In this geometric relationship we find the greatest proof of the fallacy of "fixation" characterizing early piano pedagogy. The more we attempt to restrict motion to one joint, the more the motion becomes curvilinear. The construction of the keyboard in a straight line is opposed to curvilinear movement as understood here. This condition has been responsible for building piano-keyboards that curve somewhat around the player, which have never become generally used. They use the shoulder position as the center of their arcs, attempting thereby to eliminate the trunk movements needed for shoulder movement.
From a purely geometric standpoint, any movement of piano technique, with the exception of

1. the simplest finger-stroke, and
2. an elbow extension with an absolutely vertical upper arm,

involves motion at more than one joint. When we transfer our hand from the region of middle C to an octave or two higher, the hand describes an arc, hence we may infer movement at one joint.
But at the same time the line of action in the direction of ascent is a straight line parallel to the keyboard, and this necessarily involves movement at more than one joint.

So it is with most other movements, whereas they may be curvilinear in one or more planes they are, almost always, at the same time rectilinear in another plane, and are resultants of motion in more than one joint.

1. The passing-under of the thumb in scale and arpeggio work;
2. the vertical finger-descent;
3. all cantabile touches;
4. the lateral shift of the hand beyond the five-finger position;
5. crossing of hands;
6. simple arm-drop;
7. diatonic tremolo;

In fact any pianistic touch has in it some rectilinear motion and this can be produced only by movement in more than one joint. There is no exception to this relationship between straight-line movement and activity at more than one joint and, as a result, the isolation of movement to one joint is, in actual piano-playing, mechanically and physiologically impossible.

From a purely geometric standpoint, any movement of piano technique involves more than one joint. When we transfer our hand from the region middle C an octave of two higher, the hand describes an arc, hence we may infer movement at one joint. The line of action in the direction of ascent is a straight line parallel to the keyboard. This implies movement at more than one joint.

Although the motion may be curvilinear in more than one plane, it is almost always at the same time rectilinear in another plane.

The composite motion is the resultant of motion in more than one joint. Any pianistic touch has in it some rectilinear motion and this can be produced only by movement in more than one joint. If there is a straight line movement then there is activity at more than one joint.