That does not mean her music comes from the left side - the supposedly calculative, analytical side - of her brain. Nor does it mean she writes pieces that in any way espouse a left-leaning political philosophy.

What it means, Joan Tower says, is that she devotes a great deal of energy to thinking about whatever hand-written music happens to be just to the left of the spot at which her pencil has ground to a halt.

''Some composers know in advance just where they're going,'' she says. ''I don't. In the beginning I have to spend a lot of time on where I am and where I've just been. Yeah, I'm pretty slow. I can write about 2 1/2 minutes of music per month. That's one piece a year - two if I'm lucky.''

Tower, 48, is in her second year as the St. Louis Symphony's composer-in-residence. Her music was introduced to the St. Louis audience when St. Louis Symphony music director Leonard Slatkin included here ''Sequoia'' on the opening program of the orchestra's 1984-85 subscription season. The 1981 ''Sequoia'' was Tower's first work for full orchestra. She's recently completed a second. Titled ''Silver Ladders,'' it will be premiered by Slatkin and the orchestra on next weekend's Powell Hall concerts.

''Silver Ladders'' lasts 21 minutes, and it was a long time in coming.

''I started it in December of '85 and finished it just a month ago,'' Tower said. ''For me, 11 months for a piece this length is normal. It's harder writing for orchestra, sure, because it involves more instruments. But I think it would have taken just as long if I'd been doing a piece for solo flute or clarinet.

''You see, I don't do sketches in advance. I do start out with a basic idea, but basically I'm not very 'pre-compositional' in my thinking. I used to be, but that was because I felt insecure and needed some sort of map to get me through the infinity of choices that were available. Now I'm more of an 'organic' composer.

''I start. Then I take a look at what I've done and reshape it until it's the way I think I want it to be. Then I go on. Then I take another look at what I've done. I spend more and more time reshaping, more and more time working on the music's left side.

''Every instant of music has a past, a present and a future. The present, of course, is what's happening at a given moment. The past is everything that's come before, everything that's led up to that moment. For me, it's very important that the present grow out of the past, that past and present combined contain the seeds of the future. As a piece goes on it develops more and more past; it takes on more shape, and the more shape it has the more you know about where it's headed. It's like a tree: When it first sprouts you don't know how it's going to grow, but after it's been growing for a few years you have a pretty good idea of what it will grow into.

''You have to be careful, though. Even when a piece has a lot of shape there are still many choices to be made, and you always run the risk of cutting off what you've done. So I'm always looking to the left side, making sure that the music's present is consistent both with its past and with the future I have in mind for it. It's a very painstaking kind of sculpting. Yeah, it takes a long time.''

Tower began working on ''Silver Ladders'' shortly after she replaced Joseph Schwantner as the St. Louis Symphony's composer-in-residence. Jointly funded by Exxon Corporation and the National Endowment for the Arts and administered by the New York-based organization called Meet the Composer, Inc., the residency program requires that composers produce for their host orchestras one piece during each of their residency's two - possibly three - years. Resident composers are also expected to serve their host communities in various ways as advocates for new music.

To date, Tower's most visible activity has been her involvement with the Symphony's ''Chamber Music St. Louis 'On Stage' '' series, originally presented - with listeners as well as performers on stage - at Powell Hall but lately moved to the Sheld on Memorial. As Schwantner did when he helped launch the series in the 1983-84 season, Tower selects repertoire, organizes the programs and serves as emcee during the mid- or post-concert discussion sessions.

More recently, she's helped establish a visiting composer lecture series at Washington University. Christopher Rouse, whose new ''Phantasmata'' was premiered by Slatkin and the orchestra in October, was the first guest speaker; four or five other composers will lecture during the 1987 spring semester, Tower says.

A not very visible yet nonetheless important and time-consuming part of her work as composer-in-residence is her consultation with Slatkin about new pieces. Several hundred scores and tapes are submitted to the music director each season. Tower's job is to screen them and make recommendations. During the 1987-88 season, she says, Slatkin - who makes the final decisions - will follow up on two of her suggestions; a third piece will be programmed, she says, if a suitable soloist can be found in time.

