The Limits of Logic: Structure and Aesthetics in Xenakis's Herma
Xenakis is one of the most prolific of contemporary composers. His published works number well over one hundred; of these, nearly all have been performed, many have been recorded, and some have become established classics of the twentieth-century repertoire. Xenakis is perhaps even more famous for his theoretical innovations. Here, the best-known aspect of Xenakis's thought is his use of mathematics to construct models for musical structure. Indeed, the impact of these theoretical models has been so strong that Xenakis's music has gained a reputation as highly abstract, and his compositional technique has been shrouded in the kind of mystery usually reserved for advanced physics or astronomy. Xenakis has promoted his image as a mathematician in a number of articles throughout his career. The most important of these are collected in Formalized Music: Thought and Mathematics in Composition. This book was first published in 1971; a second edition, which reproduces the first intact and adds several new chapters, appeared in 1992. As its title implies, Formalized Music concentrates on the development of mathematical techniques for the control of musical sound. It is not an easy book to read. Xenakis's explanations of the mathematics involved are often convoluted and obscure for those unversed in mathematics. Even those who have closely studied and understood the volume have criticized its obstacles; thus Jan Vriend in his illuminating essay on Xenakis's 'cello piece Nomos Alpha condemns the composer's exposition of the piece in Formalized Music as "absolutely insufficient and misleading." Despite these faults, the book has strongly influenced perceptions of Xenakis. In general, commentators have developed from Formalized Music the picture of a composer completely enthralled by mathematics and by strictly determined composition. For example, Michel Philippot in the entry on Xenakis in the New Grove writes:
If it were necessary to summarize Xenakis's technique in a single word `formalization' would be the most appropriate . . . Xenakis has in many instances taken a model developed to describe the laws operating in a physical system and then applied that model to govern the elements of an aesthetic construction.
A corollary of these views has been that details of mathematical calculations in abstract models are much discussed in the literature on Xenakis while questions of the real relationship between the model and the music are frequently ignored. This study attempts to advance a more subtle understanding of Xenakis's use of mathematics through an analysis of the problematic role of mathematics in the piano piece Herma, written in 1961. This piece offers a particularly felicitous opportunity for such a study as the account of its composition that Xenakis gives in Formalized Music allows direct access to the theory behind the writing of the piece.
The idea for Herma springs from Xenakis's distinction in musical "architecture or categories between 'outside-time,' 'in-time,' and 'temporal.'" (FM, p. 183) This distinction is one of the linchpins of Xenakis's thought and the example he uses in Formalized Music is unusually clear:
A given pitch-scale is an outside-time architecture, for no horizontal or vertical combination of its elements can alter it. The event itself, that is, its actual occurrence, belongs to the temporal category. Finally, a melody or a chord on a given scale is produced by relating the outside-time category to the temporal category. Both are realizations in-time of outside-time constructions. (FM, 183)
There are, therefore, three "architectures:" two are abstract and one is phenomenally real. As an "outside-time" architecture, a scale is a non-temporal concept, existing only in conceptual space. The pattern of soundings of a scale patterns time and creates an abstract temporal architecture because time is considered as a linear continuum; thus each pitch's moment of sounding is an event that marks a point in time. Finally, the real aural experience of hearing the scale makes the third level of structure, the "in-time."
According to Xenakis, Herma takes its inspiration from the idea that music can communicate symbolic logic through its outside-time architecture. In these terms, Herma aims to prove that "we can reason by pinning down our thoughts by means of \sound." (FM, 172) This objective completely dominates the account of the piece's composition in Formalized Music. In this account, Xenakis begins by assuming that pitch is one of the three "ultimate aspects" of all "sonic events." From this, he argues that the perception of a series of sounds at different pitch levels by an "amnesic observer"—a human tabula rasa - will evoke in him or her the notion of interval and that this process will "end in the totally ordered classification not only of pitches, but also of melodic intervals." (FM, 158) Such an ordered set of elements can be treated as a mathematical set and can take part in logical functions.
Having thus established the possibility of treating musical parameters as logical elements, Xenakis makes a number of pre-compositional decisions about the structure of Herma, explaining each decision in terms of its relationship to logic. First, he chooses the set of all pitches as the referential set R that will form the logical world for the piece. After this, he needs to create a number of sets from R as symbolic logic can exist only through a process (a proposition) in which several sets are combined. Xenakis decides to use three sets in Herma, labeling them A, B, and C. Xenakis now needs to determine which pitches of R should be members of A, B, and C. In the score of Herma the appearance of each set is clearly labelled by Xenakis and Example 1 below gives the pitch membership for each set based on these labeled appearances.
