inharmonicity formulas

[Richard Moody]

----- Original Message -----
From: Ron Nossaman <RNossaman@KSCABLE.com>
To: <pianotech@ptg.org>
Sent: Thursday, March 02, 2000 5:37 PM
Subject: inharmonicity formulas


> Hi gang,
> I've finally got you some inharmonicity formulas rounded up and looked
> over. Here's what I've got so far.
>
>
>
>
> Mcferrin
>
> Core  =(5.3*10^12)*(Dc*0.001)^2/(Freq^2*Lin^4)
>


This is only the first part (B) of the overall formula  I = B*n^2
 Instead of "core = "  McFerrin wrote  B= 5.3*10^12*d^2 / f^2*L^4
d and L are in inches.

Concerning n^ 2   he wrote, "Calculate B and multiply by the square of the
number of the harmonic whose inharmonicity is desired."

It is not absolutly clear what he meant by "number of the harmonic".   Some
say there is the fundamental, and then the first harmonic.  Others say the
harmonic (or partial when speaking of piano wire) is equal to the number of
vibrating segments.   Thus the fundamental would equal one, and the 2nd
harmonic is frequency of one of the segments of wire vibrating in two
segments.  These segment frequencies are a little more than whole number
multiples of the fundamental which is of course where inharmonicity comes
from.

Now if the fundamental is considered as partial "one" or "n" and  if  n^2,
then  I = B*1.
the fundamental should have an inharmonicity of 0 (zero) so B should equal
0.
But this is not the case.  If the fundamental is not considered a partial,
then the first partial would be the frequency of one of the segments of the
wire in the mode of vibration of 2.   The formulas that McFerrin drew from,
express  n as the "mode of vibration".  This though suggests that the mode
of vibration for the fundamental is "one" and the mode of vibration for the
next partial what ever it is called would be "two". So the lowest
inharmonicity would be B*4.

To think of inharmonicity  of partials on a logarithmic line (by powers of)
really stretches the imagination.  That would make the difference between
the second and third partial as 2 : 9.   I don't think the  third harmonic
increases that much over the second.  That would mean the fifths would tend
to be sharp to be beatless. On the other hand I have always thought fifths
(ET) are purer than the theoritical beats given.  So perhaps this is due to
inharmonicity?  ?
    The above is concerned only with stright string, not wrapped wire.
While the lure of Ih formulas for wraped wire  is appealing (and necessary
for tuning machines),  I think the quickest way is to measure the
inharmonicity of the wrapped wire in different diameters of core and wraps
and put it in a chart and work out scaling from there.  Because of the
variables of wrapped wire and the vaguries between the various formulas Ron
presented (Thank you very much) I would think such an approach would be as
informative as trying to choose which formula to go by.  ---ric