33

Theory of Variations

33.1. MUSICAL VARIATION

So back to the Flood: we discovered music, in the beginning, in the kitchen. Now we must tell the story of another, more poetic, origin. We cannot confirm that it happened like this across the ages. But the same history happened to us once upon a time.

In the confusion of a chaos of natural sounds, we imagine our early man for a moment less attentive to signs. Why should this be so, when he is usually so vigilant? Because one of these noises, still from the same cause, varies in its effects without there being any cause for alarm: the wind moaning more strongly or loudly, the wave battering the shore with more vigorous regularity, the sound of a pebble bouncing away, its echo reverberating farther and farther into the distance. If this development of sound signs were a source of information, he would listen for this. This is not the case, for the wind is always the wind, the sea goes on forever, and the distant falling of pebbles is neither here nor there; he is listening for something else: this residue is music. When effects vary and the cause is cancelled out, then the sound object comes into being, poised in readiness to be musical.

When, much later, within a settled musical civilization where music is in place and the ear conditioned, a new early man places the chaos of recorded sounds on the turntable, he has all too many reasons to decipher its contents, the natural and the cultural, in terms of one of the codes at his disposal. Whether the turntable goes round at about average speed or changes gradually or in “chromatic” leaps, there is our man, rediscovering ancient (though still highly conventional) experience; even if he is exasperated by the “wow,” or the harmonic progression the phonogène plays over anything recorded, the phenomenon deserves his attention, for it is the springboard for music: a variation that causes the contents to be forgotten.

It is important to understand that this new input into the musical gives a sort of symmetry to the first chapters of this book. We had imagined homo faber moving from the utensilar to the instrumental by a process in many ways analogous but where the emphasis was on the discovery of sound bodies, each one providing an object immediately related to structures. Such were musicianly invention and musical intention, both spontaneous. Here the birth of the musical is different: no longer from a discontinuous structure created by distinct bodies that enable objects to be easily identified but from a continuous movement coming from the object itself, revealing a form and bringing this form face-to-face with the “dimensions” of the perceptual field.

33.2. PERCEPTION OF VARIATIONS

Prosaically we rediscover the basic axiom “every structure depends on a variation,” but we are brought to this discovery through two very different experiential journeys. One discovers structure in a discontinuous configuration, a series of “musical” objects; the other sees it in the continuity of one single object.

So we should be tempted to study a glissando, for example, in terms of traditional pitch and thus to relate its value to a discontinuous calibration: it would occupy an interval of a fifth, for example. But this is a physicist’s approach. A glissando occupies pitch in a strange way: it is an original perception that has very little to do with the interval shown on the score: the glissando is the criterion or a new musical object, different in every way from the nominal interval it occupies in the symbols of traditional theory.

But the scale, the melody, the harmonic relationships seem to be indisputable referential facts. We thus need to understand how these various facts can be reconciled: the scalar structures of musical objects, on the one hand, and musical variations within objects—that is, continuous musical structures—on the other. It is commonly believed that we can relate the perception of the continuous to the discontinuous: this is a mistake often made by musicians; physicists, who sometimes also make it, have, however, identified the first two scales of sensitivity, structured differently according to whether discontinuous harmonic pitches (calibration in intervals) or continuous portions of sound (calibration in mels) (see chapter 30) were played.

An even more striking example of disparity was given in book 3, with anamorphoses: the dynamic variation in the sound of a piano note is not perceived as a variation in level but as another sound quality: an attack. An even more convincing example: if I vary the spacing of successive impacts, what physical law will enable me to predict what will happen? I simply expect a variation that conforms to a criterion of rhythm, from slow to fast. But soon the extrapolation ceases to be valid. The rhythm, now fast, changes meaning for perception: it is perceived as allure or grain. Then there comes a zone where once again we perceive a variation of the criterion of grain, from wide to dense. When the grain becomes “velvety,” this is yet another perception (which we have grouped together with grain to avoid spreading our analysis too thinly). Meanwhile, another perception has appeared concurrently, which no theory of rhythm could have foreseen and with no connection to the preceding one: pitch. We have already quoted the example of the bassoon to highlight not only this movement from one perception to another when the same physical parameter varies (frequency of a movement) but also their coexistence: this same causal parameter is perceived in two different ways at once, under two distinct criteria of the discontinuous and the continuous, grain and pitch.

