[impedance] / inharmonicity

[Richard Moody]

----- Original Message -----
From: <BobDavis88@AOL.COM>
To: <pianotech@ptg.org>
Sent: Wednesday, June 14, 2000 1:36 PM
Subject: Re: [impedance]


> Engineers,
> Doesn't the moving bridge cause the terminus of the string to "appear"
> farther away than it is? And the string therefore to act longer? And
wouldn't
> the stiffness of the sbd vary the apparent length, and therefore the
> inharmonicity of the string?
> Thanks
> Bob Davis

Accroding to McFerrin, "An interesting feature is that the inharmoncity or
coefficient of inharmonicity, varies inversely as L^4, which means that the
longer string has less inharmonicity.  Also if   f  the fundamental
frequency is increased by tightening the string the inharmonicity is
decreased. "

The formula is

    B= K*d^2 / f^2*L^4.

B  is coefficient of ih
K  is a number depending on the system of measurement  ie  inches or mm. A
rather large number.  3.4*10^13 for d and L in centemeters and 5.3*10^12 for
inches.

Actual inharmonicity in cents is gotten by.

    I = B*n^2

Which means to find the cents value from the frequency of the natural
harmonic, you first calcuate B, then multiply by the square of the number of
the partial you are seeking the inharmomicity of.
This means the higher partials have more inharmonicity.
But remember this is a value of cents, and cents is a comparative value, a
logarithmic value of the ratio of two frequecies.
To find the actual frequency (F) of an inharmonic partial you compute  using
frequency of the natural harmonic (f)        f*2^(cents/1200) = F.

If B= 1 cent for the string of A2,  then the inharmonicity of the second
partial of A2 is 1*(2^2), or 4 cents.   For the third partial it would be
1*(3^2) or 9 cents.  I would guess B to be less than 1 to get the tuning we
get on piano strings.
McFerrin's source is Robert W Young, "Inharmonicity of Plain Wire Piano
Strings, Journal of the Acoustical Society of America. Vol. 24, May 1952.