Inharmonicity

[Richard Moody]

----------
> From: Jim Coleman, Sr. <pianotoo@imap2.asu.edu>
> To: pianotech@ptg.org
> Subject: Re: Inharmonicity
> Date: Sunday, August 16, 1998 2:37 AM
> 
> 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
--------.


Jim, List, 
	A convencing argument for the spaces the nodes take up (causing shorter
segments which because they are shorter are therefore sharper) is found in
McFerrin. "The Piano--Its Acoustics." p. 39  However the formula he
gives,(from Robert Young) is not nearly as simple as the concept.


> 
> 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.

Yes but the "pivot points take up space on the wire thus begging a
conclusion that that  segments of three for example are each shorter than
1/3 length of the string, and therefore should be sharper than a harmonic
produced by a true 1/3 length of string. 

> Stiffness does affect the higher partials more than the lower partials, 

Yes but what exactly is "stiffness"?  Are wires of diff compostiton but of
the same diameter "stiffer" than others? So what is used to "measure"
"stiffness" ? 

McFerrin gives an inharmonicity formula by Robert Young (JASA 1952) which
includes Young's modulus for piano wire. 

It seems this "stiffness" factor should be the same for all piano wire,
execpt where effected by diamenter.  But the formula for computing ih
(inharmonicity) is not as simple as I would expect. 

Well in one presentation it seems simple, that is 

	I = B n^2          (n^2  means n squared) 
I = ih (inharmonicity)
B = coefficient of inharmonicity
n = number of harmonic, (does he mean partial??)

For those wondering what B is, 

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

This is so much mumbo jumbo for me who expects the partial frequency to be
a simple matter of the ratio between diameter of the wire and the partial
length. 

OK so Mcferrin gets B down to 

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

d  = wire diameter  in inches
L =  string length inches. 

as you can see the ratio relationship between diameter and ih needs a 
fudge factor.  And I dunno for sure but it looks like you could take the
square root both sides and  have 
	
	I = B(n)

hmm in that case what would 5.3 x 10^12  look like ?? Damn I forgot my
algebra and don't have time to look it up  
5.3 x 10^6  ??   TEACHER ?!?

Ric Not so Mathamatic

ps
If you are into it this far, here are the references,,, 

Railsback, OlL.  JASA Vo; 9, p 274, 1938

Young, Robert, JASA. Vol 24, p. 267, May 1952. 

A N D - - - -  These remarkable conclusions by McFerrin 
Because ih "varies inversly as L^4 , the longer string has less
inharmonicity.  Also, if f, the fundamental frequency is increased by
tightening the string, the ih is decreased." 

So I guess a tighter strings does not mean a stiffer string... 

Or howbout a 7 foot bass string at 1000 lbs ?  ; )   rm 






..........
..> 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.