33.3. VARIATION AND STRUCTURE

This section should be an important crossroads for the reader, both for bringing together the various approaches attempted in this work and for carrying out the essential synthesis of the two concepts.

This fundamental, ultimately very simple, musical truth is not particularly easy to see for two main reasons. The first, as we have said often enough, is that tenacious belief in parameters and physical laws. We must get rid of this prejudice once and for all. While an unequivocal physical law may well link a particular parameter to a resultant effect (or a number can unequivocally mark out a magnitude such as frequency or dynamic), the faculties of our ear have complete freedom to make us hear this or that as we please (we humans), and not as the logic of the particular parameter, for example its continuity, would dictate.

The other difficulty in reaching a proper understanding of music resides in the fact that, as we have just seen, it presents in two ways: through experience of the discontinuous, mentioned in the first chapters; and of the continuous, of variation, which we are about to discuss. Between the two, we have spoken at length about structure. How can we put all this into words?

We will return to the previous examples. As we have often said, physicists, perhaps without making the fundamental distinction between the criteria for the object and the structures within which it presents itself for their study, have nevertheless the merit of being the first to demonstrate the existence of two perceptual fields, one with simultaneous harmonic sounds they call harmonic, and the other melodic (an ill-chosen term) when, using quite special measures, they produce successive melodic gaps. Thus a fifth in the high register (A6–E7) would have the same “melodic magnitude” as a third in the medium register (C3–E3) on condition that they are successive notes, that there is no “tonal context,” and that (if we may be so bold) the musician who consents to take part in this experiment is happy no longer to act as a musician. This history of the mel at odds with the octave has long intrigued us, and the only explanation we can see is in a structural context. Physicists are wrong to emphasize simultaneity (harmonic) and succession (melodic). Their experience is apparently totally determined by their conditioning. But it is likely that mels (we have not had time to support these ideas with systematic experiments) are much more like continuous gaps (which disconcert the best musicians) than well-articulated melodic intervals.1

In any case, we can summarize these various experiments by placing the perceptual qualities related to the discontinuous and the continuous in two columns:

Such is the disparity of the main musical perceptions compared to what most strikes our contemporaries: the (mathematical) possibility of reducing everything to three parameters: frequency, level, and time.

How is our discontinuous sound made? From the continuity of the fragments it is composed of the continuity of rafts of frequency, dynamics, and pulsations. We can only perceive the discontinuous through the cohesion of its elements: the sensation of pitch from an A, and not the identification of 440 rhythmic pulsations; the velvetiness of a grain, and not the perception of a bow tearing off the rosin; the impact of an attack, and not the detail of the sound diminishing. The continuous is therefore the obverse of the discontinuous: it ensures it is perceived. The perceptions of these elements are therefore necessarily, naturally all different (otherwise there would be neither sound nor music).

Now we will look at what we have called the other gateway into music: through varying objects. We fundamentally change the orders of magnitude so well devised by nature. Instead of hearing sound the right way round, we examine it “back to front.” There are two ways of doing this: either, in the laboratory, with the aid of equipment unaffected by orders of magnitude, we receive revelations on what happens physically; our ear, sufficiently informed, can endeavor to hear what was masked in natural listening. Or else we transpose forms into a higher level of duration. The attack, anamorphosed, spreads out in profile. The frequency, too low, begins to float and beat right down to a sixteenth note. The glissando spreads out, different from a scale, or like a scale spun out to the point where its degrees are joined together. Now we find other perceptions, of the same order as those that underlay the objects of our listening to the discontinuous. We find them, for the same reasons, different.

