Varèse's Hyperprism and Penderecki's Polymorphia

Christopher Meister

This paper concerns itself with comparing two different pieces of twentieth-century music as self-contained and internally-referent information structures. To that end, the following questions will be addressed: how does one part of a piece compare with another; where are the significant compositional similarities, and in what ways do these similarities fashion themselves into larger structures and organizational patterns? The pieces to be examined are Hyperprism, by Varèse, and Penderecki's Polymorphia; Hyperprism will be discussed first, then Polymorphia, and as the latter is analyzed such comparisons will be made between the two as seem pertinent.

The Varèse contains a large percussion complement, and the present analysis concerns itself with the pitched instruments only. In order to examine briefly their harmonic content, this paper borrows from Larry Stempel's 1979 Musical Quarterly article "Varèse's 'Awkwardness' and the 'Symmetry in the Frame of Twelve Tones'". According to Stempel, much of Hyperprism's harmonic structure may be explained as one kind or other of chromatic segment. The simplest and least ordered of these segments are shown in Examples 1a and b.

Example 1a and b: Varèse, Hyperprism, chromatic segments

Within the overall category of chromatic segments, special treatment appears to have been reserved for chromatic trichords whose middle pitches have been displaced either up or down an octave (see Examples 2 a-e). Frequently, "up" and "down" displaced trichords will be played off against each other. Notice that this "updown" play creates a [1] loose sort of symmetry.

Example 2a through e: Varèse, Hyperprism, chromatic segments

Despite the considerable gains towards understanding Hyperprism that this harmonic analysis offers, it fails to address the matter of the composition's non-amplitudinal "dynamics." The remainder of this paper's first half sketches out a model of Hyperprism's behavior, which might be expanded to include Stempel's pitch work. This model divides Hyperprism's musical activity into five "eventcategories." (Example 3a. In this and similar examples, temporal alignment is represented vertically, and measure numbers appear above the topmost staff.) The five categories are 1) piccolo (flute)/clarinet, 2) filter, 3) tremolo/trill, 4) brass, and 5) solo. Each of these events cuts in and out of the sonic collage fairly independently of the others. Although Varèse is not particularly meticulous about an appearance's taking up the exact musical gesture that its previous disappearance had left off, the continuity is nevertheless obvious, and each event has its own [2] formal-dramatic pattern of behavior.

Example 3a: Varèse, Hyperprism, mm. 1-20 Event Relationships

b) c)

Example 3b and c: Varèse, Hyperprism, Example 3c: Varèse, Hyperprism, mm. 28-30, mm. 47 -48. Exploration of New Relationships Maximum Concrescence

The event-categories, however, do not behave in solely independent fashion: their primarily linear progressions are to some extent influenced by an interactive inclusional association among them, whereby a musical moment will appear simultaneously in more than one category (see example 3a, in which dashed lines indicate this inclusional associativity). This occurs when a section of the music contains enough traits of two (or more) categories to be included in both. The ramifications of this multiplicity of a gesture's interpretation are most profound near Hyperprism's beginning, for it is here that a single event, the filtered one, generates most of the other categories: for instance, the filter event in measure 4 begets first a constituent of the brass event, then in measure 6 the tremolo event, and later the piccolo/clarinet event in measure 14. The piccolo/clarinet event, which is itself a second-generation gesture, fathers the initial solo event, which begins in measure 18. Following this initial generation of material, the music explores new associations among the event-categories. For instance, the tremolo of measure 17 may be described as the brass gesture below and the piccolo/clarinet gesture above finding a new way of relating to each other. (So far the brass and piccolo/clarinet categories have found common ground through only the filter event at measure 14, not through the tremolo/trill).[3]

