Hexagonal error plots

I've been thinking about the graphs Paul does that show lines of equal TOP-max error as hexagons and rank 2 temperaments as lines between points representing equal temperaments. I've worked out that you can plot them using the weighted deviation in a perfect fifth (EF) and the weighted deviation in a major third (ET) for a tuning with pure octaves as

x = EF/2 - ET

y = sqrt(3)EF/2

You can draw hexagons showing the maximum - minimum weighted errors. These also show the Kees-max error and they approximate the TOP-max error.

The distance of a point from the origin tells you the standard deviation of the weighted errors of the corresponding equal temperament. This approximates the TOP-RMS error.

Equal temperaments are truly straight lines.

Paul's graphs show ratios of mappings instead of errors. If I call MO the mapping of the octave, MF the weighted mapping of the fifth, and MT the wieghted mapping of the major third, we have

MF/MO = 1 + EF

MT/MO = 1 + ET

These correspond to two of the directions in Paul's graphs, and you can plot these instead of the errors if you want. The third direction is

MT/MF = (1+ET)/(1+EF)

I don't know how he gets that to work with the other two. It's approximately

1 + ET - EF - ETEF - EFEF

the corresponding direction for my plots is

(MT-MF)/MO = ET - EF

for errors measured in octaves. You can define the other directions as subtractions as well

(MF-MO)/MO = EF
(MT-MO)/MO = ET

I don't know if this helps us to find the TOP-max errors, but it at least gives a way of visualizing the octave equivalent approximations.

Graham

>I've been thinking about the graphs Paul does that show lines of equal
>TOP-max error as hexagons and rank 2 temperaments as lines between
>points representing equal temperaments. I've worked out that you can
>plot them using the weighted deviation in a perfect fifth (EF) and the
>weighted deviation in a major third (ET) for a tuning with pure octaves as
>
>x = EF/2 - ET
>
>y = sqrt(3)EF/2
>
>You can draw hexagons showing the maximum - minimum weighted errors.
>These also show the Kees-max error and they approximate the TOP-max error.
>
>The distance of a point from the origin tells you the standard deviation
>of the weighted errors of the corresponding equal temperament. This
>approximates the TOP-RMS error.
>
>Equal temperaments are truly straight lines.

I thought straight lines were rank 2 temperaments.

-Carl

Carl Lumma wrote:

> I thought straight lines were rank 2 temperaments.

Yes, sorry. And they are straight.

Graham