Yahoo Tuning Groups Ultimate Backup tuning Re: [tuning] Inferior tuning system for J. S. Bach??

Dear Mr. Paul Erlich,

Could an inferior tuning system satisfy J. S. Bach?

Did Bach prefer a technologically inferior system (=Werckmeister III) to
"wohltemperirt"?

The system of J. S. Bach is called henceforth "wohltemperirt", as the composer
writes himself. It has 5 welltempered and 7 perfect fifths.

REFERENCES:
General reference, my website:
http://ha.kellner.bei.t-online.de
Specific reference:
Kellner, H. A.: Temperaments for all 24 Keys - A Systems Analysis. Acustica,
Vol. 52/2, 1982/83. S. Hirzel, Stuttgart. p. 106-113.
(Publication of the lecture delivered July 1980 at the Bruges 6th International
Harpsichord Week)

Did Bach prefer a technologically inferior system (=Werckmeister III) to
"wohltemperirt" ?

Structure of fifths:
Nominal Werckmeister III:
8+4 perfect resp. tempered; the latter 696,09 c

Bach/Werckmeister "wohltemperirt":
7+5 perfect resp. tempered; the latter 697,28 c

Criteria to be applied:
Size/quality of the basic tonal third C-E
Size/quality of the tempered fifths

The result (in cents) looks as follows:

Fifth Third

697,28 389,11 Bach"wohltemperirt"
696,09 390,22 Werckmeister III

1,2 - 1,1 Differences (rounded)

Bach's 5 tempered fifths "wohltemperirt" are
BETTER by 1,2 cent than the 4 tempered fifths of Werckmeister III

Bach's basic tonal third C-E "wohltemperirt"
is by 1,1 cent BETTER than this third in Werckmeister III

CONCLUSION:

Anybody accepting Werckmeister III as Bach's tempering system for all 24 keys,
ACCEPTS that Bach employed an objectively inferior system - as demonstrated
under the criteria enounced above.

Or else, what OTHER rational and more appropriate criteria should be applied,
for assessing two irregular baroque tuning systems?? (Their fifths being
structured 7+5, respectively 8+4).

REMARKS:
There is no sufficient reason that in Werckmeister III the third C-E is bridged
by 3+1 fifths, tempered, respectively perfect. Rather than by 4 EQUAL fifths
amongst themselves, as in Bach/"wohltemperirt".

To the honor of Andreas Werckmeister it must be stressed that he was already
1691 in possession - and the inventor - of the System "Bach/wohltemperirt", as
his treatise "Musicalische Temperatur" shows!!

TO BE CONTINUED
Herbert Anton Kellner

Johnny Reinhard is sure it is Werckmeister.
Barnes, > van Eck, and others have come to various other conclusions. Perhaps
> you'd care to discuss the _evidence_ behind your deduction that this
> is the temperament Bach intended?
>

🔗Afmmjr@aol.com

Dear Mr. Kellner,

Since my name has come up regarding Bach's tuning and I have the opportunity
to address you directly, I am delighted. To the non-German reading public,
it has not been easy to understand how it is that you have divined Bach's
tuning. Some of your points need to be refuted, however, or at least
challenged.

In a message dated 7/1/01 2:27:13 AM Eastern Daylight Time,
ha.kellner@t-online.de writes:

> Could an inferior tuning system satisfy J. S. Bach?

To many modern musicians, all early temperaments are considered inferior.

> Did Bach prefer a technologically inferior system (=Werckmeister III) to
> "wohltemperirt"?
>
> The system of J. S. Bach is called henceforth "wohltemperirt", as the
> composer
> writes himself. It has 5 welltempered and 7 perfect fifths.
>

Actually, Werckmeister first published the term wohltemperirt, long before
J.S. Bach. However, where did you find in Bach's lifetime that there was a
split between 5 tempered and 7 perfect fifths? Not in Bach's material, I
gather.

>
> Structure of fifths:
> Nominal Werckmeister III:
> 8+4 perfect resp. tempered; the latter 696,09 c
>
> Bach/Werckmeister "wohltemperirt":
> 7+5 perfect resp. tempered; the latter 697,28 c
>
>
Why the effort to improve by so little? Isn't splitting a comma into 5 more
time consuming to evaluate in tuning a keyboard than 4? (There goes the 15
minutes to tune theory.)

