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The Method of Abstraction: A Musical Analysis

by Stephen Schloesser

Stephen Schloesser is a Jesuit scholastic completing the philosophate phase of the normal course of studies. He received his BA. (philosophy) at the College of St. Thomas (St. Paul, Minnesota) and expects to receive his MA. from St. Louis University. The following article appeared in Process Studies, pp. 19-31, Vol. 15, Number 1, Spring, 1986. Process Studies is published quarterly by the Center for Process Studies, 1325 N. College Ave., Claremont, CA 91711. Used by permission. This material was prepared for Religion Online by Ted and Winnie Brock.


One cannot undertake the performance of a great work without first sorting out its principal trends, its architectural sense and the relation between the different elements which make up its structure. It is not that reason should be in command. It is at the basis of inspiration, which becomes, as one might say, a sort of exaltation of what has first been ordained and fixed by the intelligence.
-- Pablo Casals

If this word "music" is sacred and reserved for eighteenth-nineteenth-century instruments, we can substitute a more meaningful term: organization of sound.
-- John Cage

The only true comment on a piece of music is another piece of music.
-- Igor Stravinsky

"Philosophy," writes Whitehead, "is the welding of imagination and common sense into a restraint upon specialists, and also into an enlargement of their imaginations. By providing the generic notions philosophy should make it easier to conceive the infinite variety of specific instances which rest unrealized in the womb of nature" (PR 17/ 26).

Music theory, as traditionally approached, centers largely upon a description of Western music covering roughly the period from Bach’s chorales through the late Romantics. Such theory not only superficially treats the Greeks, the Medieval, and the Renaissance -- but has few constructs with which to examine our own century’s art (as well as the Eastern world as a whole).

A "new approach" to music, on the other hand, may approach "sonic design" or the "organization of sound" from four perspectives: musical space, time and rhythm, musical language, tone color.’ Examples may include Gregorian chant, Bach, Mozart, Beethoven, Schoenberg, Carter, as well as traditional Japanese music for the koto, the Indian Raga system, and a Sioux dance.

The latter approach, with its variety of analysis, impresses one with the truth that no conceptual description is adequate to the reality. The former approach, with its highly formalized and specialized theory -- for example, common tonal practice while treating of Bach, or Schenkerian analysis while investigating Beethoven -- yields a world of truth from a microcosm, but is proportionately abstract, artificial and limited in application. Yet again, the "organized sound" approach, while more concrete and "phenomenological," yields a highly varied but proportionately less "in-depth" understanding of the genus. Both approaches, finally, impress one with the truth that -- although nothing substitutes for the reality -- formal analysis only intensifies the reality’s depth of satisfaction.

These are the topics which Whitehead investigates in his discussion of the relevance between forms and the occasions they participate in. This essay is largely a simple, concrete, particularized application of the universal principles discussed in the chapter entitled "Abstraction" found in Science and the Modern World. I have attempted to illustrate the metaphysical traits with simple musical examples. Finally, I have drawn some conclusions respecting the relationships discussed.

Whitehead writes:

The first definition of Euclid’s Elements runs:

‘A point is that of which there is no part.’

As [this statement occurs] in Greek science, a muddle arises between ‘forms’ and concrete physical things. Geometry starts with the purpose of investigating certain forms of physical things. But in its initial definition of the ‘point’ and the ‘line,’ it seems immediately to postulate certain ultimate physical things of a very peculiar character. Plato himself appears to have had some suspicion of this confusion when he ‘objected to recognizing points as a separate class of things at all’. . . . He wanted ‘forms’, and he obtained new physical entities. (PR 302/ 460f.)

The problem with Euclid’s definition is that it begins with abstracted entities -- formulae of interrelation -- instead of the concrete actualities being related. Formulae, or "forms," as general, possible patterns of relationship, are abstracted from analogous concrete situations and explained by them. The "geometrical" relations are merely the most general cosmological instances of morphological (i.e., "formal") possibilities for internal relations binding the actual occasion into a nexus, and which bind the prehensions of any one actual occasion into a unity, coordinately divisible (PR 288/ 441).

