Yahoo Tuning Groups Ultimate Backup tuning Werckmeister III does not exist! (had "wohltemperirt"!)

Dear arl_123@hotmail.com,

arl_123@hotmail.com schrieb:
> Hello, all. According to H.A. Kellner, J.S. Bach favored an irregular
> ("well") temperament that distributes the Pythagorean comma over five
> fifths: C-G-D-A-E and B-F#. The Werckmeister III (Correct temperament
> #1) temperament distributes the Pythagorean comma over four fifths: C-
> G-D-A and B-F#. Computing note values in cents starting with C results
> in the following:
>
> Kellner Werckmeister III
> C 0 0
> C# 90.2 90.2
> D 194.5 192.2
> Eb 294.1 294.1
> E 389.1 390.2
> F 498.0 498.0
> F# 588.3 588.3
> G 697.3 696.1
> G# 792.2 792.2
> A 891.8 888.3
> Bb 996.1 996.1
> B 1091.0 1092.2
>
> Examination of the above shows that the greatest difference between
> identical notes is 3.5 cents for A. For all practical purposes these
> temperaments appear to be equivalent. Are there considerations that I
> have missed? Your comment is appreciated. Sincerely,

Dear arl_123@hotmail.com,

This contribution will show how to compare in a rational way
the systems Werckmeister III and Bach.
Thereafter, I'll demonstrate that Werckmeister did not really take
seriously the system "nominal W III", but preferred and had encoded
by baroque methods of Gematria the system now known as
"Bach/wohltemperirt" and patented in 1981.

1 Comparison of scales:
*************************************

You quantify the largest difference between the scales
of Bach and Werckmeister III amounting to 3.5 cent.
You base the result of your calculation onto equal scale-
values in cent of C=0.

However, your result comparing scales has not the slightest
mathematical significance. Proof: Let now the cent values of A
coincide, rather than those of C. Why not??

This result will look as follows:

Bach W III Diff
0,0 3,5 -3,5
90,2 93,7 -3,5
194,5 195,7 -1,2
294,1 297,6 -3,5
389,1 393,7 -4,6
498,0 501,5 -3,5
588,3 591,8 -3,5
697,3 699,6 -2,3
792,2 795,7 -3,5
891,8 891,8 0,0
996,1 999,6 -3,5
1091,0 1095,7 -4,7

The largest difference now turns up as being -4,7 cent, instead of
your calculated value of 3.5 cent!!

This proves that neither the 3.5 cent nor the above result of 4,7 cent
could possibly have any absolute or significant meaning.

In order to perform a more reasonable comparison, isn't it clear
that beforehand, the two scales must be shifted relative to each
other such that the overall deviations attain their minimum??

Obviously, one should apply a shift by about 0.392 cent upwards to
your scale of Werckmeister III to adapt these two scales mutually
to each other, BEFORE proceeding to comparing:

W IIIshif Diff to "Bach/wohltemperirt"
0,4 -0,4
90,6 -0,4
192,6 1,9
294,5 -0,4
390,6 -1,5
498,4 -0,4
588,7 -0,4
696,5 0,8
792,6 -0,4
888,7 3,1
996,5 -0,4
1092,6 -1,6

Thus, there are plenty of different deviations between Bach
and W III on various steps of the scale.
But now the maximum deviation is only 3,1 cent rather than your 3.5 cent.
The D has 1,9 cent, and e.g. the B shows a difference of -1,6 cent.

2 Werckmeister invented "Bach/wohltemperirt":
**************************************************

Let's look now into the first example for baroque GEMATRIA:

11 letters 11 letters partition according to the UNITAS
MUSICALISCH ETEMPERATUR
112 135 112+135=247
=CHRISTUS 1=UNITAS
3=THIRD in thoroughbass
5=FIFTH in basso continuo
!!!!!!!!!!!!!!!!!!!!!!!!!!!!
This is the specification of the
C-major triad "wohltemperirt"
in which third and fifth beat at
the unison: UNITAS!

247
Total "Musicalische Temperatur" 112+135=247=13*19
(mediator:) CHRISTUS = 112 1 = UNITAS
3=TRINITAS
closure of circle: 19.

This was Werckmeister; "predecessor" of JSB.
_________________________________________________
Going further in Gematria, a striking example, Bach's title:

12 letters again 12 letters : partition, UNITAS=mediator
DAS WOHLTEMPE RIRTE CLAVIER
133 133 Totalling: 266= 14*19
BACH =14=2+6+6
CHRISTUS = mediator
closure of circle: 19.

This final application below of Gematria for baroque musicology
proves beyond doubt that Werckmeister considered by no means
the "nominal system Werckmeister III" as optimal!

Could any person of esprit and competent in harpsichord
tuning take seriously a system having within C-G-D-A-E three
tempered and one perfect fifth?
Could anybody explain to me what are the sufficient reasons
NOT to equalize these four fifths??
This equalization of the three fifths leads from a temperament,
structure of fifths 8+4 to a temperament 7+5 and this is achieved,
provided the 5 tempered fifths are reduced by 370/369!
*********************************
The superparticular ratio of 369 is 370/369; (see my 2 publications
quoted in my site on the Four Duets of Clavierubung III).

