It kind of sounds like a dirge (funeral song), doesn't it? It's sad and slow, almost mournful, and Thom Yorke's voice echoes as if he's in an empty room. Thom will often carry one syllable over several notes ("girls in the ei-EI-ties/but gravity all-ALL-ways wins") and will drag it out, which adds a unique texture to the sound. A writer from The New York Times noticed this too, calling it a "pivot point," and discusses how it is used in Radiohead's first hit "Creep":
"'Creep' was the first of many Radiohead songs that used pivot tones, in which one note of a chord is held until a new chord is formed around it. (In the turn from G to B, the note B is the pivot point.) 'Yeah, that's my only trick,' Yorke said, when this was pointed out to him. 'I've got one trick and that's it, and I'm really going to have to learn a new one. Pedals, banging away through everything.'"
The music follows the progression of the characters' moods. At first it is melancholy: we begin in the key of A Major with a few lonely chords on acoustic guitar, eventually layering in some basic keyboard notes that sound electronic and alien. The first two verses alternate back and forth between A Major, f# minor, and D Major (the fourth chord in the key of A). All this flipping between major and minor chords adds a lot to the song because each line starts in a major key (happy) and ends in a minor key (sad), showing the progression of the characters' states of mind.
Johnny Greenwood used an old Hammond organ instead of a normal piano for some of the song's sonic layers, giving it a lush sound. The beginning of the song gives us a glimpse of this plastic couple's everyday life, starting with the girl. We get the sense that she's being observed by an outsider who takes pity on her situation. She goes about her business zombie-like, aware of her entrapment in a superficial world but resigned to the belief that she can't do anything about it.
When we are introduced to the male character in the next verse, the song picks up in momentum. Drums and more background keyboard notes come in, along with the beginnings of electric guitar. The notes build in intensity, with Yorke holding one exceptionally long note that is full of angst and bitterness right before singing, "She looks like the real thing / She tastes like the real thing," when the music really starts to crash and get louder. The song reaches its ultimate crescendo, appropriately, when he sings about wanting to bust through the ceiling.
Then, just as suddenly, the music drops off, gets quiet, and returns to the way it sounded at the beginning, timed exactly with Yorke singing, "It wears me out." The quick drop in sonic power perfectly mirrors the man growing tired and weary after fantasizing about his big escape. The music, and his imagination, grows more and more intense until he realizes that it's all hopeless and his world comes crashing back down around him, with even more weight than before. From here on out, it's doomsville for the guy. Yorke's voice gets softer and rises higher; a plea to the world. He ends repeating the words "If I could be all you wanted," and his voice trails quietly away, like someone who has just given up.