--- In firstname.lastname@example.org
, "Andreas Sparschuh" <a_sparschuh@...>
(That message has five pages of explaining and figuring.)
I printed out your five pages and I boiled it all down to one simple
set of by-ear instructions that I had to derive for myself...and it
has to be from a C fork, unfortunately, although I asked you
specifically for instructions from an A fork. I then set it up on a
harpsichord to give it a fair spin, and I've been playing some late
Couperin on it (which, again, is what I asked you to explain, and not
Here's what I've got. These were the most helpful several lines from
your five pages, where you allowed us to round off your 5ths to "only"
four different sizes, instead of micro-managing decimal fragments of
> F -1 C -1 G -3 D -3 A -3 E for approximation of about an ~SC
> E -0.25 B_F#_C#_G# -0.25 Eb -0.25 Bb -0.25 F in PC^(1/12) units
In other words, we're supposed to set it up almost the same as in
"Vallotti", but make the core set of tempered 5ths on the naturals
*uneven* instead of even. F-C-G get only 1/11 SC apiece, and all the
rest of the syntonic comma goes into G-D-A-E, 3/11 per. And then the
one schisma that's left over gets spread carefully across *four* of
the six remaining 5ths. Golly.
(And we're supposed to ignore the fact that for Neidhardt and Sorge,
the 18th century experts, the schisma was the smallest practical unit?
They were right: it *is* hard to control any 5ths that are less than
a schisma off pure, let alone anything as fine as 1/4 schisma...or
decimal bits and pieces grinding it even more finely than that. But,
Anyway...I did it like this:
C from fork.
E in the C-E major 3rd nudged one schisma sharp of pure 5:4.
C-F 5th one schisma narrow by knocking the F upward the slightest
audible nudge from the pure spot.
C-G 5th one schisma narrow by knocking the G downward the slightest
audible nudge from pure.
Set D and A so all three of G-D, D-A, and A-E are averaged out in
quality. They're rough. They're as bad as almost 1/4 SC each!
There's one schisma left to burn off, doing all the notes from
E-B-F#-C#-G#-D#-A#-F, where E and F are the fixed endpoints. From E,
I gave the B the very slightest smudge flatward (1/4 schisma!), which
is just about impossible to control with non-threaded pins, but my
wrist knows I did it. Then I made B-F#-C#-G# pure as prescribed.
This left D# and A# to be finessed in between G# and F, so each of the
three intervals gets another one of these 1/4 schisma smudges. It
works out, but again it's all in the wrist: amounts this tiny are just
about impossible to HEAR with any accuracy. A quarter of a schisma
And that's all 12 notes.
I played through some music from the last four Ordres by Couperin, and
I'll do some more. The results sound reasonable, although I can't say
they're any noticeable improvement ahead of normal "Vallotti". The
shape is almost identical, other than being done unevenly downtown:
too much tempering in G-D-A-E and not enough in F-C-G. The several
worst triads (B major, F# major, C# major) are still as bad as they
are in "Vallotti", give or take less than a schisma. And E major, A
major, and D major aren't as good as Vallotti's; the major 3rds are
higher, and the D-A-E 5ths are rougher. Oh well!
It's a passable sound, but I'd need to be given some compelling
reasons why I (or anyone else) should fuss with this uneven
F-C-G-D-A-E business downtown, instead of simply making all of them
the same quality as one another. Why don't we just square off all the
schismatic and sub-schismatic stuff, and give a straightforward
F-C-G-D-A-E of 1/5 PC each, instead of ratios or syntonic comma stuff?
The same major 3rds will be good or bad by virtually "the same"
amounts, when listening to and playing normal 18th century music.
And, after the harpsichord has sat there for an hour or so, it will
have drifted off spots enough already that your 1/4 schisma business
will all be negligible at best. How is anyone going to know, or care,
that the little 1/4 schisma fragments are in the right spots around
the back, and not between F#-C# or wherever?
Meanwhile, your five-page (five!) batch of reckoning still doesn't
help much with a start from an A fork, which was the question. (The
note A is in the middle of the heaviest tempering....) Assume you're
wanting some other harpsichordist to try this on a real harpsichord,
but they're less mathematically savvy than I am. They don't want to
know anything about beats, but they want to get it "right enough" so
they can go ahead and play their music WITHOUT CALCULATING ANYTHING.
They don't have Scala and they couldn't care less about any numbers.
Please give some step-by-step practical instructions that would
satisfy the accuracy you desire.