A further example of Hyperprism's concrescence may be seen in Example 3b measures 28-30, during which three separate events converge into one.This begins with the piccolo/clarinet event meta-morphosing into a filter event with the addition - on the clarinet pitch concert B-flat - of first a muted trumpet in measure 29, and later a non-muted trumpet in measure 30. This entire sonority (piccolo, piccolo clarinet, two trumpets) remains on the same pitches, but transforms itself into a tremolo event at the fermata of measure 30: all three categories, the filter, the piccolo/clarinet, and the tremolo, have at this point converged into a single gesture. The most extreme instances of interrelation among the categories are those at measures 47-48 (Example 3c) and an identical but longer passage at measures 56-59; it is here that gestures belonging to the piccolo/clarinet, filter, and brass events achieve an extraordinary level of concrescence. This happens in that together they create the impression of a single, animated texture, and each of the categories' gestures co-exists with the others in a state of near-equivalence. All of the gestures share enough similar traits to be included within the tremolo/trill category in example 3c. But, curiously, the dissipation of these categorical characteristics throughout the texture at these two moments of maximum convergence appears to have concomitantly induced an exhaustion of the will to concrescence, for after these two moments of maximum convergence the number and level of interrelationships drop abruptly. (See figure 1 which is a graph of the form of Hyperprism and figure 2 which is a graph of the number of associations per convergence.)

This model of inclusional convergences contains a place in it for the augmented triad begun by the horns in measure 15. This is an awkward harmonic gesture to explain, in that it does not fit well into the predominantly chromatic-segment harmony. However, if harmony is considered to exist within Hyperprism's inclusionallyassociative fabric, one might argue that this augmented triad's abruptness is buffered by the ensuing trumpet entrance in measure 16. This occurs in that the trumpets play only one major third, half as many as the horns, and they also repeat their interval, repetition being one of the brass event's characteristics. This presentation of the trumpets' major third (enharmonically spelled as a diminished fourth) more closely approaches Hyperprism's compositional norms than did the horns' augmented triad: even a disparate [4] harmonic undercurrent is absorbed into the mainstream of the compositional flow.

Figure 1: Varèse, Hyperprism, Form

Figure 2: Varèse, Hyperprism, Event-categories incorporated per convergence

a)b)

Example 4: Polymorphia, Mono-intervallic and Symmetrical Pitch Formations: a: no. 24, Quarter-Tones. b: no. 31, Half-Steps. c: no. 7, Three-Quarter-Tones. d: Whole-Tones. e: no. 27, Perfect Fourths

and Canadian agent for Herman Moeck Verlag, Celle.

Example 5: Polymorphia, Symmetrical Non-Mono-Intervallic Pitch Formations. a: no.59, b; no.7

(c) Penderecki POLYMORPHIA (c) 1963 by Herman Moeck Verlag, Celle. (c) Renewed. All Rights Reserved. Used by permission of European American Distributors Corp., sole U.S. and Canadian agent for Herman Moeck Verlag, Celle.

Example 6: Polymorphia. a: no. 37, b: no. 59, c: no. 60, d: no. 67. Polymorphia's Only Non-Symmetrical Pitch Formations

agent for Herman Moeck Verlag, Celle.

To sum up the discussion so far: Hyperprism operates as a collage of musical gestures belonging within one or more behavioral categories. Judging from the concrescionally-depletional effect that the two areas of maximum convergence created for the remainder of the piece, it seems reasonable to hypothesize that any gestural confluence - a musical moment's belonging to more than one category at a time - may have some sort of lesser and more localized impact upon the course of the music. In Hyperprism Varèse pared the function of harmony in order to imbue the non-pitch elements of timbre and articulation with an increased precedence in the compositional organization. Polymorphia exhibits the same emphasis of timbre and articulation at the expense of traditionally pitch-oriented compositional processes. However, there exist significant differences between the two pieces' material and organization. For instance, Hyperprism's occasional employment of "up-down chromatic [5] trichords" are its sole display of near-symmetrical chord structures. By contrast, Polymorphia's chord forms are virtually all symmetrical, some mono-intervallic (Example 4), some not (Example 5). Very few exhibit no symmetry (Example 6). In addition, Polymorphia's harmonic structures are drawn from a wider intervallic field than [6] Hyperprism's (see Examples 4 - 6).

