Werckmeister III according to Korg

Happy New Year to you all! On the first morning of the year 2000 I couldn�t
find anything better to do than to start with an appropriate musical
microtonal experience: I tuned my frets to Werckmeister III and started
working on Bach�s 3rd Suite for the Lute BWV 995, transposed to A minor to
fit the guitar fret board mechanism. In order to understand what this tuning
is about I decided to calculate the cent values myself and found that they
were different from the information provided with my Korg MT-1200 Tuner.

Korg gives:
C . D . E F . G . A . B
+12 +2 +4 +6 +2 +10 0 +8 +4 0 +8 +4

Or, transposed 12 cents down, so C starts on 0:

C . D . E F . G . A . B
0 -10 -8 -6 -10 -2 -12 -4 -8 -12 -4 -8

Werckmeister indicates that CG, GD and DA should each be a quarter-comma too
small. This is *approximately* 5.375 cents and therefore a 3,42 cents
deviation from equal temperament. This last value is 4 cents every time
according to Korg. The error of 0,58 cents cumulates three times in a row
and leads to the wrong value for the A (-12 cents). Rounded off downwards,
upwards and downwards as correctly as possible I come to the following
model:

C . D . E F . G . A . B
0 -10 -7 -6 -8 -2 -12 -3 -8 -10 -4 -6

Or, in order to program the tuner (A should be 0):

C . D . E F . G . A . B
10 +0 +3 +4 +2 +8 -2 +7 +2 0 +6 +4

Am I right? My ear tells me this sounds slightly better.

Wim Hoogewerf

Also a happy and merry 2000!
Wim, no the Korg tuner has the right values. Werckmeister meant
the Pythagorean and not syntonic comma. So his fifth is tempered
1/4 Pyth. comma which is 5.865 cents, or 3.91 cents below ET fifth.

> My ear tells me this sounds slightly better.

Probably because the fifths are a little purer.

To find more temperaments than in your tuner you can consult the
file ftp://ella.mills.edu/ccm/tuning/software/scales/contents.txt
It contains a file list for the archive which I've just updated:
ftp://ella.mills.edu/ccm/tuning/software/scales/scales.zip
The files in it are text files with scales.

Manuel Op de Coul coul@ezh.nl

Wim Hoogewerf wrote:
>Korg gives:
...
>C . D . E F . G . A . B
>0 -10 -8 -6 -10 -2 -12 -4 -8 -12 -4 -8

This matches werck3.scl in Scala, which is described as "Andreas
Werckmeister's temperament III (the most famous one, 1681)".

>I come to the following model:
>C . D . E F . G . A . B
>0 -10 -7 -6 -8 -2 -12 -3 -8 -10 -4 -6

This matches werck3_eb.scl in Scala, which is described as
"Werckmeister III equal beating version, 5/4 beats twice 3/2".

So I guess you and Korg are both right. :-)

BTW, Wim, what kind of fret tuning mechanism are you using?

Cheers,
Christopher

I believe that Werckmeister himself is the source of this confusion.
Rasch writes in his introduction to his Werckmeister edition that
W. recognizes as micro-intervals only the (syntonic) comma and
the lesser diesis. The Pythagorean comma is indicated by its
ratio and described as 'ein klein wenig mehr als ein comma'.
For the rest he neglected the difference between the commas.
But because he prescribed 8 pure and 4 equally tempered fifths
it's clearly the Pyth. comma.
I found another complete tuning instruction for W.III in Fred
Bettenhaussen: _Clavecimbel, clavichord en pianoforte. Stemmen,
stemmingen en onderhoud_, De Toorts, Haarlem, 1984. He has it
from Jan van Biezen, whose book I don't have. Bettenhausen gives
a beating frequency between c' and e' of 3 Hz, contradicting

>One begins the Werckmeister III tuning by
>setting the C4 reference note (...) and tuning E4 a perfect (zero-beat)
>major third above it.

The equal beating version which Christopher mentioned is an invention
of Owen Jorgensen.

Manuel Op de Coul coul@ezh.nl