Werckmeister vs. Kellner

Dear Herbert Anton,

I just returned from Montreal, working with a string quartet and found all
your new postings. After this morning's rehearsal with my recorder
colleague, we looked together at the print outs of your postings and we
seemed to be in agreement.

Rather than respond to each point, it may be best to summarize my responses.
Since, I have the Rasch reproduction of the Werckmeister plates, I could see
what you refer to on page 75. And yes, the "singular" title might have fit
on only 2 lines, rather than 3. But how do you know that this was
Werckmeister's doing. I am not sure it is in Werckmeister's hand. We know
that the monochord drawing was by someone other than Werckmeister, according
to Lindley.

Frankly, with so much writing, much of it verbose, or even unnecessary by
today's standards, it doesn't seem likely that Werckmeister could have, or
would have pre-planned choosing page "75" so that it would be code for an
even finer tuning than his own. As for his own favorites, Werckmeister
offers variants always, and you could even certainly suggest another as a
result of tarot cards. Still, there is only "Werckmeister III" for the full
circle of keys in his Quedlinburg publication; the others are for a swatch of
keys, not intended for full 12-note chromaticism.

This is Werckmeister's genius, to describe for the first time in history,
BEFORE J.S. Bach, that it was possible to evolve past Praetorious' meantone
and sub-semitonium workings by using pure and 1/4 comma-flat and perfect
fifths alone. It is the basis of his book and it follows Werckmeister I (of
just intonation), Werckmeister II (of quarter-comma meantone), and then,
Werckmeister III. Werckmeister IV, V, and VI are diatonic tuning variants.

Counting the letters at 11 and 11 for the title does not interest me much
either, I'm afraid. The final letter "e" of "Musicalish eTemperatur" looks
wrong as a code. It seems much more likely to me that this is the only, and
OBVIOUS title for the topic of the book. Other books by Werckmeister had
other titles. It might be useful for your position to locate Gematria
present in the other Werckmeister titles. But this book was one among many
that he authored, on different subjects in music, and he consistently
referred back to his 1691 publication for advanced expansion on temperament
throughout his life. I checked. It doesn't seem believable that
Werckmeister would encrypt the very thing that he is exposing for the first
time on the world stage.

I understand that you exposed the unequal circle idea for Bach in this
century and it is to your credit. However, your "obscure and arcane" variant
to Werckmeister III is indistinguishable from it. Working presently on a
performance of the Brandenburg Concerto No. 4, I have a laboratory for
exploring this, but I prefer the 3 identical perfect fourths of 504 cents in
Werckmeister III found in G Major, rather than the non-repeating ones of 503
and 502. The motif of the last movement needs the fullest stretch for a
perfect fifth for greatest intonational effect, to my taste. 504, no more,
and the movement has a wonderful momentum and tempo.

I congratulate you on your original thinking, but I think Werckmeister
invented the term "wohl-temperirt" not Bach. Also, Werckmeister does not
take credit for inventing the tuning, and I believe him. He certainly was
aware of other chromaticists that were using it in Thuringia, maybe back to
Michael Bach of Erfurt, known for his chromaticisms, and the first Bach
superstar.

I hope that you are not offended by my honest answers to your posts. I am
actually deeply interested in "Bach's Tuning" and have been researching a
book. The very difference between our favorites is really so very little.
There is no time to hear the beats in any live performance, after all. It is
just that Werckmeister III is not OBVIOUSLY FALLACIOUS. This overwrought
statement sends me running in the other direction. Talk of inferior makes no
sense when their is no great improvement in harmony. After all, Werckmeister
III enhances counterpoint at the expense of harmony. The separate lines are
more distinguishable from each other in Werckmeister III, making it superior.

Please let me know I am incorrect on any of this, or if there is something
more that could sway a practical interpretation towards an arcane one.