But Polymorphia's proliferation of intervallic structures is only one facet of its compositional network, in which the harmonies are treated as detachable entities to be mapped onto other, likewise detachable and mappable articulations, timbres, and shapes. For instance, a fair amount of information is exchanged just among shapes alone: the bottom of figure 3 presents two prime shapes, "band" and "glissando," and all the other shapes in this section (glissando within a band, expanding band, and band upon glissando) are built from these two prime shapes. The top of Figure 3 is a simplified graph of where these shapes occur in the total pitch-space and when in time. And the middle portions of Figure 3 tell what timbres and intervallic structures are present at what time.

Figure 3: Polymorphia., graphic reduction of no.s 1-22

Besides considering these shapes to exist within a "bandness" or "glissness" scale, one might also view some of them as generating others through a mapping process: for instance, the static band at number 5, if mapped onto the shaped glissandi at numbers 11 or 13, yields the banded glissandi at numbers 15, 18, and 19. This mapping works in a small way within the shape field (or parameter), but if one were to take the mapping procedure outside of any one particular parameter and apply it between the two parameters of shape and intervallic content - to cite but one example - the mapping applications begin multiplying themselves. For instance, an interval may be cross-mapped from one shape to another; note that the canonic glissando at number 13 and the banded glissando at number 15 both share a whole-tone harmonic structure. [7] Conversely, one shape may have more than one chord-form mapped onto it.And when timbre enters into the picture, the mapping possibilities begin multiplying quite [8] rapidly.

This appears to be a case of freer employment of a wider field of harmonic, timbral, articulatory, and even shape material than was found in the Varèse. In Hyperprism, despite any transformations created by the inclusional associativity, one could rightly speak of the "identity" of an event-category being maintained throughout, largely due to the narrower field of material and the more linearly-invariant mapping associations; but in the Penderecki, the personality metaphor seems inappropriate: instead, parametric phenomena "glom on" to each other, creating global units - bands and glissandi - recognizable in their macro-structure, but complicated to the point of being bewildering in their compositional interrelationships. Before proceeding to the remainder of Polymorphia, a few comments regarding #'s 1-22 are in order. First notice that all of the sounds are long, bowed ones: none are short. Second, all are pitched, either definitely or indefinitely. Third, within the field of intervals, all are used with the exception of the quarter-tone, major and minor thirds and perfect fourths. And finally, from the beginning to #22 more and more sounds are piled on.

Significantly, from #'s 22-44 this whole situation reverses itself (Figure 4). First, all the sounds are short ones, save those few briefly hanging over from the previous section; second, from #'s 22-38 the perfect fourth and quarter-tone, two intervals virtually ignored in the first section, play a major role; and third, unpitched tones are introduced alongside the pitched tones for the first time, thus creating a pitched-versus-unpitched argument which extends from #'s 22-44. Even though this whole section in figure 4 is limited to only short sounds, much of its interest is the result of the course of this pitched-unpitched argument: from #'s 31 to 37 the pitched side of the argument gains ground, to the point of eventually saturating both pitch-space and intervallic content between #'s 37 and 38. (See Example 6a.) But afterwards, from #'s 38-88, only the unpitched sounds are used, these becoming increasingly removed from pitched territory, to the point of not being produced on the instrument at all.

Figure 4: Polymorphia., graphic reduction of no.s 46-67

Another facet of Figure 4 which bears discussion is the existence within it of two successive accretions; the first is the pitchedunpitched argument from #'s 22 to 38, which collapses, making way for the second, totally unpitched accretion from #'s 38-44. Similarly, the initial section, #'s 1-22 of figure 3, may also be considered an [9] accretion and subsequent collapse. Since each of these large-scale accretions is built from its own set of musical characteristics, it becomes clear that a very few compositional operatives define the conditions for a considerable stretch of music. Were one to compare Penderecki's accretions with those in the Varèse, he would find that in Polymorphia, they occur for minutes at a time and are the main sections of the piece, whereas in Hyperprism the accretions appear to function as signposts of the [10] larger sections, analogous to codetta and coda of sonata form.