So how is variation related to structure? At a certain level the (isolated, coherent) object was a structure of continuous elements welded together, not perceived individually. It was not itself perceived as a structure but as an object forming part of a higher-level structure: discontinuous. What if this object happens to be expanded (or joined to others) at a level such that now this (continuous) structure itself appears within the framework of normal perceptual durations? The whole previous (masked, unconscious) lower register of perceptions comes into play; the higher-level perceptions disappear, dissolve for lack of structure: the object is its own perceptual structure. If it happens to be composed of discontinuous elements itself, it is these, in turn, that will gradually take on the register of the previous perceptions.

33.4. TYPOLOGY OF VARIATIONS

If the musical domain of continuous, more or less fixed sounds is already extensive, the domain of combined variations is intractably complex. We will therefore proceed step-by-step and endeavor to establish similarities between perceptions that are all too strange, too previously unheard.

We have therefore chosen to make our entry into musical analysis through the study of the simplest deponent examples. Homogeneous sounds gave us access to the study of fixed masses; then these same fixed masses, this time formed in duration, introduced us to the theory of dynamic forms. But already, in order to take the latest experimental data into account, we have had to associate simple variations of harmonic timbre (linked profiles) with the dynamic profile itself, while still remaining within the framework of the theory of formed sounds with fixed mass. We can easily imagine that once variations are taken into account, this framework will sooner or later be destroyed, because the ear will stop perceiving the fixedness of a mass or the general profile of a form as primary data. This is where the “fortified zone” of the theory of variations begins.

We have already besieged it from afar in chapter 14, with musical duration. As what matters to us now is no longer musical duration itself but the criterion of variation, itself affecting a particular sound criterion, our initial idea will be a variation rate, something like a speed or a “density of information.” More precisely, to repeat the observations already made in the first chapter on typology, we could say that variations will come to our ears in three ways: either they will be slow enough for us to be able to link our perceptions to fixed discontinuous values (variation of a preceding criterion), or they will be comparatively rapid and will give us an original variation criterion in a “characteristic form,” or, finally, they will be too rapid to be followed, and their form, decipherable on the bathygram, for example, will be anamorphosed. This is what these perceptual mutations may be: variation criteria. Thus a female singer’s held note has a practically nonexistent profile, but her slight inflexions of nuance will be perfectly perceived, although without these variations distorting the practically homogeneous character of the sound. A bow stroke on the violin may present a clear variation in intensity perceived as an “envelope”: this is the median example of dynamic forms; finally, at the extreme, a piano sound, given the rapidity of its (dynamic) variation, gives a new type of perception: percussion-resonance anamorphosed sounds.

So density of information is the first criterion in our typology of variations. In the first example we will use the term slow progression of sound in a given dimension—mass or intensity, for example; in the second, medium variations, we will have profiles (the prototype being the dynamic profile mentioned in the theory of forms); finally, in the third, rapid variations, we will simply use the term anamorphoses (the perception of percussion-resonance is the model for dynamic variations; the very rapid glissando, heard as the crack of a whip, is another example of anamorphosis).

But this initial analysis needs to be completed. Speed of variation is, in fact, too broad a criterion for us to be able to refer to it effectively, even in a typological classification.

Besides, it would be a mistake to dig our heels in over numerical values: the information we are talking about is tainted by subjective curiosity linked to deciphering the event, and it may be that this is all the stronger with subtle variations (the singer) than large dynamic shifts (piano). The former belong to the suspense of the tightrope walker; the latter come from clear predictability.

We can also see that the median example (the formed bow stroke) floats between two clearer extremes. In one case we pay attention to the sound all the time, and so we know what we have “our ear on” (we do say of something moving that we have “our eye on it”); in the other case we received the impact of the sound right from the beginning. These median examples are situated between the two and borrow from both. We suspect that in the first example we will find fused perceptions that will bind together the criteria for the discontinuous in a temporal unity, whereas in the extreme example other criteria are perceived. We can only find this out from experimental material. The first example, therefore, raises the question of a morphology of profiles, the latter of a temporal anamorphosis of perceptions.

So we have just found a first criterion for the typology of variations: density of information. We need another to provide a rather better explanation of the character of what varies. We do have three modules for density of information, but information about what? As we have been plunged into the abstract, it is time to go over to the concrete side: the making of a variation.