Best, Johnny Reinhard

Dear Johnny,

Thanks for your message:
I shall reply to you in due course,

Kind regards,

Herbert Anton

Afmmjr@aol.com schrieb:
> Dear Herbert Anton,
>
> I just returned from Montreal, working with a string quartet and found all
> your new postings. After this morning's rehearsal with my recorder
> colleague, we looked together at the print outs of your postings and we
> seemed to be in agreement.
>
> Rather than respond to each point, it may be best to summarize my responses.
> Since, I have the Rasch reproduction of the Werckmeister plates, I could see
> what you refer to on page 75. And yes, the "singular" title might have fit
> on only 2 lines, rather than 3. But how do you know that this was
> Werckmeister's doing. I am not sure it is in Werckmeister's hand. We know
> that the monochord drawing was by someone other than Werckmeister, according
> to Lindley.
>
> Frankly, with so much writing, much of it verbose, or even unnecessary by
> today's standards, it doesn't seem likely that Werckmeister could have, or
> would have pre-planned choosing page "75" so that it would be code for an
> even finer tuning than his own. As for his own favorites, Werckmeister
> offers variants always, and you could even certainly suggest another as a
> result of tarot cards. Still, there is only "Werckmeister III" for the full
> circle of keys in his Quedlinburg publication; the others are for a swatch of
> keys, not intended for full 12-note chromaticism.
>
> This is Werckmeister's genius, to describe for the first time in history,
> BEFORE J.S. Bach, that it was possible to evolve past Praetorious' meantone
> and sub-semitonium workings by using pure and 1/4 comma-flat and perfect
> fifths alone. It is the basis of his book and it follows Werckmeister I (of
> just intonation), Werckmeister II (of quarter-comma meantone), and then,
> Werckmeister III. Werckmeister IV, V, and VI are diatonic tuning variants.
>
> Counting the letters at 11 and 11 for the title does not interest me much
> either, I'm afraid. The final letter "e" of "Musicalish eTemperatur" looks
> wrong as a code. It seems much more likely to me that this is the only, and
> OBVIOUS title for the topic of the book. Other books by Werckmeister had
> other titles. It might be useful for your position to locate Gematria
> present in the other Werckmeister titles. But this book was one among many
> that he authored, on different subjects in music, and he consistently
> referred back to his 1691 publication for advanced expansion on temperament
> throughout his life. I checked. It doesn't seem believable that
> Werckmeister would encrypt the very thing that he is exposing for the first
> time on the world stage.
>
> I understand that you exposed the unequal circle idea for Bach in this
> century and it is to your credit. However, your "obscure and arcane" variant
> to Werckmeister III is indistinguishable from it. Working presently on a
> performance of the Brandenburg Concerto No. 4, I have a laboratory for
> exploring this, but I prefer the 3 identical perfect fourths of 504 cents in
> Werckmeister III found in G Major, rather than the non-repeating ones of 503
> and 502. The motif of the last movement needs the fullest stretch for a
> perfect fifth for greatest intonational effect, to my taste. 504, no more,
> and the movement has a wonderful momentum and tempo.
>
> I congratulate you on your original thinking, but I think Werckmeister
> invented the term "wohl-temperirt" not Bach. Also, Werckmeister does not
> take credit for inventing the tuning, and I believe him. He certainly was
> aware of other chromaticists that were using it in Thuringia, maybe back to
> Michael Bach of Erfurt, known for his chromaticisms, and the first Bach
> superstar.
>
> I hope that you are not offended by my honest answers to your posts. I am
> actually deeply interested in "Bach's Tuning" and have been researching a
> book. The very difference between our favorites is really so very little.
> There is no time to hear the beats in any live performance, after all. It is
> just that Werckmeister III is not OBVIOUSLY FALLACIOUS. This overwrought
> statement sends me running in the other direction. Talk of inferior makes no
> sense when their is no great improvement in harmony. After all, Werckmeister
> III enhances counterpoint at the expense of harmony. The separate lines are
> more distinguishable from each other in Werckmeister III, making it superior.
>
> Please let me know I am incorrect on any of this, or if there is something
> more that could sway a practical interpretation towards an arcane one.
>
> Best, Johnny Reinhard
>

Dear Johnny Reinhard,

Thank you for your message. The lines below may interest you.

As you know, it is not easy, and in fact virtually impossible to
reconstitute the temperament of an organ that has been retuned
from whatever have you, to E.T. This happened at the end of the
18th and within the 19th century to many organs.

Couldn't it appear then as madness to claim there exists an exact
mathematical/technological specification of Bach's "wohltemperirt"
harpsichord temperament???

