Yahoo Tuning Groups Ultimate Backup tuning-math Optimal Kees vs TOP-Max Tuning

I've been looking at Kees weighting for my errors and complexities PDF:

http://microtonal.co.uk/primerr.pdf

One thing I've shown is that the max Kees error for a set of weighted primes, w, is

max(w) - min(w)

and the TOP-max error (previously known as plain TOP) for the optimal stretching of w is

max(w) - min(w)
---------------
max(w) + min(w)

It would be nice if the two had the same optimal tuning, as they're obviously closely related. The Kees one should be simpler to minimize. Anyway, while searching the list for some definition of projective space, I found an old thread about the Kees tuning not being the unstretched TOP(-max) tuning. This appeared in a quote from 10th September last year:

<<3 -5 -6 -1 -15 -18 -12 0 15 18||
>
> with TM basis {56/55, 64/63, 77/75} and mapping
>
> [<1 3 0 0 3|, <0 -3 5 6 1|]
>
>The stretched TOP tuning has pure 7s, and the Kees tuning has
>the error of 3, over log(3), equal to the error of 5, over
>log(5).
>
>Stretched TOP:
>
> <1200 1915.578 2807.355 3368.826 4161.472|
>
>Kees:
>
> <1200 1915.929 2806.785 3368.142 4161.357|
>
>While the tunings are pretty close, they clearly are different.

Now that I can find TOP tunings for rank 2 temperaments, I went and checked this. I get the tuning map as

<1195.4856, 1908.3816, 2796.792, 3356.1504, 4145.8152]

This is exactly what the linear programming library returns. It seems to be of limited precision. Unstretch it and I get, to three decimal places,

1200.000, 1915.588, 2807.353, 3368.824, 4161.471

which doesn't match either of Gene's. It has a TOP-Max error of 4.513934 cents per octave. This is using the formula that's independent of the scale stretch. Gene's "Stretched TOP" map has a TOP-max error of 4.513950 cents per octave. Gene's Kees map has a TOP-max error of 4.513988 cents per octave. So, while Gene's TOP tuning does have a lower TOP-max error than his Kees, tuning, mine is still lower. So, it must be closer to the real TOP-max tuning.

So, to the Kees error, if I'm right about that. My TOP-max tuning has an error range of 9.061950 cents/octave. Gene's Kees optimal tuning has an error range of 9.060224 cents/octave. So, there you go. Gene wins.

However, we're still comparing differently inaccurate temperaments here. Gene says his Kees optimum should have equal weighted errors for 3 and 5. Well, I worked out what the generator for that should be:

3log(5) octaves
-----------------
5log(3) + 2log(5)

or 561.47065259506269 cents. The tuning map is

<1200.000000, 1915.928919, 2806.785134, 3368.142161, 4161.357027]

The TOP-max error is 4.413924 cents/octave. Funnily enough, that's the same as my previous TOP-max error. The Kees error is 9.060116 cents per octave. So, I win the appeal!

What's happened is that the TOP-max error is

5log(7) - 6log(5)
-----------------
5log(7) + 6log(5)

for all four tunings, give or take the rounding error. That's because the tunings of primes 5 and 7 are independent of the octave tuning. This is a case where the TOP-max tuning doesn't have a unique value.

Well, we always knew that there wasn't always a unique minimax tuning. This example doesn't show that the two optima can't be made to agree. So, I remain unconvinced.

Here are the weighted primes for the Kees optimum then:

<1.000000000, 1.007347134, 1.007347134, 0.999797038, 1.002418289]

They're more useful than the tuning map because you can calculate the errors from them directly, and you can still calculate the tuning map from it.

The restretched TOP-max tuning has a 1195.728754 cents and a generator of 559.358949 cents if you want to check. The tuning map is

<1195.728754, 1909.109417, 2796.794744, 3356.153692, 4146.545211]

and the weighted primes are

<0.996440629, 1.003761611, 1.003761611, 0.996238389, 0.998850309]

Graham

Optimal Kees vs TOP-Max Tuning

🔗Graham Breed <gbreed@gmail.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Graham Breed <gbreed@gmail.com>