Subject: Wilson and recurrences

>

Hello Gene!

>
> From: "Gene Ward Smith" <gwsmith@svpal.org>
> Subject: Wilson and recurrences
>
> I've been playing with these, and don't know exactly why. To me the
> most interesting thing about the Wilson fifth is that it gives a
> meantone brat of -1; however Wilson thought it was important that it
> was the limit of a linear recurrence.
>
> He also used a linear recurrence for something called "chopi", which
> may or may not be more or less the linear temperament I've been
> calling hexidecimal, with commas 36/35 and 135/128. If so, it seems we
> should call that one "chopi". He used the characteristic polynomial
> x^4-x-2, which gives
>
> C[n] = C[n-2] + 2 C[n-4]

we have meta mavila which is
C[n] = C[n-3] + 2 C[n-4]
which is the one i think you mean?

>
>
> This polynomial is reducible, equalling (x+1)(x^3-x^2+x-2), so we
> could also use
>
> C[n] = C[n-1] - C[n-2] + 2 C[n-3]
>
> This is a cubic rather than a quartic recurrence, and so would seem to
> be preferable. It may be that since if zero coefficients are ignored
> we have a three-term rather than a four-term recurrence, Wilson
> preferred his version.
>
> I've two questions:
>
> (1) What musical value attaches to these linear recurrences? As usual,
> reading Wilson is difficult for me; he seems to think it is important
> but I can't tell why.

they are ways of generating scales, that for one, can have every interval unique. this is true of the numerical
series before they converge. also due to the recurrent sequences, the difference tones produced form tones that
generate the scale itself or many of its members. the original purpose of those pursuing 'gold' and gold like scales
was to find the most disonant scales possible. the thing isis that since they do generate their own scale members.
you end up with another type of consonance. All these scales will contain triads that are equal beating.

>
>
> (2) Why did he choose the fourth-order recurrence when he could have
> used a cubic recurrence?

how would this occur in a musical context? I personally have noticed such patterns but couldn't translate them into
sound.

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