Euclid’s mistake, then, was postulating the form as the fundamental fact. Actually, the only facts are prehensions, that is, the actual occasions. The fact is that duration of self-enjoyed becoming -- that moment of passage in which the past actual world is prehended and objectified in a new way for the use of the future. The facts are prehensions interrelated by means of energy transmission. Thus, "To be an actual occasion in the physical world means that the entity in question is a relatum in this scheme of extensive connection" (PR 288/442).

"Every actual occasion is set within a ‘realm’ of alternative interconnected entities. This realm is disclosed by all the untrue propositions which can be predicated significantly of that occasion. It is the realm of alternative suggestions (SMW 158/ 228).2 The very possibility of comprehending this matrix requires a reference to ideality, i.e., to a realm of undertermined, purely possible formulae for interrelationships of actual entities. These possible formulae -- the eternal objects -- are characterized by their ability to be comprehended without reference to some one particular occasion of experience (SMW 159/ 228). Together, they form the comprehensible matrix from which the indefinite number of propositions may be derived.

An eternal object may be considered in two ways. With respect to its uniqueness (its "individual essence") the eternal object is considered as adding its own unique contribution to each actual occasion (SMW 159/ 229). But the eternal object must also be considered in reference to other eternal objects -- that is, in its "relational essence." This referential character adds what is commonly overlooked in discussion of ideality and reality -- namely, no eternal object (or ideal) is ingressed (or realized) simpliciter, but only as a relatum whose relata are simultaneously ingressed relatively (insofar as they are graded in value up or down). Put another way, ideality is never grossly imitated or mirrored, but creatively synthesized with a view toward intensity or aesthetic satisfaction.

Let us take an analogy from music, and examine the musical procedure termed "fugue." The fugue as a procedure ("formula") has its individual essence -- it is a fugue, and in a unified series (e.g., Toccata, Adagio, and Fugue) it adds its unique contribution to the whole. However, what it involves in itself cannot be divorced from its status in the universe of discourse -- there is a determinateness as to its relationships with other eternal objects. For example, the forms "Tonic, Dominant," "Subdominant," may all be included in the relational definition of the "Fugue." A "Fugue" is characterized with respect to the intervals at which the subject and its answer enter. Mathematically, the procedure’s relational essence (in a much simplified example) can be expressed:

Subject stated at 0
Answer stated at 7
Subject stated at 0
Answer stated at 5
Answer stated at 3 or 9
Subject stated at 0
(where 1 = one half-step of the diatonic scale)

Having "defined" a fugal formula, we see that it is entirely indeterminate with respect to any actual occasion. Indeed, if it were determined, there would be no possibility of more than one actual "fugue." The procedure "Fugue" would simply be grossly mirrored whenever we chose to hear it. There would be no novelty -- only a simple One. The creator must order the material -- that is, the available field of sound permitted by the available instruments -- according to the determined intervallic relationships of the formulae. Thus, as one of many possibilities, a Fugue in C major could read:

Subject stated at C
Answer stated at G
Subject stated at C
Answer stated at F
Answer stated at E or A
Subject stated at C

We can note here that the above-cited tones are also eternal objects, "C" being comparable to "red." In itself, "C" is just "C." If we attempt to define the "individual essence" of "C," we may specify 33, 65, 131, 262, 523, 1047, 2093, or 4186 cycles per second -- depending on the range of a particular "C." But even here we see that this notion of "C" is relational (i.e., to space and time) and a universal with respect to several possible instances. Relationally, "C" is fully determined with respect to the other tones. The interval C-G is an example of an interval of 7 half-tones. This relationship is built on the determined relationships of the overtone series, apart from any specifications respecting an individual instance or region of "C" or

A comparison between Gregorian notation and modern notation reiterates the point. The Gregorian notation (Figure 1) indicates mere intervallic relationships with respect to an established "Fah." No reference is made to specific tones (frequencies). Modern notation (FIGURE 2), however, goes beyond mere intervallic relations and sets up a correspondence between the notation and the well-tempered tuning system. The modern system is less abstract than the medieval in that it specifies modes (frequencies) of ingression.

