Why least-squares with sinusoidal weights gives 7 or 8 regular meantone fifths

I figured it out with old-fashioned algebra and trig.

What is determining the overall shape is the balance between major
thirds in each triple sharing a minor diesis (e.g. C,E,G#). They have
weights W1, W2, W3 equal to
W0+cos(theta), W0+cos(theta+2pi/3), W0+cos(theta-2pi/3)
where theta ranges round the circle. For a tuning 'centred' at C major
we can choose that theta=0 corresponds to C.

If for the moment we ignore fifths, the least-squares solution is that
the three deviations go as
t1=W2W3, t2=W3W1, t3=W1W2
with the sum W1W2 + W2W3 + W3W1 being equal to the diesis.

In fact this sum is identically equal to 3(W0^2-1/4).
Therefore the expected deviations of thirds are proportional to
(W0+cos(theta+2pi/3))(W0+cos(theta-2pi/3))
which turns out to be

WO^2-1/4 - W0cos(theta) + 1/2 Cos(2 theta)

To find what happens near the 'central' key (i.e. the meantone-like
part) we can expand in powers of theta. I won't go into the details,
but it turns out that when W0=2 (thus the weighting is [3,1]) the
coefficients of theta vanish up to the fourth power! So the thirds
vary *unexpectedly slowly* round the circle.

And for W0 somewhat less than 2, one has a negative theta^2 term and a
positive theta^4 term that more or less cancel - so that the thirds
stay almost exactly the same, up to theta=pi/3, that is 1/6 the way
round the circle.

To put this back in musical terms, because of the way thirds are
balanced against each other, for a range of reasonable-seeming
weightings, one obtains a run of 5 major thirds with virtually the
same deviation.

If the weighting is shifted a little round the circle so that (say) C
and G equally get the highest weight (i.e. set theta=-pi/12 for C),
the number of thirds which come out regular is only 4.

Now considering the fifths, in the parts of the circle where the
solution for the thirds is regular, they will tend to follow suit -
their influence on the resulting shape is a lot less than that of the
thirds, particularly if their deviations are smaller (i.e. if thirds
deviate more than 1/5 comma) - which turns out to be the case for
almost all choices of weighting.

Then when the solution indicates 5(4) quasi-regular thirds, we should
get 8(7) quasi-regular meantone fifths.

For example: with W0=1.65 (i.e. [2.65, 0.65]), equal weight between
fifths and major thirds, and a small offset from G-centred weights, I
get thirds

8.1, 7.9, 7.7, 8.8, 12.6, 18.2, 22.9, 24.3, 21.3, 16.0, 11.4, 8.9

and fifths

-3.4, -3.6, -3.5, -3.4, -3.6, -3.8, -2.5, 0.4, 2.0, 1.0, -1.1, -2.6

(Curious coincidence here that the 'worst' third comes in the same
place as the 'best' fifth...)

What this might have to do with musical practice is unclear - but
maybe Werckmeister knew more than he let on, when he recommended the
continuo tuning with 7 or 8 regular meantone fifths and 2 or 3 wide
ones for the 'diatonic-chromatic-genus, which is mostly used these days'.

If I can try to summarize what I think it means:

C major (in general the 'central' key, sometimes in fact G) can't be
too good, otherwise both E and Ab major will be too bad, taking
account of their non-negligible frequency of use.

G major might be a little better if one considers the less often-used
B major, but the fact that Eb major is rather more common contradicts
that. Overall (for a wide range of weightings) it's a wash. (Similarly
for F major.)

D major might be better, considering the 'worst' key F# major; but
again this is contradicted by Bb major, which should now be about
equal with D. Again, more or less a wash.

Only when we get 3 places away from the 'centre' do we reach a key
that is competing with a 'better' one: hence A major (or E major, with
the centre at G) must finally have worse intonation.

Conclusion: Mildly Modified Meantones Minimize Musical Mistuning!

... And they're easy to tune, too.

~~~T~~~

P.S. Anyone who objects to my notation can jump in a lake, for all I
care. If you start from the assumption that I don't write idiotic
gibberish, problems can be sorted out with just a small amount of
thinking.