This post applies most directly to wind players, although knowledge in this topic can be applicable to other areas of musical performance. Wind players blow air into their instrument to cause the vibration that produces sound. By making slight modifications in the way we blow, we can emphasize certain overtones more than others. Varying the emphasis of different frequencies across the overtone spectrum can change the color of a wind performer's tone. Skilled musicians use this technique to adapt their tone to the musical style that they are performing.
The modifications in mentioned in the last paragraph are made through slight adjustments in a performer's throat, tongue, mouth and jaw. Through experimentation with long tones, a performer can discover how to produce the tone that they desire with accuracy and consistency. Many wind players disagree over how many of the body parts listed above should be used in shaping ones tone and to what degree they should be manipulated. However, most agree on the concept of hearing the desired tone in one's mind and then experimenting to achieve it.
I cannot speak specifically on this topic for every wind instrument. I can, however, add specifics about the study of overtones in saxophone performance. Saxophone players often practice shifting between the overtones of a particular note while fingering the fundamental on a long tone. In addition, we practice overtone scales by using the fingerings of lower fundamentals to produce the pitches of higher notes within the sequence of a scale. The ability to eliminate lower frequencies within a pitch and bring out a certain overtone as if it is the fundamental is helpful. A saxophone player can use this ability to gain more control over the shape of their tone. We can also use this ability to increase the range of our instrument by playing harmonic pitches.
As a student, I always found it helpful to understand the application of the elements I was practicing. Overtones are one of those elements that many people include in their practice regiment without understanding the application. I hope that this post will help you to understand the significance and use of overtone studies in your practice time.