In sum, we can say that the eternal object "Fugue" is determined with respect to other eternal objects, and is indetermined, or has a "patience" for relationships to actual occasions. The indeterminateness of the eternal object "Fugue" may be solved into the determinateness of the actual performance of "Fugue in C." The actual occasion has the eternal object as an internal factor, while the eternal object maintains an external relationship. It retains, for example, its patience for ingression as "Little Fugue in G Minor."

An actual occasion synthesizes within itself the complete relatedness of "Fugue" to every other eternal object, including the eternal object "C major." This synthesis is a limitation of realization but not of content (SMW 162/ 233). For example, in the aesthetic synthesis "Fugue in C" (i.e., the actual occasion), a high grade of inclusion is assigned to the eternal objects "Fugue" and "C major," while the lowest grade of value is assigned to the eternal objects

"Sonata" and "D major." In so far as "C major" is highly valued in this particular aesthetic synthesis, it is included and with it every other eternal object with which it is internally related -- including "D major." However, in this particular synthesis, D major remains an unfulfilled alternative -- the composer has chosen not to employ it for aesthetic satisfaction.

With respect to actual occasion "a," eternal object A as not-being means that "A in all its determinate relations is excluded from ‘a’." Then "A as being in respect to ‘a’" means that A in some of its determinate relationships is included in "a." However, no actual occasion can include A in all its determinate relationships. For example, no piece can include "C" with its internal relatedness to "F" as "Dominant," and "C" with its internal relatedness to "F" as "Tonic" (or "Sub-Dominant"). Thus, with respect to excluded relationships, A will be not-being in "a," even when in regard to other relationships A will be being in "a" (SMW 163/ 234).

We are brought again to the question of how a "form" is "realized." Every occasion is a synthesis of the eternal object as being and as not-being. Forms cannot be grossly mirrored. They contain with their very internal relations incompatibilities, contraries, or -- in the mathematical language of Plato -- incommensurabilities. We can see at this point the similarities between Plato’s and Whitehead’s definitions of "Being": "the dynamis to render incommensurables commensurable" and "individual effectiveness in the aesthetic synthesis." Forms considered in abstraction contain within their relational essences contraries which cannot simultaneously "be." Considered abstractly, these forms are isolated from each other, engaging in no real togetherness. However, as realized, the ideals are graded in value with respect to what will enhance the present self-creation of the actual occasion. In this act of ingression, every relation is prehended -- some relations are prehended positively (qua being), while others are prehended negatively (qua not-being). This is not merely a matter of semantics -- to choose negatively is to choose. To understand what a creator did not choose enlarges one’s imagination and enhances the appreciation of that chosen. More generally, the mark of an advanced organism is its ability to select some things while neglecting others -- in composition or analysis.

The recognition of this complete scheme of interrelatedness provokes a new question: How is any partial truth possible?

Insofar as there are internal relations, everything must depend upon everything else. But if this be the case, we cannot know about anything till we equally know everything else. Apparently, therefore, we are under the necessity of saying everything at once. This supposed necessity is palpably untrue. Accordingly, it is incumbent on us to explain how there can be internal relations, seeing that we admit finite truths.

Since actual occasions are selections from the realm of possibilities, the ultimate explanation of how actual occasions have the general character which they do have, must lie in an analysis of the general character of the realm of possibility. (SMW 163/ 23Sf.)

The primary metaphysical truth concerning the realm of eternal objects is "that the status of any eternal object A in this realm is capable of analysis into an indefinite number of subordinate relationships of limited scope" (SMW 164f./ 236). The question remains as to how a limited relationship between objects is possible.

