error graphs of my four 1/5- and 1/6-comma temperaments

This shows the squared error of the major thirds
around the circle of 5ths in my temperaments...

http://lumma.org/stuff/squarederror.gif

The bottom end of the best keys in each scale is
currently on F. Can anyone tell me where to put it to
achieve the expected results with baroque or classical
keyboard music? If it were me I'd put it on G in each
case, but I want to do at least one thing by the
status quo.

-Carl

I wrote...

> This shows the squared error of the major thirds
> around the circle of 5ths in my temperaments...
>
> http://lumma.org/stuff/squarederror.gif
>
> The bottom end of the best keys in each scale is
> currently on F. Can anyone tell me where to put it to
> achieve the expected results with baroque or classical
> keyboard music? If it were me I'd put it on G in each
> case, but I want to do at least one thing by the
> status quo.

This new graphic trims out what I thought were the weaker
of my 1/5- and 1/6-comma scales, adds my 1/7-comma scale,
a George Secor scale, and a Johnson-Secor scale:

http://lumma.org/stuff/squarederror2.gif

Clearly I just need to bump all my scales +1 fifth.

As for the 1/7-comma scale, it looks like a good deal
as long as you don't mind F# being better than B or C#.

But lo, looking at these, I never would have guessed
that the sum of the squared error is actually:

Johnson-Secor RWT 2578
Lumma 1/5 2438
Lumma 1/6 2608
Lumma 1/7 2432
Secor VRWT 2604

Given the simplicity of the 5th-comma version, it gets
my vote. But Johnson-Secor is interesting if you think
24/19 is audibly better than other nearby sharp thirds
(I'm lookin' at you, Cameron :).

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> I wrote...
>
> > This shows the squared error of the major thirds
> > around the circle of 5ths in my temperaments...
> >
> > http://lumma.org/stuff/squarederror.gif
> >
> > The bottom end of the best keys in each scale is
> > currently on F. Can anyone tell me where to put it to
> > achieve the expected results with baroque or classical
> > keyboard music?

In the case of most of Neidhardt/Sorge's efforts, and some of
Werckmeister's, the answer is C, in that F may have a slightly sharper
third.

The thing that really gives you a handle on it is to compare D with Bb
and A with Eb. Most of Neidhardt and Sorge tend to make D(A) about
equal to Bb(Eb). In this respect the 1/7 comma one is already OK.

There is an exception in the case of Sorge's tunings intended 'for
Chorton organ'. These are flat-centered - the secret here being that
Chorton was a tone higher than the pitch of (most) chamber-music
instruments. Therefore the organ has to transpose, playing in a key a
tone flatter than the rest of the ensemble (C major for the other
instruments is Bb major for the organ).

> This new graphic trims out what I thought were the weaker
> of my 1/5- and 1/6-comma scales, adds my 1/7-comma scale,
> a George Secor scale, and a Johnson-Secor scale:
>
> http://lumma.org/stuff/squarederror2.gif
>
>
> But lo, looking at these, I never would have guessed
> that the sum of the squared error is actually:
>
> Johnson-Secor RWT 2578
> Lumma 1/5 2438
> Lumma 1/6 2608
> Lumma 1/7 2432
> Secor VRWT 2604

Sum of squared errors automatically favours the tuning which is
nearest ET, since it's equal to the sum of squared deviations from ET
plus a constant.

Apparently paradoxically, the tuning with fifths deviating most from
ET has thirds that deviate least. But that is easily explained given
that the 'best' third results from only 3 tempered fifths, and the
'worst' has 1.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> >
> > I wrote...
> >
> > > This shows the squared error of the major thirds
> > > around the circle of 5ths in my temperaments...
> > >
> > > http://lumma.org/stuff/squarederror.gif
> > >
> > > The bottom end of the best keys in each scale is
> > > currently on F. Can anyone tell me where to put it to
> > > achieve the expected results with baroque or classical
> > > keyboard music?
>
> In the case of most of Neidhardt/Sorge's efforts, and some of
> Werckmeister's, the answer is C, in that F may have a slightly
> sharper third.

Thanks.

> > But lo, looking at these, I never would have guessed
> > that the sum of the squared error is actually:
> >
> > Johnson-Secor RWT 2578
> > Lumma 1/5 2438
> > Lumma 1/6 2608
> > Lumma 1/7 2432
> > Secor VRWT 2604
>
> Sum of squared errors automatically favours the tuning which is
> nearest ET, since it's equal to the sum of squared deviations
> from ET plus a constant.

Yes.

-Carl