Ex. 1
Here’s how the geometry works for violin and viola: When the bow moves from the D string to the A string, the frog and bow hold draw a clockwise loop starting from the 12 o’clock location on the loop. The tip of the bow, meanwhile, draws a counterclockwise loop starting on the 6 o’clock location on the loop.

For cello and bass, when the bow moves from the D string to the A string the frog and bow hold draw a counterclockwise loop starting from the 6 o’clock location. The tip of the bow draws a clockwise loop, starting at 12 o’clock.

In both cases, because the point of contact of hair and string is the fulcrum of motion, the tip and the frog of the bow will always draw geometric forms that are in opposition.

When you get to the second pair of notes in the melody, you’ll start crossing from the A string to the D string.

Ex. 2
If you’re doing it right, you’ll see that everything happens in reverse; motions that were clockwise before are now counterclockwise, and vice versa, while loops that started at the top (12 o’clock), now start at the bottom (6 o’clock). Then it all switches back to the way it was in the beginning when the melody notes return to the D string. If you don’t reverse directions cleanly and without added motion, your bowing will be awkward.

Ex. 3
Now try doing this exercise in triplets. Each note of the melody becomes three: You play the melody note, cross to the next open string, then come back and play the same melody note again before going on to the next one. This is how the first note looks as a triplet, in the bass clef and then in the treble.

Here’s how the geometry works for violin and viola: As before, in Ex. 1, when you play the first triplet from the D string to the A string and back to the D string, the frog and bow hold draw one downbow in a clockwise loop starting from the 12 o’clock position. Meanwhile, the tip of the bow draws a counterclockwise loop starting at 6 o’clock.

For cello and bass, the frog and bow hold draw one counterclockwise downbow loop starting from 6 o’clock. The tip of the bow draws a clockwise loop starting at 12 o’clock.

As always, the loops should be smooth and even, each end of the bow neatly mirroring the other.

Now see what happens when you play the second triplet—it’s an upbow this time, and everything works in reverse, compared to the first triplet.

Ex. 4
A clean stop and reversal of motion will be critical for graceful bowing. Dick warns, “A lot of students get kind of tongue-tied here. When the arc changes direction, they don’t stop and retrace the path the opposite way; they put a little loop in there so they can keep going the same direction, and that makes the bow hit a neighboring string, and it sounds messy. A real slow practice tracing those patterns will clear up so much stuff. It really makes you control those arcs, and stop and reverse the arc in a straight plane rather than a curved plane.”

Scott says you shouldn’t give up if you don’t master this right away. “Most of us need years of practice in all patterns to make all string crossings feel more natural than foreign,” she says.

For now, how many times should you practice these patterns? Dick has been known to write at the bottom of an exercise, “Repeat 68 times.”

“When something isn’t working, it isn’t because your teacher hasn’t explained it enough, it’s because you haven’t practiced the physical skill enough,” he says. “I’m sort of a drill sergeant.”