Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.3 4.3bsd-beta 6/6/85; site well.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!ucbvax!hplabs!well!csz
From: csz@well.UUCP (carter scholz)
Newsgroups: net.music.synth
Subject: Re: Difficulty of synthesising piano sounds
Message-ID: <717@well.UUCP>
Date: Sat, 1-Mar-86 22:23:58 EST
Article-I.D.: well.717
Posted: Sat Mar  1 22:23:58 1986
Date-Received: Sun, 2-Mar-86 19:22:53 EST
References: <669@wcwvax.UUCP>
Organization: Whole Earth Lectronic Link, Sausalito, CA
Lines: 90
Summary: formula for overtones of a stiff string



     Someone at Whitechapel Computer Works started a
discussion of piano acoustics some time ago, asking for the
appropriate formulae.  Since no one has posted them yet, I will.
     The non-mathematical gist is this:
     As thickness and stiffness increase from zero, and as
tension decreases from infinite, the overtones of a vibrating
string depart from the "ideal" Fourier harmonic series.
     The tension, length, diameter, and material of a stiff
string determine a "constant of inharmonicity", B.
     Knowing B, you can calculte the "allowed frequencies" of
the vibrating stiff string.  This will ot tell you anything
about the time evolution of the piano tone, which is essential
for synthesis.  But it will give you an idea of how the overtones
are "stretched" from the harmonic series.
     All units are in CGS (cm, grams, seconds, Hertz).

==============================================================

        n * f(1) * (1 + B*n^2 / 2 )            pi^3 * Q * d^4
f(n) = ----------------------------- ;    B = ----------------
                 (1 + B/2)                      64 * L^2 * T

  d = diameter of string
  Q = Young's modulus of string material (1.95E12 for steel)
  L = length
  n = overtone number (NOTE: 1 = fundamental)
  T = tension on string (typically on the order of 2E12)


===============================================================

Empirically derived values for real piano strings:

for solid strings:                 for wound strings:

    3.95E10 * d^2                        4.6E10 * d^4
B = -------------                  B = ------------------ ,
    L^4 * f(0)^2                       D^2 * f(0)^2 * L^4

where d=core diameter and D=outer diameter of wound string.

B can be determined empirically by measuring the frequencies f(m)
and f(n) of two partials m & n, and taking their ratio r:

        (r*m/n)^2 - 1               (m*f(n)/n)^2 - (n*f(m)/m)^2
B = ---------------------     F^2 = ---------------------------
    n^2 - (r*m/n)^2 * m^2                    m^2 - n^2

                                                        (r-2)
When m=2n,  F = ( 8*f(n)-f(2*n) ) / (6*n) ; B = 2/n^2 * -----
                                                        (8-r)

               f(n)             1
Also: B = ( ---------- ) ^2 - ----- .
            n^2 * f(0)         n^2


NOTE: f(0) is not the fundamental.  f(1) is the fundamental.

        1            T
f(0) = --- * sqrt ( --- )   where rho = density * pi * d^2 / 4
       2*L          rho

For pinned boundary conditions:  f(n) = n* f(0) * sqrt (1+B*n^2)

For clamped boundary conditions:

f(n) = n * f(0) * sqrt(1+B*n^2) * (1 + (2/pi)*sqrt(B) + (4/pi^2)*B)

The piano string usually behaves as something between a pinned
and clamped condition, necessitating some kind of real-world
approximation between these two formulae.  One approximation is:

f(n) = n * f(1) * (1+(B/2)*n^2) / (1+B/2)

===============================================================
REFERENCES

Rayleigh, _Theory of Sound_, Dover Publications.
Morse, _Vibration and Sound_, McGraw-Hill, 1948.
Fletcher, "Normal Vibration Frequencies of a Stiff Piano String",
   _Journal of the Acoustical Society of America_ (JASA),
   v.36, n.1, pp. 203-209, Jan 1964.
Fletcher et al, "Quality of Piano Tones", JASA, v.34, n.6,
   pp. 749-761, June 1962.

===============================================================
--Carter Scholz         well!csz