Inharmonicity and humidity

[lianqii]

While considering the partial frequencies of a piano string(inharmonicity),I
would like to take into account three factors:the string vibration,the
hammer striking,and the soundboard vibration.

   1. the string:
In 1963,Mr.Harvey Fletcher studied the vibration frequencies of a stiff
piano string.He gave us a delicious formula
              F(n)=N*F(0)*(1+B*N^2)^(1/2)
where B=(3.14)^2*Q*S*K^2/T*L^2
>From the formula we can know,for an ideal steel string,the partials are
determined by the tension,the Young'modulas,the area of cross section of the
wire,the speaking length, etc.
    The real steel string is not ideal.So,"the small segment of string was
previously bent at the V-bar, but now is moved out into the speaking length
of the string because of tuning, that little bend could certainly have an
effect on the inharmonicity - even if it was only 1/16" long." -Bob Scott
    The bass string with wired string is another example.

    2.the characteristics of the soundboard:
Like string,soundboard has its own partials(we call them modal
frequencies).Ten years ago,I made modal analysis of a small grand piano and
found the "partials" of the soundboard(without strings) are
90Hz,144Hz,179Hz,230Hz,293Hz,330Hz,395Hz,413Hz,......
    The vibration of the string will "co-operate"with that of the
soundboard,Usually,the soundboard will decrease the partial frequencies of
the string.
    Humidity won't influence the string vibration a lot,but it will
influence the vibration characteristics of the soundboard.So my conclusion
is that humidity does influence the inharmonicity.But the influence is not
large enough because the string is the main factor during the "co-operation"
    3.the hammer striking:
Different strike can results in different amplitude of string vibration,and
thus,different string tension,different timber-- and different partials.
    4.the supporting of V-bar or agraffe also affect the partials in some
extent.

WISH MY LIMITED ENGLISH CAN EXPRESS MY IDEA!

Baoli Liu
Shenyang Conservatory of Music
China