Feldman, Proportions, and Turtles (A working title…)

Benjamin D. Krock

Atonal II – The Sequel!

Spring 2014, Florida State University

The music of Morton Feldman presents a variety of analytical issues, as made clear by

Catherine Hirata in her 1996 Perspectives of New Music article. Mustering the theoretical

musings of Jonathan Kramer, the compser-ly writings of Karlheins Stockhausen and Morton

Feldman, the p-space adventures of Robert Morris, and the phenomenology of Merleau-Ponty, I

hope to provide an insight into a late work of Feldman.

I – Philosophical Background: Perception, Musical Time, and Eternity

The basic premise of Jonathan Kramer's book, The Time of Music, is that "music unfolds

in time. Time unfolds in music" (Kramer 1988: 1). There are two ways in which time structures

music and music structures time: linearity and nonlinearity. The former, linearity, is based on

"the determination of some characteristic(s) of music in accordance with implications that arise

from earlier events of the piece." It is processive: earlier events imply later ones. The latter,

nonlinearity, is nonprocessive: "the temporal continuum that results from principles permanently

governing a section or a piece" (Kramer 1988: 20). There are parallels between Kramer's notions

of linearity and nonlinearity with the philosophical notions of "becoming" and "being,"

respectively. Kramer is careful to point out that we, as cultural listeners, attend to both linearity

and nonlinearity in music. "If we believe in the time that exists uniquely in music, then we begin

to glimpse the power of music to create, alter, distort, or even destroy time itself, not simply our

experience of it" (Kramer 1988: 5, italics in original). The focus of Kramer's premise is that

"time is a relationship between people and the events they perceive. It is an ordering principle of

experience" (1988: 5).

Similar to the religious scholar Eliade's notion of "sacred time" (Eliade 1971: 22),

Kramer states that "musical time exists in the relationship between listeners and music," and

specifies four types of musical time: multiply-directed time, gestural time, vertical time, and

moment time (Kramer 1988: 6-7). These types of musical time are not always complementary or

comparable; they can overlap and create a "myriad temporal experiences, best described by a

proliferation of overlapping labels and categories" (Kramer 1988: 9). Figure 1 contains brief

explanations for each of Kramer's musical times.

Figure 1 – Kramer’s Four Musical Times

Karlheinz Stockhausen has written and spoken at length about moment time as it applies

to his concept of "moment form." Appendix One contains several lengthy quotes from

Stockhausen, the essence of which I will try to convey here. Moment time is non-teleological;

the typical "beginning-middle-end" paradigm of narrative is absent. As such, moments are not

the result of prior moments, nor do they necessitate later ones. The emphasis of moment form is

on the present, skewing our perception of time into eternity: "This is not an eternity that begins at

the end of time, but an eternity that is present in every moment" (Heikinheimo 1972: 120-1). A

certain amount of homogeneity is a prerequisite for moment time. This feature, along with a lack

of hierarchy, flattens out the climaxes and leaves the whole experience equal. As listeners with

only our own, often misguided expectations for a piece, moment time plunges us into an

exploration of a single moment, does away with the concept of duration (read: "real time"), and

forces us to confront the eternity in the here-and-now.

Kramer further elaborates the idea of moment time in his chapter 8, "Discontinuity and

the Moment." Moment time pieces are made up of multiple moments, which Kramer defines as

unfolding a single harmony or as carrying out a process. Sections governed by moment time

exist independently and in a static state or as moments of process. The ordering of sections is

arbitrary, and consecutive moments are not understood as outgrowths, climaxes, or predecessors

of surrounding moments. Returning sections must appear to be arbitrary. As such, a certain

amount of consistency is required: "There can be moments which have no common elements, or

as few common elements as possible, and there are other moments which have a lot in common.

Moment-forming simply means that there is also the extreme of no common material, and that

every given moment has a certain degree of material that has been used before, and of material

that is going to be used next. And I say 'a certain degree.' And I choose these degrees very

carefully from moment to moment, between zero and maximum. So the maximum means there is

a moment so full of other influences of the past and the future that it is hard to identify this

moment" (Stockhausen and Kohl 1985: 25).