This last parenthetical remark raises the question of form in Polymorphia. Examination of the "articulatory" section of figure 5 shows that all the sounds from #46 to the end are long ones, as they were between #'s 1-22. Additionally, timbres, shapes, bands, and glissandi similar to those of figure 3 are developed throughout in figure 5; from this it seems safe to infer that Polymorphia's form is essentially an ABA in its large-scale outlines.

But besides being a recapitulation of Figure 3, Figure 5 is also a summation of the previous discourse. Note that the intervallic content of Figure 5 incorporates each of the two previous sections' sets of intervals in addition to adding some intervals of its own: clearly, the entire harmonic content of the piece is being summed up right here. And besides functioning as a harmonic summation, this third section additionally incorporates the long-versus-short and pitched-versus-unpitched arguments of the two preceding sections.

Figure 5: Polymorphia., graphic reduction of no.s 46-67, bridge pitched and unpitched sounds.

This incorporation, or "synthetic" phrase, takes place near the very end of Polymorphia in the following manner: first, note in the lower half of the timbralarticulatory portion of figure 5, that at #63 the instruments begin playing a behind-the-bridge "ugly creaking" (Penderecki's term). This sound is a continuous series of short, individual "creakettes," and as such this unpitched sound incorporates for the first time both the long and short articulatory antitheses. Immediately after beginning this behind-the-bridge creaking, the performers are directed to play on the tailpiece and on the bridge itself, thus producing the first truly long, unpitched sound: the pitched-unpitched, long-short events are being mapped onto each other in new combinations. And these new mappings find their most inclusive expression in the creaking bands begun at #62: these incorporate both long and short sounds as did the behind-thebridge creaking; and since their inharmonic (unpitched) content partially obscures the tones' pitches, these quasi-pitched, creaking bands additionally bridge pitched and unpitched sounds.

Examination of Figure 6, a three-dimensional model of information exchange in Polymorphia, provides an overall sense of how information functions throughout the piece. To explain the model briefly: time runs from left to right, and rehearsal numbers are placed below the X-axis; the Y-axis is divided into "long" and "short" halves; and the receding Z-axis is divided into pitched and unpitched halves, the unpitched portion being denoted by a "P" with a line above it. The heavy lines show when and to what degree the music is in which of the available quadrants; and the remarks interior to each of the heavy-line "large-operative" boxes show the field of mappings within each box among the parameters of pitch, timbre, and shape: a line between two parameters means that most of one parameter's elements will map onto most of another's. Comparing the relationship between the interior mappings and the "largeoperative" articulatory-pitch boxes yields significant information. For instance, in the first "long" section (to #22) we see an increase in the number of mapped elements within each parameter and a resulting increase in the number of mapping combinations among the parameters. At #22, moving into the short area and just partially into the unpitched area affects a striking reduction in the number of mappings among the parametric elements; moving entirely into unpitched territory eradicates both pitch and shape, and the only shape left is the large-scale, accretionary one.

Figure 6: Polymorphia., Long- and short range information exchange and mapping

This inter-parametric mapping has been so depleted by the move into short, unpitched territory that even the return to the long and pitched quadrant at #44 is not sufficient to ensure an immediately corresponding return to the former level of mapping activity: the number of combinations available at #44 is minimal, and it takes until #56 before the number of combinations reaches its former level of #22. And comparison of #'s 46 and 59 in Figures 5 and 6 shows that the increase of interparametric mapping is associated with a rapid turnover in both timbral and harmonic parameters; this turnover seems to create an instability within and among the pitch, timbre, and shape parameters which could well be construed as catapulting the thrust towards concrescence into the "larger operative" areas of pitch and articulation occurring between #'s 62 and 63.

Finally, comparing Figures 6 and 2 provides a sense of how concrescence fluctuates in each piece. In Polymorphia the area of maximum concrescence appears very near the end; in Hyperprism, towards the middle. This is most likely the result of each composer's approach to information, whether conscious or not: Varèse's convergences are byproducts of the interactions among the event categories, each of which is primarily concerned with maintaining its own identity in the face of the others; but in Penderecki the large-operative information itself determines the course of the inter-parametric events, which have no personal identity save that of existing within the general thrust toward concrescence.

