La emigración interestatal

If all potential divisions of a measure are not per- ceived, there is, nevertheless, incontrovertible 130 A Theory of Meter as Process

evidence for the relevance of virtual articulations if we can feel “the” meter, or a definite metrical type, in the absence of sounding pulses. Thus, in example 9.11, if we can feel “duple meter” in the bar-measure projections, we must have felt the ef- fect of articulations that are not actually sounded. Whether we interpret these bars as examples of 2/2 or 4/4 (whether our attention is focused on half-note or quarter-note pulses), we will in any case sense equality—duple, equal measures. If sig- natures are somewhat misleading if they are taken to promise homogeneous division, they do, nev- ertheless, indicate metrical, projective types that in limitless instantiations can usually be felt (though the degree of such feeling is, as I shall argue, highly variable).

I would now like to turn to the question of metrical types by considering the difference be- tween two types — duple (or equal ) and triple (or unequal ). To avoid problems of virtual artic- ulation, I will represent the pulses given in the time signatures as beats. Although in the follow- ing examples the measures we shall consider are notated as bar measures, the analysis should ap- ply equally to smaller or larger measures — mea- sures within the bar or measures comprising two or more bars.

As types, duple and triple are universals — every duple measure is an instance of duple me- ter, and, in this sense, every duple measure is the same. Thus, we might say that for a piece “in” 2/4, meter, at least at the level of the bar, is in- variant. But again, if meter is regarded as repeti- tion of the same, particularity and novelty will be ascribed to rhythm as something other than and therefore opposed to meter. Later I will argue that meter, even when viewed from the perspective of metrical type, is fully particular and never “the same.” But to argue against a re- duction of meter to type is not to dispute the re- ality of distinctions of metrical type as distinc- tions of feeling. And in the following discussion I will attempt to account for the emergence of such feelings from the standpoint of projection. We have already considered many examples of duple meter. Duple meter is created when the projected duration functions as continuation for the beginning of the projection. Since projec- tion is essentially binary and requires that the two terms be immediately successive, and since

projection results in equality, a projective ac- count of triple, unequal meter is problematic. Therefore, my discussion of metrical types will be focused primarily on the problems posed by triple meter. Instances of duple and triple meter are given in example 9.17.

In comparison to the projection shown in example 9.17a, projective potential Q is ex- tended in example 9.17b. Here there are two weak beats or two distinct continuations that prolong the activity and presence of the begin- ning (and, consequently, the projective potential that emerges from this beginning). The begin- ning of the third beat, like the beginning of the second, is denied as a beginning that would make the beginning of the nascent measure past. The third beat does, nevertheless, make the sec- ond beat past, thereby confirming the projected duration of the second beat and the completion of a projection. This might suffice for a descrip- tion of triple meter if we were to view projec- tion as an abstract scheme. Since we have substi- tuted a beginning that lasts throughout the group and continuations that “prolong” the be- ginning for Hauptmann’s binary unity of an ac- cented and an unaccented pulse, there is no sys- tematic problem raised by a third pulse (or a fourth or fifth pulse). However, the problem that

is raised is a perceptual one. If we allow a second

continuation, why can we not allow a third or a fourth? That we cannot is given by the fact that we do not. Presented with a series of “ob- jectively” homogeneous pulses (at a moderate tempo), we will spontaneously create groups of two or three (or multiples of two or three) and

The Perspective of Projective Process 131

a)

42

W Q X

œ

œ

b)

43

W Q X

œ

W

œ

œ

EXAMPLE9.17 Projected beats for (a) duple and (b) triple meter

not groups of five or seven. And if we are given a group of five (at a moderate tempo) we will hear this as a composite of duple and triple groups. Moreover, the evidence of musical practice shows that duple and triple (or equal and unequal ) measures constitute two basic forms of metrical organization. In the following section I shall at- tempt to account for this limitation of types through a systematic development of the notion of projection.

