Velocity and Metric Class as Formal Determinants in Milton Babbitt’s Third String Quartet



Analytic Chart description (figure 1, 1-10)

From top to bottom, on the left side:

1. PC Arrays (Arnold and Hair)
2. Rhythmic Arrays (Arnold and Hair)
3. Metric Class with bar numbers
4. Time Point Unit. Bracketed where more than one unit occurs in a single Metric Class (meter)
5. Exact start and stop of the time points (durations). It does not always agree with the changes in metric class (meter) and sometimes occurs within the metric class. notice the vertical lines in the 2/4 sections mm.n78-161 and 367-424. Also the lack of vertical lines in mm. 307 and 337.
6. Bridge/Overlap (my terms) A bridge, in [brackets],is an area between two time point units (notes in the rhythmic organization) where neither asserts itself, but another larger “combinatorial” unit acts as an transition. Overlap is where a time point unit (note) operates in a metric class (meter) its not suppose to.
7. Total bars of the Rhythmic Field (my term), or time point units.
8. Asserted Form (my term) for the form created by the varying intensities of the time points.
9. The Relationship between the time point units derived from my “Rhythmic fields.” The eight time point velocity set members are assigned a level number and a transposition letter (see 11, on this chart) Two exact transpositions are shown with the above sideways brackets.
10. Activity level. The hairpins “<” and “>” show the relative increase and decrease in surface activity. “A” is for active, “Less” is less, “Very” is very Active. “Mixed” has starts and stops. There is a correlation between changes in the “rhythmic field” and changes in surface activity in almost all cases. Activity includes: attack and release points(durations), sustains, and multiple entries.

addenda for Babbitt Chart

11. A key to the rhythmic fields similar to example 9.   From the original holograph chart.  The 32nd notes are 0, 16th are 2, 8th notes are 4.  Duple is a quintuplet b, triplet c, septuplet d.  The reason the 5 subdivision seems to come before  the 3rd is that this chart goes from left to right in order of duration length.  

October 6, 1991 version

Velocity and Metric Class as Formal Determinants in

Milton Babbitt’s Third String Quartet

by Philip Fried

The original topic of my paper was “velocity and meter as formal determinants Milton Babbitt’s Third String Quartet.” But since few in this audience know much about my methods and interests, I was asked by the Chair to prepare a general overview of my work. This is, of course, good for you but difficult for me as I now must present an analysis and a theory (really a theory in progress) in the same (short) time.

Such discipline, however, is good for the soul, and so I state my interests thus: it is my preference to analyze 20th Century music using a method centered on deducing formal elements from the most apparent surface redundancies, their repetitions, and transformations.

I would describe this technique as an investigation of surface content. As a method its analytic predecessor would be thematic analysis. One important point to consider, is that my surface contents method is not primarily pitch based (in the case of the Third Quartet, Babbitt himself notes the difficulty of hearing the pc arrays). What I present will see to be a rhythmic analysis of the Third Quartet, but that is only because I would start an analysis with the most sharply defined features of a given work.

Arnold and Hairi, who define the form of the Third Quartet by its pc arrays, as does Babbitt himself, also point out in their study that rather than pitch, it is “the rhythmic plane with its frequent changes in velocity that is the most sharply defined feature of the quartet.” So, the surface changes in “velocity” will be my starting point for my analysis. The problem with confronting this issue, is the fact that the pc array is deemed more important structural aspect by Babbitt himself.

However, I fee certain advantages to this method of analysis for recent 20th century music: 1) I content that surface content, if it is truly a description of the most salient aspects of a work, would be audible; 2) Since surface content does not primarily involve pitch it is confluent with pitch based analysis, both forte and 12-tone set theory.

The main pitch study of the Third Quartet, which I will refer to, is by Arnold and Hair. It is an important work, in it they elucidate the pc array, the rhythmic aggregate and the various lynes and so define the Third Quartet’s form. However, I intend to use the Meter and its related Velocities (which I state are audible) to define the form of Babbitt’s Third Quartet, hence my analysis will not necessarily differ from theirs, but rather than contradict their findings complement them. Further, since Arnold and Hair show the complete unfolding of the pc array and rhythmic aggregate, I will use their examples for constant reference, since it is not my intention to avoid comparison, but rather to show how these same elements (the pc array, rhythmic aggregate and the 8 lynes) can take up different meanings in the context of surface content (surface audibility).

Arnold and Hair (as well as Babbitt) divide the work into 4 main parts relating to the array structure “four main parts (for each of the array dimensions…)” In my figure 1: 1) the top line, represents the four pc array parts, 2) represents the rhythmic structure which is isomorphic to the pc structure, (it’s first two parts). Further, figure 1-8, and 9 shows the contrast created when the pc arrays are “draped”over the time point velocities, that causes different “rhythmic fields” to become apparent. 3) Surface content, is not a composer specific method. So my language will attempt to be general in nature.

The difficulties for a surface content approach are obvious, in that all sound compositions are known to have pitch, but not all sound compositions are known to have “thematic material.” Traditionally pitch analysis and thematic-ism have always worked together (ever Schenker referred to thematic material). Even when there is ambiguity between the two, it is always a point of passionate interest to theorists, and significant others.

Originally, I wanted to show that “Groves”was wrong about the time point system being inaudible, and that it’s audibility had important formal consequences for Babbitt’s Third Quartet. And so we begin…

The question of formal principles in Babbitt’s music, specifically in his Third Quartet, may be formulated in this manner: how is form to be perceived in a work where thematic-ism has been dissolved? In this work, Babbitt does not use the repetition of row forms to articulate macro form. As Babbitt remarks in the notes to his 1972 Turnabout recording,ii the Third String Quartet “does not instance any surface ‘formal’ pattern created by conjoined repetitions within and among [its] musical dimensions.”iii Pitch Classes, though repeating within the large arrays (P, I, R, RI)iv, have no reappearance of prime or pitch areas that would delineate from. It is a “completely, though by no means excursively, polyphonic [work],”v strictly ordered to form 12-tone aggregates (280n of them to be exact) that are not set out in any way that would make them an “event.”

