Re : Re: Werckmeister III according to Korg

(Manuel Op de Coul:)
> 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.
>
(Christopher Chapman:)
>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. :-)

Thanks, Manuel and Christopher. Of course I thought about the Pythagorean
comma. However, the book which provided my information should be
trustworthy: Fundamentals of musical acoustics by Arthur H. Benade. In
Chapter 16.3 he describes the actual tuning procedure for Werckmeister III,
transmitted to him by Herman Greunke, an associate of the tracker-organ
builder John Brombaugh. I quote: �One begins the Werckmeister III tuning by
setting the C4 reference note (...) and tuning E4 a perfect (zero-beat)
major third above it. This E will serve as a temporary (later on the E will
be tuned a perfect 5th on A, WH) check-point in setting the next part of the
scale. Following this a series of four contracted 5ths are tuned (C to G, G
to D, D to A, A to E), very much as in setting up an equally tempered scale,
except that here one must shrink the tuning of each of the four intervals
enough for the resulting E to be identical with the checkpoint E we have
already set. Each member of this set has a frequency ratio of 1,4952 (5.5
cents short),...� (Unquote) That�s definitely a syntonic comma and not a
Pythagorean. I don�t want to contest the *official* values of Werckmeister
III. Korg gives them, Jean Lattard (Musique: Gammes et temp�raments) and the
New Groves as well (chapter on *Well-tempered clavier*) even if the
description of the tuning is erroneous. �For the first four perfect 5ths
(C-G, G-D, D-A, B-F#) he used a tuning slightly tempered by a third of a
comma, while the eight other 5ths remained pure� (chapter on Andreas
Werckmeister). At least Benade tells us how to tune to this temperament by
ear. He doesn't mention that this is the equal-beating version.

(Christpher:)

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

The Vogt system with six moveable frets for each position.

Wim Hoogewerf

Manuel, could you quote some german phrases from Werckmeister concerning his
most famous temperament?

> But because he prescribed 8 pure and 4 equally tempered fifths
> it's clearly the Pyth. comma.

Does he clearly say that all four fifths should be tempered in the same way
or does he speak about beats (Schwebungen). I'm sorry, I don't have the
original german text.

Thanks, Wim