What is the subject of Beethoven's Fifth Symphony? Is it just those first four notes? Does it include the twin, transposed companion too? What of the other variations, augmentations, and inversions? Do they all stem from a single prototype? In this case, yes.

Or do they? For later in the symphony the theme appears in triplet form to serve as counter subject of the scherzo: three notes and one, three notes and one, three notes and one, still they make four. Melody turns into monotone rhythm; meter is converted to two equal beats. Downbeat now falls on an actual note, instead of a silence. With all of those changes, the themes are quite different and yet the same. Neither the form in the allegro nor the scherzo alone is the prototype; separate and equal, they span musical time.

Is there some more abstract idea that they both embody? This is like the problem raised by Wittgenstein of what words like game mean. In my paper on frames [Minsky 1974] I argue that for vision, 'chair 'can be described by no single prototype; it is better to use several prototypes connected in relational networks of similarities and differences. I doubt that even these would suffice to well represent musical ideas ; there are better tools in conceptual dependency, frame-systems, and semantic networks. (See Roads, 1980.)

What is a good theme? Without that bad word good, I do not think the question is well formed because anything is a theme if everything is music!

So let us split that question into (1) What mental conditions or processes do pleasant tunes evoke? and (2) What do we mean by pleasant? Both questions are hard, but the first is only hard; to answer it will take much thought and experimentation, which is good. The second question is very different. Philosophers and scientists have struggled mightily to understand what pain and pleasure are. I especially like Dennett's [1978] explanation of why that has been so difficult. He argues that pain "works" in different ways at different times, and all those ways have too little in common for the usual definition. I agree, but if pain is not a single thing, why do we talk and think as though it were—and then represent it with such spurious clarity? This is no accident: illusions of this sort have special uses. They play a role connected with a problem facing any society (inside or outside the mind) that learns from its experience. The problem is how to assign the credit and blame, for each accomplishment or failure of the society as a whole, among the myriad agents involved in everything that happens. To the extent that the agents' actions are decided locally, so also must these decisions to credit or blame he made locally.

How, for example, can a mother tell that her child has a need (or that a need has been satisfied) before she has learned specific signs for each such need? That could be arranged if, by evolution, signals were combined from many different internal processes concerned with needs and were provided with a single, common, output–an infant's sentic signal of discomfort (or contentment). Such a genetically pre-established harmony would evoke a corresponding central state in the parent. We would feel this as something like the distress we feel when babies cry.

A signal for satisfaction is also needed. Suppose, among the many things a child does, there is one that mother likes, which she demonstrates by making approving sounds. The child has just been walking there, and holding this just so, and thinking that, and speaking in some certain way. How can the mind of the child find out which behavior is good? The trouble is, each aspect of the child's behavior must result from little plans the child made before. We cannot reward an act. We can only reward the agency that selected that strategy, the agent who wisely activated the first agent, and so on. Alas for those behaviorists who wasted their lives life by missing this simple principle.

To reward all those agents and processes, we must propagate some message that they all can use to credit what they did; the plans they made, their strategies and computations. These various recipients have so little in common that such a message of approval, to work at all, must be extremely simple. Words like good are almost content-free messages that enable tutors, inside or outside a society, to tell the members that one or more of them has satisfied some need, and that tutor need not understand which members did what, or how, or even why.

Words like 'satisfy' and 'need' have many shifting meanings. Why, then, do we seem to understand them? Because they evoke that same illusion of substantiallity that fools us into thinking it tautologous to ask, why do we like pleasure? This serves a need: the levels of social discourse at which we use such clumsy words as 'like', or 'good', or 'that was fun' must coarsely crush together many different meanings or we will never understand others (or ourselves) at all. Hence that precious, essential poverty of word and sign that makes them so hard to define. Thus the word 'good' is no symbol that simply means or designates, as 'table' does. Instead, it only names this protean injunction: Activate all those unknown processes that correlate and sift and sort, in learning, to see what changes (in myself) should now be made. The word like is just like good, except it is a name we use when we send such structure-building signals to ourselves.

