Set up:Twin Peaks, Population 51,201, is a small set inside Washington state on the Canadian border. Pete Martell, who helps run the local sawmill, discovers Laura Palmer, dead, wrapped in plastic tarp on the lakeshore. The local law enforcement, Sheriff Harry S. Truman and his deputies, are immediately on scene.
Ronette Pulaski stumbles into town in a daze, skimpily dressed, with the same twine marks found on her wrists that Laura Palmer had.
FBI Special Agent Dale Cooper arrives, having been on a similar case where other young ladies have died. The letter J is found underneath her fingernail and implicates everyone whose name begins with J, which is nearly every character on screen.
Discussion with Spoilers:John Gardner in The Forms of Fiction argues that the form helps create meaning. This can be true if with limited utility. I'll start with the "mystery" as a form that shapes what is and isn't going on here.
We have a crime--murder--and typical clues, such as the letter J. But the show veers from the form in its unorthodox methods of detection, primarily in its use of dreams, clairvoyance, and intuition. We accept this, in part, because Sherlock Holmes our founding father of mysteries, worked with such encyclopedic knowledge that it often seemed supernatural. However, one might attribute their use to scientific observations such micro-expressions or body gestures although none of that is specifically used.
Agent Cooper accepts his dreams and hallucinations as a legitimate form of evidence, and investigates on the basis of them, which no one questions. He throws rocks at a bottle to decide whether a person whose name starts with J is related to the murder of Laura Palmer.
The letter J is not only the name of the perpetrator, but also segues into the name of a culpable place. Even clues have double meanings.
Doubles are everywhere. Nearly everyone leads a double-life relationally--a bad one and a good one. If you're blond in Twin Peaks, your primary relationship is bad. Laura Palmer leads a double life as a popular student, but also a destructive one as a professional call girl who has a cocaine habit, which she uses to force her popular, football-playing boyfriend into the double life of dealing the drug. Palmer doubles again when her cousin arrives (played by the same person, except with dark hair). At one point, the actress plays the doubled cousin doubling as her cousin.
Most of the characters have these double relationships, double employments. Benjamin Horne, for instance, not only owns an above-board lodge and department store, but also has plans for the destruction of the sawmill and its owners, not to mention some stake in the prostitution ring in the casino on the other side of the Canadian border. Canada isn't Canada, per se, but that borderland of personality where people aren't who they display on the surface.
The town and its attitudes and limited locales feel more provincial and casual than a town of 50,000. 5,000 may be more accurate. This, too, might be considered a double: plenty of employment opportunity of a larger town but with the casual and intimate relations of a smaller town.
Interestingly, the secondary double assumes the primary role within the narrative in most cases.
I'd be curious to learn what true-form mystery aficionados thought of the series when it aired. One imagines a lot of grumbling consternation.