Less officially, she spends time just ''hanging out'' with visiting soloists and conductors.

''This is one of my big shticks,'' Tower says. ''I feel that our superstar performers are not living up to their responsibilities when they ignore the music of our own time. I think that we, the composers, are obliged to make them aware of our presence. So I try to get a little dialogue going. I don't come on heavy; I mean, I don't tell them that they should do this or that piece. But if they ask me what I think about certain pieces I tell them. It strikes me that many of them don't have faith in their own opinions, and that's a sad commentary on the state of affairs.''

Asked to explain, Tower took a deep breath and said:

''I've noticed that the thinking of many big-name performers today has gotten very far away from music itself. When it comes time to evaluate a new piece, they're not sure if their opinion is just a matter of personal feeling or a real judgment. So they're not willing to put themselves on the line. If an artist decides he or she likes a piece, he or she has to be willing to put it out front. Managers, presenters and certainly audiences won't encourage them to do new music, so they have to stand alone with it, on their own.

In the case of Joan Tower, who on April 24 won the $150,000 purse for a piece she wrote for the St. Louis Symphony, it means she'll be able to get her teeth fixed.

''Really, I'm thrilled about this,'' Tower said over the telephone late one night last week. She was speaking from her home in a small New York town called Annandale-on-Hudson, where for the last 19 years she has comfortably served on the faculty of Bard College. ''This is great. My husband and I both need a lot of dental work, and this'll help pay for it. Heck, I'm just a composer; I don't have any insurance for this kind of thing. And if there's money left over, I don't know, maybe we'll take a trip or something.''

For anyone who came to know Tower during her three-year stint as the St. Louis Symphony's composer-in-residence, the response is typical. Tower is serious about her work, of course. But she seems reluctant to discuss that work, or her feelings about it, without first sending up a smoke screen of casual chatter. She is not at all inarticulate, especially when prodded with questions technical in nature. Like so many creative artists, however, she is shy; she makes jokes, I think, to postpone the difficult process of putting into words the ideas she best expresses through non-verbal means.

''No, seriously, it was very exciting to get this award,'' she said. ''I mean, this was only the second time I ever won a prize. The first time was in 1988, when I was a finalist in the Kennedy Center's Friedheim Competition. But then I just came in in fourth place. The funny thing is, the piece that got fourth place in the Friedheim was the same one that just landed me the Grawemeyer Award.''

What does this mean? In most athletic events, prizes have a meaning that is crystal-clear. The trophies go to the victors, to the individuals or teams that chalk up the most points, or run the distance in the shortest time, or do whatever else is required to prove - beyond the shadow of a doubt - their momentary superiority over the other participants. Occasionally there is a quibble, a result challenged and then reversed when it is discovered that a champion did not quite play by all the rules. Usually, though, there is no room for debate. The ultimate authority is the scoreboard or the stopwatch, and its declarations are based on irrefutable facts.

But it doesn't work that way in diving, figure skating and other Olympic-style events where gold and silver medals are handed out by panels of judges who work with subjective, arguably nebulous, criteria. It doesn't work that way in journalism contests, such as the one that recently generated the 74th annual round of Pulitzer Prizes. And it certainly doesn't work that way in the arts.

Tower's entry in both the Friedheim and Grawemeyer competitions was the same: a 22 1/2-minute piece titled ''Silver Ladders'' that was commissioned by the St. Louis Symphony and premiered in Powell Hall on a snowy evening in January of 1987. Neither in its printed form nor in its recorded realization did the music change one iota between then and now. Yet it was a mere finalist in one contest and the richly rewarded laureate in another. How does this happen? What does it mean, in a music competition, to win the prize?