Xenakis ends his discussion of the pre-composition of Herma with the choice of the logical proposition: the function through which the sets will demonstrate the symbolic logic. He decides to use Boolean algebra, which is commonly used in order to manipulate logical propositions. By reference to the mathematics of functions Xenakis generates a proposition for Herma: "A Boolean function with n variables can always be written in such a way as to bring in a maximum of operations +, ∙ , -, equal to 3n∙2n-2 -1." (FM, 173) With 3 sets in use, n will equal 3 and the maximum number of operations will be 17. The function thus generated, according to Xenakis, is:
Xenakis then complicates the situation by introducing another function, with only ten operations but with the same conclusion:
Comparing these two functions, Xenakis finds "a more elegant symmetry" in the first than in the second, but allows that the latter is "more economical." He chooses to use both of them: "It is this comparison [between the two functions] that was chosen for the realization of Herma." (FM, 175)
Example 1: The Pitch Sets of Herma, as they appear in the score.
The presentation of these pre-compositional decisions in Formalized Music thus emphasizes the direct relationship between the logic and the musical elements. From this account it is easy to assume, as most commentators have done, that Herma is a strict representation of symbolic functions and that it directly communicates the logic "behind" the piece. In his initial argument from the amnesic observer Xenakis lends support to this reading when he argues that the "observer" will perceive a series of pitch events as sets: "If the observer, having heard A and B, hears a mixture of all the elements of A and B, he will deduce that a new class is being considered, and that a logical summation has been performed on the first two classes . . . If class A has been symbolized or played to him and he is made to hear all the sounds of R except those of A, he will deduce that the complement of A with respect to R has been chosen." (FM, 271) In other words: given a distinct structure of sets ("classes" in the quoted translation) as "input," it follows that the in-time structure of the musical "output" will convey the same distinct structure as an outside-time structure to the listener. By this token, Herma is reducible to the sum of its outside-time structure which is "communicated" in the piece.
The problems attendant upon this view of Herma will become quite apparent below. But even in his pre-compositional decisions, Xenakis reveals a remarkable lack of concern for the symbolic logic; a lack that is disguised by the language of Formalized Music. For example, there is no logical reason to utilize three sets of pitches. Two sets could form propositions just as well and would be easier to identify. Moreover, there is no logical motive for the sheer number of pitches in each set, or for the idea of "comparing" two functions. Indeed, it could easily be argued that the logic would be better communicated if each group contained three or four pitches rather than twenty-odd. And if only one function were used for the piece, surely the underlying logic would be available to the listener in a much more straightforward way.
These intriguing questions unearth serious doubts about the role of the symbolic logic in Herma. But it is not necessary to discuss them in detail. For a closer examination of the score of Herma reveals an even more problematic aspect of the piece; something that appears to undermine the raison d'être of the work.
In terms of its large-scale temporal structure, Herma divides into two halves. The first of these begins with a statement of the referential set R, followed by the respective statements of the sets A, B, and C. Each set statement is succeeded by the statement of its complement. Thus, in the score, the section labeled A is followed by a section labeled , that labeled B by , and so on. This appears straight-forward until the actual pitches of each section are identified. And then it becomes clear that the sets A, B, and C are not well-defined by the pitches that appear in Herma. In each of the three sets, some of the pitches in the initial statement of a set also appear in the following statement of that set's complement. For example, a3 appears in measure 74 in the section labeled as the statement of B; the same a3, however, also appears twice in the section labeled . This pitch, therefore, has a logically impossible existence: it is both a member of a set and not a member of it. There are many such pitches in the score of Herma. Example 2 below gives lists and the relevant statistics. I have coined the term "dual pitches" for these pitches, as they (impossibly) represent two logically opposed set formations.
In the above figure the names of the "dual pitches" are followed by the number of times each occur in the statements of the classes and the statements of the complements. The pitches are notated in the system whereby "Middle C" on the piano is c1, the pitch an octave above is c2, the pitch an octave below is c etc.