Bibliography

Breeden, Daniel Franklin. "An Investigation of the Influence of the Metaphysics of Alfred North Whitehead Upon the Formal-Dramatic Compositional Procedures of Elliot Carter." DAI, 37 (1976), 678A (University of Washington).

Penderecki, Kryzstof. Polymorphia. Celle, West Germany: Hermann Moeck Verlag, 1963.

__________. Polymorphia. Cond. Henryk Czyc, Chorus and Orchestra of the Cracow Philharmonia. Philips, 839 701 LY, n.d.

Schuller, Gunther. "Conversation with Varèse." Perspectives of New Music, 3 (Spring-Summer 1965), 32-37.

Stempel, Larry, "Not Even Varèse Can Be an Orphan."The Musical Quarterly, 60 (1974),46-60.

__________. "Varèse's 'Awkwardness' and the 'Symmetry in the Frame of Twelve Tones': An Analytic Approach." The Musical Quarterly, 65 (1979), 48-66. Varèse, Edgard. Hyperprism. New York: G. Ricordi & Co. 1924.

Webern, Anton. Concerto, Op. 24. Vienna: Universal Edition, 1948.

Whitehead, Alfred North. Process and Reality: An Essay in Cosmology. ed. David Ray Griffin and Donald W. Sherburne. New York: The Free Press, 1978.

[1] In addition to the chromatic segments, Hyperprism uses major thirds, which are discussed later in the paper. (See p. 108.) Rarely, Hyperprism will employ quarter-tones, but evidence suggests that Varèse considered these part of the predominantly chromatic segment harmony.

[2] Remarks on the designation of the categories and the identifying characteristics of each: 1) The piccolo (flute) / clarinet timbre may be defined as music played by these two instruments whose rhythmic activity will in general separate it from music played simultaneously by other instruments. 2) The filter event is characterized by two types of behavior: a) a rapid accretion of several instruments (as if a gate were being opened), such as occurs between mm. 41-44, and b) a klangfarbenmelodie, such as occurs between trombone and French horn, mm. 3-12. (Varese himself speaks of filters in Gunther Schuller, "Conversation with Varèse," Perspectives of New Music, 3 (Spring-Summer 1965), 36.) 3) The tremolo/trill is identified by either rapid iteration of a single tone (French horn, mm. 6-7) or successive repetitions of a short figure having a quick enough attack rate. 4) The brass event is defined by mezzo and lower brass sonorities (generally), often with accented or pesante articulation, usually proceeding at a rate slower than that of the other categories. Additionally, the brass event is frequently characterized by several immediate repetitions of either a single pitch or an entire chord. 5) The solo category is characterized by the texture's thinning to one pitched instrument; additionally, this event's pitch structure employs both up and down displaced trichords.

If one examines each of the event categories in a strictly-linear, non-interactive fashion, some curious occurrences emerge. For instance, in their first appearances prior to the moments of maximum concrescence, the tremolo/trill events appear to define a cycle: Hns. 6 Hns., Trps., Wnds. 6 Trps., Wnds. 6 Hns.

After the first moment of maximum convergence, the cycle progressively deteriorates, in conjunction with the overall loss of concrescence throughout Hyperprism's second half. The piccolo/clarinet event seems to become increasingly animated upon each appearance; compare this to the brass event, which begins quietly, builds, subsides, and then makes its final appearance with renewed life. The most noticeably ordered of the event-categories is the filtered one. Of the timbral (klangfarbened) filter events, two pairs exist in retrograde relation to each other: 1) mm. 3-12 and 60-65, and 2) mm. 13-14 and mm. 28-30. The two accretionary events are roughly equivalent.