What it means - and all that it means - is that a group of judges put their heads together and collectively made a decision. It's possible, of course, that one of the panels was severely predisposed in favor of a particular type of music, or even in favor of a particular composer. It's possible, too, that a certain member of one of the panels was so persuasive that he or she held sway over the entire voting procedure. In all likelihood, however, the judges were not biased, and not easily manipulated. Probably they were as knowledgeable as any judges in the field can be, and probably they came equipped with as much fair-mindedness as they could muster. Neither the Friedheim panel nor the Grawemeyer panel was ''better'' - more competent - than the other. They were just different, and thus so were their final pronouncements.

''It were not best that we should all think alike,'' wrote Mark Twain; ''it is difference of opinion that makes horseraces.'' The character in the novel ''Pudd'nhead Wilson'' who speaks those words was discussing not sport but the law.

Still, the wisdom applies. Even in areas that would seem to allow no room for arbitrariness, differences of opinion are productive. They prompt people to be more than just observers of a phenomenon; they encourage real thinking about the essence of whatever it is that is up for discussion. Joan Tower says that, all kidding aside, she is deeply honored to have been given this year's Grawemeyer Award.

Established in 1984 by retired Louisville industrialist Charles Grawemeyer and recently expanded to include the fields of political science, education and religion, the award program seeks to recognize ''the world's best ideas.'' In the music category, its previous recipients have been Witold Lutoslawski (for his ''Symphony No. 3''), Gyorgy Ligeti (for his ''Etudes for Piano''), Harrison Birtwistle (for his ''Mask of Orpheus'') and Chinary Ung (for his ''Inner Voices''). Ung, a Cambodian-born composer now living in the United States, is not yet much known. But the others form a constellation of the international music world's brightest stars. Tower is proud to be in their company. And she feels especially pleased that the final round of judging was done not by a group of her peers but by a group of educated, apparently tasteful music lovers from the Louisville area.

Too many music juries are ''incestuous,'' Tower says; they get hung up on technical details, or on aesthetic ideologies, and they lose sight of the fact that music is, after all, made for the listener. As far as Tower knows, Louisville's ''lay panel'' is the only one of its kind; its vote of confidence in ''Silver Ladders, '' she says, means a lot.

At the same time, Tower has misgivings about the attention the Grawemeyer Award has brought her. ''I was out of town during the week of the announcement - in Philadelphia, for a performance - and so I was, you know, hard to get a hold of,'' she said. ''But now I'm home, and the phone is ringing like crazy. Mostly it's people asking for interviews, but I suppose it won't be long before requests for commissions start to pour in.

''I know it will happen, but it won't seem right. Prizes, I think, are not the way to generate commissions or to build a career. And they have an adverse effect on the way people regard your work. Because we live in such a PR-oriented society, people tend to be passive in their relationship to the music of 'famous' composers; criticism tends to be much more creative, and perhaps more honest, when the composer is not at all famous.

''So when my students ask me about whether or not they should enter a competition, I usually tell them 'forget it!' They don't matter, and they can give a composer a false sense of security. All a prize means is that three, or six, or nine people somewhere took a vote and decided they liked your piece better than the others that were in the running. What matters is the music, not the prize. If an orchestra is going to offer me a commission, I want it to be because they like my music, not because some prize has made me suddenly famous.''

The real prize for ''Silver Ladders,'' Tower says, came with the performance in Powell Hall three years ago. It was not until 1987, near the end of her residency in St. Louis, that she finally got a sense of what the orchestra, as an institution, was all about.

''I was just waking up to the idea that the orchestra inhabits a world of masterpieces,'' she said, ''and I was just realizing that my music had to enter this world and compete. It was pretty scary. I knew that the players wanted me to write a really good piece for them, and I needed to prove - to myself as well as to them - that I could do it. For a little composer like me, there was a lot of pressure.''

The pressure resulted in music described by this reviewer as containing ''a powerful feeling of momentum,'' a ''rhythmically volcanic'' composition that ''drives relentlessly toward its climaxes'' and features enormous increases and releases of tension. From this perspective, ''Silver Ladders'' seemed not just ''a well-built, well-crafted piece'' but also ''a dynamic contribution to the modern orchestral literature.''