speed school:

When I was a teenager I assisted to music lessons at the Conservatori del Liceu, Barcelona. At the same time, I trained track and field in a running club in Hospitalet. I was training for 100m, 200m, 400m so I was a sprinter. While in the field I was told to run as fast as possible and all the efforts were directed toward this goal, in the music school I was advised to skip the ornamentations and the fast pieces. I was even told to slow down, play slowly. Rapid technique seemed a matter of time, an adult thing. Back in the field, adults usually run faster than kids, so I supposed that slowing down at the piano made sense. But it didn't. In fact, that was total nonsense. Running, an immature body has not developed all the muscles and bones yet, so it made sense to expect more speed after some years and it made sense not to force the body to let it develop properly. However, with music, the kid's brain is probably faster than the adult's. And you will never, never, damage a brain to invite it to learn speed playing. On the contrary, the method I was being taught damaged my abilities and ultimately broke my career chances. I was doomed for the very beginning and I was paying for that. Here I will discuss some things I have learned about speed playing during the last year. I think they should be taught at the earliest age.
The first thing about speed is its definition. In physics, speed is simply length divided by time, but we are not interested in length here. Instead, we think of notes, so in fact we will work with frequency, which is the number of cycles per time unit. In music, the number of notes per second. Does this subject belong to music itself? This is the same question as to whether the mechanics of a piano hammering belongs to music. The answer doesn't matter: without a good mechanics of a piano and without a proper technique, music won't be properly played. So let's move on to what is speed in music. Speed is the ratio between one note divided by the time of the processes needed to perform that note. To play a single note, on average, we need a lot of processes. For example, read the score, analyze it, send a signal to move some muscles, actually playing the key... Let's call t_j to the average time needed to perform the action j. Then the speed or frequency nu of playing is simply

nu=(1)/(t_1+t_2+t_3+...)

This is extremely important. For example, consider that you are able to reduce to a true minimum the time execution of all processes but there is one in which you spend a lot of time. This process will dominate the denominator and will be a wall to your speed. If for example t_3 is greater than t_2 and t_1 we'll have

nu_max ~(1)/(t_3).

