THE SIGH is audible over the phone from Los Angeles. "You don't really want to know about my marriages, do you? It's all so boring. Were there seven? - Oh, hell, I lost count. When Dorothy Kilgallen died I lost my score-keeper. Besides each one was over in three months - no, make that a week. But, anyway, that's all part of the Artie Shaw business and I got out of the Artie Shaw business 23 years ago."

Artie Shaw, the swing great, the perennial husband of glamor girls, the Peck's Bad Boy of jitterbug, is alive and well and living in Los Angeles, and - writing. Writing? Writing - exactly what he says he started out to do, lo those many years ago before music, fame, women and alimony got him sidetracked.

Up until 1936 Shaw wasn't much different from any other kid who could play the clarinet and ran away from home in New Haven, Conn., to play in a band. Except, of course, that what he really wanted to do, he says, was write. So after several years of band bumming around the country he settled in New York to free-lance for radio station orchestras while attending literature courses at Columbia.

That lasted until 1936 when Shaw entered a swing contest and knocked their socks off by forsaking the usual brass section to play clarinet solo with a string quartet background.

"From that point on," says Shaw, "I got into the celebrity business. I was the Elvis of my day."

Not to mention the Don Juan. Beginning by marrying then-starlet Lana Turner ("Why did I marry Lana? Well, who wouldn't have?), Shaw's succession of wives included Elizabeth Kern, daughter of composer Jerome Kern, Ava Gardner, novelist Kathleen (Forever Amber) Windsor, actress Doris Dowling and his last wife, Evelyn Keyes, from whom he has been separated for five years.

Since then, however, Shaw has lived with a woman whose name he won't mention but who is currently studying nutrition at a California university.

Although Shaw was always trying to quit the music biz he always came back, until, that is, 1955 when "Nixon and company got the IRS on my tail and I said to hell with it and went to live in Spain." But "being a corrupt American kid who can't live far away from a hot pastrami sandwich" Shaw came back five years later, moved to Connecticut and went into the film distribution business.

He also kept writing, turning out two works. "The Trouble With Cinderella" and a trio of novellas, "I Love you, I Hate you, Drop Dead, "about the break-up of three marriages - a subject I know a lot about."

Currently, Shaw writes only fiction - "the last way to tell the truth without hurting feelings and getting sued" - even though his work is not published under his own name. "It gets in the way."

Somewhere along the way, he also had two children whom he never sees - "I didn't get along with the mothers so why should I get along with the kids? - and slipped in five years of Freudian analysis. As a result, he is today, he says, "absolutely determined not to do anything I don't want to do. I've paid dearly for that privilege."

So what might Shaw like to be remembered for? "I don't care if I'm forgotten. I became a specialist in non-specialization a long time ago. For instance, I'm an expert fly fisherman. And in 1962 I was ranked fourth nationally in precision riflery. My music? Well, no point in false modesty about that. I was the best. And when you look at those of us who were big then - Miller, Dorsey, Basie, Goodman - I think my life has turned out the best, too."

As for the women - or rather the wives - well, Shaw is philosophical about that, too. "Listen, I started getting married at roughly 21. I'm 67 now and during the time I found maybe 10 women. Now, when you think about it, that's not so may.

"The only problem was that I married most of them."