The rate of information flow is vital to the perception of musical time, but it is especially

relevant in moment time. Several important dichotomies inherent to information flow were

foregrounded in the preceding discussion: difference and repetition, stability and instability,

surface activity and stasis. While stasis is a necessity for individual moments, it does not

necessarily govern entire compositions in moment time. "The threshold of perceiving stasis

[depends not only on the absence of linear expectations, but] also on context. […] The threshold

ultimately depends on the rate of information flow. In a given context a certain amount of new

information per unit time creates a static impression, while more information produces motion"

(Kramer 1988: 210). Therefore, if a moment is defined by a single unfolding of a harmony or by

a process, any perceived motion must be, by nature, non-teleological: motion for the sake of

motion. Any perceived hierarchy is imposed by the listener and is not a result of musical

structure.

Although moments in a moment form do not exude hierarchy, there may be

"submoments" grouped together within given moment. Structural importance is not indicated by

duration in moment form. Rather, to assign formal significance to durations of varying lengths

"demands a nontraditional mode of listening" (Kramer 1988: 209). As opposed to vertical time,

in which a listener can miss entire lengths of a piece and not lose its overall effect, the moments

in moment time "are not the same, and missing some of the moments means missing something

both some of the material and some important component(s) of the proportional scheme that may

define the total coherence" (Kramer 1988: 209-10). Based on these observations, moment time

seems to be governed by the proportions of each moment in relation with those surrounding it.

As listeners, we cannot comprehend the nonlinearity of moment form until the piece "is

within memory"; in other words, after we have heard it in its entirety. The dynamism and

excitement of moment form come from the "degrees of similarity and difference between

moments located in different parts of the piece." Paradoxically, excitement is generated by

linearity: "implications can arise in a series of short moments. We can have expectations, which

may or may not be fulfilled, from a pattern of moment recurrences. […] Whether or not [these

expectations are met] does not determine the existence of our expectations" (Kramer 1988: 218).

Following this logic, there are three apparent paradoxes in the concept of moment time. These

are summarized below (Kramer 1988: 219).

1. "It is impossible to separate totally linearity and nonlinearity. […] Our listening

process is always linear, yet we always come away from a piece with a nonlinear

memory of it."

2. "Moment time also tries to defeat memory. […] If a given passage generates no

continuation, development, or return, then the music offers no subsequent

structural cues to help us remember. […] Moment music focuses our attention on

the new: it places a priority on perception above memory."

3. "But, if moment time thwarts memory, how can a moment composition rely on

remembered sections lengths in order to convey a sense of balanced proportions?

The answer is that, because of the self-containment of (usually static) sections,

moment time divorces duration from content. A moment form may challenge

memory, but it can simultaneously operate on the assumption that a feeling for the

lengths of moments is remembered even [if/when] specific materials are not."

In summary, moment time removes the emphasis from content and places it on

perception and proportions. Structural relations between moments may be inherent, but moment

time represents a denial of linear listening, rendering any relations as the listener's. As Western

listeners, we impose hierarchy and linearity, but moment time forces us out of this method and

into eternity.

Much of what Kramer and Stockhausen have to say about moment time is echoed in the

philosophies of Merleau-Ponty in his Phenomenology of Perception. A complete description of

the phenomenology of Merleau-Ponty is best reserved for philosophy dissertations, but his

chapter on temporality is especially relevant for this discussion. For instance, Merleau-Ponty

(hereafter, M-P) argues against the metaphor of time as being a river, objectively experienced by

all consciousnesses. Instead, temporality changes depending on the role of the observer: "Time

is, therefore, not a real process, not an actual succession that I am content to record. It arises

from my relation to things. Within things themselves, the future and the past are in a kind of

eternal state of pre-existence and survival" (M-P 2002: 478). For M-P, the past and future exist

in the present. Temporal relations give rise to events in time, and time is the dimension of our

being (2002: 481, 483). Applied to the idea of proportions in moment time, Merleau-Ponty's

philosophy suggests that the perception of time relies on the moment, on the present.

Stockhausen's observations about the "now" conjure ideas of a primarily nonlinear

musical time devoid of linear progression. Certain quotes from Morton Feldman's writings and

interviews come to mind, specifically in reference to linear progression. Feldman's comment that

he desires sounds "more direct, more immediate, more physical than anything that existed

heretofore" is summed up nicely in his offhanded comment about "the sounds themselves"

(Boretz 1961: 4). The implication in this quote is that individual sounds are not outgrowths of

earlier sounds and, likewise, don't imply later sounds (Hirata 1996: 6-10). This parallels

Kramer's and Stockhausen's discussions of moment time, and it is Feldman's Piano and String

Quartet from 1985 that will be used to demonstrate these notions.

