
eo
ry
th
eo
ry
th
e
DIATHROUGH
GRAM(graphein) WRITE
DIAGRAPHEINThrough Writing
DIAGRAMMAGreek Latin
Olivier, Laurence. Still from “Hamlet.” 1948.Isozaki, Arata. Fujimi Country Clubhouse . 1973-74.
a b c
a
b, c
48
49
re
arc
hite
ctu
re
In my intervention project on the Morningside Heights campus of Columbia University, Or
Not? uses the operation of axial connectivity established by the existing campus plan to question
the authority of the orthogonal axes over the interdisciplinary networking between department
buildings. In other words, the intervention overlays the existing campus plan’s orthogonal grid
with non-orthogonal lines which network between buildings. Following the enlightenment project
of Columbia’s neo-classical architecture, geometries derived from Palladio’s Villa Rotunda are
stratified along the non-orthogonal axes of the intervention. In the accompanying diagram of
Palladio’s Villa Rotunda, the multivalent readings of the building’s massing operate as a sort of
theme and variations, demonstrating an anti-genealogy which continues ad infinitum.
ec
tu
re
arc
hite
ct
ur
ea
rc
hite
ctu
a
b c
Porter, Alex. Sketches of the Subversion of Axis on Columbia University’s Campus. a Model and Isometric Drawing of Design Possibilities . bIterative Diagram . c
Rudolf Wittkower. Architectural Principles in the Age of Humanism. 1949
Palladio. Villa Rotonda near Vicenza. 1566.
Interior
Exterior
Public
Public
Public
Public
(Private)
Public
Public
Public
Public
Exposed Exposed
Public
Public
Private
Private
Private
Private Exposed
Expo
sed
Exposed
Expo
sed
Expo
sed
Exposed
Diagrams of
Block Massing Block Divided by Cruciform Subtraction of Cruciform 4-square result
Cruciform Massing Cruciform read as Two Linear Rectangles Resulting cruciform of two distinct parts Hidden and Exposed Structures Large Space MassingLinear DivisionSubtraction of Dividing LinesMassing Extends es
Block Massing with Entrance MassesConceptual Division of 9-squaresMassing Extendsare Massing
Section Reduction Stacked Linear SpacesMatrix in Section Reduction
Cruciform above Block Block Divided by Cruciform in Sectional Space Block Traced on Cruciform Cruciform Diagram overlayed with 4-square Diagram
Massing in Sectional MatrixDistortion of 9-square in Sectional MatrixMassing Extends in Sectionare h Sectional Matrix
Expansion
In two directions
53
Figure Non-figure
112th st
111th st
Park
Ave
Park
Ave
re
arc
hite
ctu
re
ec
tu
re
arc
hite
ct
ur
ea
rc
hite
ctu
In my project for a grocery market under a Metro-North track in East Harlem, Or Not? questions
the absolute condition of the column grid as a spatial force under the track. By pushing the points
of this grid away from their origin, the project separates the space it creates from that which is given.
It also relates to the disjunction in time of Feldman’s Crippled Symmetry, by pushing progressively
further from a regulated condition. The genotypic similarity in the mutual diagrams of these two is the
disjoint movement conditioned by Cartesian regulation. As Feldman curates the disparity between
parts from Cartesian time, this project does so from the Cartesian grid.
Porter, Alex. Plan Diagram for the Distortion of a Point Grid in Harlem. 2012.
54
55
The chamber piece, Diachronism, uses mathematical pre-composition to distort the pitch
content of a C-major triad in a series of 15 sounded, 8-bar groups, each compounding the op-
erations of the last. Diachronism investigates memory in time, as the pitch sets move from [C E
G] in mm. 1-8 to [B C E F F# G] in mm. 9-16 and so on. This is a little like the opening discus-
sion of Riley’s In C, in which repeated passages give way to variation in pitch, except that the
seams in Diachronism are meant to be obvious, while Riley wants to hide them. By following a
modular form, the units of Diachronism pass more distinctly into memory than the motives of In
C, and the conception of simultaneity is more readily apparent. When these repetitions are rec-
cm
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depress keys as silently as possible
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ognized, each is necessarily heard in relation to the last, provoking the first question of Or Not?
which is inherent to repetitions of difference. The present experience is a constant reference to
the past, and the authority of the present is thereby deferred to the past. It is not distant from
this proposition to conceive of such repetitions which carry on ad infinitum, as in the diagrams
of Palladio’s villa. Indeed, this is the reason that 16 measures of written music precede the first
note which the instrumentalists play, as if the piece is a selection of possibilities from a range
of available processes, without resorting to the indeterminate.
ic
mu
si
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us
ic
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i
56
57
1 S
ee
De
sca
rte
s, R
en
é. “
Dis
-c
ou
rse
on
th
e M
eth
od
”2
S
ha
kesp
ea
re, W
illia
m. A
ct
3, S
ce
ne
I, L
58
3
“An
d t
he
refo
re a
s a
str
an
g-
er
giv
e i
t w
elc
om
e.