This is also true of the larger aggregate sections as they are all integrated into the larger polyphony. Here, rhythm is controlled by a rhythmic aggregate which is isomorphic to the pc array,vi i.e., it follows the pc arrays (more on this later). If we understand rhythm as coordinated groups of pulsations,vii rhythm does not exist here, at least in the traditional sense, as Babbitt has dissolved any surface activities such as motives or developmental procedures. Therefore, how can we perceive from in this ? Babbitt himself states that the four pc array areas (P, I, R, RI) are referential: “…within the familiar transformations – the linear disposition and the ordering of the pc classes, the linear constituency of the aggregates and the order of aggregate progression of the [four] sections are identical.”viii Yet, Babbitt also states “this parallelism [between the four arrays] may not be completely obvious on first hearing because of these degrees of reinterpretation [of the musical material].” The key to an understanding of the audibilityix question as it pertain to Babbitt’s music lies in a special by-product of the time point system as it is used in the Third Quartet.

A Question of Audibility

The New Grove states that the time point system is inaudible.x The primary function of the time point system is to create durational relations between the notes that would correspond to the serial pc dimensions, i.e., the rhythmic aggregatexi. The rhythmic aggregate is a durational set, yet the time point system has the following property: it can be seen and heard as a collection of possible attack and release points within the measure. The time point system is not, then, a strictly durational set. Even in Composition for Four Instrumentsxii you can find a similar technique in that contrasting sections are built up from different modulo durations.


My Example 1 shows only five of the many possible interpretations of the rhythmic aggregate durations between time points 0 and 4 (I have excluded all other factors, such as articulation, which would complicate matters). Though each of the five parts of example 12 are allowable according to the time point system, it is difficult to hear them as equivalent because the durations are different in each case. However, there is no problem hearing these durations as being different. Example 1-5 has a special property for obscuring audible durational relationships, in that it extends time point 0 further in time, than time point 4. This happens quite frequently in the music itself, as we will see later.


If you look at my Example 2, you will see the durations of 0,4 from my example 1-4. I have listed the different durational possibilities for these time-points as they would change with every change in velocity of the Third Quartet.xiii If we assume a steady tempo,xiv it is easy to notice that events happen six times faster in the 2/8 meter than in the 6/4 meter. Metronomically speaking six times faster is a very significant change. You will hear an immediate difference, even at twice as fast. As in example 1, the time-point attacks of 0 and 4 are still a constant in each case, even though each change in the time point “velocity” has changed their duration. That means that each measure’s notated meter (with the exception of the 2/4 and ¾ meter) becomes a frame for defining the “velocity” of its time points, i.e. how the measure is divided into twelve. (The complexity of the surface would argue against describing velocities as merely tempo changes, because, velocity is not motor rhythm. As release points (durations) are variable, and further, even the slowest velocity (an eight note equals metronome 144( is so fast as to be out of character with the quarter). Furthermore, if you take a hypothetical row (assuming that we use time-point system as it is used in the Third Quartet, where meters of different length can control equivalent space) unfold it over different meters – e.g., 64 then 2/8, ex. 3 – you can also hear a difference. This is because the events (the pc/rhythmic aggregates) altered by the velocities, would happen six times as fast in a 2/8 bar than in a 6/4 bar.xv It is this audible contrast of the rate of events that is directly tied to the changes in meter and the velocities within the meters.


The us of the time-point system in the Third Quartet is a fusion of an older practice as, for example, in Three Composition for Piano, where the rhythmic sets are groupings of a constant modulo; the 16th note attack unitxvi (see West.), so are as time-point units are (here) interpreted as a single velocity. It is important to note that, in the time-point system, the use of a single rhythmic unit as a “background? For rhythmic events in no way implies a steady pulse.xvii In Babbitt’s early Three Compositions for Piano, from 1947-8, there is a constant reiteration of a modulo (16th) so that there is constant motion, pulse, and motor rhythm (really pulse rhythm) that does not appear in Babbitt’s Quartet No. 3 (of for that matter even in his Composition for Four Instruments). Though the velocities are constantly reiterated, the time-point “minimal duration” is not.

Since I propose that this aspect of the time-point system, the change in the rate of events, is audible, it will be the focus of this article to describe how the contrast and repetition of velocities and their interaction with meters (which I call “rhythmic fields,)”xviii create form.

The Rhythmic Field

I have shown the possibility of hearing contrasts among large-scale changes in notated meter, and their time-point velocities. But, what happens within these sections themselves? I propose that these sections are to be understood as a rhythmic field in Babbitt’s Third String Quartet. “Rhythmic” because it is not just made up of velocities but also of durations, as well as other factors; and “field” because of the sporadic (start and stop), but not static, nature of the musical material.

Sporadic Events

Babbitt’s variations and integration of musical materials disrupts any feeling of reference we might hear or feel. I have already spoken of the lack of coordinated surface (rhythmic) events. Further, each time-point velocity is associated with one of eight lynes of dynamics, and, as there is a constant change of dynamics, there is therefore, a constant change of accent.xix This particular technique of accent articulation cannot in any way be construed in a traditional way since it throws off any attempt to feel a regular pulse,xx since the dynamic lynes do not always coordinate with the notated meter (mm. 2-7, 9-11). Andrew Mead in his example 10, my example 4, shows a surface accent pattern (see his arrows) based on metric position (time-points 0 and 4), overlaying the notated 3/8 meter. His arrows show that the 3/8 metric patterns shift in mm. 2-5 and dissolve in mm. 5-6. this shifting metric pattern shows another aspect of the sporadic surface.