Here also we can make three completely general distinctions. We may have a variation that is felt only as an imperfection in a desired stability; we have already mentioned this type of fluctuation. It may be a progressive development. Finally, it may be a modulation—that is, a development in stages—already sketching out a scalar structure.

So our typology of variations can be summarized as follows:

33.5. VARIATION CRITERIA

Since we have already isolated the dynamic criterion, having studied it in sounds with fixed mass, it only remains to study the variation of mass criteria like the variation of sustainment criteria. It would obviously be convenient to proceed in stages: studying sounds that apparently have no dynamic profile, where the mass can be considered as fixed. Then we would study selectively the movements of this mass in the pitch field: melodic progressions, profiles, or anamorphoses. Then, taking sounds with neither dynamic nor melodic variation, we would take those that have variations of mass. This is possible in certain specific cases. Now, it is very clear that as soon as we approach varying sounds, everything is given all at once, that melodic variations are linked to dynamic variations, and that it is extremely difficult to isolate mass variations that do not have a melodic profile. Finally, a study such as this, based on deponent examples, might seem to give results, but on a very limited scale since they cannot be extrapolated without the risk of making mistakes.

It is therefore without great conviction, rather as an exercise in method and a plan for research, that we will embark on the presentation of typical variations, since we cannot take on everything at once. So we will hypothetically separate the “representative examples,” if not of the “purity” of an isolated variation of one of these criteria, at least of its “dominance.” Then we will use this main type to shed light on other examples, always expecting surprises both in the recombination of textures and the reaction of the contexts.

Before going on to the four main examples—melodic and mass variations, allure and grain variations—we will see how this will bring us back to the comparisons of type, class, genre, and species.

Our general typology of variations will obviously apply to each of these four examples. First we will have to clarify the last two lines of the typology briefly outlined in chapter 24. Morphology will cover variations where the progressions and profiles are not disrupted by too great a density of information. Apart from these we will discover new perceptions experimentally through the “genres of sounds” that make them up. As long as our extrapolation of the discontinuous retains a value, we can talk in terms of melodic or mass profiles. As soon as we hear differently and something different, we have to name the new objects given to us by the experiment. This is precisely where, closely linked, we find variation criteria forming original configurations. As for examples of species, which are already difficult to situate or calibrate in the simplest of sounds, they will obviously become all the more so. For a variation criterion can also vary. This is even the norm. A melodic or mass profile can speed up, slow down, fluctuate, or modulate in the course of its duration. These variations (in the square. . .) could then be evaluated for their importance in the same perception through a second-order analysis, inevitably leading us back to density of information again: the large, medium, or small gap (in relation to the property under consideration) linked to a slow, moderate, or fast variation duration. We will apply these general principles one after the other to mass or sustainment variation criteria, in the knowledge that they have already been mentioned in earlier chapters.

33.6. TYPOLOGY OF MELODIC VARIATIONS

The involvement of densities of information explains the difficulties we were in to classify varied sounds, which may be fast or slow, simple or compound. The three types Ȳ, Y, and Y′ clearly correspond to the three “states” of information: slow, moderate, or fast. With the simplest example, which serves as a schema for many other developments in melodic evolution, the glissando, we will say that in the first form (Ȳ) it is not perceived as such, but as a continuous gradation of pitches; in the second (Y) it is a form in the tessitura, a suitably assembled and memorized movement; in the third (Y′) it is an anamorphosis that transforms the perception into a snapping sound, the crack of a whip (the tweeting of a bird, for example). Next to the note Y, supposedly simple, occupied by the variation of a single (thick or tonic) mass in the tessitura, we will place derivative types related to much more complex amalgamations, which we want to put together for various reasons, such as the way they are made (single causality) or used (overall function). Thus we find:

The large note W, even when long, brings together linked profiles; the cell K groups together disparate and necessarily scalar impulses; the motif M, homologous with the large note, exhibits an artificial organization. We cannot take these unpredictable morphologies any further. We will be coming back, however, to note Y, which has a “reasonable” profile, to remind ourselves that it had already been considered in traditional music.