> Bach's 5 tempered fifths "wohltemperirt" are
> BETTER by 1,2 cent than the 4 tempered fifths of Werckmeister III
>
> Bach's basic tonal third C-E "wohltemperirt"
> is by 1,1 cent BETTER than this third in Werckmeister III
>
>

While 1.2 cents is perceivable to me, it is not likely to make much of a
difference in the non-recorded music of real-time J.S. Bach. More
importantly, where is there any evidence that this is what Bach did? It
seems more of a modern deduction about what Bach "should have used" than any
indication that this is indeed historically accurate for the composer in his
time.

CONCLUSION:

>
> Anybody accepting Werckmeister III as Bach's tempering system for all 24
> keys,
> ACCEPTS that Bach employed an objectively inferior system - as demonstrated
> under the criteria enounced above.
>
>
Unfortunately, you argument has not been convincing. It's hard enough for
moderns to accept that Werckmeister promoted a non-12 equal in his first
description of a 12 majory/minor key circle. Since there is a prevalence of
Werckmeister's chromatic in central Germany, Bach's distinctiveness would
require serious evidence to the contrary. I await more.

> Or else, what OTHER rational and more appropriate criteria should be
> applied,
> for assessing two irregular baroque tuning systems?? (Their fifths being
> structured 7+5, respectively 8+4).
>
If you have evidence of 7+5 tuning for Bach I'd love to hear of it.
Incidentally, Werckmeister never claimed to have invented "Werckmeister III"
or "chromatic Werckmeister." He described it as already in practice.

> REMARKS:
> There is no sufficient reason that in Werckmeister III the third C-E is
> bridged
> by 3+1 fifths, tempered, respectively perfect. Rather than by 4 EQUAL
> fifths
> amongst themselves, as in Bach/"wohltemperirt".
>

Actually there is, since Werckmeister described his temperament as a
progressive result of Praetorious's quarter-comma meantone. Werckmeister
keeps the development clear.

> To the honor of Andreas Werckmeister it must be stressed that he was already
> 1691 in possession - and the inventor - of the System "Bach/wohltemperirt",
> as
> his treatise "Musicalische Temperatur" shows!!
>
By 1681, Werckmeister had begun spreading his ideas throughout out central
Germany. I do not quite understand why you are saying Werckmeister was an
"inventor" of something called Bach/wolhtemperirt. And yes, I have read the
1691 treatise in translation.

> TO BE CONTINUED
> Herbert Anton Kellner
>
>
>

Looking forward to further understanding of your position.

Best, Johnny Reinhard

🔗ha.kellner@t-online.de

Dear Mr. Johnny Reinhard,

Thanks for your messages I greatly appreciate - it deserves a well reflected and
conscientious answer which I will do asap.
Please patient a bit in the meantime!
Kind regards,
Herbert Anton Kellner