The question concerning meter in Babbitt’s music can be divided into two parts: 1) the generative nature of time-points to meter and 2) the question of the audibility of meter itself. Meter alone does not account for all the changes in the time-point set. Even though I find sixteen velocity subdivisions in this work, as there are sixteen occurrences of meter (who’s interactions I call “rhythmic fields”), they are not are related directly to changes in meter. For example, there are divisions with a meter (2/4), mm. 78-166, 366-424, and meters that don’t assert their own time-point units, such as mm. 304, and 307.

Babbitt’s own definition of the function of meter within the time-point system is “…a measure divided into twelve equally spaced time-points with the metrical signature probably [italics are mine] determined by the internal structure of the time-point set (example 9), and with the measure now corresponding in function to the octave in the pitch class system.”xxi

This raises several points; 1) that we should examine the structure of the time-point sets (example 9), and 2) that the measure has an octave function, so that measures of different length are in most cases equivalent (see ex. 3). Babbitt’s “probably” leaves the question of the “primal” time-point set/metric relationship open.xxii However, Babbitt clearly states that there is a close relationship between the notated meter and the time-point set – “…It is necessary merely to embed it (the time-point set) in a metrical unit, a measure in the usual musical metric sense [my italics], so that a recurrence of succession of time-points is achieved, while the notion of meter is made an essential part of the systematic structure.”xxiii This does not however explain how several different velocities can inhabit a single meter and thereby create division within a single meter. Further, a surface analysis reveals transitional areas between the termination of one velocity and the start of another (see Bridges). There are also measures where velocities appear where they don’t belong (m. 307). (It’s a difficult life for a velocity.)

Metric Audibility

The question of metric audibility in Babbitt has been addressed directly by two theorists, Andrew Mead and Joel Lester.

Joel Lester. “Notated and Heard Meter” PNM, 1986, vol. 24. pp. 116-127. Lester notes the difficulty, if not the impossibility, as a performer of imparting to an audience the various set durational (distance) relationships between the notes in Composition for Four Instruments and Aria da Capo. Lester describes the duration set in Composition for Four Instruments (p. 122), and concludes that this destroys pulse and , further, (tonal?) meter itself. “…Without powerful effects on the arrival of an expected event on a predictable time-point or of the syncopated placement of strong accentuations off the bear, there is a sense in which the impulses are not moored to a larger system of measurement… I believe that the rhythmic notations of the Babbitt compositions … cannot reliably be considered accurate representations of the perceived metric structure of this music, and cannot be considered accurate representations of the perceived durations [my italics] of the individual tones.”

I agree with Lester (particularly concerning pulsations,xxiv since Babbitt does not compose “thematic event” music at t his time), but, find that I don’t require such strong medicine as Lester. The fact is that composers today use a wide variety of rhythmic styles within our notational limits, and our notation, with its inherent redundancy has the ability to allow all of them. So, specifically in Babbitt’s music I hear changes in the rate of events and intensity relating to the velocity and clusters (more on this), rather than to traditional rhythm and pulse. It is through this transformation of familiar elements that Babbitt has found a new way to use rhythm and meter, one which I call a rhythmic field.

Andrew Mead, “About About Time’s Time: A Survey of Milton Babbitt’s Recent Rhythmic Practice,” PNM 25, nos. 1-2 (Winter/Summer): pp. 182-235. Mead’s article is a direct response to Lester concerning the existence of pulse in the works of Babbitt that Lester describes in his article. Referring to the Third Quartet: “There is a single pulse maintained throughout the work, through a variety of metrical interpretations (p. 226).” Furthermore, he finds major “pulse-affirming” elements in Babbitt’s Third Quartet, i.e., the downbeat. For example, he states (p. 195), “ a high percentage of bars have clearly articulated downbeats…” On p. 227 he states “…the downbeats of many measures are clearly articulated.” This can be related to Mead’s suggestion that within Babbitt’s music there is (non-traditional?) metrical hierarchy (p. 198). “One a pulse pattern or meter is established, there is also established an a prioi hierarchy among the elements that drastically alters many of the concepts of equivalence applied to twelve-element theory in the pitch domain. This difference necessitates new ways of thinking about structures translated into the rhythmic domain.”xxv

Although I agree with Mead’s defense of Babbitt, I feel that a question must be raised: Is pulsation, or audible pulsation, necessary for musical coherence?xxvi I say no.xxvii Strong metrical regularity is not consistent with the generally acknowledged audibility problems posed by Babbitt’s Third Quartet. However, the surface of the quartet does sound sporadic and irregular (as Mead’s own example shows). If pulse or meter are so obscured by the musical surface as to be made inaudible, their traditional functionality is removed. This does not dispute Mead’s ideas about metrical hierarchy in Babbitt, but rather sows why hierarchy based on metric position has a hand in its own dissolution (as Mead says “…necessitate[ing] new ways of thinking”). It is for that reason I define rhythm as a rhythmic field, and I would suggest calling the notated meter metric classxxviii. Of course this does not preclude any composer from using traditional rhythmic means.

Tempo Factors

The quartet has only one tempo, a quarter note = ca. 72, and there are not tempo alterations such as rit. or accel. Which might alter the aural effect of pc aggregate/rhythmic field intersections. A steady rate of attack points per measure would be a minimum requirement to bring out audible changes within the rhythmic field. Obviously, if there are more attacks in faster sections, these differences would be more pronounced (Since I have no time to reference figure 1-10 Activity Levels, we must make due with the more ruthless example 5). Furthermore, the relationship between tempo and meter is analogous to the relationship of the meter and the velocity, that being one constant and one variable, constant tempo (quarter note = 72) and changing meter, constant velocity and different durations.