Let us again turn to a musical analogy. In the realm of eternal objects -- a realm in which the relationships between objects are simultaneously unselective and systematically complete -- we discover the musical tone "B." Considered purely with respect to its relational essence, "B" entertains the possibility of many relations: it may be a tonic, a dominant, a subdominant, a neopolitan, a relative minor; it may form the base of a fully diminished vii chord, or the seventh tone of a V/V chord, the third tone in a twelve-tone row, the fourth interval in a five-tone cell, and so on. In other words, we have just examined only a small number of the functions which the eternal object might play. These functions are not particular to "B" -- they are possibilities for any other tone as well, but only insofar as we do not consider the individual essence of each object. With respect to its individual essence, in the realm of eternal objects "B" simply is what it is in isolation. In FIGURES 3 and 4 below, the notation specifies the same tones: in both figures, the tones C - E - G are indicated.3 In their individualities, both clusters from a "C major triad." But in the contexts, the two clusters are functionally unrelated -- hence the difference in notation. Simply put, the two clusters are functionally unrelated, but are actually the same. Functions as pure possibilities are unlimited because they leave unaccounted the internal essences.

The inclusion of the eternal object "b" in an actual occasion "a" and the selection of its functional possibility as a "Tonic" determines its real-togetherness with the eternal object "F-sharp." Here they

escape the confined isolation of indeterminateness and exist in a determined functional relationship of individual essences. In this realization of forms, a value is achieved complete with purpose and emotion. (For example, Bach intuited the cosmic feeling of divine pathos in the key of B-minor, as exhibited in the "Mass in B Minor" and "St. Matthew’s Passion.") The value is shaped by the definite eternal relatedness -- the eidos -- or, that to which the creator "looks" for guidance. The emergent actual occasion is the superject of informed value.

The difficulty with respect to finite relations is evaded by two metaphysical principles:

i) the relationships of any eternal object A, considered as constitutive of A, merely involve other eternal objects as bare relata without reference to their individual essences, and

ii) the divisibility of the general relationship of A into a multiplicity of finite relationships of A stands therefore in the essence of that object. (SMW 165/ 238)

With respect to (i), the pure possibilities mentioned above for "B" do not specify the related eternal objects -- indeed, the more highly abstract possibilities could not be applied generally if specifications were demanded. The scheme, then, is general and not nearly exhausted by our examples, so that with respect to (ii), the scheme is analyzable into a multiplicity of limited relationships which may be known without referring to the whole. I can understand "B as Tonic" and "B as Dominant" as two finite relationships comprehensible without reference to each other. Indeed, considered with respect to individual essences, they are incommensurable and could not both be ingressed (qua being) in one and the same actual occasion.

It will be seen by now that "B as Tonic" -- as a limited set of two eternal objects -- is itself a complex eternal object made up of the related components "B" and "tonic." Likewise, "B major as Tonic" is a complex eternal object analyzable into components and derivative components. Thus, we can analyze as in FIGURE 5.

The complex abstract eternal object "B major as Tonic is an "abstract" situation in two senses. There is first an abstraction from actuality -- we are considering "B major" without reference to any concrete actual occasion. Second, we are considering "B major as Tonic" as abstracted from possibility. For example, both "B" and "B major as Tonic" are abstractions from the realm of possibility. "B" refers to "B" in all its possible relationships in the realm of eternal objects, among them "B major as Tonic." Also, "B major as Tonic" must mean B major as Tonic" in all its relationships. But this meaning of "B major as Tonic" excludes other relationships into which "B" can enter. Thus, as we increase the complexity of the eternal object, we are also abstracting in progressively higher levels from the realm of possibility.

Whitehead terms this progression in thought through successive grades of increasing complexity "an abstractive hierarchy" (SMW 167/ 241). The base of an abstractive hierarchy is a set of objects of zero complexity. Let us consider formally the abstractive hierarchy based upon the following group (g) of simple eternal objects:

g = B, D-sharp, F-sharp, Dominant, Tonic, 0 : 4 : 7, Major

The abstractive hierarchy based on g is a set of eternal objects such that:

1) {B, D-sharp, F-sharp, Dominant, Tonic, 0 : 4: 7, Major} are the only simple eternal objects in the hierarchy.