Afmmjr@aol.com schrieb:
> Dear Mr. Kellner,
>
> Since my name has come up regarding Bach's tuning and I have the opportunity
> to address you directly, I am delighted. To the non-German reading public,
> it has not been easy to understand how it is that you have divined Bach's
> tuning. Some of your points need to be refuted, however, or at least
> challenged.
>
> In a message dated 7/1/01 2:27:13 AM Eastern Daylight Time,
> ha.kellner@t-online.de writes:
>
>
> > Could an inferior tuning system satisfy J. S. Bach?
>
> To many modern musicians, all early temperaments are considered inferior.
>
>
> > Did Bach prefer a technologically inferior system (=Werckmeister III) to
> > "wohltemperirt"?
> >
> > The system of J. S. Bach is called henceforth "wohltemperirt", as the
> > composer
> > writes himself. It has 5 welltempered and 7 perfect fifths.
> >
>
> Actually, Werckmeister first published the term wohltemperirt, long before
> J.S. Bach. However, where did you find in Bach's lifetime that there was a
> split between 5 tempered and 7 perfect fifths? Not in Bach's material, I
> gather.
>
>
> >
> > Structure of fifths:
> > Nominal Werckmeister III:
> > 8+4 perfect resp. tempered; the latter 696,09 c
> >
> > Bach/Werckmeister "wohltemperirt":
> > 7+5 perfect resp. tempered; the latter 697,28 c
> >
> >
> Why the effort to improve by so little? Isn't splitting a comma into 5 more
> time consuming to evaluate in tuning a keyboard than 4? (There goes the 15
> minutes to tune theory.)
>
>
> > Bach's 5 tempered fifths "wohltemperirt" are
> > BETTER by 1,2 cent than the 4 tempered fifths of Werckmeister III
> >
> > Bach's basic tonal third C-E "wohltemperirt"
> > is by 1,1 cent BETTER than this third in Werckmeister III
> >
> >
>
> While 1.2 cents is perceivable to me, it is not likely to make much of a
> difference in the non-recorded music of real-time J.S. Bach. More
> importantly, where is there any evidence that this is what Bach did? It
> seems more of a modern deduction about what Bach "should have used" than any
> indication that this is indeed historically accurate for the composer in his
> time.
>
> CONCLUSION:
>
> >
> > Anybody accepting Werckmeister III as Bach's tempering system for all 24
> > keys,
> > ACCEPTS that Bach employed an objectively inferior system - as demonstrated
> > under the criteria enounced above.
> >
> >
> Unfortunately, you argument has not been convincing. It's hard enough for
> moderns to accept that Werckmeister promoted a non-12 equal in his first
> description of a 12 majory/minor key circle. Since there is a prevalence of
> Werckmeister's chromatic in central Germany, Bach's distinctiveness would
> require serious evidence to the contrary. I await more.
>
>
> > Or else, what OTHER rational and more appropriate criteria should be
> > applied,
> > for assessing two irregular baroque tuning systems?? (Their fifths being
> > structured 7+5, respectively 8+4).
> >
> If you have evidence of 7+5 tuning for Bach I'd love to hear of it.
> Incidentally, Werckmeister never claimed to have invented "Werckmeister III"
> or "chromatic Werckmeister." He described it as already in practice.
>
>
>
> > REMARKS:
> > There is no sufficient reason that in Werckmeister III the third C-E is
> > bridged
> > by 3+1 fifths, tempered, respectively perfect. Rather than by 4 EQUAL
> > fifths
> > amongst themselves, as in Bach/"wohltemperirt".
> >
>
> Actually there is, since Werckmeister described his temperament as a
> progressive result of Praetorious's quarter-comma meantone. Werckmeister
> keeps the development clear.
>
>
> > To the honor of Andreas Werckmeister it must be stressed that he was
> already
> > 1691 in possession - and the inventor - of the System "Bach/wohltemperirt",
> > as
> > his treatise "Musicalische Temperatur" shows!!
> >
> By 1681, Werckmeister had begun spreading his ideas throughout out central
> Germany. I do not quite understand why you are saying Werckmeister was an
> "inventor" of something called Bach/wolhtemperirt. And yes, I have read the
> 1691 treatise in translation.
>
>
>
> > TO BE CONTINUED
> > Herbert Anton Kellner
> >
> >
> >
>
> Looking forward to further understanding of your position.
>
> Best, Johnny Reinhard
>
>

🔗ha.kellner@t-online.de

Dear Mr. Charles,
Thanks for your message. I am pleased to learn that his work earned Mr.
Sparschuh a golden tuning fork.

In October 2000 already Mr. Sparschuh and myself went out and we had together
coffee and cake; I am aware of his work and he will have to continue the work
with his results and further publications and await further reception of his
ideas.

About 300 organs do exist already with the system "Wohltemperirt"/Bach I
established in December 1975 and ever since I have been lecturing and publishing
on this subject.

Best regards,

Herbert Anton Kellner

Charles schrieb:

"herbert anton kellner" <ha.kellner@t-online.de> wrote in message
news:3B3E1C71.EC603D7D@t-online.de...
> Could an inferior tuning system satisfy J. S. Bach?
>
> Did Bach prefer a technologically inferior system (=Werckmeister III) to
> "wohltemperirt"?