About the Rhythmic Aggregate

Example 6a shows the first three pc/rhythmic aggregates of the Third Quartet, and how they are translated into music surface, ex. 6B, in measures 1-18. Note that ex. 6B, in order to conserve space, is not to the exact proportions; ex. 6B is marked by measure numbers which coincide with the rhythmic aggregate on the left side, exactly, and then move to the right. Some measure numbers are added underneath certain rhythmic aggregate members as well. The eight orchestrational/registral lynes of the pc aggregate are not translated into eight dynamic lynes of the rhythmic aggregate (see left hand side of the example). The notes that are boxed are the “wrong notes” mentioned by Arnold and Hair,xxix (refer back to example 6a).


Comparing examples 6a and 6b, one is struck by the amount of repetition of the rhythmic aggregate members, harking back to my example 1-5. For instance; in rhythmic aggregate 1, time-point 4 extends past time-point 9. Arnold and Hair point out that the rhythmic aggregate unfolds at a slower rate than the pc aggregate. This is facilitated by consistent repetition of rhythmic aggregate members (and their eight related dynamic lynes) in a way not found in the PAC As Babbitt says,xxxi “Repeated time-points…must not be regarded as analogous with pitch repetitions. (Pitch repetition is not a pitch procedure, but a temporal one.) The repetitions… are analogous to the representation of a pitch class by different ‘registral’ members of that class.” It is this constant repetition of the rhythmic aggregate members by “register” that interferes with the time-points being aurally understood as distances between the rhythmic aggregate members.

Hearing the Rhythmic Aggregate

The rhythmic aggregate (as the pc aggregate) is not set out in any way that would make it an aural event. Though rhythmic aggregate 2 (6a, 6b), with its stronger dynamic package, appears on time-point 0 or m7, ff, it cannot be construed as an audible “event” because stronger dynamics have just taken place in m. 6 (that is where, rhythmic aggregate 1 ends on time-point 11 fff). The point being that rhythmic aggregate 2 is not differentiated by dynamics or the previous orchestration which continues. Further, note that in m. 11 aggregate 3 overlaps aggregate 2 so there is a smooth transition between them.

Aural Changes within the Fields-Clusters

I propose that it is possible for several different lynes of the serial dimensions, to come into such close proximity that they seem to work in concert to create an audible “event,” which I call a “cluster.” I contend that it is not any one event or lyne, or any coordinated predictable rhythmic events that is important, but rather a group of related events or lynes that become referential entities, that is they stick out.

Just as the change in metric class and velocity create contrasting elements within the macro form, so do clusters of high or low intensityxxxii within fields become apparent on the small scale. For example, the opening of the Third Quartet (mm. 9-11 as far as its time-point 4) may be analyzed as a cluster of high intensity. Intensification of dynamics is accompanied by an increase in the time-points attacked (unison count as one) per measure (see example 7), followed by a cluster of low intensity.


Clusters can appear on the pc plane as well. Even though repetitions of the twelve notes are hidden by continuously varying their register and orchestration, even at the start, Babbitt’s Third Quartet has repetitions of pitch classes in the same register, by the same instrument, an with the same orchestration. These repetitions are mostly hidden by the complex surface, yet, when the surface clears, they can become audible as a cluster of p(itch)/r(egistral) reiteration (see examples 8 – Violin II, mm. 51-54; Violin I, m. 61, mm. 70-71, and mm. 190-192.)


The Time-Point Set


If we look a little closer at the time-point system, we can examine the eight time-point (set) units themselves, and see their inter relatedness. This explains why, in example 9, I have organized the time-point set into “rhythmic levels,”xxxiii or by 2/1 relationships (vertically), and by sequential “rhythmic transpositions” (duple, quintuplet, triplet, septuplet, left to right).xxxiv Further, each velocity is related to a particular metric class, as I show in example 10.

For ease of discussion, I have translated the time-point set of example 9, using a “movable do” concept in example 10. I translated the thirty-second note to a0, since it is the first time-point subdivision (“a” for first, and also since it is in this work, the largest durational type), and “0” for the smallest level. For those reasons I have given it priority for this quartet (other works would have other criteria. I call the sixteenth note a2 (or “a” twice as slow(, the eighth note a4 (four times as slow as a0), all or these having the 2/1 “rhythmic level” relationship, and, for that reason, all appear in the same column in examples 9 and 10. I call the triplet thirty-second note c0, the triplet sixteenth, c2. Etc. Again, this shows the relationship between the “rhythmic transpositions”: slower to the left, faster to the right – sixteenth note, sixteenth note quintuplet, sixteenths triplet, sixteenth not septuplet, or a2, b2, c2, d2. These relationships have some meaning as Babbitt uses b2, c2, and d2, all in the same 2/4 metric, class showing a close association of the sixteenth note level.


However, a few more aspects of the relationships between the time-point set units need answers. Are all these units expressions of a single rhythmic idea? To argue this point (affirmatively), I would point out that the sixteenth note (a2) is transposed three times to b2, c2, d2, and it is the only time-point transposition to appear on rhythmic levels b and d. It also appears in three different rhythmic levels, and has a habit of showing up where it “doesn’t belong,” such as mm. 307, 434, etc. On the other hand, it is true that the thirty-second note a0 is our first time-point unit.