I said earlier that in example 9.17b the third beat, like the second beat, functions as a continu- ation. However, it must be noted that the initia- tion of the third beat functions as a more com- plex denial of ending than did the second beat.

The third beat cannot function exactly like the second beat simply to continue the duration be- gun “before” there were any beats, for now that there is a second beat there is also a real potential for projecting a half-note duration (the potential Q in example 9.17a). In order to function as a continuation, the beginning of the third beat must deny this potential. In contrast, the begin- ning of the second beat denied no potential — rather, it created one projection and the potential for another.

It may be helpful here to review this process in detail. (See example 9.18.) The beginning of sound is a potential for an as yet indefinite dur- ation. As potentially reproducible, a presently 132 A Theory of Meter as Process

a) ( P X)

œ

| d) W Q X ( P ) b) X

œ œ

¿

| \ d)c) R W

œ œ Œ

| \

œ

| d)

œ œ

W Y

œ

e) Q W

œ œ

Q'

œ

| | f) Q R W W

œ œ¿ œ

*

EXAMPLE9.18 Projective decisions for equal and un- equal measure

emerging duration is also potentially projective, and (in the absence of a prior projection) this projective potential is also durationally indefi- nite. In example 9.18a an indefinite projective potential P is indicated in parentheses. With the beginning of a second sound (example 9.18b) there is a projection. Since the second sound pre- sents a new beginning, a projective potential ini- tiated with the first sound is realized, and this potentiality is realized before the projected dura- tion is realized. If the second sound is also a con- tinuation, the beginning it continues retains pro- jective potential because this beginning is still “present.” But now it is a definite potential (Q in example 9.18b) because the projected dura- tion of the second sound is definite as a (more or less) definite potential. And yet, to the extent that the dominant beginning has the potential for re- maining active beyond Q, there is also a greater and still indeterminate projective potential P. If, as in example 9.18b, nothing happens after the second sound, Q will not be realized, P will be denied, and the beginning will have become past and inactive. However, as example 9.18c shows, we may have to wait some time after the second sound has ended to be sure that the beginning is really past (and that the potential P is really dead). And, in general, it seems that we hold onto beginnings for as long as we can.

In example 9.18d, with the introduction of a third sound, two possibilities are presented. If, as in example 9.18e, the third beat emerges as a new beginning (rather than a continuation), the projective potential Q will be realized, and the beginning of the third beat will project the du- ration of a half note. If, as in example 9.18f, the beginning of the third beat is perceived as a con- tinuation, the projective potential Q will be de- nied and will be replaced by the projective po- tential R. (The possibility for an R was shown in example 9.18b by P.) This is the denial of a defi- nite potential and the affirmation of a potential beyond Q that becomes definite only with the new beginning *.

Since the denial of projective potential shown in example 9.18f is a special sort of denial — dif- ferent from continuation as a denial of ending and different, too, from the denial of the projec- tive potential Q shown in example 9.18b — it will simplify our discussion to give it a name.

“Deferral” seems an appropriate word since it implies postponement, delay, putting off to a fu- ture time, and also the renunciation or the yield- ing of a claim. Deferral involves the cancellation of a prior and definite projective potential (Q in example 9.18f ). Since there is a postponement of a decision that would create a definite projec-

tive potential from Q to R (or a yielding of Q to

R’s projective claim), I will call this characteris- tic of triple meter “the deferral of projective po- tential.” But this is only one aspect of deferral and cannot in itself account for the phenome- non of triple meter. The other aspect of deferral directly involves not the expansion of projective potential, but the expansion of a projection. Of these two aspects of deferral, the second — ex- pansion of a projection (or what I shall call the deferral of projected potential ) — is conceptually the more difficult to grasp and will therefore re- quire closer analysis. Again, we must consider both what constitutes the event’s self-fulfillment and what the event can offer beyond itself as datum for a successor.

