Hi @pickfly . Sorry I couldn’t respond to this question sooner, I’ve been very ill this last week and I’m just starting to improve. I hope you can forgive any typos, etc.
I very strongly recommend that you aim to make large, powerful movements which change direction rapidly and which feel effortless.
The usual idea of economy of motion is that smaller movement are faster or more efficient is incredibly naive. It is totally incompatible with the physical and motor control realities of the situation.
In a physical sense, our picking hand can be (roughly) modelled as a forced oscillator. The equation which describes the amplitude of a forced oscillator is given below:
With terms
A = amplitude of oscillator
F_max = amplitude of driving force
w_d = frequency of the driving force
w = natural frequency of the oscillator
m = mass
b = damping
It’s important to understand that the frequency of the forced oscillator is equal to the frequency of the driving force w_d. In movement, this frequency is neurological in origin; it is the result of our ability to rapidly fire the neurons which trigger the contractions of our antagonistic muscle pairs. The other factor we have control over in this equation is the amplitude of the driving force F_max. The mass m and the natural frequency w are not adjustable parameters in our context.
Now, here’s the critical thing. The amplitude A is linearly proportional to the amplitude of the driving force F_max, but almost inversely proportional to the square of the frequency of the driving force w_d. What this means, is that with the same driving force, the amplitude A (movement size) will inevitably get smaller as the frequency increases.
If we begin with a movement which is small and lacking in force at a lower frequency (that is, tempo), the movement will simply be too small or too weak to be functional at a higher frequency (tempo).
The only way which we can deliberately make a movement smaller at a given frequency is to increase our damping coefficient b. This can only be done through muscular exertion and absolutely does not result in greater efficiency or frequency. Damping is to be avoided.
Small movements aren’t fast. Fast movements are small. The direction of implication that everybody teaches is completely backwards. It’s an overly naive reduction which doesn’t hold up to rigorous examination. It’s like saying that rich people buy Lamborghinis, so if I buy a Lamborghini I’ll be rich.
So, our goal is to make an oscillator which remains functional at our highest potential driving frequency w_d (which again, is neurological in origin). This means that we need to ensure that we have sufficient force production to drive the oscillator and that we have minimal damping. Essentially, if we’re practicing at a lower tempo, our movements should be large and powerful, but they should feel almost effortless.
This is the physical reality of periodic movement and it’s absolutely inarguable. I have heard some people try to find an “out” for the conventional ideas of economy of motion. Their argument being that focusing on making smaller movements is still (somehow) the best way to learn to play faster, despite the physics being backwards.
That’s an absolutely ridiculous claim based purely on the underlying physics. Beyond that, it’s completely incompatible with the realities of motor control. Specifically, the idea violates a principle in motor control called Fitts’ Law, which is the most well established and tested law in the field.
The idea is actually quite brilliant. Fitts used ideas from information theory to describe movement difficulty. His metric is ID, the index of difficulty, given by
Here, D is the distance to the target and W is the width of the target. Think of W as a tolerence which determines a “hit” or a “miss”.
Fitts also created a metric IP called the index of performance, which relates ID to the movement time MT and which is relatively constant in any task:
Then, we find a linear regression model:
So what does this all mean?
Well, it’s important to understand that whether we play faster or slower, our accuracy is subject to a target width W which is essentially constant.
Suppose we practice slowly with small movements. Now, D is small, so ID is small. Infact, it’s too small, because ID (index of difficulty) for this type of practice is far below a realistic ID for fast playing. Since MT is also large, the index of performance IP is also very small. As such practicing slowly with small movements does not in any way test that our index of performance is sufficient for fast playing.
With the understanding that IP is relatively constant, the only way we can make slower playing representative of faster playing is if the index of difficulty ID is greater for the slow playing than the fast playing. Thus, as we play faster and MT decreases, ID decreases proportionally and IP remains constant. Now, slow practice is representative of the actual difficulty of fast playing.
But how can we do this? Well, there’s really only one answer. Since the target width W cannot be adjusted, we can only increase ID by increasing the distance D to target.
So, when practicing slowly, we again want our movements to be large, powerful and easy.
The final element is that guitar playing requires not only spatial accuracy, but temporal accuracy also. For this, it has been demonstrated that faster movement speeds result in greater temporal accuracy. So, when practicing slowly (lower tempo), our hands should still be moving quickly. We should focus on making fast movements which accelerate and change direction rapidly.
Listen to this podcast for more information.
@Ruefus has already summarised my position on this exercise. Unless you love chromatic lines and want to optimize your playing around them, don’t bother.