It is indispensable to have the facsimile of Werckmeister's
"Musicalische Temperatur". The editor of this facsimile observed
correctly that on p. 75 there is the ONLY chapter, 28, of the
******
book that bears an explicit heading.
(28 is the second "perfect number"). The setup is:

GEMATRIA-Values
-(75.) -
Das XXVIII. Capitel 114
Von der Temperatur insge= +255 **(=369)**
mein + 39 (=408)

Unfortunately, the Editor of that facsimile was not sufficiently
interested in the subject matter as to let the ONLY explicit
chapter-heading undergo the Gematria-conversion!

However - having the facsimile - this line=
separation was NOT necessary due to rather wide margins in
Werckmeister's treatise. Hence cutting this line was unjustified!
What, then, are these particular reasons for Werckmeister's
strange page-setup??

These reasons derive from the fact that, in view of baroque musical
acoustics and mathematics, one worked in terms of
superparticular ratios:

The superparticular ratio of N is defined as (N+1)/N.
The simple intervals
octave, fifth, fourth and major third are the SUP. RATs of
1 2 3 4 being
2/1 3/2 4/3 5/4.

The Tempering, for what represented to Werckmeister "MEIN
e Temperatur", (last and 3rd line of chapter-heading)
is the SUPERPARTCULAR RATIO of 369. Thus 370/369.
********************

I studied practically all of Werckmeister's writings, such that
this analysis of the only chapter's title in that treatise, on
page 7 5 could not surprise me.
7 perfect fifths
5 fifths "wohltemperirt"!

For the completely elaborated paper, which furnishes proof that
Werckmeister was familiar with "Bach's" system wohltemperirt,
one might consult:

Kellner, H.A.: A propos d'une rï¿½impression de la "Musicalische Temperatur"
(1691) de Werckmeister. Revue de Musicologie Vol. 71, 1985, page 184-187

More general, about Werckmeister's state of mind as regards
gematria and his spirit fond of enigma:

Kellner, H.A.: Is there an enigma in Werckmeister's "Musicalische Temperatur"?
English Harpsichord Magazine, Vol. 3, No. 7, 1984, page 134-136

Idem: One typographical enigma in Werckmeister, "Musicalische Temperatur".
English Harpsichord Magazine, Vol. 3, No. 8, 1985, page 146-151

Idem: Did Werckmeister already know the tuning of J.S. Bach for the "48"?
English Harpsichord Magazine, Vol. 4, No. 1,1985, page 7-11

CONCLUSION:
************
In some sense, as Werckmeister himself did not take seriously his
obviously fallacious system W III,
WERCKMEISTER III (nominal) DOES NOT EXIST.
and should be replaced in all cases by
the system "Bach/wohltemperirt". Except, if one prefers a
technologically inferior system that Werckmeister himself "rejected" as he
proposed something more reasonable. Werckmeister himself
must have been the inventor - and has encoded it via gematria
combined with his peculiar page-setup.
Perhaps also for pedagogic and didactic reasons, but more certainly,
I believe, in order to pose a riddle. A pastime appreciated by
baroque musicians.

Best regards,
Herbert Anton Kelner

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Dear arl_123@hotmail.com,

1 Comparison of scales:
*************************************

You quantify the largest difference between the scales
of Bach and Werckmeister III amounting to 3.5 cent.
You base the result of your calculation onto equal scale-
values in cent of C=0.

However, your result comparing scales has not the slightest
mathematical significance. Proof: Let now the cent values of A
coincide, rather than those of C. Why not??

This result will look as follows:

The largest difference now turns up as being -4,7 cent, instead of
your calculated value of 3.5 cent!!

This proves that neither the 3.5 cent nor the above result of 4,7 cent
could possibly have any absolute or significant meaning.

In order to perform a more reasonable comparison, isn't it clear
that beforehand, the two scales must be shifted relative to each
other such that the overall deviations attain their minimum??

Obviously, one should apply a shift by about 0.392 cent upwards to
your scale of Werckmeister III to adapt these two scales mutually
to each other, BEFORE proceeding to comparing:

W IIIshif Diff to "Bach/wohltemperirt"
0,4 -0,4
90,6 -0,4
192,6 1,9
294,5 -0,4
390,6 -1,5
498,4 -0,4
588,7 -0,4
696,5 0,8
792,6 -0,4
888,7 3,1
996,5 -0,4
1092,6 -1,6

2 Werckmeister invented "Bach/wohltemperirt":
**************************************************

Let's look now into the first example for baroque GEMATRIA:

247
Total "Musicalische Temperatur" 112+135=247=13*19
(mediator:) CHRISTUS = 112 1 = UNITAS
3=TRINITAS
closure of circle: 19.

This was Werckmeister; "predecessor" of JSB.
_________________________________________________
Going further in Gematria, a striking example, Bach's title:

12 letters again 12 letters : partition, UNITAS=mediator
DAS WOHLTEMPE RIRTE CLAVIER
133 133 Totalling: 266= 14*19
BACH =14=2+6+6
CHRISTUS = mediator
closure of circle: 19.

This final application below of Gematria for baroque musicology
proves beyond doubt that Werckmeister considered by no means
the "nominal system Werckmeister III" as optimal!

Could any person of esprit and competent in harpsichord
tuning take seriously a system having within C-G-D-A-E three
tempered and one perfect fifth?
Could anybody explain to me what are the sufficient reasons
NOT to equalize these four fifths??
This equalization of the four fifths leads from a temperament,
structure of fifths 8+4 to a temperament 7+5 and this is achieved,
provided the 5 tempered fifths are reduced by 370/369!
*********************************
The superparticular ratio of 369 is 370/369; (see my 2 publications
quoted in my site on the Four Duets of Clavierubung III).