para-inharmonicity and tuning curves

[Richard Moody]

----------
> From: Jim Coleman, Sr. <pianotoo@imap2.asu.edu>
> To: Richard Moody <remoody@easnet.net>
> Cc: pianotech@ptg.org
> Subject: Re: para-inharmonicity and tuning curves
> Date: Sunday, May 23, 1999 7:04 PM
> 
> 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.

Ah... so C4 was 2.1 cents from ET on the piano (your L) when you measured
it. Because of the temperament you had tuned. Right?  Which illustrates
the beauty of cents as the great equalizer. (Sorry I couldn't resist) 
Anyhow hats off to A. J. Ellis, translator of Helmholtz who brought into
our discussion of tuning, the term "cents" 

And thanks to your explanation below. The only thing to make it more clear
is call the intervals by letter name ie  "fifth" and the partials by
number,, ie the coincedent partials of the fifth are the 3rd (of the
fundamental) and the 2nd (of the fifth).    The 5th partial sounds as a
seventeenth (two octaves plus a third) above the fundamental. Also  the
double octave can be called the fifteenth. The double octave + fifth 
would be the nineteenth. Which is where the 6th  partial resides.   Which
is immediately evident with the Coleman "Beat Locator" placed on a
keyboard.  Thank you much again. 
 
> 
> In regard to your comment about flat partials, it IS possible to to have
> flat partials due to many perhaps unknown-as-yet reasons. 

That would be interesting for a fire side chat. It would seem a law of
acoustics would have to bend over backwards, or waylaid by a mysterious
stealth of para-inharmonicity or it would be para-harmonicity
itself........ ?

Richard M.





>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 {from the fundamental} . This machine reading notes
normally makes the {fifth} (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 {which forms a third} 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.
>