> But we teach them in a
> wrong way just fingerings instead make them able to
> understand the harmonic series & the relations of the tones
> each other.

I remember at about age 10 working out the basis of the Bb side
fingerings by myself and carefully drawing a chart that showed how the
Bb fingerings were off from the F fingerings by a 4th. I took it to
show my teacher, who didn't seem to be impressed, and as far as I can
remember thought I had misunderstood what the Bb side was about and
how the fingerings worked. He had simply been teaching me fingerings
out of the book and seemed uninterested in how the harmonic sequence
controlled things.

I'm pretty sure I did get the chart right, because I don't remember my
understanding of the harmonic sequence and of fingerings changing
radically since. It has been refined a bit by a better understanding
of which harmonics are a bit out of tune relative to equal
temperament, but the basic understanding has not changed.

Regards
Jonathan West
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