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I often hear poeple talk about how to "improve your strength" in piano technique by doing exercises, Bach, or whatever, but I've never seen an actual definition of the word strength in this context. It would be useful to know exactly what we're talking about.

In sports science, muscular strength refers to the ability to generate maximal force for a particular task. So strength is how far a thrower can put a shot or how much weight a power-lifter can bench press. If that definition were applied to piano, strength would refer simply to how loud you can play a note or a chord (since the key strike force translates into hammer speed which translates into string amplitude which translates into sound volume). Perhaps that is one small aspect of it, but I doubt that is what we really mean by strength in piano technique.

Is their a serviceable definition of strength in piano available? What are the specific technical elements of strength in piano playing?

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I am still trying to develop a clear idea myself. This summarises where I have got to in my researches (from respected piano technique books and some YouTube piano teachers plus a laws-of-physics/engineering mind). Mine is an amalgam of views that made sense to me.

I think the finger strength idea comes from the finger independence school of piano techniques. The premise is: to play precisely you need to control the velocity of the key down action precisely. So, playing a legato passage very quietly requires each key to launch the hammer at the strings so that each hammer touches delicately and the key arrives at the key bed before releasing the key to apply the key damper.

Having strong playing fingers while being able to relax muscles in arm, shoulder, back and in the lower arm for non-playing fingers yields more control - once you get the hang of it. Weak fingers are likely to have less control so will recruit compensating actions from muscles in the wrist, arm weight and back. In one book I use, the author says some piano players were said to have fingers of steel. (I think that was Horowitz - I might be wrong). That strength would allow them to play fast pp and ppp passages evenly with ease. I think they must use their fingers to press down the keys more than they use wrist flexibility, rotation and arm weight.

Next comes the matter of equalising the finger strength across the hand vs. choosing the fingering to take advantage of the different qualities of each finger. You will find lots of exercises that focus on developing fingers 4 and 5. Mine are weak so I do these exercises from time to time. Presently, I doubt going overboard with finger equalisation exercises will do me any good. One reference book I use says develop finger strength but don't expect equalisation, so choose your fingering accordingly.

To put this in contrast, Dorothy Taubman developed a different approach. She looked at child prodigies who at a young age were playing very advanced pieces very well. Their fingers were not strong, so how did they play so well? It appears they used arm rotation actions, wrist flexibility and arm weight to a greater extent. These techniques offered control but in a different way. The Taubman techniques are not just about rotation, but it features.

Some piano technique experts argue that one is better than the other. There are some strong opinions on this from highly skilled pianists. As far as I can see, both pedagogies have merit and can be used together.

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The strength is the forte of the attack, which is depending of the velocity of the attack.

The velocity is depending of your movement of the fingers - using them as hammers bowed up and down crabbing or by a stretching movement, both technics need some muscle strength but even more a speedy neural reflex.

Mind that we don’t play only with the fingers! We use our arm, shoulders and the whole body - that also young children and people with thin fingers can play with a strong attack = forte.

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