Discrepancies in the second
part of the score further compound these problems. In terms of the logic, the second part of the
piece involves combinations of the sets already sounded—following the functions
Study of the score reveals that many of
these set-combinations involve pitches that were not stated initially in the
(labeled) statements of the sets being combined; they therefore cannot be
members of the set-combinations. These
pitches, originating outside the logical functions, I have termed
"additional pitches." Example
3 gives details of their appearances in the set-combinations.
Example 3: "Additional pitches" in Herma.
These facts of pitch distribution threaten the entire theoretical basis outlined above. If the sets are not well-defined by pitch, then the "outside-time" architecture of Herma cannot represent the symbolic logic. Why does Xenakis undermine his own structure in this way? Why, in short, do these "dual" and "additional" pitches exist?
One possible answer deserves immediate consideration: might the problems in Herma be solved through an investigation of the publication history and the subsequent literature on the piece? This will ultimately prove a false trail, but following it provides substantial illumination on the issues of the piece.
The score of Herma exists in only one edition, published by Boosey & Hawkes in 1961. Are the pitch discrepancies the products of faults in this edition? Two pieces of evidence make this unlikely. A first proof that the "dual pitches" exist at the manuscript level can be found, ironically enough, in the errata slips which accompany the 1961 edition. These hand-written slips correct printing errors contained in the score, but do not touch on the existence of the "dual pitches." Indeed, the slips themselves include examples of some of these pitches in both set and complement.
A second piece of evidence for the fidelity of the Boosey & Hawkes edition to the manuscript of Herma comes from the French pianist Claude Helffer. Helffer, one of the foremost performers of Xenakis's music in the world today, studied with Xenakis and has written several articles on his music. In private correspondence he has confirmed that the manuscript does not differ from the Boosey and Hawkes edition of 1961: "Le manuscrit dont j'ai une photocopie n'est pas différent de l'edition Boosey and Hawkes, la seule existante." Without a detailed study of the manuscript, final proof that the pitch discrepancies do not stem from publishers errors cannot be provided. However, in view of the two pieces of evidence offered, it seems safe to say that it is extremely unlikely that the "dual" and "additional" pitches in Herma do not stem from Xenakis's pen.
At present, there is only one English-language source that deals with the matter of the pitches in Herma. This is a doctoral dissertation by Rosalie Sward: "An Examination of the Mathematical Systems used in Selected Compositions of Milton Babbitt and Iannis Xenakis." This dissertation includes extensive discussion of Herma together with much useful diagrammatic information; it represents, in fact, the most complete treatment of the work in English to date.
Sward acknowledges the existence of the "dual pitches" and in an appendix to her dissertation provides a list of corrections to pages 1-12 of the score; this covers the statement of R and the statements of the three sets and their complements. Example 4 gives the sets, revised to take account of these corrections, a comparison with Example 1 is useful.
An examination of this list brings up several problems. Comparing the list with the score, it is plain that several of the "dual pitches" are not eliminated. Thus, in set A the F1 at measure 30 remains as a member of A, while the same pitch also appears in at m. 64; the f<2 remains at m. 46 in A and also sounds 4 times in ; and the "error" of including c3 in A at m. 48 is mentioned in the list but no "correct" note is supplied. Even if these are dismissed as minor matters, which in logical terms they cannot be, the disturbing fact remains that the list only goes as far as page 12 of the piece. This means that no "corrections" exist for the second half of the piece, yet it is precisely here that some of the most serious problems concerning pitch occur, as Example 3 testifies.
It seems that this list cannot provide a satisfactory resolution to the problems of pitch in Herma. This is so not only because the list leaves "dual pitches" behind and extends only to p.12 of the score. What makes it even more difficult to accept the list as a solution to the problem of the dual and additional pitches is that it does not provide any coherent explanation for the existence of these pitches in the first place. The provision of such an explanation becomes a compelling question because of the stress Xenakis lays on the logical structure of the "outside-time" level and of the denial of that logical structure in the "dual pitches." What may be termed the historical evidence concerning the pitches of Herma is inconclusive and unsatisfactory precisely because the "why" of the "dual pitches" has been left unanswered.
In search of an answer to this question, a deeper analytical investigation of the piece will be attempted. At this stage, it is useful to recall Xenakis's comment in Formalized Music concerning the role of the musical parameters of Herma: "timbres, attacks, intensities and durations will be utilized in order to clarify the exposition of the logical operations and relations which we shall impose on the set of pitches." (FM, 170) It is not clear from this whether we are to take these parameters as forming "outside-time" structures as the pitches do. At least, however, the tracing of these parameters in the music should illuminate the relationships between the aural surface and the logic.