[3] The term "concrescence" is borrowed from Alfred North Whitehead's philosophy of organism. The following quotes on this term, from Alfred North Whitehead, Process and Reality, ed. David Ray Griffin and Donald W. Sherburne (New York: The Free Press, 1978) are but a few of the many which can be culled from Whitehead's own writings: "The coherence, which the system seeks to preserve, is the discovery that the process, or concrescence, of any one actual entity involves the other actual entities among its components. In this way the obvious solidity of the world receives its explanation." (p. 7) And the following, from p. 26: "In a process of concrescence, there is a succession of phases in which new prehensions arise by integration of prehensions in antecedent phases. ...The process continues till all prehensions are components in the one determinate integral satisfaction." A most succinct description of Whitehead's notion of concrescence within the context of metaphysics appears in Daniel Franklin Breeden, "An Investigation of the influence of the Metaphysics of Alfred North Whitehead Upon the Formal-Dramatic Compositional Procedures of Elliot Carter," DAI, 37 (1976), 678A (University of Washington). Breeden writes: "Whitehead defines existence as an organic 'process,' where the changes which occur from moment to moment are determined by the positive or negative 'prehension' (acceptance or rejection) of environmental data surrounding an experiencing 'entity.' This prehension is determined by the entity's "subjective aim' or basic disposition, and results in various 'subjective forms,' or emotive states. The goal of existence is 'concrescent satisfaction,' or a temporary oneness with an environment. These moments give meaning, a Being, the Becoming of existence."

[4] The major third's permeation into the compositional fabric does not end with its present excursion and subsequent mollification within the brass category. In m. 17 it has infiltrated the texture so far as to be included within the tremolo/trill category, as well. In doing so, however, this interval seems to have overextended and spent itself, for its next significant appearance is its final one. This occurs at mm. 38-39, and compared to its situation in m. 17, the major third exists in reduced circumstances: first of all, it appears in inversion as a minor sixth; and second, it belongs not within two categories, but only within the brass event, just as it did upon its introductory appearance.

[5] The rhythmic patterns accompanying Varèse's chord structures do not point out Varèse's symmetrical arrangement of the pitches. (Compare this, for instance, with the beginning of Webern's Concerto for Nine Instruments, wherein the pitch symmetry is made clear by rhythmic stratification and segmentation.)

[6] Hyperprism restricts itself to chromatic segments and major thirds, whereas Polymorphia employs quarter-tone, half-step, threequarter-tone, whole-tone, major-minor-third, and perfect fourth harmonic structures.

[7] Compare the half-step structure of the canonic glissandi at #11 with the whole-tone harmony of the identically-canonic glissandi at #13.

[8] The tasto of the #5 band maps onto the highest pitch at #10 and later the banded glissando at #15. Likewise, the ordinario of the sinusoids onto the banded glissandi at #'s 18 and 19; the ponticello of the canonic glissandi onto the bands at #'s 14 and 16.

[9] The accretion from #'s 1-22 was built from that section's interplay among shapes, timbres, and intervals; from #'s 22-38 the accretion was formed by the interplay between pitched and unpitched "short sounds," and culminated in the hegemony of the former over the latter between #'s 37 and 38; the accretion between #'s 38 and 44 was based upon unpitched "short sounds."

[10] As to form in Hyperprism, a few comments seem pertinent (see figure 1): 1) Although it may not be correct to speak of the area of maximum concrescence as a true development section, it is developmental in the sense that this is the section of the piece during which the maximum number of interrelationships among the categories is explored. 2) The first of two roughly-equivalent accretionary moments occurs just before the area of maximum concrescence, and the second at the end of the piece: together their placement within Hyperprism appears analogous to codetta and coda of sonata form. 3) The opening klangfarben (mm. 3-5) iterates the pitch C-sharp; and immediately after the moments of maximum concrescence, a gesture unmistakably similar to this opening klangfarben returns (mm. 60-65) on the pitch F-sharp: again, the similarities to sonata form are striking. These remarks ought not be construed as the author's claiming Hyperprism to be modeled on sonata form; rather the point is that these are the only observations about Hyperprism's form that this writer has been able to make and, instead of demonstrating an historical continuity (keeping in mind Varèse's remark: "Beware traditionalists from the left."), seem indicative of an abstruse sense of return, development, and closure.