In examples 9.18e and 9.18f it was assumed that the beginning of a third beat offered the possibility of beginning a new “half-note” mea- sure. Now it may be fairly asked why we should imagine such a possibility for a third beat. Ear- lier, I suggested the following reason. Just as the second beat makes the first beat past, the third beat ends the second beat and makes the second beat past. However, since the second beat is a continuation that completes a projection, the third beat in making the second beat past also functions to make this prior and completed pro- jection past.

But in triple (or unequal ) meter the third beat does not function in this way — it is, in fact, a continuation of the nascent measure, and not a new beginning that promises to reproduce the duration of a completed two-beat measure. If, in a perception of triple meter, the third beat does not begin as a reproduction of a two-beat unit, what does it reproduce? Clearly, it reproduces the duration of the second beat. If, as in example 9.19a, the third beat, C, is not a new beginning or “accented” in relation to a continuative, “un- accented” second beat, we can regard each of the first two beats as projective.We will then recog- nize two projections: S – S ' and T–T'.

It will be remembered from the section of this chapter on division that an “overlapping” of this sort would be suppressed if a two-beat pro- jective potential (Q in example 9.18e) were to be realized in a projection (Q – Q '). However, in this case, since no projection Q – Q ', in fact, emerges, C can reproduce the duration B as B reproduced A. In view of this “transitive” rela- tionship, it should be possible for C, in reproduc- ing B, to reproduce something of B’s special re- lationship to A — namely, the function of con- tinuing a dominant beginning (d in example 9.19a). Thus, it would appear that what is given for a newly emerging C is not simply the pro- jective potential (T) of B, but also a completed projection (S – S ') in which B has functioned as continuation of a greater duration begun with d. If C is also continuation, it will therefore be the reproduction of a continuation (B in example 9.19b). Furthermore, by reproducing B as con- tinuation, C will, presumably, reproduce some- thing of the specific form of this continuation — a continuation that completes a projection. As in example 9.19b, I will use the symbol \ — \ to indicate this reproduction of function. To sum- marize: in functioning as continuation (rather than as a new beginning) C defers the completion of a projection S –S' to open the new projective potential R shown in example 9.19b.

As a result of deferral, the projective situation is complicated. A duration A – B has been cre- ated — there is now a unit A – B (| _{\), which is}

completed when C appears, and since A – B is past it is irrevocable and necessarily given for C (just as the duration A is past and given for B as B’s projected potential ). C reproduces B and continues the beginning of the nascent measure. But what does C’s inheritance mean for the

larger measure’s projective potential? I have speculated that by reproducing B, C can also re- produce B’s function of completing a projection. In this case, C is nothing apart from this projec- tion and will, in this sense, belong to a single projection A – B – C as an extension of continu- ation through a reproduction of the projected duration already realized by B. If C reproduces the projected duration of B, C is not merely added or tacked onto the completed A – B. Its novel contribution is to alter the entire projec- tive situation. C results in an enlargement of the projection and a deferral of its completion. Fi- nally, if the ending of B is now no longer the ending of a projection and C is not a beginning that will make the projection past, then C will have reproduced a result of B’s adjacency to A — that is, the duration of B as projected dura- tion (something that B inherited from A as a re- sult of projection). In this case, deferral means deferral of the realization of a projection or, to abbreviate this formulation, a deferral of pro-

jected potential — a denial that the projection is

completed with the ending of B.

That there should be two aspects to defer- ral — the deferral of projective potential and the deferral of projected potential — follows from the dual nature of projection, which necessarily in- volves both a projective and a projected compo- nent. These two aspects of deferral are fully complementary. By reproducing B’s realization of a potential for duration that would complete the projection (B’s promise for an end of the projection) and thus deferring the completion of the projection, C also defers the projective potential of A – B as a unit (Q in example 9.18e). By deferring the projective potential of A – B, C also becomes a continuation (and not a 134 A Theory of Meter as Process

a) A D d a S W

œ

B b S' T W

œ

C c T'

œ

|

|

|

\

| b) A R X

œ

B

––

œ

C

œ

|

\

\

EXAMPLE9.19 Deferral

beginning) that can reproduce the continuation B and, hence, B’s function of completing a pro- jection. These aspects of deferral are inextrica- ble. Their difference emerges as a difference of perspective — the difference between whether we regard the event for itself (the creation of a unified duration) or beyond itself (its potential for reproduction).

