para-inharmonicity and tuning curves

[Jim Coleman, Sr.]

Hi Richard:

In your last email to me, you seemed confused about the linear
relationship of inharmonicity. It is exponential, or you might think
of it as logarithmic. Here is a very simple example of plotting
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 you take 5mm graph paper, turn it sideways and on the left column
number cents from 0 up to 20 for each of the 5mm divisions, and
number across the bottom from 0 to 64, you can plot the first partial
above the number 1, plot the second above the number 4 (that's 2 squared).
Plot the 4th partial above 16 and plot the 8th partial above 64.

You will see that you have a relatively straight line when you
connect the dots.

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

Now if you plot these numbers on the same graph, they will be all over 
the place. They must be adjusted in relation to harmonic partials
instead of equal temperament readings. From the 3rd partial subtract
1.95, from the 5th partial add 13.687, from the 6th partial subtract
1.95 (same as 3rd partial), from the 7th partial, add 31.?? (I can't 
find the exact number and don't have time to hunt right now.)

If you make the above corrections, you will find that the 3rd through
7th partials will fit right into you previous straight line when you
plot the 3rd partial above the 9th division, the 5th partial above
the 25th division, the 6th partial above the 36th division and the
7th partial above the 49th division.

Anymore questions?

Para-inharmonicity will be noticed more among the 1, 2nd, 3rd and
occasionally the 4th partials. Most everything else will line up
fairly well on the straight line.

Jim Coleman, Sr.