Positional Visualization vs. Thumb Crossings
Here's how to be done with "thumb crossings" forever and to FORGET THEY EVEN EXIST. I call the technique "positional visualization".
Play the notes C E G as a block chord. Now play the same chord an octave higher.
Now play them back and forth: BANG BANG BANG BANG ...
You can do that pretty fast, can't you?
Now, instead of playing those two chords as a block, simply ROLL the notes up, move up to the next position and ROLL them up again.
No stupid "thumb crossing", no discomfort, just a smooth arpeggio. Instead of pumping your fingers up and down, and straining and stretching to get the thumb under the 3rd or 4th finger, you are effortlessly playing the notes in sequence in a relaxed manner which can nonetheless be made to happen as rapidly as needed. AND YOU DIDN'T EVEN NEED TO PRACTICE IT!
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Here's how to use the "positional visualization" technique to play scales at the speed of light with ease and total comfort.
Let's pick a "hard" scale. How about Gb major?
Your first position is F Gb Ab Bb (fingers 1 2 3 4). We'll call this the "4 finger position". Hit all those notes at once as if they were a block chord. (Yes, I know it sounds nasty).
Your second position is B/Cb Db Eb (fingers 1 2 3). We'll call this the "3 finger position". Go ahead and hit all those notes at once as if they were a block chord.
Alternate banging all the notes at once in the "4 finger position" and the "3 finger position" back and forth under you can do it with total ease (which should be almost immediately).
Now roll the "4fp" up followed by rolling the "3fp", then immediately roll the "3fp" down followed by rolling the "4fp" down. Easy, huh?
Naturally you'll need to take care that each of the notes has an equal value.
Once you've mastered that continue with:
1. "3fp" up - "4fp" up - "4fp down" - "3fp" down
2. "4fp" up - "3fp" up - "4fp" up - "4fp dn" - "3fp" dn - "4fp" dn
3. "3fp" up - "4fp" up - "3fp" up - "3fp" dn- "4fp dn" - "3fp" dn
How long did that take? 5 minutes? And you can now play the scale all the way up and down the keyboard with no sweat as fast as you like.
The same principles hold true for ALL chords and scales (except the chromatic scale).
Forget about "thumb crossings" completely. They are useless, they slow you down, and they are psychologically crippling.
To sum up:
If you shift your thinking to "positional visualization" the thumb does not cross under. The entire hand simply moves to the next position.
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Here is the oeuvre which supports the "Thumb Over"
fingering technique described above Fundamentals
of Piano Practice by Chuan C. Chang (pdf)
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Just for Fun: A parody of the "Positional Visualization" article