Another question re: Subharmonics - undertone singing clips

> From: "Rosati" <dante@pop.interport.net>
>
> Jim-
>
> Excellent! I've downloaded the clips and had a quick look at some
> spectrograms. While I need to study them further, there was one thing I
> noticed right away. Instead of generating subharmonic partials, what is
> happening is that the main note you are singing is becomming the third
> partial of a series with a fundamental a twelvth lower. What shows up on the
> spectrogram is both a fifth and a twelvth below the principal sung note.
> Fascinating stuff!
>
> dante

Hello dante,

I forgot to ask last night if on the spectrogram there is the octave below the
principal sung note. I hear it and esp. before the octave and fifth below
principal note sung (hereafter "pns") kicks in. This is part of the reason I am
still seduced by the idea of these being part of a subharmonic series - because
that structure of intervals appear from the singing - it seems to unfold
naturally - that is, from pns a note an octave below appears as I'm singing,
then a bass note a fifth below that comes (although I'm not always able to
clearly hear the octave with this clearly), etc. - perhaps I need to put another
clip up showing this transition to demonstrate what I mean. What puzzles me
also is that I don't hear "sub 4" (two octaves below pns) while progressing from
sub 3 to sub 5.

Thanks for the help...

~Jim

Jim Cole wrote,

>I forgot to ask last night if on the spectrogram there is the octave below
the
>principal sung note. I hear it and esp. before the octave and fifth below
>principal note sung (hereafter "pns") kicks in.

You would usually hear the octave below before hearing the twelfth below
because period-2 behavior, followed by all powers of 2, usually occur before
period 3 in the progression from linear to more and more nonlinear dynamics.
You might still hear it, if you're trying to, after the twelfth below kicks
in, since many of its harmonics are still present and strong, as harmonics
of the new fundamental.

>What puzzles me
>also is that I don't hear "sub 4" (two octaves below pns) while progressing
from
>sub 3 to sub 5.

This too is explained by the fact that all powers of 2 usually occur before
any powers of any odd numbers. All powers of 2 can fit within a finite
portion of parameter space (along the axis which defines the "degree of
nonlinearity") since each power of 2 occupies a range of parameter values
that is 1/4.67 times smaller than that occupied by the previous power of two
(at least analytically in the typical case of a quadratic nonlinearity and
experimentally in many physical situations). See
http://www.cis.ohio-state.edu/hypertext/faq/usenet/sci/fractals-faq/faq-doc-
11.html for more info.