Joe,

The more that I stared at my "correction" in message #1926, the more I realized
that I (erroneously) added something (2^) to the solution for Z as a function of
X. With the understanding that the dependent variables Y and Z (as well as the
independent variable X), being in the *exponents" of your (nicely done)
diagram's algebraic identity [which was (3^(X/3))*(5^(X/6)) ~= (3^Y)*(5^Z)]
represent quantaties in the "logarithmic domain" (and not the "linear domain"),
the correction below should work well for the such determinations of Y and Z as
functions of X as (it would seem) you have requested.


--- In tuning-math@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > Hi J,
> >
> >
> > > From: unidala <JGill99@i...>
> > > To: <tuning-math@y...>
> > > Sent: Sunday, December 16, 2001 7:56 PM
> > > Subject: [tuning-math] Re: formula for meantone implications?
> > >
> > >
> > > J Gill: Monz, it sounds like you want to build a machine
> > > than can "think" (like people do)! I guess if you can
> > > define a set of JI ratios (which you like, or which meet
> > > some "man-made" criteria for the numerical size of the
> > > numerator/denominator involved, etc.), you could write
> > > a program to "decide" which of those ratios your meantone
> > > pitch value is "closest" to [by some pre-determined measure
> > > such as RMS error in deviation from a function such as
> > > 2^(pitch/reference)].
> >
> >
> > Not at all! It's much simpler than that.
>
> JG: So, you *do not* have a predetermined finite set of JI scale pitch-ratios
in mind with which to compare with your mean-tone scale pitches? Your response
(above) *could* indicate that you do *not* want to limit the possible JI
scale-ratios to a finite set???
>
> > I'm just looking
> > for an elegant mathematical formula to explain what I'm showing
> > on my lattices.
>
> JG: In message #1924 (corrected and revised in mesaages #1925 and #1926) I
algebraically re-arranged the "mathematical formula" which your diagram showed,
in order that you would have the identities necessary in order to solve for Y or
Z as a function of X:
>
> Solving for Y (with Z held as a constant value between X2 and X1):
>
> Y = (((LN of (5^(X/2)) - LN of (5^(3*Z))) / (LN of(3)) + X) / 3
>
> Solving for Z [WHERE YOU INSERT Y AS = (Z + (dY/dX)* X) in each
> of the three equations for the three (isolated sections) falling
> between Xmin and Xmax, where dY/dX above is the log-log slope and Z
> and X are in *octaves*] will require using a "solver" program
> (since the "independent" variable Z to be determined appears in >*TWO*
> places in this restated identity, thus requiring an "iterative" >solver (OR
PERHAPS, LINEAR ALGEBRA, WHICH IS NOT MY STRONG POINT)
> (Mathematica should have such capabilities):
>
> Z = (LN of (5^(X/2)-(3*Y + X)/(LN of 3))) / (3 * (LN of 5)))
>
>
> The (combined, AND CORRECTED) form for Z is as follows:
>
> Z=(LN of(5^(X/2)-(3*(Z+(dY/dX)*X)+ X)/(LN of 3)))/(3*(LN of 5)))
>
> I guess I'm still confused as to what you want you algorithm to do...
>
>
> > The only measure I'm using is simple closeness
>
> JG: of your mean-tone pitches to *what* (finite or infinite) set of your
intended JI pitch-ratios to be compared to your mean-tone pitches?
>
> > in pitch-height.
>
> > The only reason it gets complicated and requires two solutions
> > sometimes is because some meantone pitches are exactly midway
> > between the two closest implied ratios.
>
> JG: That "special case" could be dealt with, it seems.


Sincerely, J Gill :)