The parameters mentioned above by Xenakis are all prominent structural elements of the first half of Herma. Their functions in the music from measure 1 to m. 135 are outlined in Example 5. In this opening section each of the three presentations of the sets divide between two "qualities," labeled in the score as linéaire and nuage. Neither "quality" has a truly consistent aural identity, as will become clear below. However, their prominence in the score demands their inclusion in the analysis.
Example 5: A Parametric Analysis of the Opening of Herma, mm.1-34
Figure 5 (cont'd): A Parametric Analysis of the Opening of Herma, mm.34-48
Example 5 (cont'd): A Parametric Analysis of the Opening of Herma, mm.48-62
Figure 5 (cont'd): A Parametric Analysis of the Opening of Herma, mm.62-76
Example 5 (cont'd): A Parametric Analysis of the Opening of Herma, mm.76-90
An examination of Example 5 shows that the dynamic parameter supports the llinéaire / nuage division as each linéaire statement is sounded f or ff, and each nuage statement p or pp. The relationship between dynamic and the set structure is less clear but, in broad terms, the complements are sounded louder than the set statements. The parameter of speed reinforces this distinction as the complements are relatively faster than the sets. In addition, in set A the three different speeds, 0.8 s/s, 3.3 s/s, and 10 s/s, can be understood as representing the three different sound-qualities linéaire, nuage, and complement. However, a departure from this system comes in measure 82 during the statement of B, where the speed of the linéaire increases from 1.8 s/s to 5 s/s and that of the nuage from 3.3 s/s to 5 s/s. This sudden acceleration is quite at odds with the preceding music, and the acceleration confuses the identity of B as a unit. Moreover, it means that the combined speed of linéaire and nuage reaches ten sounds per second, equal to that of .
The use of the pedal in the extract is ambiguous. In the statement of A, the joint statements of the linéaire and nuage are pedaled while the linéaire on its own, and the complement are not. In the statement of B, however, this "rule" is broken: pedal is applied to the first half of the initial statement but no pedal is used in the second half. Unlike the case of, the pedal is here used in the statement of the complement of B. In the statement of C the pedaling then returns to that used in A: pedal is only used with the nuage quality. Clearly this use of the pedal cannot "clarify" the logical structure in any way.
Finally, the durations of the set statements in seconds can be calculated as A: 40 (22); B: 24 (18); C: 18 (28), (the parenthesized numbers represent the durations of the complements). There is no particular relationship revealed by these figures: can there be a "logical" motive why any statement should be longer than another? Perhaps the most obvious feature is the length of the statement of the complement of C and this corresponds to no feature of the symbolic logic.
This analysis shows that, contrary to Xenakis's claim, there is no consistent relationship between these musical parameters and the logical structure of Formalized Music in this opening section. Different and contradictory aural clues are contained within the music; while some parameters support the logical structure at least some of the time, such as silences and dynamics, others obscure and even contradict the logic, such as the changing speeds, the use of the pedal, and the duration of the sections. Whether these parameters form real "outside-time" structures or not, they do not unambiguously serve the strictures of the logic but combine to create their own patterns and conflicts, both inside and outside time.
The second part of Herma, from measure 136 on, "compares" the two functions, ending with the statement of the final set F. In this part, the ambiguity between the symbolic logic and the music is even more apparent. Xenakis's own diagram from Formalized Music, reproduced as Example 6, shows the formal relationship between the two logical functions, including the dynamic levels.
Example 6: The Realization of the Two Functions of Herma. Courtesy of Pendragon Press, Stuyvesant, NY.
In this diagram, the logical expressions are differentiated from each other by assigning to each two distinct dynamic levels: f and fff for the first plane, ff and ppp for the second. Example 7 shows how the occurrence of each distinct dynamic in Herma is distributed with respect to the set-combinations.
While most of the set combinations have a fixed dynamic with which they are associated, two factors weaken this association. First, the use of fff for the final combination F re-uses a dynamic that had already been assigned to some of the set-combinations on the first plane: this must imperil the independence of F as a fully independent statement. Second, the use of ff for the set gives a logically unwarranted emphasis to this set. All the other sets on this plane are sounded ppp and thus it is at least possible that the listener might mistake as a part of the first plane. A little reverse logic clarifies this problem: if it were the sole task of the dynamics to differentiate the planes, why use two dynamic levels for each plane? Surely one dynamic level each (say f set against pp) would be a better choice? Broadly, the dynamic functions of the second part of the piece do not contradict the structure of symbolic logic, but the relationship is not at all as transparent as Xenakis's words imply.
