Sigfrid Lundberg writes, concerning the method used by Jan Ling and
Margareta Jersild, 1965. "A method for cataloguing vocal folk
musik". Meddelanden fren Svenskt Visarkiv 21:
>
>The authors claim that this "spectral analysis" of tunes offers an
>excellent methods for finding similar tunes. It is sort of related to the
>methods you were thinking of, Laurie.
I think this is evidence for my claim that any reasonable method
of identifying tunes should work reasonably well...
Sigfrid Lundberg also writes, concerning tune searching using polynomial
approximation:
>Isn't it easier to use the squared distances between matching pairs of V?
>You don't need the polynomials to measure the distance. However, there is
>more to simililarity than this. I like Phil Taylor's protein metaphore,
>since tunes evolves somewhat like proteins.
In the end, it all comes down to measuring the distance between
vectors--the polynomials are there as a visual aid, so to speak. (I'll
explain the details in another post.)
>Nice idea - but surely powers of x are not the best choice
>for basis functions. How about instead trying Tchebychev
>(or however you spell that - I haven't got Cyrillic here)
>polynomials or (as you say) Fourier components?
>
Aha! You too learned about (Ts)Chebyshev polynomials before they
changed the transliteration of Russian. In fact you are right in the
sense that when you get down in the trenches and actually decide how to
find the polynomials, you find use polynomials which are ortho-normal
relative to the particular weight function chosen. They won't be the same
as the classic orthonormal polynomials such as the Chebyshev or Hermite
polynomials (different weight functions) but the idea is the same.
>For the polynomial, the first term will tell you whether
>the tune is essentially flat or whether it goes up on average
>from beginning to end or down. The second term will
>describe whether it sort of goes up in the middle then
>down or down in the middle then up. Higher terms
>get harder to describe. My guess is that this sort of
>description would tend to work better on a short
>stretch of music (say the A music or the B music or
>4 bars). I feel that Fourier components are a better
>bet for the whole tune.
>
It's a bit more complicated than that. Or maybe more simple,
depending on your point of view. I'm not trying to use the individual
coefficients of the polynomial, but rather its global nature: the fact
that a kth degree polynomial can have a limited number (k-1) of points
where it changes from increasing to decreasing, so the method is
guaranteed to smooth out the tune, while one can approximate any
continuous function arbitrarily closely by a polynomial, so it gives a
large enough class to do the approximation. (Of course, the degree must be
high enough.) I think that the method should work for whole tunes, not
just snippets, tho the matrices get larger as the tune lengthens, and the
numbers get much larger, which might be the worst problem. The method can
be used virtually without change for Fourier components, and for several
other equally reasonable alternatives. (According to my claim, all these
alternatives should work reasonably well!) I think that choosing the best
one would be a matter of experiment.
Of course, the question is, "Best for what?" There are a number of
possible uses. I'd expect that if one were actually trying to use such
algorithms in a serious way, one would use a number of different
ones, each having different known strengths and weaknesses. (And, of
course, finally, it would come down to the slowest algorithm of all, but
the only one which counts in the end: listening with a human ear.)
Cheers,
John Walsh
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