# Re: Definition and Measurement of Harmonicity

```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.

>-- Original-Nachricht --
>Date:         Fri, 21 Jan 2005 13:17:39 +1100
>From: Harvey Holmes <H.Holmes@xxxxxxxxxxx>
>Subject:      Re: Definition and Measurement of Harmonicity
>To: AUDITORY@xxxxxxxxxxxxxxx
>
>Chris and Others,
>
>[...]
>
>Thus, Reinhart Frosch's formula only applies for a particular model of
>thick plucked or struck strings.  It also doesn't give an overall measure
>of inharmonicity, since it only applies to the individual partials.  In
>addition, it doesn't take into account the relative strengths of the
>various partials (e.g. a signal would still be almost perfectly harmonic,
>even if some of the partial frequencies are grossly in error, provided that
>the corresponding partial amplitudes are very small).  It takes no account
>of noise of any sort in the signal, which is often the reason that a signal
>is less than perfectly harmonic.
>
>
>         Harvey Holmes
>
>At 05:23 15/01/2005, Reinhart Frosch wrote:
>>The inharmonicity of piano strings is treated in
>>section 12.3 of the book "The Physics of Musical
>>Instruments", by Fletcher and Rossing (Springer,
>>2nd ed. 1998).
>>
>>The basic equation for the frequency of the k-th
>>partial tone is:
>>
>>f[k] = f[1i] * k * (1 + k^2 * B)^0.5 ;
>>
>>here, f[1i] is the fundamental frequency of an
>>idealized string that has the same length, mass
>>and tension as the real string but is infinitely
>>flexible (i.e., has no stiffness).
>>
>>B = 0 corresponds to a string without stiffness
>>and thus to a harmonic complex tone;
>>B is an "inharmonicity coefficient".
>>
>>Reinhart Frosch,
>>(r. Physics Dept., ETH Zurich.)
>>CH-5200 Brugg.
>>reinifrosch@xxxxxxxxxx
>>
>> >-- Original-Nachricht --
>> >Date:         Thu, 13 Jan 2005 14:50:20 +0000
>> >From: Chris Share <cshare01@xxxxxxxxx>
>> >Subject:      Definition and Measurement of Harmonicity
>> >To: AUDITORY@xxxxxxxxxxxxxxx
>> >
>> >Hi,
>> >
>> >I'm interested in analysing musical signals in terms of their
>> >harmonicity.
>> >
>> >There are numerous references to harmonicity in the literature
>> >however I can't find a precise definition of it. Is there an
>> >agreed definition for this term?
>> >
>> >If someone could point me to some relevant literature it would
>> >be very much appreciated.
>> >
>> >Cheers,
>> >
>> >Chris Share

```