A plethora of pentatonics!

Carl wrote,

>Regarding your entropy minimizer algorithm... have you reached any
>conclusions as to how much the initial scale choice influences the
>relaxed result? Wouldn't it be possible to find, for a given number
>of tones, the one scale with the least total dyadic harmonic entropy?
>I understand that this might not always be desirable -- when looking
>for a local minimum around a certain chord, for example. But for
>finding new scales -- how much did the initial choice of n-tET
>influence the results?

For 3 and 4, we already know that the ET did not lead to the global minimum.
For higher ETs . . .
I promised another List member to seed the program with random scales, and I
suspect that would address your concerns. Well, for 3 and 4 notes, we
already know most of the terrain, so I'll start with 5. Throwing out the
results with unisons, shifting to most compact mode, and sorting by
discordance (which comes out to 21.092 for 5 of the same pitch),

scale discordance #observations
0 195 390 699 891 39.506 3
0 192 501 696 891 39.506 2
0 112 316 498 814 39.829 2
0 316 498 702 814 39.829 2
0 118 314 619 815 39.99 2
0 196 501 697 815 39.99
0 197 383 502 698 40.01 3
0 196 315 501 698 40.01 2
0 196 315 501 699 40.01
0 115 317 702 816 40.048
0 114 499 701 816 40.048 3
0 314 497 628 812 40.193 2
0 184 315 498 812 40.193 2
0 311 389 702 887 40.228
0 185 498 576 887 40.228 2
0 184 388 572 886 40.243
0 197 501 586 699 40.296
0 113 198 502 699 40.296 2
0 91 205 589 703 40.393
0 229 317 498 814 40.398
0 316 497 585 814 40.398
0 314 390 702 812 40.411 3
0 386 499 701 814 40.412
0 113 315 428 814 40.412 2
0 268 498 703 885 40.473
0 112 235 498 814 40.476 2
0 203 387 586 702 40.478 2
0 116 315 499 702 40.478
0 113 428 611 815 40.484 3
0 388 702 780 886 40.49
0 125 312 623 703 40.505
0 313 497 624 703 40.533 2
0 201 316 584 701 40.548 2
0 116 431 701 818 40.551
0 81 390 498 703 40.572
0 205 313 622 703 40.572
0 86 316 583 703 40.605
0 202 431 700 817 40.629 3
0 313 497 628 886 40.681 2
0 386 498 615 703 40.703 2
0 268 384 499 767 40.712
0 117 223 498 616 40.719
0 112 424 499 611 40.796 2
0 187 498 581 812 40.816 2
0 313 426 626 811 40.822
0 313 426 625 811 40.823
0 127 312 441 625 40.86
0 184 313 498 625 40.86
0 234 497 623 812 40.879
0 237 386 622 883 40.895
0 125 314 440 702 40.944
0 129 311 392 626 40.972
0 271 387 586 770 40.998
0 311 438 621 702 41.05
0 127 234 314 625 41.051
0 183 267 497 581 41.06 2
0 117 233 431 616 41.082
0 102 183 319 497 41.098
0 178 314 395 497 41.098
0 232 387 498 620 41.119
0 75 185 388 499 41.156
0 264 388 497 581 41.198
0 258 320 496 577 41.225
0 80 235 392 498 41.239
0 88 193 272 585 41.346
0 229 311 394 623 41.457

That's probably missing a bunch, but at this point, if the "diatonic"
pentatonic in pseudo-meantone isn't the global minimum, I'll eat my shoe.

6&up . . . later

>For 3 and 4, we already know that the ET did not lead to the global minimum.

Think I missed that. I'm really getting bad in my old age.

>For higher ETs . . . I promised another List member to seed the program
>with random scales, and I suspect that would address your concerns.

A statistical approach to a brute-force approach. I don't know anything
about multi-dimensional optimization problems, but isn't it possible to
actually calculate the global minimum at each cardinality? Even something
like John's spring model. . .?

>That's probably missing a bunch, but at this point, if the "diatonic"
>pentatonic in pseudo-meantone isn't the global minimum, I'll eat my shoe.

Right on, right on. But what does the "#observations" column do?

>6&up . . . later

Rockin'.

-Carl

--- In tuning@egroups.com, Carl Lumma <
CLUMMA@N...> wrote:

> >For higher ETs . . . I promised another List member to seed the
program
> >with random scales, and I suspect that would address your concerns.
>
> A statistical approach to a brute-force approach.

It's known as a "Monte Carlo method" --
indispensible to scientists.

> I don't know anything
> about multi-dimensional optimization problems, but isn't it
possible to
> actually calculate the global minimum at each cardinality? Even
something
> like John's spring model. . .?

John's spring model works because the objective
function (the function being minimized) is
quadratic in all the input parameters. The full
harmonic entropy curve is clearly not quadratic;
hence this wouldn't work. Based on my education
as a physicist and my profession as a statistician/
financial engineer, I can tell you that global
optimization of functions with many local optima is
a very icult problem to attack rigorously and is
typically approached with Monte Carlo methods.

>
> >That's probably missing a bunch, but at this point, if the
"diatonic"
> >pentatonic in pseudo-meantone isn't the global minimum, I'll eat
my shoe.
>
> Right on, right on. But what does the "#observations" column do?

In this case, where they range from 1-3, not a
whole lot. With a great many more observations,
you'd get a sense of the "field of attraction" of
each scale . . .

[Paul Erlich:]
>It's known as a "Monte Carlo method" -- indispensible to scientists.

The name being taken from gambling, the random throw of the dice or spin
of the roulette wheel. I have personally experienced the power of
programs written using this method: they can get a lot done in a hurry!

[Carl Lumma:]
>>I don't know anything about multi-dimensional optimization problems,
>>but isn't it possible to actually calculate the global minimum at each
>>cardinality? Even something like John's spring model. . .?

[Paul:]
>John's spring model works because the objective
>function (the function being minimized) is
>quadratic in all the input parameters. The full
>harmonic entropy curve is clearly not quadratic;
>hence this wouldn't work.

That's close to correct. In fact, through I'm CURRENTLY using constant
spring constants, which is one-to-one with quadratic energy functions,
my model per se would be applicable to more general curves, including,
(with programming enhancements!) Paul's general harmonic entropy curve,
by non-linearizing the springs. The actual relaxation process would not
be made much more difficult.

I'd have to provide for "rewiring" a spring when its deflection crossed
a point of local harmonic entropy maximum, something else I'm not
currently doing (I'm considering alternate tunings of intervals before
the springs are wired, something that will eventually need to change!).

BUT, if I did that, I'd fall into the same pockets Paul is falling into,
because I'd wire the springs based upon the starting points (and the
points they eventually settled into), and I would get stuck in higher
energy traps just as Paul is (assuming Paul's code is bug-free, which I
have no particular reason to doubt).

[Paul:]
>Based on my education
>as a physicist and my profession as a statistician/
>financial engineer, I can tell you that global
>optimization of functions with many local optima is
>a very [diff]icult problem to attack rigorously and is
>typically approached with Monte Carlo methods.

Very true, to the best of my knowledge as well.

JdL