Petrof Inharmonicity

[Delwin D Fandrich]

Stephen Birkett wrote:

> Del wrote:
> > No, you don't "hear" inharmonicity as such. As Bill points out, it will
> > have an effect on how the piano is tuned in that a piano with "high"
> > inharmonicity will be tuned somewhat sharper in the treble than will one
> > with "low" inharmonicity. However, inharmonicity by itself has no effect
> > on what you hear in terms of tone quality.
> >
> Beg to differ on this point Del, on two counts.

OK. It won't be the first time we've disagreed and we'll both survive.


> There is a tone difference
> that can be heard...and can be easily demonstrated with a spectrum
> generator. Set up a pure spectrum with maybe 50 partials and vary B
> (inharm.  coeff.). The change in tone is very noticeable as B is increased
> from zero to typical piano maximum.

I'm familiar with the demonstration. All it proves, though, is that when you set up a pure
spectrum on a spectrum generator, feed it through an amplifier and into a speaker and then
vary B, that you can change the sound you hear from the speaker. While the piano string
is, in a sense, a spectrum generator, a spectrum generator is not a piano string hooked up
to a piano soundboard.


> > All three strings have a tension of 160 lbs. But each of the other
> > parameters varies. The harmonic structure of each of these strings will be
> > different when struck, but not "because" of inharmonicity. It will be
> > different because each string has a different speaking length and a
> > different tension....etc
> >
> Inharmonicity affects the behaviour of the pulse created by the hammer
> contact.

String tension, mass and stiffness affects the behavior of the pulse created by the hammer
contact. Inharmonicity is a by-product of string tension, mass and stiffness.


> The pulse travels to the close end of the wire, is reflected
> back toward the hammer which it "pushes" off the string.

This is affected more by the strings tension, the hammer's strike point along the speaking
length of the string and by the strike point's proximity to the termination point than it
is by the strings stiffness. It is not a function of inharmonicity


> More
> inharmonicity increases dispersion on the string, causing the pulse to
> change its shape as the higher frequency harmonics move ahead of the lower
> ones. So, already before the standing wave is even set up on the wire,
> inharmonicity has had an effect.

What?


> The standing wave on the wire is different for two independent reasons:
>  -  that variation of the pulse from hammer excitation
>  -  the direct effect of inharmonicity on the vibrating wire
> These result in a different harmonic structure, with different partial
> energies, before it is even passed to the soundboard system. Bringing the
> soundboard into this is really just confusing the real issue (although I do
> agree with your impedance description etc.).
>
> Stephen

I didn't say that the varying characteristics of piano strings do not have an effect on
inharmonicity. They obviously do. And the physical characteristics of the piano string --
its length, mass, stiffness, tension, etc. -- do have an effect on the energy level of the
various partials that are excited within the string on hammer impact. And, yes, those
partials are enharmonic. But...

What I said was "we don't hear inharmonicity." In fact, we don't hear the energy spectrum
within the string at all. Except in a VERY minor way, we don't hear anything at all
directly as a result of energy movement along or within the piano string independently of
the soundboard assembly. What we hear is the effect of the soundboard assembly vibrating
in response to all that energy input from the string. While the soundboard is incapable of
vibrating independently of the string -- that is, it doesn't just sit there and vibrate
all by itself, it requires some energy input from the string to make it move -- the fact
is that it does not have to respond uniformly and equally to the full spectrum of energy
that is present in or along the string. It can be -- and is -- frequency selective.

The physical characteristics of a piano string certainly do have an effect on the energy
content of the various partials that are excited when a hammer impacts it. However, within
the normal bounds of normal strings in normal and practical pianos, it is primarily the
design -- that is, its mass, stiffness, static crown, string loading (i.e.,
"downbearing"), etc. -- and functioning of the soundboard that defines the tone character
of every given piano. The inharmonicity coefficient of a couple of test strings as
measured by some electronic tuning device has precious little to do with it.

Regards,

Del