Meanwhile, we must add some finishing touches to the typology in chapter 26, which only dealt with fixed masses or a simple development of mass (Y row) and relegated run-of-the-mill variations to a fourth row. What were this simple and these run-of-the-mill variations? On the one hand, surely our “fixed masses” already allowed for some fluctuation? On the other hand, our variable masses only suggested generally coherent developments resulting from the determinism of a causality, going from the very simple sound object Y to the large note W. Now we must take into account a “modulation” type, hence scalar, variation, the result more often of the will of a composer than an instrumental causality. We said rather hastily in section 28.7 that our theory of the object must be as free as possible from authorial intentions, since it is a study of material. The time has come to say that this is not always so easy for two reasons or in two cases. The first, which we find in classical music, produces visibly modulated objects, easy to break down but whose overall structure we may wish to consider. We therefore need a term to describe them: they are groups of notes. The second occurs in new musics, beyond the concept of the note. As long as there are notes, and instrumental notes, we know what we are talking about. As soon as there are new sound objects, more or less distinct or more or less fused together, merging into each other, we no longer know what to say. It does not matter whether they are composer’s objects, fashioned deliberately or chosen from a wide sample; they are already too specific to constitute a general material: we have said that we will call them motifs. This could well be condemned as a system in which every object would be so original that no commonality would be possible. What happens is different. Instead, we have a new style of musical perception, continuous now, that fuses the criteria formerly dispersed in notes into one object. So there is no reason to be alarmed if such objects present themselves as too original for typology. This means that they are already musical motifs, whether an object has been made like this or modulated intentionally, or whether a naturally occurring development or modulation has been chosen for its originality.

To sum up: the development-modulation pair can be applied in two contexts—the purely morphological context of a transition between continuous and discontinuous, and the context, this time musical, of the perception of an intention, the recognition of originality, whether in a natural development (large note W) or an artificial modulation (motif M).

So from these various approaches in our brief establishment of order, to complete the panoply of classification terms, we will retain the new definitions of the group G and the motif M. We will also keep in mind that the weft T denotes the slow development of scarcely differentiated structures, whereas the ostinato P can give rise to the development, also slow, of structures organized as groups.

We will then find a typological regrouping that may be summarized thus:

33.7. MUSICAL TRADITION OF MELODIC VARIATIONS: NEUMES

If it is true, as Armand Machabey writes, that neumes are “simply an aid to memory referring only to the movements of the voice, to the exclusion of the value of intervals and the absolute pitch of sounds and their duration,” if they “perhaps correspond to chironomic gestures . . . when the melody included melismas of two, three or four notes on the same syllable,”2 then we may conclude that for several centuries the musical unit was the varied musical object and not its breakdown into notes, an object that finds its equilibrium in music and text chanted syllable by syllable in this way.

Thus the aim of musical symbols was initially to express the movement and continuity of figures rather than the discontinuity of values. Soon condemned as superfluous, replaced more effectively by the notation of the result to be obtained, this type of figuring nevertheless reveals spontaneous musical intuition: the equilibrium of a gestural and a perceptual structure. So, for a very long time we have looked to letters (or other signs) to indicate values and to neumes to represent objects: when writing in neumes, which was linear, gave way to a notation in degrees on the staff, “tied notes remained,” and these continued to emphasize pitch relationships (intervals) and typological units simultaneously. Hence, for example, the clivis (descending interval) and the podatus (ascending interval); so we expect a symmetrical notation: now, the clivis is notated by a displacement, with the podatus superimposed on it; the former is feminine, the latter masculine; finally, the variation in tessitura doubtless also linked to the accent and the rhythm. Neumes thus brought features together and created figures full of meaning, the product of a highly evolved morphology.

