Inharmonicity in strings

[Jim Coleman, Sr.]

Hi Bill:

On Sat, 13 Jun 1998 Billbrpt@aol.com wrote:

> In a message dated 6/12/98 11:10:54 PM Central Daylight Time,
> imatunr@primenet.com writes:
> 
> << Bill,
>  Is not the fundamental also able to cause a Beat?
>  --
>  Joe >>
> No, unless they areat the same approximate pitch.
> Bill
> 

I beg to differ with you on this last answer. The subject of difference
tones and "Beat Frequency Oscillators" says otherwise. I play violin.
When two strings of a violin are accuractely tuned, it is possible to
hear the difference tone which is an octave lower than the lower note
of that 5th interval (ie when A and E are played together, you can hear
the A an octave lower. Here is the math: 660-440=220)

Pipe organs depend upon this to simulate extremely low tones.

There are also summation tones. For example 660+440=1100 where A4 and E5
will produce a faint tone at C#6. C#6 is not a harmonic of A4 nor of E5.
Paul Hindemith a great German composer utilized these principles in many
of his compositions.

A very practical application of this matter of difference tones occurs
in the Bass strings of pianos. Did you ever wonder why it is so difficult
to get an accurate assessment of the pitch of the fundamental of a spinet
low Bass note? When you consider that each of the partials (not harmonics)
of the note A0 produce difference tones with their neighboring partials.
Let me first illustrate this by pretending that they are harmonics instead
of partials. The first two harmonics would be 27.5 and 55 Hz. 
55 - 27.5 = 27.5. So you see, the difference tone reinforces the 
fundamental. Likewise the difference between the 2nd and 3rd harmonics
again equals the fundamental.  82.5 - 55 = 27.5. This same principle holds
for all the succeeding pairs of partials. All of them reinforce the 
fundamental.

However, when it comes to real frequencies of piano strings which have a
great amount of inharmonicity, and since the difference between successive
pairs of partials increases on an exponential basis as you consider higher
and higher pairs of partials,
the answers to all the math comparisons yield different values for these
differences at approximately the fundamental pitch. Now you know why it is
so confusing to listen to spinet pianos in the low Bass. You are 
actually hearing many representations of the fundamental. This explains
our difficulty in listening to the pitch of fundamentals of spinets. The 
same thing happens on Concert Grands, but to a much lesser extent.

Thanks for triggering me on this subject. I haven't written about this
since I wrote about it to Virgil Smith several years ago. this incidentally
forms some of the basis for his theory about listening to the fundamentals
only when tuning, or more correctly, listening to the whole tone.

Jim Coleman, Sr.