Re: Definition and Measurement of Harmonicity (Harvey Holmes )


Subject: Re: Definition and Measurement of Harmonicity
From:    Harvey Holmes  <H.Holmes(at)UNSW.EDU.AU>
Date:    Sun, 23 Jan 2005 10:44:35 +1100

Jim, Reinhart, Chris and Others, I think Jim's comment below (under the new thread: Inharmonicity definition and measurement) is probably what Chris intended, and my contribution was attempting to show ways of doing this, with measures of degree of harmonicity that could be derived from the signal itself without reference to a physical production mechanism (e.g. vibrating strings), interesting though the production mechanism may be in its own right. I mentioned two basic harmonicity measures (with variations, including another one below), but there are many other conceivable measures, such as those in the survey by W.J. Hess that I quoted, plus a great many contributions in the speech coding literature of the last 10 or 15 years. However, I think the term "tonality" should be used with care, since this and several similar words (below) have a number of different but related meanings, largely depending on who is using them. You can check this by doing Google searches and looking at how the words are used. In addition, most of these meanings relate to psychoacoustic perception, which is much more than just harmonicity. In the first place, "tonality" means something else entirely in music theory. In addition, it is often also used to mean the same as "tonalness", which is a purely perceptual concept, referring to the sensation of pitch of a sound complex. Pitch perception is much more complex than the question of the degree of harmonicity. For example, pitch can be heard in sounds that are far from being harmonic, as explained in the articles by E. Terhardt on pitch that are available on http://www.mmk.ei.tum.de/persons/ter.html or by R. Meddis and M.J. Hewitt (JASA, 89 (6), June 1991). There have been attempts to predict tonalness based on various theories, such as Terhardt's virtual pitch concept or the Meddis and Hewitt temporal approach (q.v.). I haven't seen them, but I believe that there are even standards about this: ASA 118-1995 and DIN 45681 (the latter still in draft form a few years ago, but may be final by now). Another similar concept with a similar name is "tonality measure", which is used when deriving auditory masking thresholds (also a perceptual concept) for use in audio coding algorithms such as MPEG. This refers to the degree to which the individual sine wave components stand out above the noise floor. Masking models often treat "tonal components" differently from others when calculating the auditory masking threshold. Also, in some speech coding work these or similar terms may also refer to the degree to which individual partials of a tonal complex can be "heard out" individually (usually only the lower partials), still another perceptual concept, and similar to (but different from) the masking concept above. If partials can be heard out, they are sometimes coded differently from those that can't be heard individually. The fact that these terms (tonality etc.) are often used to mean different things by different authors is alone a good reason to avoid them unless they are clearly defined when used. I therefore think that degree or measure of harmonicity (or similar) is a better term when referring to the degree to which a signal is harmonic. This is a relatively straightforward physical concept (though with many possible ways of defining or estimating it), and is much simpler than the concepts underlying the other terms (tonality, tonalness, virtual pitch etc.), which are mostly perceptual in nature and have shifting meanings depending on who uses them. ******************** While still on this topic, another variation of my first harmonicity measure H1 is obtained with a different definition of ACF: RB(k) = SUM (x(n) * x(n+k)), where the sum is taken over all n in the range [0, N-1], with values of x(n) outside this range being set to zero. (This definition is the one used by the Matlab function xcorr.m. It applies a window to the signal, and, apart from a constant scale factor, gives a biassed estimate if certain statistical assumptions are made, unlike the previous definition.) The resulting harmonicity measure is then H11 = MAX (RB(k)) / RB(0), where the maximum is taken over k in the range [1, MAX], where MAX should be larger than any likely period of the harmonic component. (Incidentally, I should also have written [1, MAX] instead of [1, N-1] for this range earlier, since both R(k) and RB(k) are unreliable for large lags k, though for different reasons.) Harvey Holmes At 09:45 22/01/2005, Jim Beauchamp wrote: >----- >That said, it turns out that this is not really what Chris >was interested in. 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


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