Most of the "uses" of music mentioned in this article–learning about time, fitting things together, getting along with others, and suppressing one's troubles–are very "functional, but overlook much larger scales of "use." Curtis Roads remarked that, "Every world above bare survival is self constructed; whole cultures are built around common things people come to appreciate." These appreciations, represented by aesthetic agents, play roles in more and more of our decisions: what we think is beautiful gets linked to what we think is important. Perhaps, Roads suggests, when groups of mind-agents cannot agree, they tend to cede decisions to those others more concerned with what, for better or for worse, we call aesthetic form and fitness. By having small effects at many little points, those cumulative preferences for taste and form can shape a world.

That is another reason why we say we like the music we like. Liking is the way certain mind-parts make the others learn the things they need to understand that music. Hence liking—and its relatives—lies at the very heart of understanding what we hear. 'Affect' and 'aesthetic' do not lie in other academic worlds that music theories safely can ignore. Those other worlds are academic self-deceptions that we use to make each theorist's problem seem like someone else's.

Note: Many readers of a draft of this article complained about its narrow view of music. What about jazz and other "modern" forms. What about songs with real words, monophonic chants and ragas, music made with gongs and blocks, and all those other kinds of sounds? And what about those listeners who claim to be less intellectual, to simply hear and feel and not to build those big constructions in their minds? We can't discuss here all those things, but we can ask how anyone could be so sure much about what their minds do. It is ingenuous to think that you "just react" to anything a culture works a thousand years to develop. In any case, because it's not my purpose here to define boundaries, it's better to focus in on something that we all agree is musical – and that is why I chose this Symphony. For what is music? All things played on all instruments? Fiddlesticks. All structures made of sound? That has a hollow ring. The things I said of words like 'theme' hold true for words like 'music' too: that word is public property, but not all the senses of its meanings to each different listener.

Theories about children need not apply to adults because (I suspect) human minds do so much self-revising that things can get detached from their origins. We might end up liking both The Art of Fugue and The Musical Offering, mainly because each work's subject illuminates the other, which gives each work a richer network of "significance." Dependent circularity need be no paradox here, for in thinking (unlike logic) two things can support each other in midair. To be sure, such autonomy is precarious; once detached from origins, might one not drift strangely awry? Indeed so, and many people seem quite mad to one another.

In his book Sentics [l978], the pianist-physiologist Manfred Clynes describes certain specific temporal sensory patterns and claims that each is associated with a certain common emotional state. For example, in his experiments, two particular patterns that gently rise and fall are said to suggest states of love and reverence; two others (more abrupt) signify anger and hate. He claims that these and other patterns–he calls them 'sentic'–arouse the same effects through different senses–that is, embodied as acoustical intensity, or pitch, or tactile pressure, or even visual motion–and that this is cross-cultural. The time lengths of these sentic shapes, on the order of 1 sec, could correspond to parts of musical phrases.

Clynes studied the "muscular" details of instrumental performances with this in view, and concluded that music can engage emotions through these sentic signals. Of course, more experiments are needed to verify that such signals really have the reported effects. Nevertheless, I would expect to find something of the sort for quite a different reason: namely, to serve in the early social development of children. Sentic signals (if they exist) would be quite useful in helping infants to learn about themselves and others.

All learning theories require brains to somehow impose "values" implicit or explicit in the choice of what to learn to do. Most such theories say that certain special signals, called reinforcers, are involved in this. For certain goals it should suffice to use some simple, "primary" physiological stimuli like eating, drinking, relief of physical discomfort.

Human infants must learn social signals, too. The early learning theorists in this century assumed that certain social sounds (for instance, of approval) could become reinforcers by association with innate reinforcers, but evidence for this was never found. If parents could exploit some innate sentic cues, that mystery might be explained.

This might also touch another, deeper problem: that of how an infant forms an image of its own mind. Self-images are important for at least two reasons. First, external reinforcement can only be a part of human learning; the growing infant must eventually learn to learn from within to free itself from its parents. With Freud, I think that children must replace and augment the outside teacher with a self-constructed, inner, parent image. Second, we need a self-model simply to make realistic plans for solving ordinary problems. For example, we must know enough about our own dispositions to be able to assess which plans are feasible. Pure self-commitment does not work; we simply cannot carry out a plan that we will find too boring to complete or too vulnerable to other, competing interests. We need models of our own behavior. How could a baby be smart enough to build such a model?