2) the components (e.g. B,D-sharp,F-sharp) of any complex eternal object (e.g. B major) in the hierarchy are also members of the hierarchy.

3) Any set of eternal objects belonging to the hierarchy are jointly among the components or derivative components of at least one eternal object which also belongs to the hierarchy. (SMW 167f./ 242f.)

This third condition is the condition of connexity. The abstractive hierarchy "includes" (i.e., "has as a subset or is equal to") every progressively higher grade of abstraction. The hierarchy is also internally "connected" by the reappearance in higher grades (e.g., "B major") of any set of its members belonging to lower grades (e.g., [B,D-sharp, F-sharp]) in the function of components or derivative components. FIGURE 6 illustrates an abstractive hierarchy.

This figure illustrates the commonplace that there is an inverse proportion between intension and extension. That is, as the content of the concept increases in specification (and hence, complexity), and a progressively large number of relationships is excluded, the number of members decreases to a unitary set at the level of maximum complexity. This member satisfies the condition of connexity in its function of synthesizing the interrelationship of every member of the hierarchy within itself. FIGURE 7 offers an alternative perspective.

The maximum level of complexity is the "point" -- which is the same as saying that it is the maximal limit of abstraction. In its function as the sharpest focus of definite connectedness, it is likewise at the maximal remove from pure possibility. Paradoxically, this point defines the starting point of our analyses. Beginning with the "point" of maximal abstraction (i.e., the "vertex"), we descend through grades of less complexity whose members are the components of the higher grade objects. Descending from the vertex, we re-collect the component members of the "first proximate’ (i.e., proximate to the vertex) grade, then the members of the "second proximate" grade, and so on, until we arrive at the grade of simple objects. Using our previous example, instead of beginning with the setg and arriving at "B major as Tonic" (i.e., the method of composition), we begin with the complex "B major as Tonic" and arrive at the set g (i.e., the method of analysis). Throughout analysis, Whitehead cautions, we should remember that we are entirely in the realm of possibility -- that is, the "cuts" we make (as in later Platonic dialectic4) are only selections among many possible analytic "cuts." In abstraction, the eternal objects are devoid of any real togetherness -- they remain "isolated" (SMW 169/ 244).

Only ingression qua-being enables eternal objects to escape their isolation and enter into real relationship. In any actual occasion "a," there is a group of eternal objects ingredient in that actual occasion. Since any given group of eternal objects may form the base of an abstractive hierarchy, there is an abstractive hierarchy associated with any actual occasion "a." "This associated hierarchy is the shape, or pattern, or form, of the occasion, insofar as the occasion is constituted of what enters into full realization" (SMW 170/ 245).

The abstractive hierarchy is both the bane and wellspring of our ability to explain any concrete reality (e.g., a musical event) using concepts, images, or language. The infinity of the hierarchy assures that we can never exhaust an actual occasion’s potentiality for description. There are always more interrelationships, more components and derivatives possible. However, by virtue of the condition of connexity, partial grasping of interrelations is possible by virtue of the inclusion of the more abstract subset by the less abstract, and by the "reappearance" of the components and derivatives in the complexes.

By way of comparison, the corresponding geometrical abstraction is the convergence of the geometric "point." Every "base" (i.e., "abstractive set") has an associated geometrical element. The geometrical element called the "point" is that which does not include another element. It is the "point" (see FIGURE 8) of maximal complexity, of sharpest convergence in abstraction (PR 299/ 456).

