May the 4ths be with you

[Richard Moody]

.
> >
> > What is the formula?
>
> 2^(((N-1+(I/100))/12)*27.5 * n
>
> where N is the note number, I is the inharmonicity constant
> and n is the partial you want to find the frequency for.
>
> RicB

This   one is from McFerrin which was developed by Robert W Young
in the 1940's. I think you may have it or it was posted before.

I = B*n^2

Where B  is the coefficient of inharmonicity
n is the mode of vibration or partial number counted from the
fundamental being 1

B is given in a beginning formula as

I = (1731*pi^2*Q*K^2*S / 2*L^2*T )*n^2 = B*n^2

    I    inharmonicity in cents
    Q  Young' smodulus for piano wire
    K    radius of gyration of a vibrating wire about its neutral
axis of cross                 section
    S    cross section of wire
    T    tension
    n     mode of vibration or partial number
    L    speaking length of string
    B  coefficient of inharmonicity


"The value of B, the coefficient of inharmonicity can be changed
to"

B = 1731*pi^2*d^2*Q / 2*64F^2*L^4*D

    d =  diameter of string
    f = frequency of fundamental
    D = density of steel in the string.

"If the value of Q and d {misprint shouldn't it be D?} and if d
and L are in centimeters, then.....

B=3.4*10^13*d^2 / F^2*L^4

For inches

B=5.3*10^12*d^2 / f^2*L^4

To Actually compute I  use...
 I = B*n^2

This will give you I  in cents......

If you need to know how much the frequency of the actual partial
is increased by I then you need to compute how far sharp the cents
value make the partial above its theoretical frequency.   It might
be interesting to compute the actual frequencies of partials of an
octave and compare the 2:1, the 4:2, the 6:3  and the 10:5 even.

I suppose some brave soul might attempt this on a spread sheet ?
?


You can see that the values of Q and D are presumed to be
constant.  I have absolutely no idea what K or "radius of gyration
about its neutral axis of cross section.  It must be a constant
also.
  I read the original
publication of Young ("Inharmonicity of Plain Wire Piano Strings."
Journal of the Acoustical Society of America. Vol. 24, May 1952,
Page 267)  and must have missed the radius of gyration.

Of interest in the article was the "spread" of readings of the
actual frequencies of the inharmonic partials.  They "lined up"
best on a log graph so that is where the n^2 comes from.   The
formula gives only the cents value for the amount of frequency
increase of inharmonicity over the value of the theoretical
partial frequency.   That is where the 1731   comes in.  That is
an approximation to give the logs of frequency ratios less than 20
cents or 20 hundredths of a semitone.   So what ever you get for I
will be cents, not an actual frequency.    ---ricm