Here is a clarinet overtone exercise I got from Caroline Hartig. She is a complete and total clarinet badass. I studied with her for a couple semesters when I was at Ball State before she made the move to teach at Michigan State. I really like this exercise because it forces you to keep your air moving as you cross registers and it’s great for working on voicing and tongue position.
Of course, the overtones on the clarinet are different than on the other woodwinds. For a very, very detailed explanation of this phenomenon, check out this page.
This exercise covers the fundamental tone and the 1st and 2nd overtones of the clarinet. The exercise, as written, goes straight through each group with only a breath mark separating one from the next. I practice it like this sometimes, but most of the time I take a few beats after each group to prepare for the next one. Also, I would suggest playing around with the tempo and rhythm of the exercise. I think it is better to err on the side of going slower than the marked tempo rather than going faster.
As with any overtone exercise, the concept of prehearing is very important. When I say prehearing, I am referring to the ability to hear a pitch before you play it. To help with this process as you work on this exercise, first play each group twice. On the first time, use the standard fingerings for the written pitches. When you play the group a second time, use the overtone fingerings (Those are the notes with the diamond noteheads.). Over time, you will be able to hear the pitches without the need of playing the standard fingerings before trying the overtones.
A final note – If you look at the exercise, you’ll see that each group has the same intervallic content. Consider the first group (E, G, C). The 1st pitch descends a Major 6th to the 2nd pitch which descends a Perfect 12th to the 3rd pitch. If you move reorder the pitch names from (E, G, C) to (C, E, G) it spells a major triad. This is true for all of the groups except for the last one. The group built on the low E has (G, B, E) which reordered is (E, G, B), a minor triad. I don’t know the science behind this change, but the exercise sheet is correct. The top note of the last group should be a G natural.