Unequal temperament through spontaneous symmetry breaking

There was the Nobel Prize in physics today, and I'm going to apply
aspects of it to circulating temperaments!

The theme is spontaneous symmetry breaking (SSB). This means that you
have a physical situation that is initially symmetrical, where the
laws have the same symmetry, but its preferred state (eg least energy)
is actually asymmetrical. Sounds strange? Think of a cone balanced on
its point - the whole situation has rotational symmetry about the
axis, but the final situation will be the cone falling down and
pointing in one direction.

The symmetry in the case of temperaments is transposition round the
circle of fifths. Only equal temperament(s) respect this symmetry
completely. In fact one can 'derive' 12-et by minimizing a function of
the deviations of all fifths and thirds, subject to the condition that
the deviations of 5ths add up to a Pythagorean comma.
Of course this function has to respect the symmetry also, ie it has to
treat all thirds, resp. all fifths, equally round the circle. This is
the analogue of the physical laws having some symmetry.

But the conclusion of 12-et doesn't always hold, even if my function
is totally symmetric! Least-squares minimization *does* always give
12-et, that can be proved, because it is a linear problem ...
essentially because the function is simple enough.

However if I take a different function of the thirds, a strongly
nonlinear (but still transposition-symmetrical) one, I can minimise it
and get an *unequal* temperament!

How could this work musically? In an unequal temp you have better
thirds and worse thirds, and even if you play them equally often, it
should give a 'better' result than having all thirds the same. This
means that beyond a certain amount of impurity, increasing the
deviation should lead to progressively *less* penalty. Basically the
purer thirds should more than compensate for the impurer ones.
Mathematically, this means our function of the third-deviations should
not be convex: it should have bulges on the hillsides, as well as a
central valley around the pure interval.

I chose a function which is the total over every major third of
Exp(-1/|D3/t|)
where D3 is the deviation in cents and t is a constant. I chose values
of t between 4 and 12, this is an arbitrary choice that models the
point at which an extra deviation starts to become less important...
or the width of the 'flood plain' at the bottom of the valley.
Basically smaller t means pickier about the purity of the good thirds.
Then I just add another weight which is the sum of squares of fifth
deviations D5, times a constant 'f', and stick the whole thing into
Mathematica's numerical minimization routine.

The thing actually exhibits first-order phase transitions! Given any
value of t, for large values of f (modeling situations where pure
fifths are more desirable) the minimum configuration is 12-et.
But for smaller values, you jump discontinuously to a situation where
fifths on one side of the circle are narrower than ET, and wider on
the other. In other words a classic 'Well Tempered' situation - but a
result that dropped out "automatically" from a setup that treated
every key the same!

Moreover by adjusting 'f' and 't' you can model different types of
'historical' circular temperament...

For example with t=6, f=0.0116 I get the following sequence of fifths:
-3.9,
-4.7,
-7.4,
-4.7,
-3.9,
-2.9,
0.7,
1.6,
1.9,
1.6,
0.7,
-2.9
a strongly unequal modified meantone.

With t=12, f=0.0115 I get
-3.6,
-4.4,
-4.4,
-3.6,
-2.5,
-1.2,
-0.4,
0.0,
0.0,
-0.4,
-1.2,
-2.5,
which is a WT very much like 'Young I'.

But for t=12, f=0.012 I get 12et again...

There are further phase transitions between results with *one* single
best (and worst) M3, and results with *two* equally best (or worst)
M3's, but I'll spare you further details just now.
It was some time ago that I thought some sort of SSB for temperaments
should be possible, I'm quite pleased that it turned out to work
rather quickly.
~~~T~~~

> There are further phase transitions between results with *one* single
> best (and worst) M3, and results with *two* equally best (or worst)
> M3's, but I'll spare you further details just now.
> It was some time ago that I thought some sort of SSB for temperaments
> should be possible, I'm quite pleased that it turned out to work
> rather quickly.
> ~~~T~~~
>

Great work. But you don't have to spare us the details.
Let 'er rip! It would be helpful for me if you could post
the best and worst 3rd and 5th for each temperament.

-Carl

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> For example with t=6, f=0.0116 I get the following sequence of fifths:
> -3.9,
> -4.7,
> -7.4,
> -4.7,
> -3.9,
> -2.9,
> 0.7,
> 1.6,
> 1.9,
> 1.6,
> 0.7,
> -2.9
> a strongly unequal modified meantone.

I used an approximation where the Pythagorean commas is 24 cents and
syntonic 22. Best and worst major 3rds here (two of each) are
386+{1.3,27.9}. Minor thirds are 316-{5.2,27.2}. Worst fifth is
702-7.4 (the code seems to like to make the worst 5th quite bad in
meantone-like tunings), purest are 702+0.7, widest 702+1.9.

> With t=12, f=0.0115 I get
> -3.6,
> -4.4,
> -4.4,
> -3.6,
> -2.5,
> -1.2,
> -0.4,
> 0.0,
> 0.0,
> -0.4,
> -1.2,
> -2.5,
> which is a WT very much like 'Young I'.

Major 3rds range from 386+6.0 to 386+21.4 (one of each); minor from
316-9.6 to 316-21.7 (two of each), worst fifth is 702-4.4, best is
pure (up to stated precision)...

I don't think the numbers themselves are all that significant, more
the fact that an unequal temperament results spontaneously, and has
the 'bipolar' form we know from history.
~~~T~~~