The solution to this problem is surprising and simple, provided one
knows something about Bach's method of composition. It is clear, once
Bach's system is known, one can establish mathematically the relevant
set of numbers pertaining to that tuning system.
Wouldn't it seem natural to any composer and in no way far-fetched,
to utilize, to profit from these numbers to structure the music my
means of them? Unifying thereby the harmonies via the numbers of the
temperament with the music's architecture.
What are such numbers? The octave must remain perfect and the fifth is
the next simple interval - it can be tempered. You know that the old
music theory worked via monochord lenghts. We take the reciprocals,
for the intervals as (N+1=)/N.
Such that N=1 means octave 2/1, N=4 the major third 5/4, etc.

Temperings, as Werckmeister wrote, also were expressed as such ratios;
see your facsimile.

For the (N+1)/N of 369, then results 370/369. This, as mathematics show,
is the tempering for Bach's fifth.
The most simple numbers pertaining to "Werckmeister/Bach/wohltenmperirt"
are therefore are: 7, 5 the fifths, and 369, etc.

The 4 Duets, before my results impossible to explain, have ***369 *****
bars; their midpoint being musically distinguished - look it up.
That midpoint of the entire structure falls into Duetto II, its bar 112.

369= 185+1+185

As Duetto I has 73 bars, we have 73+112=185 for that central bar
of the overall structure.
Gematria converts CHRISTUS, overall central bar into 112 - the "mediator"!

The Duetto II itself,
149 bars, is structured 37+75+37 bars.
7 perfect fifths+
5 fifths ""wohltemperirt".

The 75 central bars are structured 75=31+13+31 - tri-unitary. In baroque
times the triad was a musico-theological symbol. In particular, by
Werckmeister's musico-theological speculations.

The 37 framing bars are structured, again symmetrically, as 37=9+19+9:
Therein, 9 is the trias trinitatis per additionem, 9=3+3+3, and
19 intervals, 12 fifths and 7 octaves in the reversed sense, close
the circle.

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 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).

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"!

> Dear Herbert Anton,
>
> I just returned from Montreal, working with a string quartet and found all
> your new postings. After this morning's rehearsal with my recorder
> colleague, we looked together at the print outs of your postings and we
> seemed to be in agreement.
>
> Rather than respond to each point, it may be best to summarize my responses.
> Since, I have the Rasch reproduction of the Werckmeister plates, I could see
> what you refer to on page 75. And yes, the "singular" title might have fit
> on only 2 lines, rather than 3. But how do you know that this was
> Werckmeister's doing.

*It is most likely and acceptable that the engraved plate was done by an
*engraver or a draftsman. As you know, in some of his publications
*Werckmeister complained about deficient skill of the work of his draftsman.
*The text itself was set by the printing workshop. Werckmeister's
*instructions and commanding must have driven the printers close to crazy.

I am not sure it is in Werckmeister's hand. We know
> that the monochord drawing was by someone other than Werckmeister, according
> to Lindley.
>
> Frankly, with so much writing, much of it verbose, or even unnecessary by
> today's standards,

Here, one should better think about the FINALITY of acting! In order
to make sure arriving at p.75 with chapter 28(=secundus numerus perfectus)
Werckmeister had to fill up a sufficient number of his treaises' preceding
pages with no matter what! With unconspicuos matter and, in the limit, with
blabla that just serves to push "Das XVIII. Capitel" onto page 75! If no
really substantial material is available, and you work under the FINALITY to
attain with Cap. 28 the page 75 of 7 perfect and 5 well-tempered fifths,
haven't you then necessarily to be verbose???

My lifetime is, like of anybody else, limited; and so, at best will and
intentions, I cannot retype here my thoroughly elaborated text of :

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

> it doesn't seem likely that Werckmeister could have, or
> would have pre-planned choosing page "75" so that it would be code for an
> even finer tuning than his own.

*I respect in this exchange, your opinion, as that of any contemporary!

> As for his own favorites, Werckmeister
> offers variants always, and you could even certainly suggest another as a
> result of tarot cards.

Sometimes, in fact, I eat carotte-tarts.

> Still, there is only "Werckmeister III" for the full
> circle of keys in his Quedlinburg publication; the others are for a
> swatch of keys, not intended for full 12-note chromaticism.

For not-full-12-note chromaticism, may I recommend mean-tone!!!