Example 7: Set-Combinations and Dynamics in Herma.
Example 8: Set-Combinations and Speed in Herma.
The part played by speed in the second part of Herma is rather less clear-cut than that of dynamic. Example 8 shows the distribution of speeds in regard to set formations; a comparison with Example 7 is worthwhile. There seems to be no regular scheme or pattern in the use of speed. In fact, there is a degree of contradiction and overlap between the planes, and this is most clearly exemplified in the comparison between the set of the first plane and . These sets share their three characteristic speeds of 1 s/s, 3 s/s, and 5 s/s; thus, in terms of speed, there is no way of distinguishing between them. Even if two statements of different sets are not actually confused, the use of the same speeds for both must tend to obscure their separate identities. The role of speed in this section is akin to the role of duration in the opening section: it tends to blur rather than to flatly contradict the outside-time structure.
Example 9: Set-Combinations and rappel in Herma.
One more feature of the second half of Herma merits consideration: the technique of repetition or rappel. There are fourteen distinct sets used in Herma from measure 130 on, all being formed by operations on the opening three sets. There are thirty-two distinct musical statements of sets in these same measures; this fact comes about through rappel, as Xenakis labels it, whereby a set that has already been sounded is replayed. This idea of "recalling" sets is not generated by the symbolic logic: in neither expression is there any logical need to retrace operations. Example 9 below details the occurrences and re-occurrences of the set-combinations in Herma.
Using Example 9 together
with the previous two examples, it is clear that the statements of the sets
vary widely in the contexts
in which they occur. There seems little similarity among the three
, in measures 156-159, mm. 166-167
189-190 respectively. The two
contexts in which the set is heard in its first appearance—alone with pedal,
and together withwithout pedal—are not repeated.
The speed, moreover, is not held constant among statements: on one appearance the set is played at
half the speed and pedaled
together with ; in its next
The Herma which emerges from this analysis is a piece of music shot through with ambiguities between the musical "architectures" and the supposed logical structure given by Xenakis. The problems of pitch distribution discussed above now appear as part of a larger problem encompassing the genesis and the existence of the whole composition. From all kinds of perspectives, the work is far from being a representation of logic. It cannot be the case that Herma "communicates" the functions of symbolic logic and the piece is not reducible to the abstract, "outside-time" architecture that Xenakis seemed to have prepared for it. What, then, is the logic of Herma? What can be made of this piece which, it seems, is so patently not what its composer intended it to be?
Herma is not the only piece of Xenakis's in which difficulties arise between the realities of the score and the composer's explanation of it. In a penetrating analysis of the 'cello piece Nomos Alpha, Jan Vriend has exposed the astonishing number of ways in which the actual finished music differs from the account given by Xenakis of its composition in Formalized Music. Xenakis, when taxed by Vriend with these inconsistencies, gave three reasons for their existence: slips of the pen; theoretical or computational errors; and details changed for the ear. It is difficult to give the first two reasons much credence in the case of Herma. The sheer volume of "dual pitches" in the score would require an outrageously careless pen. And it seems even less likely that "theoretical errors" lie behind the pitches: it would be incredible to suggest that an engineer of Xenakis's achievements could be guilty of distributing the same pitch to both a set and its complement through mathematical mistakes. One motive remains, then: the alterations in the logical structure were made "for the ear."
If Xenakis altered the mathematical structure for aural reasons, did this create a strong and coherent aural identity for the piece? Did he exercise a traditional composer's prerogative in "bending" the rules of formal structure to allow for musical "expression"? Initially this seems plausible. According to several commentators, one of the main features of Herma is the impression of directed motion. Writing in the notes accompanying the CD of Herma issued by disques montaigne, Harry Halbreich states that the work "relentlessly builds up its tensions until the climax of the last page." Sward even compares the overall form of Herma to traditional concepts of sonata form with the statement of F operating as a "sort of Grand Coda." ("Exam," 393-394) And Xenakis himself, in response to Vriend's analysis of Nomos Alpha, has referred to "continuity" as a factor influencing his compositional decisions. Such musical continuity and driving progress in time would, in Xenakis's terms, produce a regular "temporal" architecture with a smooth patterning of time by the musical events.