Here there is a profound conflict between two ways of representing objects but also of thinking about music; there would be much to say on this subject. Ultimately, one of them had to win, not because it was right in essence but because it was practical, accurate, and it worked. On this utensilar level the other could not compete, but it alone holds the secret of Gregorian plainsong. It alone might have safeguarded the essential key, otherwise lost, to grasping hold of music directly, at the level of the object. Music has of course progressed, thanks to the precision of its notation and a necessary separation of the abstract and the concrete. But at the same time it has lost a certain sensitivity to the real and a certain inspiration linked to the direct involvement of the senses in the symbols of notation.

33.8. CLASSES, GENRES, AND SPECIES OF MELODIC VARIATIONS

Neumes, although intended to represent variations in a specific source (the voice), can provide us with a model. The notation (in relation to progressions and simple melodic and developing profiles of Y and ) is general enough for us to draw out our main classes of variations from it: podatus, clivis, torculus, and porrectus. We still have to move on to genres and think of a way to calibrate examples of species.

In music of the Western tradition melodic variation is usually inopportune: portamento is in dubious taste, as is the Hawaiian guitar—at least to our ears. In fact, it is in primitive or non-Western musics that we find this type of sounds, deliberately varied in tessitura: drawn-out sounds due to slack strings, giving rise to sounds where the continuation is consistently lower than the attack; the progressive rise in pitch of Japanese instrumental groups (Noh psalmodies); the “come hither” of Indian musics, which play with pitch in an orgy of erotico-mystic approaches and invitations. We will not go any further in our evaluation of possible genres of melodic variations.

We will be no less careful in our attempt to define species of melodic variations. We simply suggest adopting as definitive the rough pointers we have used before, which consist in evaluating the module of the variation in relation to its speed:

Different combinations of these numbers enable us to present the melodic profile, which may occupy the whole or only part of the duration of the sound, in diagrammatic form.

33.9. MASS VARIATIONS

The above domain was already more or less unexplored; this one is even less so. Electronic music thought it could approach it a priori, while it was done empirically by electroacoustic musics. We cannot really draw any conclusions from this, given the confusion that prevails in both the concepts and the applications. We can, however, understand the interest of such a study, which probably has greater potential than the study of melodic variations.

We have already touched on the subject in chapter 31, while examining certain variations in harmonic timbre. How does the mass profile differ from this? Neither more nor less than mass itself from its harmonic timbre. We have seen how much better we can distinguish a mass from its timbre as the ear is trained and also how we can act on each selectively, depending on the objects and using new procedures. We have already given examples of inversion of the harmonic profile: we know how to enrich it whereas normally it becomes poorer in the course of the sound. But these easy interventions, made with the potentiometer and filters, tend only to “sculpt” into the mass itself as a secondary process, either electronically or by electroacoustic manipulation.

Examples: a tonic sound can evolve into a thick sound, or vice versa: a resonant sound very loaded with harmonics or partials and relatively thick is gradually reduced into a tonic. We should again note that, conditioned by traditional listening, we are likely to hear only developments in timbre here; a more experienced ear will perceive the development of mass, independently of or linked to the development of timbre.

Here the distinction between development and modulation is also valid. The progressive organ arpeggio already mentioned is, in fact, a varying aggregate of low thick sounds to the unpracticed ear. In a typological context it is similar to a large note, but its scalar variation organizes its structure, and what we finally perceive is a variation of mass forming a motif. Many figures in modern music that use this type of mass effect (blocks of piano or other instrumental sounds) warrant the same analysis, which we introduced in sections 21.12 and 21.15: the fundamental variation-texture relationship.

33.10. SUSTAINMENT VARIATIONS

The violinist who allows his instrument to resonate after bowing it produces a sound that is initially sustained by rubbing and has a perceptible allure, then is prolonged by resonance: it is a type of sound to which we are so accustomed that we find both the change of grain and the change of allure quite natural. We are more taken aback by less familiar or more complicated sounds. Suppose we listen to the prolonged creaking of a door. Apart from the dynamic, melodic, or mass profiles, we have, of course, various grains one after the other: rough at the beginning of the sound, then smooth if the movement is more rapid, they may, if the door opener is skilled enough in his final deceleration, go all the way back down the scale of calibrations until at the end they are heard as iterations or separate impacts. The reader to whom this trivial example may seem very far removed from music will recall that it is no different in the orchestra, when the long final chords of a symphony are played by the brass section: the cymbal’s tremolo does not just give a dynamic crescendo but also a more and more violent scintillation—that is, the strongest variation of grain.