These time-point units are only a “tool box,” and not the only durations in each rhythmic field. The time-point velocity only controls attack points, except when a duration is equal to a time-point set unit. The only rule is that the point of attack and the cutoff must coincide with the “minimal duration.” For example, Violin II, mm. 75-78, starts as an eighth and ends as a triplet. There is, in fact, a much larger group of related durations which can be of a 2/1 relationship (a0, a2, a4, a6, a8), or various additive types such as dotted figures. The only consistency is to be rhythmically erratic, i.e. sporadic. The rhythmic sections are, therefore, an accumulation of durational “fields” of that particular time-point set unit.


Each rhythmic field is set off by the boundaries created by the change of metric class, with the exception of the 2/4 metric class in mm. 78-165, and 366-242 which has groups of 14, 15, and 21 – which are present in this composition, and fit into this system by using mod-12 for the extra notes (see Mead). In many cases, there are notes held over sections and sustained (mm. 52-55, 75-78, and 265-66), making the rhythmic changes around them more audible. Changes of “rhythmic fields” always occur on the level of the measure and never in the middle of a measure, even in 2/4 sections (mm. 81-165, and 366-424).

Durational “Combinatoriality” – A Speculation

A time-point velocity has a dual character in that they are an attack point with a variable release (within limits, yet, they can also be interpreted as a “minimal” duration, equivalent to a time-point set unit. Therefore, it is possible to combine different velocities of various durational compatibility. On the surface then, time set velocities can combine in two different ways: 1) through “durational Combinatoriality,” or 2) “points of intersection.”

Durations combinatoriality refers to the fact that some time point velocities can combine easily with larger ones. For example, c2 and its metric class can combine with a4 on its time-points 0, 3, 6, and 9. Babbitt exploits some of the “combinatorial” possibilities by extending the eighth notes from the 6/4 section into the triplet 2/4 section – they appear after one bar in mm. 79-82 ff. A further example of this kind of durational combinatoriality occurs in a surface “bridge.” What I call a bridge is an area between two metric classes where neither rhythmic field is articulated, but some larger “combinatorial” duration is used. This is what happens at mm. 162-165 where the septuplets are not stated, but the combinatorial quarter is (this also occurs in m. 226, and m. 461).

It is also possible that a velocity smaller than a rhythmic field’s minimal duration can appear. I will refer to the metric class time-point positions where different (smaller) time-point set velocities can cohabit as “points of intersection” These would occur on what used to be known as “beats.”xxxv all of the different time-point set units, as attack points, can combine on “points of intersection” of any metric class; various kinds of articulation could make the durational difference inaudible, or allowable. This can explain the occurrence of two different time-point velocities in mm. 295 and 307 (c4 and a2), and in mm. 434-461 (a2 and a4).

I have found that Babbitt does take advantage of the possibility of duple and triplet subdivision in a single metric class.xxxvi This can happen by the methods I just described. It becomes most pervasive in the final 2/4 section where a2 is “rhythmically combinatorial” with c0, in the two bar group mm. 495-496. In fact, sixteenths continue all the way form m. 434 into the final section where the last attack is on an eighth-note pizzicato. The pervasiveness of the duple proportion, in what should be a totally triplet field, could be seen as a rapprochement of the different metrical proportions three and two.

Different time-point units always occur in succession, not simultaneously, except in m. 307 where triplets and sixteenths are used. For this reason, when I first examined the work I suspected that the form was ABA’ Coda, because of the interpenetration of the sixteenth notes, which could be interpreted as velocities into a triplet field, rather than what is now shown n my chart, figure 1-8, ABA’B’. It did not occur to me then, as it does now, that the return of B could also function as a coda.


My chart, figure 1-9, shows 16 subdivisions related by the interaction of the velocity and metric class, “rhythmic field.” I divide the work into four parts ABA’B’ (figure 1-8). the sixteen metrical divisions note the division of the 2/4 meter when the velocity changes within it: in section A, mm. 78-131, 132-161, (162-5 overlap), and in its analogous place in the section, mm. 366-400, 401-424. This is balanced by two isolated ¾ measures, 307 and 337, which do not assert their own time-point units. Direct contrast between my surface content method, and Arnold and Hair’s rhythmic sections can be seen in my and their example 11 (I show my idea of macro form) to the left of their example. I will now refer to my chart, figure 1-8, and 9.

The A section is chosen for several reasons, not the least of which is that it repeats, in a compressed form, with a change of order. In my chart, “asserted form/fields,” the A section, (mm. 1) a0-a4, is followed by c2, and d2. In A’ section (m. 338) a0 is followed by c2 (an inverse relation of a4-c2 or the A section), then b2, an element missing from the A section appears, followed by a4, completing the return of the A section, in the A’ area. The metric class follows the 2/4 metric class (mm. 425-) in the A’ section, and within those 2/4 sections, b2 (quintuplets) replaces d2 (septuplets), mm. 401-424. This is done, I feel, to create a slight winding down of intensity, because the quintuplets are less active than septuplets. I would also point out that M. 338 is as close to a thematic return (the three note motive in the Cello repeats a similar formation in Violin II, m. 1) as one can get in this kind of music. Since the d2 figures are very active (14 attacks in m. 337) just before the return to a0 (m. 338), the thirty-second notes are brought into strong relief and, therefore, create audible form.

Why not take the A section into mm. 166-199, where I start the B section? Because the “bridge” area (mm. 162-5) sets this off as a new sections. The “bridge” implies a change in the activity level, a slowing down of the surface activity. My B section (m. 166) starts with a repetition of the A section (see figure 1-9, subdivisions of the asserted form), a4-c2, form A, becomes a2-c4 to start the B section. Further, in the B’ area, another bridge occurs in the analogous place, m. 461. Obviously, these two sections are being related. In the B section, a2 is followed by c4, in the B’ section (mm. 461-), a2 exact repetition of metric class), is followed by c0, creating an exact relationship with the analogous B section. The B section also has other symmetries within it. In mm. 266-306 we have a rhythmic transposition of the opening a0-a4 as c0-c4 (see square brackets in figure 1-9, subdivision/fields.

