RE: [tuning] Re: well-temperment wazoo

Jason Yust wrote,

>Your last set of imputs to the minimum-finding
>algorithm showed a large number of local minima in the region of 12-tET.
>If the graphs are this bumpy everywhere, then looking for minima tells us
>very little.

I think I showed that one can take a smooth ride through these
well-temperaments "downhill" to 12-tET -- its just that the optimizer got a
little lost.

>But maybe the graphs drop off in other regions, and we might
>find that, while in the region of 12t-ET, there are a number of minima,
>most other regions contain few minima, for instance.

Well we do know that if you start just a little too far from 12-tET, you
drop into one of those regions where two of the notes converge into one,
making an 11-tone scale. Also, while I found a few truly different 10-tone
minima, for 12, so far, I've only found one real one (and some fake ones).

>Perhaps there is
>something to be seen by looking at the problem with a coarser resolution
>which would smooth out the smaller lumps in the graph. I'm thinking of
>something like having the algorithm consider the results of a step of 5
>cents in some direction, rather than of .01 (or whatever) cents. Then you
>could see if any other regions of the graph are worth considering.

Well I could always use various different starting points in order to find
that out, and as I mentioned, I tried quite a few. Any suggestions?

>Also:
>can you find an absolute minimum given some number of tones?

For up to 4 tones, this is feasible, since I can just generate a big array
of values over a pretty wide range of intervals, and I already reported the
results found from these arrays. For 5 or more tones, memory limitations
kick in, so optimization algorithms are really my only hope.

Manuel wrote,

>By the way, when I feeded

fed

>12-tone equal temperament with one
>randomly changed tone to my least-squares method, it also produced a
>well-temperament.

Can you get a smooth range of well-temperaments using this method, with the
ones closer to 12-tET having smaller sums-of-squares? If so, your situation
is probably analogous to mine: our optimizers are getting "stuck".

>Paul, the well-temperament that you found is close to Schlick (1511)
>and Lambert (1774).

Manuel, do you see the pattern I'm referring to?

By the way, with 3 tones, the situation is quite different: starting at
random points near 3-tET, you either get to 3-tET or to a mode of 0 389 778
-- and the latter actually have a slightly lower total discordance than
3-tET. Unlike the 12-tET case, there are no "slides" connecting these points
-- you actually have four discrete local minima: 3-tET, and the three modes
of 0 389 778.

Actually, you occasionally get spun out all the way to a 5-odd-limit triad
-- which is of course even lower in discordance.

Paul wrote:
>Can you get a smooth range of well-temperaments using this method, with
the
>ones closer to 12-tET having smaller sums-of-squares? If so, your
situation
>is probably analogous to mine: our optimizers are getting "stuck".

In my case optimization is only a matter of solving a set of linear
equations.
The result is often that only one 5/4 is taken into account. Or, with less
restriction maybe also a 6/5 and 9/5. The resulting temperament then has
four
consecutive +/- 1/4-comma flat fifths.

Manuel Op de Coul coul@ezh.nl

--- In tuning@egroups.com, <
MANUEL.OP.DE.COUL@E...> wrote:

>
> In my case optimization is only a matter of solving a set of linear
> equations.
> The result is often that only one 5/4 is taken into account. Or,
with less
> restriction maybe also a 6/5 and 9/5. The resulting temperament
then has
> four
> consecutive +/- 1/4-comma flat fifths.

Well then I don't understand your last comment --
that moving just one note away from 12-tET leads
your optimizer to a well-temperament.

Paul wrote:
> Well then I don't understand your last comment --
> that moving just one note away from 12-tET leads
> your optimizer to a well-temperament.

Suppose I take 12-tET with M3 of 395 cents instead of 400.
And this set of consonant 5-limit intervals is used:
6/5 5/4 4/3 3/2 8/5 5/3 9/5
Then I set the maximum deviation for the intervals to become
"eligible" for optimization to 10 cents. That makes the 395 cents
interval within the range of 5/4. Then optimization marks these
consonant intervals a number of times:
6/5 : 0
5/4 : 1
4/3 : 12
3/2 : 12
8/5 : 1
5/3 : 0
9/5 : 0

The resulting scale is:
0: 1/1
1: 93.779 cents
2: 195.023 cents
3: 296.267 cents
4: 390.046 cents
5: 498.756 cents
6: 592.535 cents
7: 697.512 cents
8: 795.023 cents
9: 892.535 cents
10: 997.512 cents
11: 1091.291 cents
12: 2/1

Or, as chain of fifths:
0: 1/1 0.0000 0 commas
7: 697.512 cents -4.4434
2: 697.512 cents -8.8868
9: 697.512 cents -13.3301
4: 697.512 cents -17.7735
11: 701.244 cents -18.4843
6: 701.244 cents -19.1951
1: 701.244 cents -19.9060
8: 701.244 cents -20.6168
3: 701.244 cents -21.3276 10/11 Pyth. commas
10: 701.244 cents -22.0384
5: 701.244 cents -22.7492
12: 701.244 cents -23.4600 Pythagorean comma, ditonic comma

A simple well-temperament. If more than one note is changed from
12-tET, or a wider allowed deviation is used, then the outcome isn't
necessarily a well-temperament.

Manuel Op de Coul coul@ezh.nl