Researchers

In my opinion,

It is almost incredible that two chords can be analyzed through such mathematical tools, instead I always thought that optimal chords are formed by simple rules dictated by nature so should these procedures bring us to undeniable and useful conclusions, Carl Lumma and Steve Martin would have done a remarkable work.

I avail this message to inform you that in a matter of days I will get the plots of C major and F major from the JI seven tone, 5 limits scale proposed by Aristoxenus before Christ. Since its C major frequency ratios are 1, 5/4, 3/2, it is expected that the plot will show an asthetic and periodic response which are signs of optimal harmony as happens with the thre Piagui responses.

Should the JI C major plot is an aesthetic and periodic response this result would confirm that the procedure I apply to get the chord wave peaks reveals the harmony degree of the plotted chord.

Thanks

Mario Pizarro

February 05
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From Carl:

Steve wrote:

> * distances crop up in two places - the Gaussian function "height"
> depends on the distance between the actual chord and the candidate
> JI chord; and this function needs to be integrated over an area
> "belonging" to the candidate JI chord.

It's the former I think I got wrong. The latter is resolved
because we estimate this integral by multiplying the 'height'
of the Gaussian at the location of the candidate by the inverse
geomean of the candidate's identities (which approximates the
area 'belonging' to it).

> * initially I thought it was going to be essential to "plot"
> the points in this space in order to determine the neighbours
> and hence the Voronoi cells correctly; however, with the area
> approximation given by Carl, there is no need to do this.

Right. However it is still a good plot to view the results,
e.g. in a heat map or 3-D landscape plot.

> * therefore, I simply express the distance in terms of the
> x and y differences - in effect, use x^2 + y^2 + (x+y)^2 in
> the Gaussian.
>
> How does this sound?

I will have to review The Geometry of Triangular Plots
myself... how did you get the (x+y)^2 term?

-Carl

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Mario wrote:

> In my opinion,
> It is almost incredible that two chords can be analyzed through
> such mathematical tools, instead I always thought that optimal
> chords are formed by simple rules dictated by nature so should
> these procedures bring us to undeniable and useful conclusions,
> Carl Lumma and Steve Martin would have done a remarkable work.

Thanks, Mario. Hopefully something comes of it. And I should
point out that most of the work was done by Paul Erlich.

-Carl