Inharmonicity

[Jim Coleman, Sr.]

Hi Gary:

Once when I worked for the Baldwin Piano Co. I received a letter addressed
to Jim Coleman, Inharmonicity Department, Baldwin Piano Co., so I think this
may be my authorization to answer your question. (actually there was no
inharmonicity department, just a joke)

The word harmonic is a mathematical term meaning whole number relationships.
The prefix "in" makes it negative, ie not a whole number relationship. In
normal musical instruments we expect the octave harmonic to have twice the
frequency of the fundamental and the double octave harmonic to have 4 times
the fundamental frequency. Now, due to the stiffness factor of steel strings
the second partial (now not a true harmonic) will usually have a frequency
that is more than twice the fundamental frequency. Likewise, all of the
other parts (partials) of a tone will have greater than harmonic 
relationships to the fundamental. An example might be where the note A3 on
a piano has a 2nd partial which is twice the frequency of the fundamental
plus .5 cents. Therefore, if we wished to tune the A3 to A4, it would be
necessary to tune the A3 lower than 220 in order for its 2nd partial to
to be in unison with the fundamental of A440. However, it gets more 
complicated than that, for the 4th partial of this note A3 may have as much
as 1.7 cents beyond the theoretical 4 times the fundamental. This would 
cause the 4th partial of A3 to be sharper than the 2nd partial of the A4.
For this reason, it is considered prudent by most tuners to tune the A3 even
a bit flatter in order to get a compromise between a 2:1 match and a 4:2
match. Other tuners knowing that there are many more places where coincident
partials of the two notes will interact, may wish to make a compromise where
the 4th partial of the A3 will be flatter than the 2nd partial of A4. The
brighter the tone, the more prominent will be the higher partials. So, for
tuning a concert piano, one might wish to match the 6th partial of the A3
with the 3rd partial of the A4.

Please forgive me for going on and on about this if this is more than you
really wanted to know. There is much more.

Jim Coleman, Sr. 



On Wed, 12 Aug 1998, Gary Isaac wrote:

> From time to time I read about inharmonicity of piano strings on this 
> list.  I would be grateful if someone would clarify for me exactly what 
> inharmonicity is and how it affects piano tuning.  
> 
> 
>