Are you aware of the idea of Andreas Sparschuh of the Technical University
Darmstadt. He apparently gave a talk which earned him the "golden tuning fork"
of the German tuner's association (I understand an abstract was published in
the
"Deutsche Mathematiker Vereinigung Jahrestagung 1999", Mainz). His thesis is
that the design at the top of the autograph of the Well Tempered Clavier Book
1
(1722) represents a memory aid for tuning.

Any thoughts on this?

NO THOUGHTS. BUT DON't YOU DOUBT AS WELL, THAT JSB MUST HAVE HAD A PRETTY
WEAK MEMORY IF HE NEEDED FOR HIMSELF A SECRET "memory aid for tuning" ???
THAT HE EVEN HAD TO PUT FOR HIMSELF CAUTIOUSLY ONTO THE TITLE PAGE OF HIS
AUTOGRAPH??
Any thoughts on this?
h.a.kELLNER

Charles

🔗John A. deLaubenfels <jdl@adaptune.com>

🔗ha.kellner@t-online.de

Dear Mr. John A. deLaubenfels,

For your interesting statements you sent, I should like to thank you
cordially. Let me remark beforehand that you that you do have, rationally,
intuitively, or else, emotionally, the appropriate attitude towards the
"problem" at hand: You speak, as concerns your result, about DEPRESSINGLY close
to 12-tET.

Concerning the CRITERIA I took recourse to for assessment:

1001$ are more than 1000$ - nobody would contest that.

An enlarged major third smaller by 1 cent than another one, is BETTER.
A reduced fifth larger by 1 cent than another one, will be BETTER.

Your contribution leads me to asking myself the question: are the criteria I
proposed, appropriate to assess, judge and compare "wohltemperirt" versus
Werckmeister III? From the mathematical or physical standpoint,
undoubtedly. But what about the musical point of view? Dies this really
differ from Maths or Physics? Obviously, however, my criteria applied are not
really far fetched, and do seem reasonable. But are they significant to study
the situation? Could they lead to an OBJECTIVE concluding statement?

Similarly, one might pose the question as concerns your approach and method of
analysis:
What does "most consonance" within the entirety of the structure of the WTC
mean? What is the exact definition of this notion?

You certainly know the statistical study of my late friend John Barnes which
indicates an unequal temperament for the WTC and which, due to statistics, seems
to coincide with my "reconstitution": within its error margins.

Regardless, whatever "most consonance" may signify, can such procedure lead to
an objective statement that Bach's preferred and applied system would have been
E.T?

For your valuable and thought-provoking contribution/objection I am grateful. As
we are in natural English, (American, if you like), I heartily invite you to
consult:
Kellner, H. A.: J. S. Bach's Well-tempered Unequal System for Organs. THE
TRACKER, Journal of the Organ Historical Society Vol. 40/3, 1996, page 21-27.

As concerns the approach I follow in this paper, I would profoundly appreciate
your opinion, as seen from your standpoint and and background.

Kind regasrds and all my best wishes,
Herbert Anton Kellner

John A. deLaubenfels schrieb:
> I have read with interest the lively discussion of Bach's well-tempered
> tuning choices. I look forward to more clarification about the
> historical evidence that may help elucidate what Bach actually used.
>
> On the subject of history I have nothing to offer to this discussion,
> but I _do_ have an opinion as regards to the "objectively best" vs.
> "inferior" tuning.
>
> I recently did an interval analysis on Bach's "Well-Tempered Clavier",
> from the standpoint of calculating what fixed tuning (with no more than
> 12 pitch classes per octave) would result in the most consonance. The
> results are depressingly close to 12-tET.
>
> I do not dispute the statement that "For baroque harpsichord music, the
> sound of E. T. is a disaster, an insult to hearing ears." I pretty much
> _hate_ 12-tET in any context (my ears are sated now with the consonance
> that only adaptive tuning can bring).
>
> But when it comes to arguing which well-temperament is "objectively
> best" for the WTC, I'm sorry to report that they're all worse than
> 12-tET by any rational measure I have seen to date.
>
> JdL
>
>
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>
>