A prime example of this kind of directed motion in Herma comes in the opening measures, from m. 1 to m. 30. This introduction involves a statement of the referential set R, containing all the pitches on the piano. This set is heard in a gradual crescendo from ppp to fff coupled with an accelerando and a gradually thickening texture; in all, a marked impression of directed motion arises. This motion culminates at measure 30—which is the beginning of the statement of A. This statement of R is therefore a fine example of unity between the logical and the musical. The sounding of the set of all pitches is coupled with the sounding of all available dynamics and also covers all the available speeds: the presentation of the (supposed) outside-time basis for the piece is thus united with the presentation of its basis in temporal musical function. Here there is no contradiction between logical structure and musical motion. Clearly, progression and direction in the music do not necessarily imply a negation of the logic. From this example, therefore, Xenakis is not faced by an absolute dichotomy between logic and musical motion: the two may easily co-exist.
The continuation of Herma does not maintain this type of temporal "architecture." Taking the piece as a whole, the impression of directed motion is present only spasmodically throughout the piece. For example, the opening statement of the classes A, B, and C, each together with its complement, includes three aurally distinct units through the manipulation of speed and silence: taking these three parts as a whole, it is hard to find any way in which measures 30-131 could be heard as a single musical and temporal progression; here repetition and the creation of separate musical units seem to take primacy over progression.
The most striking example of discontinuity in the music occurs in measures 81-84, and it is worth considering these three measures in more detail; the relevant part of the score of Herma is reproduced in Example 10.
The silence in measure 81 cuts across the music and creates a significant disruption. This silence is broken by a sustained four-note chord which is in turn succeeded by silence. Both the texture and the position of this chord form a stark contrast with the music that surrounds it: it is perhaps the most vivid gesture of the piece so far. The musical fabric is decisively fragmented—but this is not the product of mathematical concerns; the emphasis created by this event has no rationale in terms of the logical structure of Herma. Indeed, the silences and the sustained chord contradict the logic. For these events occur in the middle of the statement of B, and can only serve to prevent the apprehension of this set as a whole. As in the example of the opening, musical motion and logical structure are in fundamental sympathy. Xenakis's disruption of the representation of the symbolic logic in the "in-time" also disrupts the musical motion of the piece. It follows from these two examples that Xenakis's alterations and the ambiguities of structure in Herma are not merely aimed at musical "continuity."
This is not all, however. For the analysis of the events above highlights the disjunctive nature of the "temporal" structure of Herma. The opening motion to m.30 is followed by the suddenly static music of the set statements, in which Xenakis's stochastic mechanisms prevent pitch patterns emerging and only broad dynamic and textural changes give variety. No distinct "temporal" structure emerges from these statements, the passing of time is marked only by a blur. But the regularity of this haze is itself broken by the events of measures 81 to 84. No temporal logic appears to motivate these disruptions - just as no outside-time logic motivates the presence of the "dual" and "additional pitches." Instead, both of the "abstract architectures" are fundamentally and radically ambiguous.
The problems of pitch distribution in Herma are thus symptomatic of a much larger process of subversion that is characteristic of the entire work. In this process, all levels of musical structure are challenged and the relationships between them blurred. The discussion of the composition of Herma shows how the crisp logic of the abstract structure is altered and transformed by Xenakis's compositional decisions at every level, so that arbitrary decisions, inconsistent with and unmotivated by the logic, are admitted. A similar process is at work in the development of the musical, "temporal" structure. There is no one, unified, and coherent musical world in Herma. Instances of apparently unmotivated events such as the chord of measure 82 are understandable only as the subversion and denial of a cohesive structure. In the same way, the existence of the "dual pitches" and the "additional pitches" deny any unified existence to the "outside-time" architecture. Through this subversion, however, the realities of both structures have a partial existence. Some of the parameters of the piece support the logic and generate an outside-time structure that could support the original logic. And, similarly, many parameters generate unified, progressive, temporal architectures. Moreover, these parameters are not necessarily opposed, and often overlap.