Innate sentic detectors could help by teaching children about their own affective states. For if distinct signals arouse specific states, the child can associate those signals with those states. Just knowing that such states exist, that is, having symbols for them, is half the battle. If those signals are uniform enough, then from social discourse one can learn some rules about the behavior caused by those states. Thus a child might learn that conciliatory signals can change anger into affection. Given that sort of information, a simple learning machine should be able to construct a 'finite-state person model." This model would be crude at first, but to get started would be half of the job. Once the baby had a crude model of some other person, it could be copied and adapted in work on the baby's own self-model. This is more normative and constructional than it is descriptive, as Freud hinted, because the self-model dictates (rather than portrays) what it purports to portray. With regard to music, it seems possible that we conceal, in the innocent songs and setting of our children's musical cultures, some lessons about successions of our own affective states. Sentically encrypted, those ballads could encode instructions about conciliation and affection, or aggression and retreat—precisely the knowledge of signals and states that we need to get along with others. In later life, more complex music might illustrate more intricate kinds of compromise and conflict, ways to fit goals together to achieve more than one thing at a time. Finally, for grown-ups, our Burgesses and Kubricks fit Odes to Joy to Clockwork Oranges.

If you find all this farfetched, so do I. But before rejecting it entirely, recall the question, Why do we have music, and let it occupy our lives with no apparent reason? When no idea seems right, the right one must seem wrong.

Compare a sonata to a teacher. The teacher gets the pupils' attention, either dramatically or by the quiet trick of speaking softly. Next, the teacher presents the elements carefully, not introducing too many new ideas or developing them too far, for until the basics are learned the pupils cannot build on them. So, at first, the teacher repeats a lot. Sonatas, too, explain first one idea, then another, and then recapitulate it all.

(Music has many forms and there are many ways to teach. I do not say that composers consciously intend to teach at all, yet they are masters at inventing forms for exposition, including those that swarm with more ideas and work our minds much harder.)

Thus 'expositions' show the basic stuff–the atoms of impending chemistries and how some simple compounds can be made from those atoms. Then, in 'developments', those now-familiar compounds, made from bits and threads of beat and tone, can clash or merge, contrast or join together. We find things that do not fit into familiar frameworks hard to understand–such things seem meaningless. I prefer to turn that around: a thing has meaning only after we have learned some ways to represent and process what it means, or to understand its parts and how they are put together.

What is the difference between merely knowing (or remembering, or memorizing) and understanding? We all agree that to understand something, we must know what it means, and that is about as far as we ever get. I think I know why that happens. A thing or idea seems meaningful only when we have several different ways to represent it–different perspectives and different associations. Then we can turn it around in our minds, so to speak: however it seems at the moment, we can see it another way and we never come to a full stop. In other words, we can 'think' about it. If there were only one way to represent this thing or idea, we would not call this representation thinking.

So something has a "meaning" only when it has a few; if we understood something just one way, we would not understand it at all. That is why the seekers of the "real" meanings never find them. This holds true especially for words like 'understand'. That is why sonatas start simply, as do the best of talks and texts. The basics are repeated several times before anything larger or more complex is presented(l. No one remembers word for word all that is said in a lecture or all notes that are played in a piece. Yet if we have understood the lecture or piece once, we now "own" new networks of knowledge about each theme and how it changes and relates to others. No one could remember all of Beethoven's Fifth Symphony from a single hearing, but neither could one ever again hear those first four notes as just four notes! Once a tiny scrap of sound, these four notes have become a known thing–a locus in the web of all the other things we know and whose meanings and significances depend on one another.

Learning to recognize is not the same as memorizing. A mind might build an agent that can sense a certain stimulus, yet build no agent that can reproduce it. How could such a mind learn that the first half-subject of Beethoven's Fifth–call it A–prefigures the second half–call it B? It is simple: an agent A that recognizes A sends a message to another agent B, built to recognize B. That message serves to "lower B's threshold" so that after A hears A, B will react to smaller hints of B than it would otherwise. As a result, that mind "expects" to hear B after A; that is, it will discern B, given fewer or more subtle cues, and might "complain" if it cannot. Yet that mind cannot reproduce either theme in any generative sense. The point is that inter-agent messages need not be in surface music languages, but can be in codes that influence certain other agents to behave in different ways.