II – Feldman’s Piano and String Quartet (1985), P-space, Difference, and Repetition

Morton Feldman's Piano and String Quartet was completed in 1985, near the end of his

life. It was written from the pianist Aki Takahashi and the Kronos Quartet, and their recording of

it runs just shy of an hour and twenty minutes. Given the length of the piece, one might expect a

highly varying musical surface that would present a set of implications about form and teleology.

However, the piece is formally divided in two sections of roughly equal durations as shown in

Figure 2. Within these two large sections, I have marked various formal boundaries, but they fall

into the category of "submoments" a la Kramer.

Figure 2 – Timeline of Feldman’s Piano and String Quartet (1985)

The first of these sections begins with a rolled piano sonority set against a dyad of string

harmonics in the viola and cello. These sounds sustain for a measure of 3/2, followed by an 8/8

measure of rests before the piano sonority is repeated, this time with a dyad between the viola

and the second violin. A pattern appears to be emerging at this point: a specific sustained piano

sonority set against a background of sustained strings is followed by a measure of rest. In fact,

this repeats a total of ten times before the piano sonority changes. By the time a new piano

sonority is introduced, a minute and a half of "absolute" time has passed, providing the listener

with a chance to absorb the sound itself. Additionally, its repetition creates an expectation in

regards to the importance of the sonority.

The sustained strings of the first large section of the piece are largely made up of quiet

harmonics and high pitches, which are not easily identified aurally. As such, I've opted to discuss

only the piano sonorities, since they are the primary source of perceived pitches. Furthermore,

my discussion will take place entirely in p-space. Figure 3 provides a score excerpt of the first

piano sonority of the piece accompanied by five sets of data: its set-class and Forte-number, the

ordered members in pc-space (pcset n), the intervals between members in pc-space (INT(n)), the

ordered members in p-space where C4 = 0 (pset n), and its spacing in p-space (SP(n)). To my

ear, the first large section of the piece seems largely concerned with the type of sonorities heard

and their specific voicings. Furthermore, members of the same set-class are not considered

equivalent unless they are also voiced the same way. Therefore, each spacing in p-space will be

assigned a generic name based on the lowest pitch heard. For Figure 3, the name assigned is D#-

1.

Figure 3 – D#-1 in p-space

Over the course of the piece, D#-1 (and several variations of it) returns, but it is easily

recognizable even among the wealth of hexachordal sonorities throughout the piece. Arguably,

its most salient feature is the high register ic2 dyad { F#6 Ab6 }. In order to maintain that

identity even when it is slightly altered, I opt to think of D#-1 as the combination of two

trichords, as shown in Figure 4. The upper structure, D#-1/2 { 19 30 32}, is a member of set-

class [012] with the specific SP(X2) = [ 11 2 ]. It will serve as the strongest identity function for

D#-1.

Figure 4 – Trichordal Partitioning of D#-1

The second sonority heard in the piece, Bb-1, is of the same set-class as D#-1, but their

voicings in p-space are different. Figure 5 shows the p-space features of Bb-1. The distinctive

trichordal partitioning of D#-1 is gone, the outer pitches are different, and none of the intervals

between consecutive pitch members are retained. As a way of expressing these differences, I've

coined a quasi-similarity measure for comparing voicings in p-space. Figure 6 compares the p-

space spacings of D#-1 and Bb-1, their outer pitches, and the total span of each sonority. For D#-

1, these elements are expressed as [ 7 6 3 11 2 ], { 3 32 }, 29. The bracketed numbers are SP(D#-

1), the ordered dyad are the outer pitches of D#-1, and the single number at the end is the

intervallic span between the two outer pitches. Spacings are compared by extracting the

difference between intervals in each spacing and plotting it into the respective place. I call this

comparison the Spacing Difference Vector (SDV). Formally, this is expressed as SDV(D#-1, Bb-

1). Outer pitches are compared in the same way, resulting in what I call the Boundary Dyad

(BD). Finally, the SDV is summed to express the Indexical Difference (ID) and is paired via a

forward-slash with the difference between consecutive outer pitches (Span Difference, or SD).

The output of this string of comparisons (which I have opted not to name…) does not provide us

with a single number: [ 1 2 4 7 9 ], { -5 -2 }, 23/3. Rather, it is a comparison of three elements of

the sonorities, as I'm not concerned with making large claims about the universal usefulness of

this method. These differences will, however, be the cornerstone of my analysis.

