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Inharmonicity definition and measurement

Reinhart, Harvey, Chris, and others:

My original formuala for the inharmonicity constant for freely
vibrating strings, e.g., piano tones, was based on Rossing's
formula (Rossing, The Science of Sound, 2nd ed, p. 291, which is:

  f[n] = n*f[1]*[1 + (n^2 -1)*A],                       (1)

where A = pi^3*r^4*E/(8*T*L^2) and
r = string radius, E = Young's modulus, T = tension, L = string length.

A nice feature of this formula is that when n=1, you have f[1] = f[1].
>From it and taking B = 2*A, I get

  B[n] = 2*(f[n] - n*f[1])/((n^2-1)*n*f[1]).            (2)

The reason for taking B = 2*A, goes back to Harvey Fletcher et al.
(JASA, 34. 749-761 (1962)) and Fletcher (JASA 36, 208 (1964)). In the
1964 paper Fletcher gives the formula

  f[n] = n*F*sqrt(1 + B*n^2),                           (3)

which for small values of B*n^2 can be approximated as

  f[n] = n*F*(1 + (B/2)*n^2).                           (4)

Setting n=1 and A = B/2, we have f[1] = F*(1 + A), so that Eq. 4

  f[n] = n*f[1]*(1 + A*n^2)/(1 + A).                    (5)

This can be further approximated as

  f[n] = n*f[1]*(1 + A*n^2)(1 - A)

       = n*f[1]*(1 + A*(n^2 -1) - A^2*n^2)              (6)

Again, if A << 1, we can drop the last term, and Eq. 1 results.

The equation in Neville Fletcher and Thomas Rossing's "The
Physics of Musical Instruments (p. 61 of their 1st ed) is
taken from Morse's "Vibration and Sound" (1948), and is similar
to Eq. 1 but different. Harvey Fletcher's equation apparently
came from an article by Robert Young (JASA 24 (1952)).

Over the years several researchers have used either Eq. 1 or
Eq. 3 to compute inharmonicities of piano tones. I have tried
to do this on a computer and have found that, using my Eq. 2,
B varies with partial number. However, there is usually a range
of partials where B is quite stable. Again, it only works for
freely vibrating string tones.

That said, it turns out that this is not really what Chris
was interested. He is interested in something called "tonality",
which is something that has been mentioned in the audio
literature many times, but I have not seen a simple definition.
But basically if a signal is composed of harmonic or quasi-harmonic
sinusoids, it is "tonal". The other extreme is a noisy, random
signal. And, of course, signals can be combinations of both.

Jim Beauchamp
Univ. of Illinois at Urbana-Champaign

Original message:
> From: Reinhart Frosch <reinifrosch@xxxxxxxxxx>
> Date: Fri Jan 21 08:00:19 2005
> To: AUDITORY@xxxxxxxxxxxxxxx
> Subject: Re: Definition and Measurement of Harmonicity
> Comments: To: Harvey Holmes <H.Holmes@xxxxxxxxxxx>
> My posting of January 15 was a reaction to that of
> Jim Beauchamp, and to the original request,
> by Chris Share, to be pointed to relevant literature.
> The piano-string equation in the second edition (1998)
> of "The Physics of Musical Instruments" by Fletcher
> and Rossing differs from the equation in the second
> edition (1990) of "The Science of Sound" by Rossing.
> Admittedly the the piano-string equation is far from
> representing a complete definition of harmonicity.
> Reinhart Frosch,
> CH-5200 Brugg.