Posted by Gerald on December 15, 1998 at 19:23:41:
In Reply to: Re: tuning tendancies of the harmonic series posted by phil sloffer on December 14, 1998 at 10:49:21:
Pianos are tuned with stretch, that is each octave is a little wider than a exactly twice the fundamental, because as the string has to stretch to vibrate, it's next overtone simultaneous with the fundamental is a little higher than exactly two times the fundamental. Its not a linear system. The same thing happens on stringed instruments. And the bowed harmonic of a bass isn't the same pitch as the harmonic of the bowed fundamental and neither is n times the fundamental pitch of the string.
To add complication, the width of the bore of a horn such as a tuba affects the pitch. And on harmonics of the fundamental, the effect is different for each harmonic because of the taper. Probably to be most adverse, the harmonic resonances are a little compressed, just the opposite to the piano's stretch, but I'm not sure of that, I've no good mathematical analysis of the vibrating air column. No doubt as the air in the pipe is compressed and expanded each cycle of oscillation, that makes a nonlinear effect too. This gets more complex as parts of the tuba's bore is straight, and part tapered and the effect on pitch depends on where the first wave ends. No doubt some rotten notes are caused by that wave ending at the gap in a valve or a step in tubing size on a tuning slide or a bend. Too deep thought can warp the mind, just pick a horn that for every note you want to play, plays on pitch, with a pleasing harmonic content, and its as perfect as you can get. Different plumbing details, including transistions from straight to taper and bends, solder lumps at fittings, size changes at inner ends of tuning slides, paths through valves, all are going to change harmonic content and harmonic pitch.
Its these variations in harmonic detail that make different instruments (whether of the same family or diverse) sound unique (e.g. have character) and the blending them in ensemble often results in a pleasant sound.