How many of you taught yourself to play?
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Another after thought is that there can be of some valuable networking from whom you take lessons. I wouldn't let that be my main reason for choosing a teacher but it can be a spin off.
For example, through my H.S. and college trumpet teacher, I got recommended to play in the Royal Hawaiian Band. Through my first alto sax teacher, a reference to one of the best composers in Japan. Through another sax teacher, I got picked to play Lead Alto and then Solo Tenor in one of the workingist jazz bands in the region, and from my composition teacher a reference to one of the best composition jobs in the country.
In each case, it paid off financially and professionally. But that's not why I picked any of them.
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I think that a good teacher teaches you how to learn, or in other words, they teach you how to teach yourself.
Nonetheless, for the purposes of this discussion, I am not a self-taught trumpeter. I have had several quality teachers.
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I agree with ROWUK. I started playing trumpet in third grade (no lessons) - gave it up but took up piano lessons -- resumed trumpet in 5th grade (no lessons) - continued both piano and trumpet --- between sixth grade and junior high school (6 grade to 7) i had my first lessons -- took lessons in both piano and trumpet throughout junior high and first two years of high school. Then moved to a nearby city - no lessons. Bought an illegal "fake book" and took a few lessons on piano from a teacher who for the first time in my "education" introduced me to the circle of fifths. No more trumpet lessons for twenty years. Then, perhaps one year of lessons from the local symphonic principal. He introduced me to long tones and breath attacks.
Otherwise - it is the love of music and the drive to get better at what you do.