I have no clue of the name, and this is a bit long, but...
From the Wiki:"Musical score: Main article: Vertigo (film score)
In a 2004 special issue by Sight and Sound devoted to Film Music, Martin Scorsese described the qualities of Herrmann's famous score.
S&S: What is your favourite film soundtrack music and why do you like it so much?
A big question. There are so many, and they all work so differently ? from a big, beautiful score for full orchestra like Jerome Moross' for Wyler's The Big Country or David Raksin's for Force of Evil, to a more modern score with very spare instrumentation, like Giovanni Fusco's for L'Avventura or Hans Werner Henze's for Resnais' Muriel. I suppose that if I were hard-pressed to answer this question ? and I suppose I am ? I'd have to say Bernard Herrmann's score for Vertigo. Hitchcock's film is about obsession, which means that it's about circling back to the same moment, again and again. Which is probably why there are so many spirals and circles in the imagery ? Stewart following Novak in the car, the staircase at the tower, the way Novak's hair is styled, the camera movement that circles around Stewart and Novak after she's completed her transformation in the hotel room, not to mention Saul Bass' brilliant opening credits, or that amazing animated dream sequence. And the music is also built around spirals and circles, fulfilment and despair. Herrmann really understood what Hitchcock was going for ? he wanted to penetrate to the heart of obsession.
And from the article on Herrmann:
Herrmann is most closely associated with the director Alfred Hitchcock. He wrote the scores for every Hitchcock film from The Trouble with Harry (1955) to Marnie (1964), a period which included Vertigo, Psycho, and North by Northwest. He oversaw the sound design in The Birds (1963), although there was no actual music in the film as such, just electronically created bird sounds.
His score for Vertigo is seen as just as masterful. In many of the key scenes Hitchcock let Herrmann's score take center stage, a score whose melodies, echoing Richard Wagner's Liebestod from Tristan und Isolde, dramatically convey the main character's obsessive love for the woman he tries to shape into a long-dead, past love.
A notable feature of the Vertigo score is the ominous two-note falling motif that opens the suite ? it is a direct musical imitation of the two notes sounded by the fog horns located at either side of the Golden Gate Bridge in San Francisco (as heard from the San Francisco side of the bridge). This motif has direct relevance to the film, since the horns can be clearly heard sounding in just this manner at Fort Point, the spot where the character played by Kim Novak jumps into the bay.
Bernard Herrmann said in a Q&A session at the George Eastman Museum in October 1973, that unlike most film composers who did not have any creative input into the style and tone of the score, Herrmann insisted on creative control or he would not score the film at all. Herrmann said:I have the final say, or I don?t do the music. The reason for insisting on this is simply, compared to Orson Welles, a man of great musical culture, most other directors are just babes in the woods. If you were to follow their taste, the music would be awful. There are exceptions. I once did a film The Devil and Daniel Webster with a wonderful director William Dieterle. He was also a man of great musical culture. And Hitchcock, you know, is very sensitive; he leaves me alone. It depends on the person. But if I have to take what a director says, I?d rather not do the film. I find it?s impossible to work that way.
Herrmann stated that Hitchcock would invite him on to the production of a film and depending on his decision of the length of the music, would either expand or contract the scene. It was Hitchcock who asked Herrmann for the "recognition scene" near the end of Vertigo(the scene where Jimmy Stewart's character realizes in a sudden instant Kim Novak's identity) to be played with music.