para-inharmonicity and tuning curves

[Jim Coleman, Sr.]

Hi again richard:

The fundamental was not tuned to 0.0 cents when I read it. These deviations
are deviations from 0.0 cents at each equal temperament note location where
the partials are. To get the amount of inharmonicity of each partial, you
would have to subtract 2.1 cents from each to get the picture in relation
to 0.0 cents deviations.

In regard to your comment about flat partials, it IS possible to to have
flat partials due to many perhaps unknown-as-yet reasons. The negative
numbers of the odd numbered partials below however are due to the fact
that the numbers were read at note locations which on the SAT are
normally read as equal tempered relationships. The 3rd partial is read
from a note location in equal temperament which is sharper than would be
the 3rd partial. This machine reading notes normally makes the 5th interval
narrow according to the requirements of equal temperament, but the 3rd
partial of a string is not tempered, so it always shows sharper (by
at least 1.95 cents) when measured by an equal tempered machine. Likewise
when measuring a 5th partial with a machine which is built upon the
principle of measuring tempered intervals (which are normally 13.687 cents
wider than pure), the 5th partial will appear to be flat of the reference
note by at least 13.687 cents.

Jim Coleman, Sr.

On Sun, 23 May 1999, Richard Moody wrote:

> Hi Jim
> 	I wanted to ask you but held off, thinking it would come up.   You
> mentioned....
> 
> 
> > the cents deviations of actual octavely related partials taken from
> > my Steinway L, note C4.
> > 
> > 1st partial   2nd partial   4th partial   8th partial
> > 
> > 2.1           2.8           6.2           20.2  in Cents
> > 
> 
> If I read it right does it say the first partial has a "cents deviation"
> of 2.1 ?   
> 
> I can't figure out how the fundamental (commonly called the first partial)
> can deviate by more than 0.0 cents let alone 2.1. 
> 
> Regarding the 20.2 cents for the 8th partial, I assume that is twenty
> cents sharp. Does that mean the freq measured for the 8th partial of C4 is
> 20.2 cents sharp from theoretical.  (8 times the fundamental   F*8 )  ?  
> 
> 	I know you know I need this like a (another) hole in the head, but why
> does the SAT need to odd partials compared to ET?.  Because of this
> comparison is that why  some of the readings have negative cents which
> could be interpreted in real terms as a partial that is flat which is
> acoustically impossible? 
> 
> Ric  Wondering 
> 
>  
> > In anticipation of your next question, when you plot for, 3rd, 5th,
> > 6th, and 7th partials, you must make a correction for equal temperament
> > being the source of the readings. Here are the readings taken with
> > and electronic tuning machine:
> > 
> > 3rd partial   5th partial   6th partial   7th partial
> > 
> > 5.9          -4.6           13.6         -15.4
> 
> 
> 
>