We see, then, how finite truths are possible. The description of an actual occasion "a" grows in accuracy in proportion to the complexity of the predicated member of the associated hierarchy. Predicating the vertex may be predicating the most abstract element with respect to pure possibility -- but it is the most concrete element with respect to the actual occasion. If students in my ear-training class correctly identify a chord as "Dominant F-sharp Seventh chord with third in the base," they have described it more accurately (albeit more abstractedly) than if they had said simply "a V7 chord."

Our temptation is to believe that by the addition of concepts we can describe an actual occasion fully, as if we could know any "thing in itself." The reason for the impossibility of such knowledge (as hinted in Plato’s Seventh Epistle5) lies in the infinite relations in an associated hierarchy. Perhaps this is what Plato meant in his alleged remark, "The point is a fiction." The "point" -- that is, the point of maximal convergence in any given hierarchy -- is a limit notion which defines the association between occasion and form, but not the occasion or the form. The "rubbing" (tribe in) in dialectic of the occasion against its associated form continues to yield partial relations describing the occasion more fully in its relational essence.

In any event, we can see the source of Plato’s suspicion concerning the point, and Whitehead’s need to derive the point from the more general. In postulating the vertex of an abstractive set, Euclid had begun with the form instead of the fact.

The fact remains the actual occasion -- the ingression of the associated (infinite) hierarchy together with various finite hierarchies in the modes of memory, imagination, and anticipation (SMW 170f./ 246). In the realm of music, the fact is always the present organized sonic-event. However, a review of the metaphysical foundations of the relationship between form and fact reminds us of the need for the formal. Ifthe aesthetic synthesis is to be one which is comprehensible and deeply satisfactory, the variety sought must be "the variety of structures, never of individuals" (PR 319/ 485). "Form" provides the very intelligibility of the experienced. Without a reference to the realm of eternal relatedness, not only music but all reality is incomprehensible.

As we have seen, increased complexity yields less extension (e.g., the number of pieces analyzable with the aid of common tonal practice is limited) but increased information in a microcosmic setting. An investigation into the generic traits of existence reveals to the musician (both as composer and as analyst) the nature of creativity as varying form and not individuals -- and creativity’s need for prior synthesis of the tradition.

On the other hand, the generic traits reveal the truth that the "womb of nature" is infinitely richer than we can imagine. Metaphysical considerations free one from preoccupation with the "point" -- whether it be the "Baroque," the "Romantic," the "Avant-Garde" -- and open one’s imagination (and ears!) to the continuum of space, time, and cultures. Approaching Western tonality as one of numerous available divisions of the sonic continuum brings a renewed realization of how rich and simultaneously arbitrary our peculiar organization of sound is.

In any event, the fact remains the aesthetic synthesis in prehension, and the guiding principle "depth of satisfaction in harmonized contrast."

 

Notes:

1Elliot Carter: "The authors of Sonic Design have made a pioneering effort to view the large field of music we live in today as a whole and to derive general concepts and principles that describe and explain methods of each style, age, and people. The development of such a comprehensive view has long been a need, for it has become clearer and clearer as we have become familiar and involved with a constantly widening horizon of different musical aims and practices, that the old ‘common practice’ theories of harmony and counterpoint could no longer be overhauled or extended, but had by necessity to be replaced by a way of description and analysis that treated the ‘common practice’ of Western music from the late seventeenth to the end of the nineteenth centuries as only one instance of a much wider musical method and practice that could be applied to all of Western music, from its origins to the present, as well as to music of other cultures." From the foreword to Robert Cogan and Pozzi Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs: Prentice-Hall, 1976), p. ix.

2 Where two references are given for Science and the Modern World, the first refers to the Free Press edition of 1967, the second to the 1926 edition.

3Assuming that the tones are produced on a well-tempered keyboard.

4 Plato, Statesman; cf. the first four "cuts" dividing "art," 218b6-226a8.

5 Plato, Epistle VII, 342a-344b.

6 Cf. "Variety is valid only as a means of attaining similarity." Igor Stravinsky, Poetics of Music, trans. A. Knodel and I. Dahl (Cambridge: Harvard University Press, 1947), p. 37.


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