Most of the times, your maximum speed is very low because you are performing one or several slow processes, and most of the times they are completely unnecessary processes to do.
To play slow is one of the most common barrier building processes that you can face. Of course, there are slow pieces in music, but from a technical point of view they are not the most difficult. You'll never be a sprinter if you train for marathon! And that's what they were teaching us as kids: run marathons so when you grow you can run 100m. Weird, false, shaming. The processes involved in slow and fast playing are totally different. If you learn slow and then try to speed up using the same technique you rapidly face time barriers. If you suffered the same school as me, just forget everything. Speed has to be the protagonist of your study from the very first day, even if you are a very small child. Specially then! The less time spent creating unnecessary time-consuming processes the better. We have to face the reality of doing many unnecessary things and accept that we have to unlearn them.
Strain is another frequent time consuming process. Usually teachers strain you and at the same time advice you to be relaxed. Never trust a teacher that advices you to relax: the translation is simply 'Play well with no further information despite my counter-teaching'. Nonsense. Other typical slowing processes are playing too hard, non-analytical playing (play the whole piece again and again without analysis and dissection), reading a score... and many more. But even if you reduce the time of all of them you still need to discover one of the secrets of piano technique: hardware acceleration. Yes, like in a GPU. Your brain won't play smoothly unless you incorporate in it some accelerators. Let me explain further.
It's very inefficient to process information every time we play something. Playing music needs automatic gestures to be hard-wired in the brain. I think that when playing a classical piece in front of an audience we need to hard wire all the piece a priori. Otherwise we'll fail in our performance. Thinking is an extremely slow process. Learning is slow too. When we play a motion that is new to our brain, the prefrontal cortex will take care of it, with great energy consumption and low efficiency. Very slowly too. After a lot of repetition you will be able to delegate the process to the basal ganglia, where efficiency is much greater. This takes time, but essentially is the key of piano technique learning: we need to hard wire in our brains all known difficulties in their fastest expression, i.e. with the minimum time configuration. So if we play a new piece we will have a great deal of their motions hard wired from our experience and we will only have to learn a handful new processes. The speed school begins here.
The first thing you need to learn is: why play from memory? Because the score will slow you, a lot. You'll need the score for learning and analyzing, and you need to go back to it many times. With good scores there are always new things to learn and discover each time you go back to them. But not for performance. You can consider that you don't have time to memorize everything, but the truth is that memorizing is an essential part of learning. In fact, is the first thing you need to do along with fingering a score.
The second essential thing to learn for speed playing is: who's playing? who's in charge of the speed? Your fingers? Your talent? OK, some genetic talent helps, and of course fingers have muscles and it's better for a sprinter muscle to train in speed or it will develop slow fibers. You don't want a marathon finger, this is for sure. But I don't think this is an essential aspect to focus on, as it will be worked along without noticing with proper training. And neither is talent a major thing. Maybe your talent is 90 and mine is 60, but how many people develop themselves to reach their limits? I haven't known anyone knowing their limits when pushing hard to find them. So if I will hardly touch my 60-limit ceiling, why should I worry or complain on the fact that I'm not 90? If you think talent in terms of speed learning, you can now consider how even a very talented student like me can be completely slowed to a halt by horrible teachers like mine. If you need 1 or 5 years extra to learn piano compared with a talented other, is there a problem when piano careers need decades to develop? You will not touch your ceiling and clever strategy for learning is all you need to become a master. If you are lazy then you have a problem of a different kind, and even then your laziness may come from lack of fun, which equals lack of progress and which equals bad teachers.
The answer is that the true responsible of your speed playing is your brain. Your neurons need to fire a proper chain of events that allow you to play fast. But don't fall into a paradox: your neurons are as fast as anyone's and they will fire as fast as hell either for fast or slow playing. What we need is to teach the brain to fire the proper actions. In order to achieve this, we need to spend a lot of time away from the score and away from the instrument itself!
OK, we are ready for our first speed exercises. My opinion is that we don't need any exercise at all but simply good scores to work with. And then to convert each of its details into a multiple, never-ending exercise. However, in order to illustrate my methodology let me develop a score-independent exercise that is really useful. As I have advanced before, we need to detach from the keyboard. Use instead a soft and elastic surface like a small plastic ball that fits in a hand. I use one that is about 9 cm wide in diameter (it changes a lot when inflated or deflated). Its softness helps our articulations to preserve their youth so we don't damage ourselves, our instruments, our ears and also our neighbors' patience. This method is neighbor friendly indeed. At least for now. The elasticity of the ball helps to feel each impact of the finger on its surface. In a silent place you can even listen the impact, but listen is slower than feeling, so focus on feeling first. This ball will be your keyboard for a long time, but not your only one. The ball helps a lot as a tool, not as a substitute of the instrument. However, it is so useful that it can become addictive. You can take advantage of the ball exercises to practice walking, exercising, traveling, waiting wherever you are, etc. The amount of time you can extract is amazing is you will to do it.
Remember my teachers' advice ornamentations? Well, let's take the opposite. Let's consider ornamentations first. In fact, let's consider everything an ornamentation. Ornaments are passages that are to be played extremely fast, so they are systematically ignored or underestimated at schools. But if we want to play fast, the fastest passage of a piece determines the maximum velocity at which we can play it! So let's begin by the most difficult part and let's not be happy until we can play all the piece at ornament speed! Ornaments can be composed of many notes, specially trills. In order to begin hardwiring the essential actions we'll consider three note cells.
Each 3 note cell needs to be studied in two modes: with the accent on the first and with the accent on the last. Each mode is to be played as fast as we can, just using finger motion. With one hand we hold the ball, with the other, having the wrist at a close distance to the ball surface, move just the fingers. It doesn't matter whether the fingers are in a flat position or more curved like. In piano, a curved position is usually taught, but during fast passages, many professional pianists adopt a flat approach. The idea is to play soft and as fast as you can, in a gentle way. Try to analyze strain, which is unavoidable at the beginning, and avoid it as your brain learns to play the action. It's a thing between your brain and your fingers. They are to be tightly connected in an isolated environment. Avoid distractions, silent your cell phone, you PC, the TV, everything. Going for a long walk in the mountain, in the beach, a good idea. Spend the minimum time seating. Take care of your back and body health. The ball method allows you to learn while being gentle with your body. Even a Yoga position is perfect for practicing. Piano study needs to become a way of meditation, with lots of benefits apart from the musical ones. As you practice in a proper way, taking care of isolation, depth and body-brain dialogue, you will notice a spiritual lift. In any case, technique is our first target, so let's focus on it.
First motions: a-b-a. Three note cells in which the last note is the same as the first note. Fingers are numbered as 1 [thumb], 2 [index], 3 [middle], 4 [ring] and 5 [pinky]. Remember that each pattern is to be practiced with the emphasis on the first note [let's call it A mode] and with emphasis on the last note [B mode]. Mode A is like the pattern begins at the beginning of a measure and mode B is like all the pattern before the last note is an anacrusis. Try to play them as ornaments played by a master player. Don't be happy with a lower speed. If you cannot speed it, think about what's slowing you. Never practice this slow. Let's begin with the basic cell involving fingers 1 and 2. Here's how I practice all its variations:
Right hand: 121[A],121[B],212[A],212[B]. Left hand: the same sequence. Then I move to 2-3, then 3-4 and then 4-5. In short, 121,212,232,323,343,434,454,545. It takes a while to complete it. The first thing you will notice is that to play fast it's impossible and undesirable to understand and discriminate each individual note. For example, when playing 232 in B mode, the first 23 is like a chunk. In fact, all the pattern should be understood as a chunk. The idea is to store this chunk in the brain as a single bit of information to be retrieved. Notice how modes A and B feel like different chunks. You can practice with high finger displacement and elevation of with lower one, with more strength or with less. The important thing is to hardwire your brain to these 3-cells at Glenn Gould speed. Of course, the more you approach high number fingers the more difficult it is. One is tempted to practice 454 very slow. Don't do it. Focus on speed and don't store in your brain the slow information. Practice as well the rest of 3-cells involving just two fingers:

