Inharmonicity

[Jim Coleman, Sr.]

Hi Richard:

No one has shown any evidence that the nodes of a piano string take up
space thereby shortening the segments of the various partials. Here is where
stiffness comes in. In a previous post I showed how thicker stiffer wire
acts more like a vibrating rod. Thicker wire is harder to bend. I also 
showed that there is no bending at the nodes, but only pivoting. The bending
takes place at the loops. It is the loops which cause sound. The loops show
more rod like characteristics the shorter they are. Both thicker diameters
and shorter lengths increase inharmonicity. The nodal influence is just an
idea, a way of explaining inharmonicity. It is not a fact. Stiffness is
resistance to bending. The nodes do not bend, they pivot. Tension of a 
string remains constant throughout the string. Each segment of a string 
is at the same tension.
Tension and stiffness are the restoring forces which cause a string to
quit vibrating. Stiffness causes the higher partials to die out sooner than
the lower partials because stiffness has a greater affect upon the shorter
higher partials. It is quite obvious that it is harder to bend a short
piece of wire than it is to bend a longer piece of wire. If we could have
a perfectly flexible wire, the partials would be true harmonics. Since we
have shown previously that in tuning forks which have only stiffness as
the restoring force (no tension) the second partial is way sharp. My
A4 tuning fork has a second partial which is at F7 plus 31.9 cents. That is
extreme inharmonicity. Piano strings have moderate inharmonicity (the second
partial of A4 is only a little over 1 cent sharper than a true 
harmonic). In piano strings the affect of tension is far greater than the 
affect of stiffness, therefore there is only a small amount of inharmonicity 
compared to bars, rods or forks.

My primary argument is that stiffness affects the loops, not the nodes since
it is the loops which bend and are resisted by stiffness thereby causing
inharmonicity. Violin strings, guitar strings String Bass strings 
Harpsichord strings  etc. are much more flexible than piano strings and 
hence exhibit much less inharmonicity (ignoring the control of bowing).

The Inharmonicity formulas which you showed in your last post do not 
completely describe what goes on in piano strings. For those who have 
the RCT, the Pianalyzer shows varying amounts of inharmonicity for different
partials (especially for the lower order partials). Dean used standard
formulas similar to the authorities cited in your post to calculate the
inharmonicity constants derived from successive pairs of partials.. 
There are yet unknown factors which are needed to account for these 
variations. At least, these formulas get you in a good ballpark.

Jim Coleman, Sr.