I often get asked how I can 'move my fingers so fast'. I don't feel that my fingers are fast: listen to Artur Schnabel or Vladimir Horowitz - I grew up with their sound. Anyway, the simple answer to what I can do is practise - practise until all movements seem entirely memorised by fingers and arms, so that I do not have to be consciously aware of them. But there are others.
It was from my violin teacher Artur Garami that I first learned how important it is to avoid muscle tension. During all my lessons with him, half his time was spent circling me, watching with eagle eye for any hint that one muscle was fighting another. At first, I would have to stop playing, put my fingers on my muscle with his, feel the tension, then concentrate on it while thinking "relax - relax ...", even force it to extend with an opposing muscle. Eventually, I was able to tell almost every muscle in my body to relax as directly as I could tell it to contract. Tension kills the sound of a fiddle - its sound must lift the bow off the strings, its body must move air, not be clamped by the player's body.
Playing a keyboard, the benefit is different. If a muscle starts to get tense, the opposing muscle has to work harder to move. Then that muscle has to relax further or the return motion is impeded still more. Tension kills speed, and it kills control.
Place one hand in normal playing position, on a table. Concentrate on relaxing it, totally. Use the other hand to lift a finger, then release it. Watch the exact motion the finger makes when it falls: the base joint returns to normal position first, then the mid one, while the end joint will probably not return to its relaxed angle until after your finger tip has contacted the table. Then use the hand's own muscles to pick the finger up. Then, relax it and watch the motion - it should be exactly the same as when the other hand lifted it. If it isn't, you aren't relaxing the finger muscles fast enough, so keep working at it. Practise this with every finger, wrist, arm and shoulder muscle. (Pick your location for this - I nearly got expelled from school in grade 5 for 'fiddling with my fingers' !)
Approach trills the same way - with one finger at a time first. Using two fingers, it's easy to hide tension - playing repeated notes with one finger makes tension obvious. Once you can play a note repeated with any single finger at half the note speed of a scale or two-finger trill, you can be satisfied that you are relaxing your muscles totally between each motion. The finger should never leave the key, but just move the pluck depth of the plectrum. It can't be done that way on a piano action, and it drives pianists nuts! Scarlatti obviously did this, cf. K.141 or K.455. K.141 in particular can be used to display the technique, with each finger in turn, two-finger 'trill' fingerings on the same note, then with all fingers flailing as pianists have to.
The goal is to be able to relax every muscle used in playing as quickly and as directly as it can be contracted.
Inertia is the resistance of something to a change in motion. Our body has inertia - changes in motion must be as smooth as possible to minimize effort.
To play an ascending C major scale with the right hand, we use 1-2-3-1-2-3-4-... Simplify it for study: just move the 2nd finger as your hand moves up the keyboard - D-G-d-g-d1-g1... Concentrate on moving that finger in a perfectly smooth wave, every knuckle moving smoothly straight up and down. Your hand must stay level and move smoothly from left to right. Repeat with the 3rd finger alone. Now, practise the thumb - its end must move smoothly in a circle. No abruptness in speed, the simplest possible path of motion.
Now put them together, two fingers at a time, then all three, only then add the thumb. Gradually, your note spacing will become even - the cycle of movement of the fingers will become slightly slower over f-b than it is over c-e, and your hand will move at a perfectly constant speed over the keyboard. The thumb will move in a circle a bit taller than wide over c, a bit wider than tall over f.
Continue, of course, for the left hand, for descending scales, then for all keys. Always, at the slightest hint of tension, stop and totally relax all playing muscles before starting again.
To play really fast scales with one hand, in places where control may be safely subordinated to raw drama, the smoothest possible motion is to turn the hand completely sideways to the keys, then to 'run' with the fingers along the keyboard. Actually, the fingers move vertically as fast as possible, and the knack is to move the hand at the correct speed so that the scale notes are hit in sequence. 5-4-3-2-5-4.. with my right hand is much the fastest, and I use it for the cadenza-like descending scales of Scarlatti (cf. K.124). A few skipped notes are not audible in performance.
The fastest scale of all is a glissando, angling the thumb so that its nail depresses each key in turn as the hand is moved, without the thumb moving up and down. Although easiest with an all-naturals scale, it can be used with one sharp key per octave, flicking the third finger down and the thumb up as the hand passes the key in question. The sound this makes is far too rough for frequent use, but is very attention-getting when used once per concert (cf. K.487). However, Scarlatti noted a dozen of them in K.379!
To play Scarlatti, you have to maximize the speed at which you move a hand reliably from one note to another far away. Mostly, this just requires awesome shoulder-muscle memory - practise. However, there is one way in which feel can work fast enough to aid in this. If your hand lands at the front edge of the sharps, non-playing fingers can be extended ahead of the finger that is to press the key. The feel of sharp key patterns is sufficiently distinct that, with practise, the playing finger can adjust the tiny bit necessary to hit the center of the desired key.
For example, if a left hand leap to a low G is required, hold the 3rd finger a bit low. It will run into the edge of the Bb, with practise leaving the 5th finger in perfect position for G. For A, use the 4th finger the same way, for F the 2nd finger. For low sharps, I prefer to play with the 4th finger, and use the 2nd and 3rd fingers to feel. E is the most awkward key to hit - the only way I find useful is to place the 2nd finger raised beside the 3rd so that the pair fit on G and Ab, but it is not as reliable as the others.
Of course, the fingers that are used to locate the hand must be relaxed enough that no keyboard noise results - 'run into' means that the finger feels, not that the hand is stopped by the key. The method gives just a little extra accuracy, that can either breed sureness for sight reading, or a crucial extra turn of speed for Scarlatti.
The muscles of our hands are interconnected to aid in holding and manipulating things - harpsichord playing had no role in its evolution! So, our playing must adapt to our hands.
I often am asked whether my use of double-third trills in Scarlatti (cf. K.450) is appropriate, by those who misquote Couperin to argue that it is 'impossible'. (Actually, in the original French, Couperin just said that he didn't think it worth learning at his age, but recommended that young students try.)
I believe that Couperin was talking of a trill played with fingers 2 and 3 in parallel with another played with 4 and 5. Anatomically, that is indeed next to impossible, even slowly. (In fact, I find even the sustained trill of Scarlatti's K.357 with it's simple 5th finger obligato extremely awkward.) But, pairing the thumb and 5th finger, to alternate with paired fingers 2 and 4, fits our hand very naturally, especially with 2 and 4 both on sharp keys.
I'm certain from his notation that Scarlatti knew this technique, but I confess to finding few places in Couperin where it works. Double trills are a very powerful sound, best used sparingly.