131,313,242,424,353,535
141,414,252,525
151,515

Notice how the greater the span, the greater the wrist involvement. Once you have gone through all the previous cells I propose the rest of the 3-cells to that are possible with 5 fingers. You need to practice all of them until they are all hardwired, played at almost infinite speed, with no effort,i with no thinking. The rest of the cells involve at least 3 different fingers. Play each combination with 3 fingers and then move one. Play always A and B modes. If the three fingers are a,b,c play

abc,bac,cba,bca,acb,cab

It's better that you understand how to generate all the combinations. Don't use a paper to write all of them: that would be a score and you need to get used to think. I know it hurts, but is how you improve. Don't memorize all the cases either: understand how to generate them to obtain:

123,213,321,231,132,312. 234,324,432,342,243,423. 345,435,543,453,354,534.

Other patterns, without explicit development:

124,125,134,135,145,235,245

You will probably find that these last patterns, with change of direction, are most challenging. This is how speed school begins. It's hard, but follow this path and you will end playing really fast. Soon I'll be back to develop this topic further. For now, play these patterns daily, in the ball and in the keyboard (you don't need a real piano yet), but mostly in the ball. Let your brain and your fingers speak silently to each other.
There is a final type of patter in the 4-cell world. The ones where you can repeat a note. Play the patterns

aab,bba,abb,baa

for 12,23,34,45,13,24,35,14,25,15, each one in A,B modes with both hands. Alternate hands frequently to minimize fatigue. If you feel tired or the slightest pain, stop and rest. If you feel mentally fatigued, don't rest: get used to it. Finally you can play the patern aaa for all 5 fingers. It's hard to repeat the same note with the same finger but the effort pays off.

Summarizing, play all combinations each day. Devote yourself to 3-cells for some time. Fragment study time to reduce fatigue. Use mainly the ball, not the keyboard. Imagine your fingers as drum players, with no tuned sounds. Always play A and B modes and alternate hands after each pattern or after a small group of them. It's not necessary to touch the ball with the non-playing fingers. Focus on speed an try to avoid strain. At the beginning is normal to feel fatigue: the muscles will be exhausted after all this workout. Your mind too. Repeat each exercise at least 8 times. If you can play it smoothly go on: if not, spend a bit more in it. Try to make the repetitions in a rhythmic way. The emphasized note, the first in A mode and the last in B mode, should not be played with more strength. It's just a mental picture of where the weight of the cell goes. In A mode, the notes after the first are like echo. In B mode, the whole things is falling to the last note which is ground. Your focus always puts weight in the principal note, the other being simply a bunch of rapid but almost indistinguishable notes. But don't hit these main notes, just put some mental weight in them. The summary of patterns is:

aba,bab

for 12,23,34,45,13,24,35,14,25,15. Then go to

abc,acb,cba,bca,acb,cab

for 123,234,345,124,134,235,245,125,135,145. Finally,

aab,bba,abb,baa

for 12,23,34,45,13,24,35,14,25,15 and

aaa

for 1,2,3,4,5. In total is 2*10+6*10+4*10+1*5=125. This agrees with a Variation of n elements taken in groups of m allowing Repetition, so we have a total of VR_m^n=n^m=5^3=125 (VR = variation with repetition). For 4-cells we will have 625 possibilities and for 5-cells 3125. A lot of work. For now, 125 patterns with modes A and B played on both hands are 125*2*2=500 different exercises. If you spend about 10" in each one it means 1h and 25' approx. At the beginning, with many patterns shortcircuiting your head will be much more. This is real music training with a single purpose: becoming a real master of the keyboard. Are you worried about not using your piano or your keyboard much? Do you think that the ball is not a realistic instrument? I wouldn't worry at all. Think that this training prepares you for playing in any keyboard: piano, harpsichord, organ, electric keyboards, electric pianos... All of them are very different, have different touches, different key widths, different everything! So stay open and flexible to change and adapt to every keyboard you have. It's true that if you encounter a bad keyboard you will not able to play a trill very easily. It's true that every time you adapt to a new gauge you will feel clumsy, but this is another part of the learning process. Think that the patterns in 3-cells could be applied to different notes. For example, 151 could C3G3C3, C3C4C4,C3C4D3, and many more. This is why the best approach is to use real scores and take each motive there to build an exercise. This is what we will do, but for completeness, for intensive training, for fun, and especially for the sake of real mastering we complement score working with these permutation patterns. And remember, this is only a tiny beginning!