Re: Definition and Measurement of Harmonicity (beaucham )

Subject: Re: Definition and Measurement of Harmonicity
From:    beaucham  <beaucham(at)MANFRED.MUSIC.UIUC.EDU>
Date:    Thu, 13 Jan 2005 11:08:58 -0600

Hi Chris,

This is a subject that I'm interested in. However, I'm not quite sure
what you mean by "harmonicity". Are you thinking of chords or are
you thinking about "the degree to which single sounds contain only
harmonics"? In the latter case, an inharmonicity coefficient can be
defined as
     B[k] = 2*(f[k] - k*f[1])/((k^2 - 1)*k*f[1]),

where k = partial number and f[k] = partial frequency. If B[k]
follows a certain law which is approximately true for string tones,
it will be independent of k. However, it real life you have to
average it over k to get a single value. (For details, see, for
instance, Rossing's The Science of Sound.) Typical B values for the
piano are .0001 to .01.

Physically, it depends on tension, mass-per-unit-length, and Young's
modulus (stiffness).

I don't know that it gives reasonable answers for instruments other
than free-vibrating string instruments, but the idea is to compute
an average deviation from a strictly harmonic relationship and
normalize it according to frequency somehow.

Best,

Jim Beauchamp
University of Illinois at Urbana-Champaign
jwbeauch(at)uiuc.edu

*************************************************************************
Original message:
>From: Chris Share <cshare01(at)QUB.AC.UK>
>Date: Thu, 13 Jan 2005 14:50:20 +0000
>To: AUDITORY(at)LISTS.MCGILL.CA
>Subject: Definition and Measurement of Harmonicity
>
>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