Inharmonicity

[Jim Coleman, Sr.]

Hi to all:

One of my pet peeves is the passing on of information saying that the higher
partials run sharp because of the space which the nodes take up. Such an
article ran in the 1949 series of the Piano Technician by George --------.

If you simply look at the nodal point of any sine wave, the bending takes
place at the loops, not at the nodes. The nodes are the pivot points.
Stiffness does affect the higher partials more than the lower partials, but
it's not because of the nodes.  The very high partials are more rod like 
than string like. I used to demonstrate this by having a very flexible felt
temperament strip in my hand. Its longer segments (partials) would vibrate 
quite freely and for a longer time when held between my two outstretched 
arms (tension being the chief restorative force).
But, if I had just a short segment (1/2 inch) protruding from between two
fingers, I could flip it with the other hand and it would vibrate like a
rod (stiffness being the only restoring factor)

The stiffer a segment is, the more like a rod will it vibrate. The second
partial of a rod (tuning fork) is more than 2 1/2 octaves above the 
fundamental. Check it out. The clang tone of an A4 fork is sharper than E7.
Of course piano strings are not nearly that stiff, but it is a good example
of what stiffness can do.

In piano strings we have two restorative forces acting at once. Tension
and stiffness. After striking a string, the tension tends to bring it 
eventually to a place of rest. But, stiffness also plays a very small part.
Flexible strings such as guitar or violin strings have little
or no measureable inharmonicity. With piano strings, you can measure it.
Bowed piano strings do not show inharmonicity according to experiments which
others have done (I tried, but without rosin, I could not get a steady
state going). My violin strings do exhibit a strange phenomenon in that the
fundamental of a bowed string gives a higher pitch that the same plucked
string (drives me bananas). Or, if I bow a string and lift the bow, the 
residual pitch immediately drops. The only explanation I have for this is
that when the bow is lifted, the additional tension provided by the
bowing is removed. But, I don't think it has anything much to do with 
inharmonicity.

If inharmonicity only is considered in the scaling of a piano, one could 
end up with a very weak but clear sounding piano. Tension and projection
are equally important. Early pianos were designed with no consideration
of inharmonicity. Some made no sense at all. Others were designed with
very smooth tension as the primary consideration. The resultant 
inharmonicity jumps made it difficult to keep the 3rds, 10ths and 17ths
running smoothly up the scale without distorting the octaves and vice versa.
The early Yamaha and Kawai grands had generally lower inharmonicity than 
they do now. Their designers apparently feel that power is more in demand
today than before. They still maintain fairly smooth scales.

The effects of poor inharmonicity show up more in small scaled pianos which
in order to maintain power have thicker strings in the tenor and/or resort
to having wound strings placed in that area to help maintain power. This is
not a bad idea, but they are just seldom  properly balanced. If one used
the longest plain strings in the tenor which would not crowd the soundboard
perimeter too closely, and then had foreshortened wound strings blended in
to fit with the tension and inharmonicity of the plain strings, those scales
would be more easily tuneable and the voicers would have fewer 
difficulties in blending in the tone going across the break. Of course, this
would require some changes in the tenor bridge to accommodate the shorter
wound strings. The Walter piano is a good example of proper blending across
the break (the Verticals or the Grands). It should be noted that they wisely
avoided wound strings on the Tenor bridge and provided the proper length 
differential between the upper Bass wound strings and the lower Tenor plain
strings.

Jim Coleman, Sr.