In this reading of the piece, Herma represents a meeting and co-existence of symbolic and musical logics and an exploration and a subversion of their respective limits in such a relationship. The music plays around the subverted structures, sometimes offering a logic based on mathematical symbols, and sometimes one based on temporal motion, while denying full coherence to both. In this play of abstractions, Herma challenges the listener, forcing a search for an explanation of the piece. Such an "explanation" can be found only in part, in the outside-time structure of symbolic logic or in the temporal, linear progression of the music. In the in-time actuality of the work, each structure makes its own claims and each explores the limits of its own logic in relation to the other. Ultimately, the single reality of either of these structures is illusory; in the context of Herma only the dual existence and interaction of the two logics is real, and only in their subversion do they exist.
This is not a view of Herma which sits well with the more positivist passages of Formalized Music. But large portions of Xenakis's writing reveal him to be concerned fundamentally with the aesthetic effect of his work, rather than its grounding in mathematical axioms. It is only from this perspective that the ambiguities of Herma become comprehensible.
Xenakis's dominant ideal is the union between music and science, a union which will affect the basis of human intelligence and thus of existence. Toward this end, he states that "each one of my works proposes a logical or philosophical thesis." (IX, 19) These "theses" are made possible by "a new relationship between the arts and sciences . . . where the arts would consciously 'set' problems which mathematics would then be obliged to solve." (Alloys, 3) He has spoken of his work as a striving to fill a "philosophical space with an intelligence which becomes real by the colored pebbles which are my musical, architectural and visual works and my writings. These pebbles . . . have found themselves brought together by . . . forming figures of local coherencies and then vaster fields summoning each other with questions and then the resulting answers" (Alloys, 6). Example 11 below, which reproduces part of a table given in Alloys, shows how literally Xenakis addresses this concept of his work as a series of answers.
existentiality ST/10-1, 080262
in-time, outside-time Nomos gamma
causality ST/10-1, 082062,
Tourette Convent (facades)
Example 11: Xenakis's works as Questions and Answers.
This idea of the work of art as an answer to a question is very relevant to Herma. In these terms, the piece is not built from the processes of symbolic logic; rather, it poses a response to the questions of determinism that are at the heart of any logic. The motive behind the title then becomes clear: the name Herma "means 'bond' but also 'foundation', 'embryo' etc" and in this work Xenakis is addressing the idea of a "bond," or a causal link, through symbolic logic, which represents and distills the essence of causality in human thought processes. The issue at stake is therefore that of causality in general, as the basis and "foundation," of all thought. In Herma logical causality meets up with another type: the causality of musical logic, which forms the "embryo" of all musical statements. The work functions as the meeting place for these two logics, it is the result of their collision and it delimits the mutual boundaries between them. In doing so, it serves as a "foundation" for Xenakis's works and theories by showing how music is susceptible of the expression of conceptual notions, and to what extent "we can reason by pinning down our thoughts by means of sound." (FM, 172)
From this perspective, the logical structure that Formalized Music prescribes for Herma sets up the rules of logic for their subversion in the piece. In much the same way as a post-structuralist painter might ground his work through a reference to established culture and proceed to destroy this foundation, so Xenakis in his writing sets up the logical scaffolding that is to be undermined in Herma. This outside-time procedure has its exact equivalent in the temporal plane. As discussed above, the opening thirty measures establishes a perfect example of temporal progression and logic. And the same logic will then be subverted throughout the piece. If this were not enough, the opening twelve pitches of the piece spell out a full chromatic collection, as if establishing a set. In the context of what follows, this is a delicious example of post-structuralist irony.
This view of Herma echoes the opinions of François Bernard Mâche in his discussion of Xenakis's relationship with myth: "The composer has, more than once, recognized that Reason never has the last word." When Xenakis writes of composition, his approach is solidly mathematical and technical but, as this study has argued, his compositions are far from straight-forward. Xenakis has concerned himself with the gap between his writing and his music: "If I try to explain my ideas in books and articles or in lectures on this or that technique, it is because I can easily speak of these things. Or, when I teach, it's to incite others to delve into these same questions. But I don't say everything, even if I sense or perceive it, because I don't know how to say it. Therefore, eventually, I have the students listen and see the results." (Alloys, 20) Only through a detailed analysis of his music, then, can Xenakis's art be properly illuminated. The analysis of Herma above functions as a step towards such an enterprise. In Herma, the meeting of the two logics produce an "in-time" that is a new sound-space. Herma locates its aesthetic identity in this space, and thus becomes a vital key to the theoretical and musical thought of Xenakis.
