Mitsuko Uchida spots the helmet under my arm and greets me with a smile that lights up her whole face.

"Cyclist! How wonderful!" she exclaims. The great concert pianist, it turns out, is an avid follower of the Tour de France.

In days gone by she used to go on cycling holidays and read L'Équipe to keep up with the latest commentary. Lately she has given up riding at home in London because of the risk to her hands, but she still loves the sport and spends the first 15 minutes of our interview discussing the various merits of this year's winner Chris Froome ("he waited to finish the Champs-Élysées with his teammates - delightful!"), the undoing of Lance Armstrong ("the scoundrel") and the charms of "our mighty Wiggo".

Uchida is one of the musical nobles of our age. There are few performers who command as much authority, grace and stripped-back eloquence as she does at the keyboard. Born near Tokyo, raised in Vienna (her father was Japanese ambassador to Austria), the 64-year-old is a peerless interpreter of Bach, Mozart, Beethoven, Schubert and Schumann, and brings a rare lyricism to music of the Second Viennese School.

In person as in performance, she radiates the kind of warmth that puts you instantly at ease. She laughs easily and talks at a rate of knots when something catches her imagination, as it often does.

Last week Uchida played at the Proms for the first time in 19 years. I tell her I was stopped in my tracks by the simple beauty of her encore alone, the Sarabande from Bach's G-major French Suite. "Ah yes," she says, "that is one of the most beautiful pieces there are." She doesn't like playing encores in the middle of concerts - "isn't it impolite to the orchestra and conductor? I feel like we've been making music together so I shouldn't steal the moment" - but that night she wanted to give a little extra.

"In the morning I went to rehearse at the Albert Hall and saw a queue snaking around the block. People told me they had waiting since 4.30am. So I felt I owed it to them to play at least well enough."

Musicians often talk about being able to sense the mood of an audience, and I ask Uchida whether she is affected by atmosphere in the hall. "Very much. Mostly, I feel the quality of the silence. For example, in the G Major concerto" - Beethoven's Fourth, which she played with the Bavarian Radio Symphony Orchestra at the Proms and again on Sunday at the Usher Hall - "there is an extraordinary silence before I start. That's when I feel it most." On Sunday somebody in the audience dropped something at the last moment and the silence was lost.

It is wrong to describe Uchida as a perfectionist: it's too clinical a term for someone so impassioned, so spiritual about her art. But she does cares acutely about the sound she draws from the piano.

That's why, whenever possible, she tours with one of her own four Steinways. She refers to the instruments as her family. Number One, "the Oldie", dates from 1962 and doesn't travel much these days except for recordings. "Older instruments need more care; it's the same with human beings." She uses Number Two Boy (they're all boys) for chamber music and daily practising, while Number Four "is just about learning to exist. He's the youngster. I might take him for a serious ride next spring, when he's ready."

Number Three, "the Munich Boy", has come to Edinburgh. "He was born around 1995 and if he was in a concert hall he would be a doddery old man, but in my hands he is still a fresh instrument."

Why is it so important to play her own instrument?

"Look at it this way. A great Formula One driver wouldn't race on an unknown car that a local garage man had prepared. He might die. Concert pianists aren't going to die if they play a bad piano, but it is infinitely better if you know your own instrument and how it has been set up."

She giggles fondly. "A good car driver will know if someone else has driven their car. It's the same for me: if someone has touched one of my pianos, I will know instantly, because the piano will be cross!"

Tonight Uchida is back in the Usher Hall for a recital whose programme she describes as "a strange one". The first half contains two Bach preludes and fugues and Schoenberg's Six Little Piano Pieces Opus 19. The second half is all Schumann: the darkly enchanted Waldszenen, the blustery Second Piano Sonata and the tender, anguished Gesänge der Frühe. "I put the kindest, gentlest, most user-friendly Schoenberg with the strangest Schumann because I wanted people to notice how beautiful and friendly Schoenberg can be, and how peculiar Schumann can be."

The Gesänge, which Schumann wrote shortly before his attempted suicide, is particularly peculiar. "Harmonically it's horrendously difficult to make sense of," says Uchida, "and the tempo markings are extreme. At this troubled point in his life, Schumann's sense of time had shifted. And yet there's something very, very deep about its beauty. If I feel that, maybe other people will feel it too. That's why I play it."

Mitsuko Uchida is at the Usher Hall tonight