I received a question about how to practice octaves, ie. thumb on the lower string and the 3rd finger on the upper string. Octaves are difficult because the spacing between the thumb varies depending on where you are on the fingerboard, and because the thumb moves at a faster rate than the 3rd finger as you go up the fingerboard. There are hundreds of exercises, all of which are useful. Below are some ideas that I have heard over the years:
1. The primary note in the octave is the lower note. This is the bass of the "chord" and is the note that the listener tends to focus on. So you must make sure your lower note is in tune. Practice ascending and descending scales with your thumb only, keeping your other fingers in position on the upper string, which is to reduce stress on the thumb. I would also practice arpeggios with the thumb. Get used to how far the thumb must shift. On the descending scale, lighten up your arm, otherwise you will tend to not shift back far enough.
2. Always keep your non-playing fingers in position. There are many ways to practice this, including playing major scales as you shift up one half-step at a time. Another good exercise is to play one octave scales from thumb on the lower string to the 3rd finger on the upper string, and then shift your hand up to the next note in the scale degree, and play the scale again though starting on the second note of the scale. For example, start with the thumb on A on the D string and play an A Major Scale. Then shift the thumb to B on the D string, and play the A Major Scale again, but starting and ending on B. Then shift to C#, etc. Note how your finger configurations must change depending on what scale degree you start on. This will help with intonation in thumb position too.
3. Practice octaves by playing arpeggios in octaves as well as other intervals like 4ths and 5ths. Get a sense of where your 3rd finger should end up after the shift. Remember, your thumb is the primary note and that your 3rd must adjust to it, not the other way around. The key is to have a destination in mind when you play octaves. For instance, if I am playing an ascending scale in octaves, like in the first movement of the Dvorak, determine the beginning and ending spacing of your hand. Then, as you go up, shoot for that spacing at the end of the scale. You may even break up this long scale into a few destinations notes. But lead with the thumb.
4. Practice octaves by slowly increasing the interval you are shifting. For example, start on A and shift to B-flat. Shift back to A. Do this until you get it in tune. Then shift from A to B-natural. Then A to C. And so on.
5. You can play scales with your third finger alone, keeping the thumb silent. You can play broken octave scales ie. D-D-E-E-F-F. There are tons of [Miscellaneous] exercises.
6. Some teachers recommend bunching your non-playing fingers against your 3rd finger to give it added support. This may work, but it increases tension in the hand and can cause cramps. So I am not sold on it.
7. If you are playing thumb position in the lower position zone, ie. along the neck of the cello, be sure to bend your first thumb joint in order to maintain the thumb perpendicular to the string.
8. Keep your wrist perpendicular to the fingerboard as you ascend. To do this, you must lower your entire arm, adjusting your entire body position.
9. Practice octaves in "single line studies," where you sound just one line while only lightly fingering the other, progressing to breaking the double stop from the bottom, then to breaking from the top.
10. Be sure to keep your wrist flat to prevent carpal tunnel syndrome. I didn't while practicing Popper #13 and ended up being incapacitated for months.
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