It is mandatory to enter, by thorough and exhaustive study, into
the mentality of that baroque author. The result of my work in this
direction I put into:

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

>
> This is Werckmeister's genius, to describe for the first time in history,
> BEFORE J.S. Bach, that it was possible to evolve past Praetorious' meantone
> and sub-semitonium workings by using pure and 1/4 comma-flat and perfect
> fifths alone. It is the basis of his book and it follows Werckmeister I (of
> just intonation), Werckmeister II (of quarter-comma meantone), and then,
> Werckmeister III. Werckmeister IV, V, and VI are diatonic tuning variants.

Werckmeister IV, V and VI is nothing more and else than than SPAM. And
even more surely, if compared to the qualities of W III, filling up by
Werckmeister IV, V and VI the pages upto 75.

> Counting the letters at 11 and 11 for the title does not interest me much
> either, I'm afraid.

I respect your personal opinion - although this is not a very
constructive one and, above all, - IMPO leads you to miss the essential
point - assures to render yourself blindfolded!

> The final letter "e" of "Musicalisch eTemperatur" looks
> wrong as a code.

*This remark I do not understand at all. What could it possibly mean??
*Where is there any "code"; what, explicitely, is wrong or looks wrong
*here??**********************************************************

> It seems much more likely to me that this is the only, and
> OBVIOUS title for the topic of the book.

*I fully agree; that title therefore ought to be analyzed by advanced
musicology pusjing all aspects, up to its ultimate consequences and
implications.

> Other books by Werckmeister had
> other titles. It might be useful for your position ++++++++

I feel perfectly happy with my position!

Under the hypothesis, Werckmeister intended to encode at an unique
opportunity his preferred system (ref. my EHM-articles above on
Werckmeister), I would waste my lifetime to check into titles of his
other treatises. I suggest you do it yourself and report here your
observations and any results that you deem significant.
Or do you really expect he would have encoded a second time something as
interesting as the system "Werckmeister/Bach/wohltemperirt"? Could it
possibly be an everydays' event to encode such features this subtle way
into a title?

> ++++++++++to locate Gematria
> present in the other Werckmeister titles. But this book was one among many
> that he authored, on different subjects in music, and he consistently
> referred back to his 1691 publication for advanced expansion on temperament
> throughout his life. I checked. It doesn't seem believable that
> Werckmeister would encrypt the very thing that he is exposing for the first
> time on the world stage.

You do certainly know "QUAERENDO INVENIETIS" and thus, are familiar with
baroque mentality. Also, Kuhnau's riddle in Gematria, from the preface
of his Biblische Historien Sonaten has now been common lore for some time.
Isn't it? Kuhnau utilizes, incidentally, the same gematria as Werckmeister
or J. S. Bach.

>
> I understand that you exposed the unequal circle idea for Bach in this
> century and it is to your credit. However, your "obscure and arcane" variant
> to Werckmeister III is indistinguishable from it. Working presently on a
> performance of the Brandenburg Concerto No. 4, I have a laboratory for
> exploring this, but I prefer the 3 identical perfect fourths of 504 cents in
> Werckmeister III found in G Major, rather than the non-repeating ones of 503
> and 502.

*The system "wohltemperirt" is the authentic one for "Das wohltemperirte
*Clavier". You claim, it is indistinguishable from Werckmeister III.
*Don't you realize that you are in CONTRADICTION with yourself, when
*you "prefer" Werckmeister III - didn't you just claim, they are
"INDISTINGUISHABLE"?

> The motif of the last movement needs the fullest stretch for a
> perfect fifth for greatest intonational effect, to my taste.

"de gustibus non est disputandum "

504, no more,
> and the movement has a wonderful momentum and tempo.
>
> I congratulate you on your original thinking, but I think Werckmeister
> invented the term "wohl-temperirt" not Bach.

I fully agree. And, in view of the properties of Bach's title,
"Das wohltemperirte Clavier," seen together with its unexpected
properties, parallel to Werckmeister's title - as a predecessor -
"Musicalische Temperatur", it would be a vain effort to expect here any
"technological" or specialized-jargon-experession.

> Also, Werckmeister does not
> take credit for inventing the tuning, and I believe him. He certainly was
> aware of other chromaticists that were using it in Thuringia, maybe back to
> Michael Bach of Erfurt, known for his chromaticisms, and the first Bach
> superstar.
>
> I hope that you are not offended by my honest answers to your posts.