(Andor Kovach pointed out to me that composers do not dare use this simple, four-note motive any more. So memorable was Beethoven's treatment that now an accidental hint of it can wreck another piece by unintentionally distracting the Listener.)

If sonatas are lessons, what are the subjects of those lessons? The answer is in the question! One thing the Fifth Symphony taught us is how to hear those first four notes. The surface form is just: descending major third, first tone repeated thrice. At first, that pattern can be heard two different ways:

But once we have heard the symphony, the latter is unthinkable–a strange constraint to plant in all our heads! Let us see how it is taught.

The Fifth declares at once its subject, then its near-identical twin. First comes the theme. Presented in a stark orchestral unison, its minor mode location in tonality is not yet made explicit, nor is its metric frame yet clear: the subject stands alone in time. Next comes its twin. The score itself leaves room to view this transposed counterpart as a complement or as a new beginning. Until now, fermatas have hidden the basic metric frame, a pair of twinned four-measure halves. So far we have only learned to hear those halves as separate wholes.

The next four-measure metric half-frame shows three versions of the subject, one on each ascending pitch of the tonic triad. (Now we arc sure the key is minor.) This shows us how the subject can be made to overlap itself, the three short notes packed perfectly inside the long tone's time-space. The second half-frame does the same, with copies of the complement ascending the dominant seventh chord. This fits the halves together in that single, most familiar, frame of harmony. In rhythm, too, the halves are so precisely congruent that there is no room to wonder how to match them–and attach them–into one eight-measure unit.

The next eight-measure frame explains some more melodic points: how to smooth the figure's firmness with passing tones and how to counterpoise the subject's own inversion inside the long note. (I think that this evokes a sort of sinusoidal motion-frame idea that is later used to represent the second subject.) It also illustrates compression of harmonic time; seen earlier, this would obscure the larger rhythmic unit, but now we know enough to place each metric frame precisely on the afterimage of the one before. Then,

Now it is the second subject-twin's turn to stand alone in time. The conductor must select a symmetry: he or she can choose to answer prior cadence, to start anew, or to close the brackets opened at the very start. Can the conductor do all at once and maintain the metric frame? We hear a long, long unison F (Sub dominant?) for, underneath that silent surface sound, we hear our minds rehearsing what was heard.

The next frame reveals the theme again, descending now by thirds. We see that it was the dominant ninth, not sub dominant at all. The music fooled us that time, but never will again. Then, tour de force: the subject climbs, sounding on every scale degree. This new perspective shows us how to see the four-note theme as an appogiatura. Then, as it descends on each tonic chord-note, we are made to see it as a fragment of arpeggio. That last descent completes a set of all four possibilities, harmonic and directional. (Is this deliberate didactic thoroughness, or merely the accidental outcome of the other symmetries?) Finally, the theme's melodic range is squeezed to nothing, yet it survives and even gains strength as single tone. It has always seemed to me a mystery of art, the impact of those moments in quartets when texture turns to single line and fortepiano shames sforzando in perceived intensity. But such acts, which on the surface only cause the structure or intensity to disappear, must make the largest difference underneath. Shortly, I will propose a scheme in which a sudden, searching change awakes a lot of mental Difference-Finders. This very change wakes yet more difference-finders, and this awakening wakes still more. That is how sudden silence makes the whole mind come alive.

We are "told" all this in just one minute of the lesson and I have touched but one dimension of its rhetoric. Besides explaining, teachers beg and threaten, calm and scare; use gesture, timbre, quaver, and sometimes even silence. This is vital in music, too. Indeed, in the Fifth, it is the start of the subject! Such "lessons" must teach us as much about triads and triplets as mathematicians have learned about angles and sides! Think how much we can learn about minor second intervals from Beethoven's Grosse Fuge in E-flat, Opus 133.