/ T
he
re
are
m
ore
th
ing
s in
h
ea
ve
n
an
d e
art
h, H
ora
tio
, / T
ha
n a
re
dre
am
t o
f in
yo
ur
ph
iloso
ph
y.
Bu
t c
om
e,
/ H
ere
, a
s b
efo
re,
ne
ve
r, so
he
lp y
ou
me
rcy,
/
Ho
w
stra
ng
e
or
od
d
soe
’er
I b
ea
r m
yse
lf
/ (A
s I
pe
r-c
ha
nc
e h
ere
aft
er
sha
ll th
ink
me
et
/ To
pu
t a
n a
nti
c d
isp
o-
siti
on
on
)” S
ha
kesp
ea
re,
Ac
t I,
Sc
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e V
, L 1
67-
173
)4
De
rrid
a,
Ja
cq
ue
s.
“Dif
-fé
ran
ce
” p
. 6
De
rrid
a
is
spe
cif
ica
lly r
efe
rrin
g t
o H
ei-
de
gg
er’
s o
nto
log
ica
l d
iffe
r-e
nc
e, b
etw
ee
n B
ein
g (
ge
ne
r-a
l, S
ein
), a
nd
be
ing
s (s
pe
cif
ic,
Se
ind
es)
.5
I
bid
. p. 2
06
Se
e
Fo
uc
au
lt,
Mic
he
l. “W
ha
t is
Cri
tiq
ue
” p
. 4
2 a
nd
K
an
t, I
mm
an
ue
l. “A
n A
nsw
er
to t
he
Qu
est
ion
: ‘W
ha
t is
En
-lig
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nm
en
t?’”
The challenge to presence which a diagrammatic framework asserts is the consequence
of Deleuze and Guattari’s initial invocation. Seen in retrograde, the passage of concept made
manifest in an object /to be/ from an object /To Be/ comes together with a question of power. At
the least, the diagram elucidates the metaphysical indeterminacy of a particular object whose
concept can be repeated. The practice of Or Not? is like Foucault’s “critical attitude,” or Kant’s
Aufklärung before this,6 a daring to subvert the authority of assumptions. As with the other
passages of this document, the question is a shape to indicate the meta-diagram of performance,
a diagram which comes before composition and before design. In such a framework, a diagram
emerges as a provocation of the movements, energies, and differences shaping presence.
cm
us
ic
mu
si
cm
us
ic
mu
si
c
The diagrammatic synthesis which Hamlet represents generates
the material for my one-act opera, Broken Images. This music is woven
of seams and masks, addressing dialogue in the libretto (a collage of
modernist and post-modernist texts), diaphora in the form (four parts
repeating parallel programs of events), and diachronism in the loose
manipulation of a 12-tone row. For obvious reasons, a passage from
Hamlet is the host for techniques of montage and pastiche which unfold
the question of the Northern Prince’s presence.
The four characters, Hamlet, Horatio, the Ghost, and the Author, define
Cardinal points of a non-Cartesian coordinate system. The scene takes place
after Hamlet has seen his father’s ghost. Hamlet, the lead, would seem to be the
“object” presented, were it not for the Ghost after whom Hamlet repeats, subvert-
ing this possibility’s authority. Horatio, the companion, would occupy the same
conceptual plane as Hamlet, were it not for the blatant disjunction between these
two characters. The Author, then, would then seem the architect of the whole, but
this role too is denied by the ambiguous power relations between the Author and
the three players. Or Not? works against each character’s univocal representation
to make clear their diagrammatic basis. Multivalence is the first indicator of the
diagram’s authority, and the authority of Or Not?
NO
TE
S
&?
44
44Piano nbb
Œ œ. œ. œ. œ. œb . œb . œb .œb
3 b bŒ .œ œ. œ œ. œb . œb . œb .
.œ .œ œbb
Jœ .œb œ œ
Frag (ile shapes in develop) mentanonymous
Score
Author is seated at a tableat the back of the stage, working at a typewriter.
i. a heap of
Broken Images
Porter, Alex. Concept Diagram for “Broken Images.” 2014. aPorter, Alex. Opening Line to the Score of “Broken Images.” 2014 b
a
b
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58 59
re
arc
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ctu
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or
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rBroken ImagesFRAG (ile shapes in develop) MENT
anonymous
Cast: 3 Players and AuthorHamlet: Soprano
Horatio: Mezzo-SopranoGhost: BaritoneAuthor: Soprano
The action occurs in the space between the pen and paper.
Author:
(Come in under the shadow of this red rock),
And I will show you something different from either
Your shadow at morning striding behind you
Or your shadow at evening rising to meet you;
I will show you fear in a handful of dust.
i i : words[The Author crosses to Hamlet, and gives him a slip of
newsprint, from which he reads]Hamlet:
My mother is a fi sh.