* Let's keep calm and not emotional in our sometimes heated discussion!!

> I am actually deeply interested in "Bach's Tuning" and have been
> researching a book. The very difference between our favorites is really
> so very little.

Bach has selected, so to speak, nothing less than "wohltemperirt" as
the basis for his personal signet. (This was No. 2 and the definitive
one of his signets. Its first usage we are aware of, is on 15. 3. 1722,
a document to Erfurt).

Furthermore, upon inspection, Bach has structured his ENTIRE musical
oeuvre utilizing the set of numbers belonging to "wohltemperirt". But
to reveal and document this fact, will be occupying students of musicology
desperate for subjects suitable for doctoral thesis' work during several
decades, after other topics for investigation are exhausted. I have
managed to study up to now only a small sample of Bach's work and will
wind this activity up soon.

And if you prefer in "Auff�hrungspraxis" Werckmeister III, that could
certainly not provoke any ploblem, let alone, catastrophy. But in case
one is interested to discover the arcane methods of how J. S.Bach he
structured his compositions, one will be startled - provided one
possesses that that mathematically derived "set of numbers" By their means
one should look into the structure of Bach's compositions.

I refer to my about 50 references to my publications in:

htt://ha.kellner.bei.t-online.de

> There is no time to hear the beats in any live performance, after all.
> It is just that Werckmeister III is not OBVIOUSLY FALLACIOUS. This
> overwrought statement sends me running in the other direction.

W III in live performance, might, admittedly, be MARGINALLY different
from Werckmeister's "wohltemperirt". And yet, organ builders do prefer
the system for their baroque-style organs that have between C and E,
FOUR equal fifths. Each one of them is better than the Werckmeister-fifth
of 8+4. It is clear, because "wohltemperirt" is structured 7+5! There are
by now, 26 years after 1975, about 300 pipe-organs tuned this way. WHY???
How many W III organs, in contradistinction, do there exist! WHY???

> Talk of inferior makes no
> sense when their is no great improvement in harmony. After all, Werckmeister
> III enhances counterpoint at the expense of harmony. The separate lines are
> more distinguishable from each other in Werckmeister III, making it superior.

The technological reasons, why "wohltemperirt" is superior to W III, are laid
out in:

Kellner, H.A.: Temperaments for all 24 Keys - A Systems Analysis. Acustica, Vol.
52/2, 1982/83. S. Hirzel Verlag Stuttgart. Seite 106-113. Publication of the
lecture delivered July 1980 at the Bruges 6th International Harpsichord Week.
>

> Please let me know I am incorrect on any of this, or if there is something
> more that could sway a practical interpretation towards an arcane one.
>
> Best, Johnny Reinhard

*It took about 257 years to realize Bach's system was not E.T.

*It may take other 257 years for the genaral understanding and
*acceptance that W III wasn't taken seriously by Werckmeister himself.
*As I describe in my publications, Werckmeister was an
*amateur-mathematician in his leisure-time.

*What do you think about the parallel between Werckmeister's and
*Bach's title "Das wohltemperirte Clavier"?

*Finally, it is clear that it is somewhat more easy and agreable to
*proffer opinions - to which everybody is, of course, entitled.
*Explore your conscience - what part of my publications had you
*read already; better, STUDIED already, because this topic isn't an
*easy go - you are forwarned!

*But as you are pofoundly interested in the subject-matter, I can
*only recommend heartily to get a somewhat more complete and useful
*list of my publications, (see YO TOMITA; better than in my own web-page)
*and to study, study, study - as you are really an interested personality
*of critical faculty!
*Do not base yourself upon hearsay! Thereafter do then criticize
*specifically any one of my writings in detail.
It is, finally, amusing that Lenin said "utchitcyia, utchitcyia,
*utchitcyia", which means "learning, learning, learning"!!!

Kind regards,

Herbert Anton Kellner

--- In tuning@y..., Afmmjr@a... wrote:

> I understand that you exposed the unequal circle idea for Bach in
this
> century and it is to your credit. However, your "obscure and
arcane" variant
> to Werckmeister III is indistinguishable from it.

Except for the fifth A-E, which is quite distinguishable.