Metric class is sometimes used as a transition: the two 3/4 bars, mm. 307 and 337, appear as isolated metric classes, as the time points from the previous section continue. Therefore, they mediate between different rhythmic fields, as they have no time-points of their own.


Many persons of note have spoken of the difficulties of hearing Babbitt’s music. To explain these aural difficulties is not the same as finding fault with the music. Rather, observing these difficulties can lead to what Mead call “new ways of thinking” about the musical (rhythmic) surface. The sporadic, but not static, nature of the musical surface sets the stage for the changes in velocity to become important “events,” because other competing factors (metric position, accents, motives, thematic events) are minimized or absent. Even so, within the rhythmic fields, clusters of intensity, and reiteration reveal the richness of Babbitt’s musical surface. So, I must argue that it is the interaction of these “rhythmic fields” with the pc arrays which create an “asserted” from, asserted by the contrasting and repeating intensities of the fields themselves. Since the Third Quartet is governed by a constant tempo, the level of activity change within the contrasting rhythmic fields and their repetition become the overriding determination of form.

Appendix 1

Questions of Choice

Some of the surface effects are directly related to the chosen set dimensions, that is the pc/rhythmic aggregates and their attended lynes (levels). Other effects are not. For example,, and not in an y particular order:

  1. The number of musical repetitions of each member of the rhythmic aggregate.
  2. The number of attacks per measure.
  3. The number of pitch classes (or instruments) that may enter at a given time-point, (for example 3 instruments enter point 0 in m. 1, 2 instruments in point 0 m. 2).
  4. Silence and rests
  5. The time-point set
  6. Meter and its repetition
  7. coordination of dynamics


Therefore, it can be argued that within the serial elements of the set(s), some dimensions are at the composer’s discretion. Even if it were discovered that none of the above are in fact discretionary, that would not alter their effect, which is: that it is only through the interaction of the rhythmic/pc aggregates and the above “independents,” that form becomes clear. As we will see in the concept of “clusters.” For example, the first cluster of intensity is still enhanced by a “wrong” note in m. 10. This is the sort of thing that Ralph Shapey would call a “perversion” (not to be confused with those “degenerate hexachord”). The point is, this is not composition by “number.” Aspects of choice, within the set elements remain.xxxvii


Perhaps a clue for these procedures can be found in Arnold and Hair’s statementxxxviii “The the rhythmic domain is independent, the pitch domain remains more important,” i.e. the rhythmic domain requires a less rigorous (serial) approach. Yet, the effect of the rhythmic domain is crucial to the form, otherwise the music would be just a collection of associated, but unrelated, orbits.” As John Rahn syas “anyone who has “12-counted” any Schoenberg must be aware that ht e”row” is not the composition.” In complex “serial compositions the importance of details, and inconsistencies cannot be stressed enough, for that is the difference between a work of art and an analysis.

example 11


Appendix 2

The following is a list of other related papers:


William Marvin Johnson. “Time-Point Sets and Meter,” PNM, Fall-Winter, No. 23 (1984) pp 278-293.

This is an excellent commentary on Babbitt’s “Twelve-Tone Rhythmic Structure and the Electronic Medium.” Johnson does not directly address questions of audibility. He states “In this paper I propose no fundamental changes in Babbitt’s general conception of rhythmic organization.” However, he does offer two compositional extensions: “1) through the interpolation of additional time-points between consecutive members of a given time-point set by repeating the prior of the two time-points modulo, an inferred beat structure, and 2) by establishing that a C12 partition may be perceived as a projection of a partition less than twelve, or that a partition less than twelve may be perceived as a contraction of the C12, and that either may serve as a representation of the original time-point set.”


John Rahn. “On Pitch and Rhythm: Interpretations of Orderings of and in Pitch and Time,” PNM, Vol. 13, No. 2 (1975), pp. 182-203

Rahn’s article is a recipe for his composition Peanut Butter Defies Gravity (a song which is published with this article). Besides the added “wrinkle” of register to the use of time-points, the interpretations of orderings refer to the fact that sets, modulo 12, can be subjected to a multiplicity of different interpretations. Referring to P. 186: “The…interdependence…of interpreted orderings is the fact that any ordering of pitches in time inevitably implies some ordering in time-points in pitch, and conversely any ordering of time-points in pitch implies some ordering of pitches in time.

Rahn devises four levels of order interpretation. The following refers to Rahn’s ex. 11 on p. 187:

T(ime) C(lass)/T(ime) the time-point attacks: 0, 1, 3

P(itch) C(lass)/P(itch) pitches in registral order, bottom to top: 0, 3, 4

P(itch) C(lass)/T(ime) the notated pitch succession: 0, 4, 3

T(ime) C(lass)/P(itch) time-points in registral order, bottom to top: 0, 3, 1

On p. 184 Rahn states: “…every sound has associated with it as least one pitch, but at least two time-points – the time of attack and release” [i.e., time-points as attacks and durations]. The meaning of rhythm, vis-à-vis the time-point system, for his song is clear in that the time-point unit is taken as a constant durational pulse. Therefore, Peanut Butter Defies Gravity” is much closer to Babbitt’s earlier music, sch as Three Compositions for Piano, in that it has a steady pulse, and motor rhythm in mm. 1-3, and mm. 8-48.


Peter Westergaard. “Some Problems Raised by the Rhythmic Procedures in Milton Babbitt’s Composition for Twelve Instruments,” PNM, Vol. 4, No. (1966), pp. 109-18.