 This study has its origins in my thesis for the degree of Master of Music from the University of Massachusetts, Amherst. I presented an earlier and shorter version at the joint conference of Music Theory Mid-West and the Society of Composers International held at the University of Iowa in April 1995. I would like to thank both Gary Karpinski and Christopher Hasty for their comments and conversations on the subject of this study.
 Iannis Xenakis, Formalized Music: Thought and Mathematics in Composition (Harmonologia Series No.6. Stuyvesant, NY: Pendragon Press, 1992). From here on all references to this book will be made in parentheses in the text, in the form (FM, p. #).
 These new articles presume the broad theoretical span of the earlier writings. As Xenakis himself remarks in the preface to the 1992 edition: "the theories which I have presented in the various chapters preceding this new edition have always been present in my music." (FM, x)
 For a more detailed discussion of this tri-partite division of time, see Ellen Rennie Flint, "Metabolae, Arborescences and the Reconstruction of Time in Iannis Xenakis' Psappha." Contemporary Music Review, Vol.7 pt.2, 1993, pp. 221-249.
 It would be possible to explain the occurrence of the "dual pitches" through reference to a fuzzy logic in Herma. There are at least two objections to this, however. First, Xenakis makes no reference to any such concept of logic in his explanations; the sets of Formalized Music are thoroughly crisp. Second, the statistics in Example 2 show that many of the "dual pitches" have occured equally in both a set and its complement. Thus, even adopting a fuzzy standard does not allow us to distinguish certain pitches as "more" members of one set than another, and so the problems remain.
 The slips may well be in Xenakis's own hand; the notation and writing on these slips are similar to those examples found in other of Xenakis's scores which have been published in manuscript form, of which the score of Akrata provides a good example.
 Rosalie La G. Sward, "An Examination of the Mathematical Systems Used in Selected Compositions of Milton Babbitt and Iannis Xenakis," (Ph.D. dissertation, Northwestern University, 1981). Hereafter referred to in parentheses in the text as ("Exam," p. #).
 Unfortunately I have not been able to examine two unpublished French dissertations on Herma: one by D. Sevrette, "Itude statistique sur Herma," presented to the Schola Cantorum, Paris in 1973; the other a ThPse de doctorat presented to the University of Paris in January 1993 by GJrassimos Solomos, "A propos des premiPres oeuvres (1953-69) de I. Xenakis."
 Sward quotes the source of the list as the aforementioned Claude Helffer, without any further indication as to its origin. In the correspondence referred to above, M. Helffer did not reveal anything about the provenance of this list; as Sward's dissertation was completed in 1984, the list presumably dates from Helffer's early interest in the pitch problems of Herma.
 Sward acknowledges this: "The other Boolean relationships were calculated by the author without the use of the score, but rather as mathematical operations of Union, Intersection, and Negation of the pitches contained within the sets A, B, and C." ("Exam," 384) This is perhaps an understandable step, given that Sward is concerned with the mathematical processes behind the composition of the work, but it obscures the delicate and contradictory relationship between Xenakis's compositional strategies and their realization in the music, and thus misses one of the most interesting facets of Herma.
 In actual fact, as Sward points out, the opening statement of R omits 3 pitches from the total 88 on the keyboard. As it is extremely unlikely that any listener could hear this omission, the temporal reality of the statement of R can still serve as a metaphor for the logical status of referentiality.
 This interpretation of Herma fits well with post-structuralist concepts of subversion in art. In this sense, the "limits" of my title echoes Foucault's use of the term. Recent work by Paul Attinello has emphasized the post-structural aesthetic of much music in the late 1950's and early 1960's in the context of a rebellion against the "high modernism" of the post-war years. (Paul Attinello, "Rebellion in Darmstadt: The Anti-Serial Reaction, 1958-68." Paper read at the SMT Annual Conference, New York, Nov. 1995). The view of Herma put forward in this study would place Xenakis at the forefront of this aesthetic.
 Iannis Xenakis, Arts/Sciences: Alloys. The Thesis Defense of Iannis Xenakis before Olivier Messiaen, Michel Ragon, Olivier Revault d'Allonnes and Michel Serres, Sharon Kanach (tr.), New York: Pendragon Press, 1985. Henceforth referred to in the text as (Alloys, p. #).
 From Alloys p.7. The facades of the Convent at Tourette, mentioned as an "answer" to the "question" of causality, represent one of Xenakis's most outstanding architectural achievements. Another of these was the creation of the Philips Pavilion. The other works are musical works.