Horatio:
[baffl ed] O day and night, but this is wondrous strange.
[Horatio abandons Hamlet]
Hamlet:
And therefore as a stranger give it welcome.
“My nerves are bad to-night. Yes, bad. Stay with me.
[to audience]
“Speak to me. Why do you never speak. Speak.
“What are you thinking of? What thinking? What?
“I never know what you are thinking. Think.”
Ghost Enters]
Ghost:
Think of the Unreal
I : a heap of[The author sits alone at a table, near the back of the
stage, typing at a typewriter]
Hamlet:
[Hamlet enters. He has just seen his father’s ghost]O all you host of heaven! O earth! What else?And shall I couple hell? Hold, hold, my heart
[enter ghost]“Are you alive, or not? Is there nothing in your head?”
Ghost:I am one thing, my words are another.Now you are aware of me! Now I am something in your secret and selfi sh life, who have marked your blood with my own
[ghost disappears]Hamlet:Ay, thou poor ghost, whiles memory holds a seatIn this distracted globe.
[enter Horatio]Horatio:
My lord, my lord!
Hamlet:
How say you, then? Would heart of man once think it?
Horatio:
These are but wild and whirling words, my lord.
You cannot say, or guess, for you know only
A heap of broken images, where the sun beats,
And the dead tree gives no shelter, the cricket no relief,
And the dry stone no sound of water. Only
There is shadow under this red rock,
[The Author rises from the back. The players do not know the source of the voice]
[Horatio Enters]
Horatio:
“What is that noise?”
Author:
The wind under the door.
Ghost:
Unreal, when I learned that
Horatio/Hamlet:
“What is that noise now? What is the wind doing?”
Author:
Nothing again nothing.
Ghost:
when I learned that words are no good; that words I ever fi t even what they are trying to say.
Horatio:
“Do
Hamlet:
“You know nothing? Do you see nothing? Do you remember
Horatio:
“Nothing?”
Ghost:
Nothing. But I had been used to words for a long time. I knew that that word was like the others, just a shape to fi ll a lack; that when the right time came, you wouldn’t need a word for that anymore.
That man would be erased, like a face drawn in the sand at the edge of the sea
But
i i : waiting[The players freeze in place. The author ceases typing,
rises, and approaches the frozen scene]Author:
O O O O that Shakespeherian Rag—
It’s so elegant
So intelligent
More matter, with less art.
What do you read,
Horatio:
my lord?
Hamlet:
Words, words, words.
Author:
just a shape to fi ll a lack
Hamlet:
Between who?
The players retreat to the background]
Author:
Though this be madness, yet there is method in ‘t.
“What shall I do now? What shall I do?”
“I shall rush out as I am, and walk the street
“With my hair down, so. What shall we do to-morrow?
“What shall we ever do?”
[Hamlet runs forward]
Hamlet:
HURRY UP PLEASE ITS TIME
[Ghost comes forward slightly]
Author and Ghost:
He who was living is now dead
We who were living are now dying
Hamlet:
HURRY UP PLEASE ITS TIME
With a little patience
we must wait for the future to show
[beat, blackout. Ghost and Author exit]
iv: for time[Horatio and Hamlet are alone on a dark stage. A single light
shines towards them]Horatio:
“It’s almost too dark to see,”
Hamlet:
“One can hardly tell which is the sea and which is the land,”
Horatio:
“Do we leave that light burning?”
[a second light]
Hamlet:
ic
mu
si
cm
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ic
mu
60 61
or
yth
eo
ry
th
eo
rr
ea
rc
hite
ctu
re There are more things in heaven and earth, Horatio,
Than are dreamt of in your philosophy.
Horatio:
Who is the third who walks always beside you?
When I count, there are only you and I together
[Author enters. A third light]But when I look ahead up the white road
There is always another one walking beside you
[Ghost enters. A fourth light]
Ghost and Author:
These fragments of the fi rst letter I have shored against my ruins
Hamlet:
Rest, rest, perturbed spirit.—
With all my love I do commend me to you,
And what so poor a man as Hamlet is
May do t’ express his love and friending to you,
Ghost and Author:
I am one thing, my words are another / A shape to fi ll a lack
Hamlet:
God willing, shall not lack. Let us go in together,
[Author returns to typing]
Ghost and Horatio:
I am one thing, my words are another / A heap of broken images
Hamlet:
And still your fi ngers on your lips, I pray.
The time is out of joint. O Cursed spite
That ever I was born to set it right!
[all exit. Fade to black]
Works Cited
Allen, Stan. Diagrams Matter. in Cynthia Davidson. ANY 23: Diagram Work. June 1998
Allen, Stan. Points Lines: Diagrams and Projects for the City. New York: Princeton Architectural, 1999.
Print.
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Alex Porter Candidate for Bachelors of Music and Architecture, Columbia University, Columbia College 2014Advisor: Joseph Dubiel
