Three term recurrences

I was complaining that the use of recurrences wasn't clearly motivated, but just now looking at the characteristic polynomials Jacques lists I keep finding three term polynomials, which define three term linear recurrences. These lead to situations where a difference tone between two terms are related to a third. Does anyone have an opinion on the audible significance of this? If it is significant, generators which are good ones for linear temperaments and which derive from a suitable three-term polynomial would be particularly interesting, though I still don't see why you need the recurrences and don't simply use the positive root > 1 of the polynomial as a generator.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> I was complaining that the use of recurrences wasn't clearly
> motivated, but just now looking at the characteristic polynomials
> Jacques lists I keep finding three term polynomials, which define
> three term linear recurrences. These lead to situations where a
> difference tone between two terms are related to a third.
> Does anyone have an opinion on the audible significance of this?

Yeah, it's not significant. Unless you do something very special
to make it significant.

-Carl

On sat May 1 2010 Carl wrote :

> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> > wrote:
> >
> > I was complaining that the use of recurrences wasn't clearly
> > motivated, but just now looking at the characteristic polynomials
> > Jacques lists I keep finding three term polynomials, which define
> > three term linear recurrences. These lead to situations where a
> > difference tone between two terms are related to a third.
> > Does anyone have an opinion on the audible significance of this?
>
> Yeah, it's not significant. Unless you do something very special
> to make it significant.
>
> -Carl

Well, if it's audible, it will have a musical signification.
(did you experiment difference tones with double ocarinas, Carl ?)
Then difference tones are not audible with all instruments, and especially with speaker amplification.
But my experience, after years of practice of integration of the difference tones in tuning creation,
numerous collective psychoacoustic workshops on the subject, blind tests etc.,
is that differential coherence has a musical signification, whether difference tones are audible, or not.
I don't have the full explication, but it is my experience and the experience of many others.

- - - - - - -
Jacques