zilla@nyit.UUCP (John Lewis) (02/13/86)
[void] The stretched harmonics in the piano have their origin in the wave equation, modified to include stiffness and damping. Both stiffness and damping contribute to the stretching in different ways. The effect of stiffness was studied empirically by Young in JASA (52?). The amount of deviation increases non-linearly with the harmonic and if I recall it becomes more than a semitone above about the 12th harmonic. The stretching is greater in the non-wound strings and probably accounts for some of the tinny piano sound in the highest octave. The effect of an assumed linear damping factor is to linearly shift the whole spectrum down slightly, which also stretches the harmonics. A number of other deviations from the ideal wave eq model are known which probably contribute to the piano sound, for example, modes which are not present in the original 'excitation' (hammering of the string) appear later in a nonlinear model (transfer of energy between modes). Also the pairs or triplets of piano strings transfer energy and this apparently is responsible for the piano envelope (articles in Scientific American on the piano sound). What is interesting is that stretched harmonics are found in any instrument described by the wave equation, though they are more audible in struck-string instruments. The non-harmonic spectrum may account for the warmth or harshness of real vs. synthesized musical sounds; most of the existing synthesis techniques produce a strictly harmonic spectrum (including sampled wavetable techniques). j.p.lewis nyit computer graphics lab
knudsen@ihwpt.UUCP (mike knudsen) (02/14/86)
> The stretched harmonics in the piano have their origin in the > WAVE EQUATION, modified to include stiffness and damping. > > about the 12th harmonic. The stretching is greater in the NON-WOUND > STRINGS and probably accounts for some of the tinny piano sound > in the highest octave. The effect of an assumed linear damping > > What is interesting is that stretched harmonics are > found in ANY instrument > described by the wave equation, though they are more audible > in STRUCK-STRING instruments. The non-harmonic spectrum may account > for the warmth or harshness of real vs. synthesized musical sounds; > j.p.lewis > nyit computer graphics lab I agree with the above, but this raises a question: Should the stretched harmonics be even more pronounced on a HARPSICHORD, because (1) the tone is brighter, with stronger high harmonics, and (2) even the bass strings are not wound. Are plucked strings less "stretched" harmonically than struck ones? Bass harpsichord strings tend to be of brass or bronze (less stiff than steel) but under much less tension. I should check my Zuckermann book (The Modern Harpsichord) to see whether harpsichord tuning is stretched like pianos'. I don't see how it could be avoided. mike k PS: I find the Harpsichord patches on most synths to be too harsh for more than a few bars of use, tho I doubt it is related to the stretching issue. PPS: Stretched tuning is neither needed nor desirable on continous-toned instruments, like organs, since their wave equation has no damping. (Not that anyone implied otherwise...)