Figure 5 – Bb-1 in p-space

Figure 6 – Comparison of D#-1 and Bb-1; SDV, BD, ID/SD

The smallest possible ID between two sonorities is 0, which would be the result of

comparing two sonorities with the same spacing. Their membership in the same set-class would

be guaranteed, but their outer pitches could potentially differ. Figure 6 gives the analyst a

starting point for measuring her perception of difference between sonorities, but a certain amount

of subjectivity could still hypothetically override this seemingly objective output. For instance,

an ID of 23 might seem like a vast difference, but as will be shown, even ID's between similar

sonorities tend to be on the high end.

After Bb-1 is repeated five times, a new sonority is heard: D#-2. Its construction is highly

similar to that of D#-1 in that the upper trichord partition is the same: { 19 30 32}, and its outer

pitches, { 3 32 }, are the same. Its p-space features and its comparison to the previously heard

Bb-1 are shown in Figures 7 and 8, respectively. Between Bb-1 and D#-2, the output of our

similarity measure is as follows: [ 3 2 2 7 9 ], { +5 +2 }, 23/3. SDV(Bb-1, D#-2) is somewhat

similar in that the last two digits are displaced by the same difference as in SDV(D#-1, Bb-1).

However, the first three digits have changed. The BD in Figure 8 shows a correction of the { -5 -

2 } shift of Figure 6, and the ID/SD is the same between both comparisons. In this way, the

similarity measure (still unnamed) suggests a close relation between D#-1 and D#-2; Figure 9

compares the two.

Figure 7 – D#-2 in p-space

Figure 8 – Bb-1 and D#-2

Figure 9 – D#-1 and D#-2

Such a comparison might seem contrary to Kramer's notion that moment time erases the

listener's memories of previous material, but it is founded in the fact that, after three repetitions,

D#-1 is heard again. In the 2 minutes and 39 seconds of "absolute" time so far, there has been 19

events made up of only three distinct piano sonorities. The next sonority is quite different,

causing a retrospective formal division; or rather, the end of a moment. But before I move on I'd

like to reflect on how the analysis so far ties in with the philosophical discussion with which I

began. The incredibly slow information flow of the first 159 seconds of the piece allows the

listener to get lost in "the sounds themselves," a la Feldman. Further implications of moment

time, as discussed by Stockhausen and Kramer, have been realized. Difference and repetition

have played a large part, in that sonorities are repeated yet even the changes between consecutive

sonorities were relatively small. My prose might lead one to hearing the return of D#-1 as a

goal—a goal which seems to be implied by the piece itself—yet the similarities between all three

sonorities flattens out any climactic expectations. Likewise, the stasis and similarities shown

suggest that this first section falls under the category of "unfolding a single harmony."

As stated before, the return of D#-1 is followed by a completely new sound colour built

on the same lowest note: D#-3. The openness of D#-1's voicing is collapsed into a clustered

mess, as shown in Figure 10. D#-3 is a member of set-class [012345], the chromatic hexachord,

and its voicing in p-space is semitonal with a whole-step and a major-7th. Figure 11 highlights

the stark contrast between D#-1 and D#-3, yielding SDV(D#-1, D#-3) = [ 6 5 2 9 9 ] with a BD

of { 0 -13 } and an ID/SD of 31/13. Three repetitions of D#-3 are heard before a re-ordering

appears: D#-4. Figure 12 compares the two.

Figure 10 – D#-3 in p-space

Figure 11 – D#-1 and D#-3

Figure 12 – D#-3 and D#-4

Referring back to the timeline of Figure 2, my second formal division happens at the 5:21

timepoint. This boundary is not based on sonorities as the first was, but rather on the rate of

information flow. The musical surface of the piece so far--sonority sustained for a measure

followed by a measure of rest--has had only one slight hiccup at 4:57 when a new sonority

arrived in the measure right after another. This event slightly denies the expectations of the

piece-so-far, but is not drastic enough to merit a new formal boundary. Three chords after the

metrical hiccup is a much more unexpected moment. While the pitch material of the piano is still

a hexachord, it has been separated rhythmically into trichord partitions. Similarly, the string

quartet's entrances are divided and rhythmically offset. Up until now, the piano and string quartet

have been completely in sync, but for these four measures they are at odds. The last of these four

measures reaches the zenith of the piece-so-far before the piano abandons hexachordal

sonorities. My next statement is going to seem hypocritical in terms of my overall point, but this

moment is so beautiful to me. The ensemble seems to fall apart, and the piano switches to single

pitches in an attempt to reckon itself with the strings. After four reassuring sonorities, the

original musical surface returns with D#-3 (the chromatic cluster). This is shown in Example 1.

