Saxophone repairman Matt Stohrer made this video to answer a question but ended up decided to just publish it because hey, its been a while. In this video, you will see a very simple example of a bent hinge tube and how it is fixed. Bear in mind that often, hinge tubes have multiple bends and can also have dings that impinge on the inner tube- so this example is extremely simple in comparison to what is often found in the real world. But hopefully by seeing this you can get an idea of the very basic mechanism behind one way (not the only way) to fix a bent hinge tube on a saxophone key.
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Also check out the Open Source Saxophone Project: http://www.OpenSourceSaxophoneProject.com