Petrof Inharmonicity

[Delwin D Fandrich]

Mike Swendsen wrote:

> I know there has been a lot of talk about inharmonicity, it being higher or
> lower than average, and quite frankly, I think that what is being listen to
> is not the in harmonicity of the instruments is question.
>
>
> If you listen to the tone of a S&S for instance it is quite complex ( read
> many harmonics, and most of them towards the high end) and with a Petrof
> those very high harmonics are not so prominent,  on the other hand the lower
> harmonics are.  This gives the piano a very clean and clear sound, I have
> heard some people say 'bright' but that isn't quite correct.  A Steinway or
> a Mason and Hamlin are bright, I.e. there is a strong sense of  higher
> harmonics.
>
>
> It would be interesting to have someone with a good frequency analyzer test these
> pianos, and show the results.
> As far as the 'stretch' numbers on Petrofs go, they almost always fall
> between 5.5 and 6.0
>
>
> C. Mike Swendsen RPT

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Mike,

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.

Inharmonicity is a calculated function of a stretched string. For example, look at just
one string, in this case C-40:
    Length = 610 mm    Dia. = 0.042"    Tension = 160 lbs.    In = 0.42
    Length = 640 mm    Dia. = 0.040"    Tension = 160 lbs.    In = 0.31
    Length = 674 mm    Dia. = 0.038"    Tension = 160 lbs.    In = 0.23
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. In the examples given above -- if everything else
remains the same -- when struck, the longer string will have a bit more energy in the
fundamental and lower partials and the shorter string will have a bit more energy in the
higher partials. Inharmonicity is simply a by-product and it tells you how the piano
should be tuned.

The harmonic structure excited in each of these strings will also be affected by the
elasticity of the hammer, and the by the exact point of excitation, i.e., the hammer
strike point.

None of which really tells you how the piano is going to sound.

Regardless of the harmonic structure that is excited within the string when struck, what
you actually hear is determined mostly by the relationship of that string to the
bridge/soundboard/rib structure. That is, the relative amounts of each excited harmonic
within the string that are actually allowed into the soundboard, and the rate at which
they are allowed in. This energy transfer rate is determined by the mass and stiffness
characteristics of the soundboard. The mass and stiffness characteristic make up what is
referred to as a soundboard's characteristic impedance. Impedance is simply a measure of
the soundboard's reluctance or willingness to be moved when some external source of energy
attempts to move it. The problem is complicated a bit by the fact that the soundboard is
more or less reluctant or willing to be moved depending on the frequency of the energy
that is trying to move it.

So, if a string's motion is made up of more than one frequency -- And it is. The string's
motion is highly complex. -- then we find that the soundboard is only willing to respond
easily to some of those frequencies. Because of the complex nature of soundboard
impedance, energy at some of the frequencies within the string will not be readily
accepted into the soundboard, while at others it will be welcomed with open arms. Those
that are readily accepted by the soundboard will make up the tone character of the initial
attack and for the first few milliseconds of the sustaining tone, those are not so readily
accepted will make up the tone character of the sustained sound. The rate at which the
overall energy package within the string is accepted into the soundboard determines the
overall sustain of the tone.

It is this complex relationship between the energy spectrum within the string and
characteristic impedance of the soundboard assembly (including the design of the bridges,
the soundboard panel itself, the ribs, the crown characteristic of the assembly, the
mounting system of the assembly, etc.) that determine the tone characteristic of any given
piano.

So, is everything now as clear as mud?

Regards,

Del