Quantz formula challenge

Since Brad thinks that it's impossible to have a modified meantone
more unequal than 1/6-comma in which Quantz' remarks can be literally
true, here's one simple counterexample.

C 0
G 697
D 194
A 891
E 388
B 1085
F# 582
C# 84
G# 790 or 791
Eb 294
Bb 1001
F 503

Here, as in fact described in Marpurg's Principes du Clavecin, seven
fifths from F round to F# are made flat, the remaining ones wider or
at least more pure. (Nothing in Quantz forces you to tune regular
meantone from Bb round to C#...)

I allow myself a small (6 cents or so) fudge factor, inside which the
impurity of a unison between the keyboard third and the solo
instrument, though possibly audible, will not be worth mentioning. I
think it is reasonable to allow a slightly larger fudge (half a comma
or so) for the minor third, since that is not pure even in 1/4-comma
meantone.

All the major thirds from Bb-D round A-C# are then compatible with a
pure third from the solo instrument. All minor thirds from G-Bb round
to C#-E also satisfy the relevant criterion.

However C-Eb, F-Ab, Bb-Db, Eb-Gb, Ab-Cb are Pythagorean minor thirds
or worse. Of these, Quantz happened to mention C-Eb, Bb-Db and Ab-Cb,
which span the set of five, perhaps because these three lie the
highest on the harpsichord inside the normal range of continuo chords.

Considering the major thirds again, all of them from E-G# round to
Eb-G are worse than 12-ET, some indeed beyond Pythagorean. Of these,
Quantz made examples of E-G#, B-D#, F#-A#, and C#-E#.

However, this is only by way of counterexample. I don't really believe
it can be justified to make this kind of pipe-dream deduction from a
clearly incomplete and inexact (though perfectly clear as to general
principles) text, a text which was never even about how to tune a
keyboard.

Actually, any modified meantone in which *most* of the seven thirds
Quantz specifically mentioned are obviously impure, and probably some
others too, but in contrast the diatonic and 'central' thirds such as
Bb-D or D-F# are not obviously impure, is consistent with the text
making sense.
~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> Since Brad thinks that it's impossible to have a modified meantone
> more unequal than 1/6-comma in which Quantz' remarks can be literally
> true, here's one simple counterexample.
>
> C 0
> G 697
> D 194
> A 891
> E 388
> B 1085
> F# 582
> C# 84
> G# 790 or 791
> Eb 294
> Bb 1001
> F 503
>
> Here, as in fact described in Marpurg's Principes du Clavecin, seven
> fifths from F round to F# are made flat, the remaining ones wider or
> at least more pure. (Nothing in Quantz forces you to tune regular
> meantone from Bb round to C#...)

Ummmm....if the point was to present a counter-example that actually
makes musical sense in the context of that book by Quantz, why does
this temperament make the harpsichord sound ludicrously bad in the
exemplary composition printed right there in Quantz's chapter for
keyboard players? The "Affettuoso" in E-flat major? Is it possible
that this temperament wasn't tested before being posted?

And does Marpurg's Principes du Clavecin really say that those seven
fifths are supposed to be made *that* flat? Please show us the
details, if that's being used as support.

Brad Lehman