Westergaard explains Babbitt’s use of durational sets of the Composition for Twelve Instruments. In example 13 (p. 117), he raises questions of perception: though we might be able to hear mod. 12 pitch relationships, can we “…hear durational relationship mod. 12…” On p. 118 “I see no way for the ear to distinguish attacks which define durations for P0 and those which define R12. Thus, I see no way for the ear to perceive either order or content.”

This article concerns music written before the time-point system was invented. As he states, “….this problem [i.e., the relationship between durational sets and metric class] … has since been solved by Babbitt in his more recent procedure in which metric position corresponds to pitch number, and hence, duration to interval” [the time-point system].

Joseph Straus. “Listening to Babbitt,” PNM Spring-Summer 1986, vol. 24, No. 2, pp. 10-25.

Straus states that “The relationships described by most analysis of Babbitt’s music are hard to hear.” Straus, therefore, presents a guide for Babbitt String Quartet No. 2, not using the aggregates or sets relationships, but by demonstrating the surface activities that are relatively easy to hear, such as register, dynamics, rhythm, articulation. He shows how they illuminate a “rich network” of surface activity.

iStephen Arnold and Graham Hair, “An Introduction and Study: String Quartet No. 3” Perspectives of New Music 14, no. 2/15 no. 1 (1976, 155-86 See p. 158 In order to avoid confusion with tonal rhetoric, lines of music “are described as layers to avoid any connotations that the word ‘line’ might invoke.”

iiOn TMK(S) Turnabout, performed by The Fine Arts Quartet, Vox Productions, Inc., 1972.

iiiArnold and Hair, figure 1 gives a list of all the aggregates, pp. 173-86

ivFor more information concerning arrays, see Milton Babbitt, “Since Schoenberg,” PNM Vol. 12, pp. 3-28

vOn TMK(S) Turnabout, performed by The Fine Arts Quartet, Vox Productions, Inc., 1972.

viSee Arnold and Hair. The pitch numbers are translated as any of the twelve time point positions; i.e., pitch-0 is time point-0, etc.

vii“Noted and Heard Meter,” Joel Lester PNM, 1986, Vol. 24 pp. 116-127. Referring to an earlier work: Babbitt’s Composition for Four Instruments (1948), “ Shorter rhythmic values do not occur in extended series; as a result, these pulses are established on l intermittently’… Further, “…without the powerful effects of the arrival of an expected event on a predictable time-point, or of the syncopated placement of strong accentuations off the beat, there is a sense in which the impulses are not moored to a larger system of measurement.” Also see Westergaard, Peter. “Some Problems Raised by the Rhythmic Procedures in Milton Babbitt’s Composition for Four Instruments.” PNM, Vol. 1 (1966) pp. 106-18 for an example of Babbitt’s Three Compositions for Piano, where pulse by this definition (extended use of a single rhythmic value or modulo) does occur.

viii On TMK (S) Turnabout, performed by The Fine Arts Quartet, Vox Productions, Inc., 1972

ixFor more about the “audibility” question in Babbitt, see Lester, Westergaard, and Straus

xPaul Griffiths, “Serialism,” The New Grove Dictionary of Music and Musicians, edited by Stanley Sadie, vol. VII. London Macmillan and Co., 1980, p. 167.

xiAs Babbitt says in his, “Twelve-Tone Rhythmic Structure and the Electronic Medium,” Perspectives of New Music 1, no. 1 (Fall-Winter 1962) pp. 49-79, “a Twelve-Tone ‘rhythmic system’ can be viewed either as a reinterpretation of pitch numbers so as to assure isomorphism between the two systems [as in the third Quartet] or as assigning temporal [durational] interpretations to the uninterpreted terms of the finite numerical equal difference structure [the time- points] of which both the pitch and rhythmic systems will be exemplifications.” See his examples 13-17 etc. Commentary articles (Lester, Westergaard) have generally focused on the durational sets and pre-time point music.

xiiSee Westergaard

xiiiThe time-point system divides a measure into twelve attack points, which has a variety of compositional uses. See Babbitt, “Twelve-Tone Rhythmic Structure and the Electronic Medium,” Perspectives of New Music 1, no. 1 (Fall-Winter) 1962; 49-79

xivMore program notes from record, “there is a single tempo to the work, within which changes in velocity – the number of attacks per unit time… [I understand velocity to mean the rate of the time point unit.]

xvThis kind of contrast has been noticed before by Peter Wetergaard in his “Some Problems Raised by the Rhythmic Procedures in Milton Babbitt’s composition for Twelve Instruments.” PNM, Vol. 4, no. 1 (1966) pp. 109-118. “Note, however the prevalence of the eight-note durations which provide an element of contrast between this section, and, say the one combining [the two row forms] I3 and RI0 with its prevalence of sixteenth note durations” viz., his footnote 13.

xviSee, Westergaard, Peter. “Some Problems Raised by the Rhythmic Procedures in Milton Babbitt’s Composition for Twelve Instruments.” PNM, Vol 4, no. 1 (1966) \: 109-18, his footnote number five, pp. 113

xvii The steady, pulse-like use of a single rhythmic unit (or several) to create subtempi is to be found in many of the works of Elliot Carter, and Henry Weinberg (such as the string quartets).

xviii Rhythmic, because they are not just made up of velocities, but also of durations, not durational sets (see Westergaard and Lester) and other factors such as dynamics and other levels of activity.

xixArnold & Hair’s ex. 10 p. 170 shows that some aspects of duration, and therefore accents, are controlled serially as part of the 8 registral/orchestrational levels, that is Arco & pizz-which have a direct effect on duration. Further, each notes intensity is further expanded when combined with the rhythmic aggregate, and dynamic lynes. However, a look at any measure will show that not all pizz’s are short, not all arco’s are long (not to mention places where longer notes are out accented by surrounding shorter ones, say m. 16-Violin I, 19 Violin I, or m. 62-3 Viola). To add to our dislocation of dynamic accent, occasionally there is a coordination of dynamics between different time-points whose durations overlap, and sometimes not, for example in m. 1, Vl I is forte all the way through even as other dynamics occur, yet in m 2, its ppp is coordinated with a cres. To p on time-point 9, with the Viola.