Since I have characterized deferral negatively, as denial, I wish to remind the reader that the reinterpretation effected by deferral involves more than denial or negation (as the delay or postponement of projective completion). Defer- ral also involves the creation of a novel projec- tion and a novel projective potential (through the renunciation of a more limited, “duple” claim). Likewise, the continuations created by the second and third beats are denials in that they deny the possibility of making the begin- ning past or inactive for the creation of duration. But continuation (as denial of negation) is obvi- ously creative — as continuations, the second and third beats keep the beginning present and ac- tive. Like continuation, deferral is a definite de- cision, a decision against realizing a definite po- tential, and, at the same time, a decision for the creation of new potential. Thus, I suggest that the third beat has two functions — it functions, like the second beat, as a continuation, but in order to do so it must also function as a deferral. And continuation, like deferral, must be under- stood in terms of both what the event is for itself and what the event is beyond itself.

The reader may have had the suspicion that the preceding account of deferral is an attempt, like Hauptmann’s, to fit the round peg of triple meter into the square hole of Paarigkeit, or an at- tempt to reduce triplicity to an underlying du- plicity. In defense of this account (or at least to keep the question open), I assure the reader that I would not present the hypothesis of deferral if I did not think it justified primarily on experi- ential grounds. However, I must concede that such a suspicion is not entirely unjustified; for, although I do not claim (as Hauptmann did) that a triple is created from the superimposition of two discrete duples, I cannot deny that the gov- erning hypothesis of projection is grounded in the twoness of immediate succession. If the no- tion of deferral is not to be regarded merely as a

methodological convenience, it must be asked if there is any evidence for the claims made by the hypothesis of deferral.

In the succession of three quarter notes in example 9.18d, if we hear triple meter, we do not first hear duple meter (example 9.18e) and then a change from duple to triple (example 9.18f ). In fact, the reader should find it virtually impossible in this example to hear both triple meter and the projection of a half-note duration initiated by the third beat, simply because once deferral has happened there is no possibility for such a projection. Since the potential for pro- jecting a half-note duration is not realized, there is now no feeling of duple meter; and now that there is a completed projection involving three beats, this projection was incomplete “before” there were three beats. Evidence for deferral cannot be adduced by treating a denied possibil- ity as if it were a realized actuality. But this does not mean that the denied possibilities involved in deferral are unreal. Moreover, the possibilities denied by deferral are definite possibilities that must make a difference in, or contribute to, the particularity of what is realized. If there is evi- dence of deferral, it will come not from a direct perception of what is denied, but indirectly from a perception of the particularity of what is actu- ally created.

In the case of continuation versus beginning, there is a feeling of weak versus strong: for duple meter, a succession strong-weak; for triple meter, a succession strong-weak-weak. However, I do not think that the third beat in triple meter is necessarily felt as “weaker” than the second. In certain contexts such a feeling may arise. For ex- ample, as Printz (quoted earlier) observes, if a triple measure lacks a first beat, the third beat can sound weak in relation to the second (as the first sound of the measure). However, there are also contexts in which such a distinction is far from clear. Thus, it is possible to perform exam- ple 9.19b without feeling that the third beat is definitely weaker (or “more continuative” ) than the second. And for this reason I maintain that categories of accent cannot account for the par- ticularity of triple measure.

My introduction of the notion of deferral is an attempt to account for the special feeling of triple meter or the difference in character be-

tween duple and triple. Traditionally, this has been called the difference between equal and unequal measures (temps egaux/inegaux, gerade/ungerade

Taktarten, battuta eguale/ineguale, tactus inaequalis).

Although this difference could be conceived simply as the difference between even and odd numbers, “equal” and “unequal” have referred to an essentially qualitative distinction. To use other words, we might say that duple sounds balanced,

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