What do grain variations consist of? According to the examples, in going from rubbing to iteration or resonance; a typical variation of allure, going from a living to a mechanical or disordered sustainment. As for genres of grain or allure variations, we touched on these in the previous chapter and summarized them in boxes 63 and 73 in the general diagram.

Variations in amplitude and module in both grain and allure in the course of the duration of the object have already been shown (boxes 68, 69, 78, 79) in a schema similar to the diagram in section 33.8.

33.11. STRUCTURES OF VARIATIONS

If we reflect on sounds in general, we are obliged to think of situations with sound that cannot be contained in the above schemas, even if every effort has been made to include a first then a second degree of variations in them. In a cell of short, chance sounds, just as in an ordinary “sample” coming from the clumsy scraping of a bow, there are more sounds, more objects than we can describe. Hence two approaches intended to get rid of troublesome sounds: one is not to consider them at all, the other to go into great analytical detail and to spend more time on these sounds than would be needed to compose a study.

How can we keep the theory in a compromise that is neither simplistic nor excessively complex? Is the musical helped by the possession of such complex objects? Surely originality should be sought elsewhere?

Here we are reminded of the old gestaltist precept: a strong form takes its virtue from the clarity of its elements (and vice versa). As long as harmonic pitches are clearly perceived, pitch structures are strong. As soon as harmony mingles degrees and timbres, the distinctness of the pitches weakens, along with the structures. Serial structures, which often rest on playing around with musical writing and are not certain to be perceived, hang together by a thread. This thread is easily broken, and we come back to aggregates of notes, those “varying schemas” (Boulez) that appear as macro-objects or continuous structures and thus belong to a new type of musical analysis, a new theory, since we are dealing with new perceptions no longer taken into account by traditional notation.

What are these variation structures, of varied objects, in the real world? Quite simply structures of glissandi, variable masses, variously accentuated profiles. What is the mistake not to be made? The mistake of believing that we can tell from a first-order variation, at the level of the varied criterion, what the organization of the structure on the higher level will be. It is probably rather the opposite, and this is, indeed, the final stage of our approach. The making of varied objects must be considered experimental, and the approximations to “deponent” variations attempted under this name, which we know do not recombine automatically, are of little importance. In fact, ultimately the perceptual sanction will be brought in at the higher level of structures (not conventional, certainly, but also not premeditated, not calculated), which will be formed by assembling objects experimentally. Not only will we hear if such structures have meaning, but it is they that in their turn will draw out the dominant value of their constituent objects. Previously, they were simply materials whose characteristics had been analyzed, a bundle of sound criteria, of variations in particular. At this point the attention we gave them determined whether any of them seemed interesting, potentially musical. But should an otherwise disparate collection of them take on a structure by chance, this is where the musical appears, in accordance with the notorious permanence-variation relationship, on which rests the musicality of the structure linked to the musicality of its component objects. What will this element of permanence (of the object) that varies (in structure) be? In this chapter it is the variation criterion itself that ensures this permanence.

This concept is perhaps difficult to understand or give an account of, but it is very apparent to the ear. A structure of glissandi reveals the criterion “glissando” as permanent and gives its (musical) meaning to a variation of these glissandi: a variation of a variation. We can see that it is the mathematical formulation that leads us astray by its complexity, for, although rationally precise, it does not tally with the simplicity of musical experience. A glissando is no more complex to the ear than a frequency—probably less so. A piano note does not have varying characteristics, even though it is principally a dynamic form evolving in duration.

This said, theory stops here. Music begins. In fact, the movement from object to structure, and the meaning structure gives to the object, is the true birth of music. In traditional music this is called the Theory of Music, basically the theory of scales. We have said that we cannot go as far as this.