The number of instruments that enter on any time-point will also effect our perception of accent, and this too is constantly varied. For example in m3, will not time-point 4, because of the 3 entrances and higher register (as well as the pizz’s) seem more accented then time-point 0, or using the same rhythmic aggregate; the fff in m. 4, time-point 10 seems stronger than the fff in m. 4, time-point 10 because of the number of entrances.

xxFor other views of audibility of pulse, see Mead, Andrews, Mead, “About About time’s Time: A survey of Milton Babbitt’s Recent Practice,” Perspectives in New Music 25, nos. 1-2 (Winter/Summer 1987); 182-235. In his footnote .H no 16. pp. 233-4, for his ex. 10, Mead replies to Joel Lester concerning what is aurally perceivable in Babbitt’s music. Mead states, “I hear the “broad pulse,” the “big duple division of the broad pulse.” Lester contra-wise states: “…without the powerful effects of the arrival of an expected event on a predictable -point or or the syncopated placement of strong accentuations off the beat, there is a sense in which the impulses are not moored to a larger system of measurement.”

xxiMilton Babbitt, “Twelve-Tone Rhythmic Structure and the Electronic Medium,” Perspectives in New Music 1, no. 1 (Fall-Winter 1962) 49-79.

xxiiThis is a question best answered by Milton Babbitt as he could have made the notated meter non-repeating (for example ¾ could become 6/8 and still contain the same time-point velocity) if he had so desired.

xxiii ibid.

xxiv However, the coordination of large scale events and points of arrival seem to be related to the rhythmic, rather tan the pulse, sphere.

xxv Mead’s example 11 (p. 198), shows that his metrical hierarchy for Babbitt, is very close to traditional tonal metrical hierarchy in that it is based primarily on metric position or “beats.”

xxvi Obviously, a conductor must impart a “beat” to the performers. As to whether it is a musical requirement or an artistic necessity for the audience to be made aware of that “beat” is open to question.

xxvii Historically, we know that musicians only gradually accepted the idea that non-tonal music could be coherent without being tonal. The same will no doubt be true for the rhythmic sphere.

xxviii There might be some precedent for this term in Babbitt’s idea of “octave equivalence of the measure” since; ¾, 6/8, and 12/16 would imply different traditional metric patterns, but in the time-point system could imply the same velocity (a sixteenth note).

xxix Arnold and Hair a Study…, in reference to footnote 8, p. 168. Referring to a missing mute: “It seems to represent a more problematic inconsistency than the occasional rhythmic/dynamic discrepancy, such as occurs on the first page of the score [italics min]. It would be helpful to have the composer’s view of these questions.” In a question of theory and practice, at least for me, the composition and the composer are the final arbiters. Also,, see Babbitt’s criticism of the European serialists for the “harmony by fortuity,” in “Some Aspects of Twelve-Tone Composition,” in The Score and IMA Magazine (June 1955).

xxx As Arnold and Hair state: “…the rhythmic aggregates unfold at the ratio of exactly 1:3 in relation to the pitch aggregates, as far as pitch aggregate 27/rhythmic aggregate 9 (ending in bar 53); thereafter, the ratio increases immediately to under 1:2…”)

xxxi Viz., footnote 11

xxxii For more clusters see: a cluster of low intensity mm. 166 (time-point 11), – 172 (time-point 7) (alterations of clusters mm. 162-66). Generally a change to a slower velocity would mean less intensity. However, the change in field from 3/8 to 6/4 has a change in mm. 51-54, time-point 4, from a cluster of low intensity to high intensity forte mm. 54 – time-point 5, this field change is further marked, in mm. 54-55, by a crescendo, decrescendo (dynamic) imitation Violin I and Violin II, and continues into mm. 57-58. (Also see mm. 394-400, low velocity to high velocity mm. 401.)

xxxiii The idea that a 2/1 relationship between rhythmic values represents an expression of a single rhythmic idea, is similar to Henry Weinberg’s idea of “rhythmic registers” related to a single tempo.

xxxiv It is possible that a chart of relative rhythms (or tempi) like the one shown in example 9 might be of use in understanding other recent 20th Century rhythmic phenomena.

xxxv For more on how beats relate to the time-point system see William Marvin Johnson, Jr, “Time-Point Sets and Meter,” PNM, Fall-Winter, No. 23, 1984, pp. 278-293.

xxxvi Further examination for the purpose of defining all the various properties of durations and the confluences with the time-point units will not be carried out here, but I will point out all that actually occur in the music.

xxxvii Arnold and Hair, viz., footnote 1, in reference to their footnote 8, p. 168, refer to a missing mute: “It seems to represent a more problematic inconsistency than the occasional rhythmic/dynamic discrepancy, such as occurs on the first page of the score. It would be helpful to have the composer’s view of these questions.” In a question of theory and practice, a least for me, the composition and the composer are the final arbiters. Also, see Babbitt’s criticism of the European serialists for the “harmony by fortuity,” in “Some Aspects of the Twelve-Tone Composition,” in The Score and IMA Magazine (June 1955).

xxxviii See endnote #1.