Example 1 – Disturbance, Dissension, and Reckoning from 4:47 to 6:08

The disturbance at 5:21 creates a retrospective section from 2:39 to 5:21…or 162

seconds. In comparison to the first section, the second section differs by only 3 seconds. When

these two sections are combined, they divide the piece-so-far in two roughly equal sections.

Given Stockhausen's comment that, in moment time, form is generated by "the proportional

interrelationships between moment lengths," and that formal coherence is best understood in

these terms, it seems that this piece is largely governed by "roughly equal proportions" at

multiple hierarchical levels1.

III – Proportions, Turtles, and Conclusion

At the formal level mentioned before, the first 321 seconds of the piece are divided into

two equal halves, as shown in Figure 13. Referring once again to the timeline in Figure 2, the

entire piece is partitioned into two large, roughly equal halves. At the biggest scale, the first large

section lasts for 44:10 (2650 seconds) and the second lasts 35:37 (2137 seconds): the piece lasts

79:43 (4787 seconds). Section I takes up roughly 55%, leaving the remainder (45%) to Section

II. The perception of this ratio is outside of my comfort-zone, but the fact that the piece is

divided in such a way lends credence to Stockhausen’s statement.

1 See Appendix One for the full quote.

Figure 13 – Equal Proportions from 0:00 to 5:21

On a smaller scale, the first of the two formal units discussed in Section II of this paper

also divides into two roughly equal halves. Figure 14 shows a screenshot of the wave form of the

first 159 seconds (through the return of D#-1). The first shaded area covers the ten repetitions of

D#-1; likewise, the second shaded area includes the five repetitions of D#-2, the three of D#-3,

and the return of D#-1. D#-1 takes up roughly 58% of the time, followed by roughly 42%. While

not exact, these proportions are echoed at multiple formal levels of the piece, up to and including

its entirety. Just as D#-1’s prominence in the first 159 seconds can be the source of listener

expectations, the proportions experienced and remembered in the same time set up expectations

for the rest of the piece. Figure 15 shows that “it’s turtles all the way down.”

Figure 14 – Roughly Equal Proportions from 0:00 to 2:39

Figure 15 – Multiple Formal Levels and Their Respective Proportions

As I hope to have shown, Feldman’s Piano and String Quartet is a prime example of

Kramer’s moment time. Bolstered by p-space analysis and similarity relations, the lack of

teleology, information flow, difference and repetition, flattening out of climaxes, memory, and

eternity all feature in my reading of the piece. Echoes of Stockhausen’s writings and Merleau-

Ponty’s phenomenology resounded in my approach, and the typical analytical issues presented

by Feldman were overlooked in favour of an approach based on perception.

Appendix One – Stockhausen on Moment Time

"[Moment forms] do not aim toward a climax, do not prepare the listener to expect a climax, and

their structures do not contain the usual stages found in the development curve of the whole

duration of a normal composition: the introductory, rising, transitional, and fading stages. […]

They are forms in a state of always having commenced, which could go on as they are for an

eternity…"

"Every present moment counts, as well as no moment at all; a given moment is not merely

regarded as the consequence of the previous one and the prelude to the coming one, but as

something individual, independent, and centered in itself, capable of existing on its own. This

concentration on the present moment--on every present moment--can make a vertical cut, as it

were, across horizontal time perception, extending out to a timelessness I call eternity. This is not

an eternity that begins at the end of time, but an eternity that is present in every moment. I am

speaking about musical forms in which apparently no less is being undertaken than the

explosion--yes--even more, the overcoming of the concept of duration."

(Heikinheimo 1972: 120-1)

"There can be moments which have no common elements, or as few common elements as

possible, and there are other moments which have a lot in common. Moment-forming simply

means that there is also the extreme of no common material, and that every given moment has a

certain degree of material that has been used before, and of material that is going to be used next.

And I say 'a certain degree.' And I choose these degrees very carefully from moment to moment,

between zero and maximum. So the maximum means there is a moment so full of other

influences of the past and the future that it is hard to identify this moment."

(Stockhausen and Kohl 1985: 25)

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