🔗JoJoBuBu@aol.com

In a message dated 7/1/2001 3:47:56 PM Eastern Daylight Time,
ha.kellner@t-online.de writes:
>
>
> Dear Mr. John A. deLaubenfels,
> For your interesting statements you sent, I should like to thank you
> cordially. Let me remark beforehand that you that you do have, rationally,
> intuitively, or else, emotionally, the appropriate attitude towards the
> "problem" at hand: You speak, as concerns your result, about DEPRESSINGLY
> close
> to 12-tET.
>
> Concerning the CRITERIA I took recourse to for assessment:
>
> 1001$ are more than 1000$ - nobody would contest that.
>
> An enlarged major third smaller by 1 cent than another one, is BETTER.
>

Just a thought on this matter.

Suppose that for the sake of argument your results showed a major third
smaller by 0.00001 cents. By your criteria one would think this to be better,
but in reality this this is inaudible to most, and very probably all,
listeners ears. Even with 1 cent it is at least debatable whether that
interval is hearable to anyone ... let alone in a real musical performance.

Even if perhaps a few individuals could hear this extremely tiny difference
it would be the extreme minority of individuals with this ability ... if any
individuals. This is not even to mention whether by ear somone tuning these
instruments really could have made the very fine adjustments and been able to
tell the difference.

I suppose it is "possible" that 0.00000001 percent of the population could
have heard the difference, heard that these intervals were better by your
criteria, and successfully tuned the instrument according to this "better"
criteria. I suggest, however, that when intervals are as small as the
differences are here that it becomes less and less important to make
distinctions about one system or interval, or etc, as being better at all.

To sum up for an interval to be a better interval three criteria must be met
as oppossed to just your one criteria.

The first is that you must show via some logical method that the interval is
better, and these one cent differences you speak of seem to do this fine(in
this isolated instance).

The second point however is that these differences must be audible to human
beings. In this case I think it is a fair argument to say that "if" these
differences are audible, that only the extreme minority can hear them, if
anyone at all. If the differences between these intervals are not hearable
then the tuning system is not better or worse but exactly the same to the
listener. (In this context listener includes the person tuning the
instrument, the composer, and the audience)

Lastly it must be possible to tune instruments correctly to this very small
interval distance "by ear." Certainly electronic instruments could do this
interval correctly, but to tune live instruments like this by ear would be
difficult, especially if criteria two is not met and the interval differences
are not possible to hear or only possible to be heard by the extreme minority
of people.

This thread has been very interesting.

Cheers,
Andy

🔗John A. deLaubenfels <jdl@adaptune.com>

[Herbert Anton Kellner wrote:]
>For your interesting statements you sent, I should like to thank you
>cordially. Let me remark beforehand that you that you do have,
>rationally, intuitively, or else, emotionally, the appropriate attitude
>towards the "problem" at hand: You speak, as concerns your result,
>about DEPRESSINGLY close to 12-tET.

Oh yes, 12-tET depresses me from several standpoints.

>Concerning the CRITERIA I took recourse to for assessment:

>1001$ are more than 1000$ - nobody would contest that.

>An enlarged major third smaller by 1 cent than another one, is BETTER.
>A reduced fifth larger by 1 cent than another one, will be BETTER.

Well... see Ed Foote's recent post. On this subject I tend to look at
consonances, but I do not want to make the claim that this is the _only_
important quality.

>Your contribution leads me to asking myself the question: are the
>criteria I proposed, appropriate to assess, judge and compare
>"wohltemperirt" versus Werckmeister III? From the mathematical or
>physical standpoint, undoubtedly. But what about the musical point of
>view? Dies this really differ from Maths or Physics? Obviously,
>however, my criteria applied are not really far fetched, and do seem
>reasonable. But are they significant to study the situation? Could
>they lead to an OBJECTIVE concluding statement?

Perhaps I was mistaken in thinking you were making such a claim? It is
only to that claim that I wish to respond (if you did make it); I do not
want to come off as saying that consonance is the only important thing
in music.

>Similarly, one might pose the question as concerns your approach and
>method of analysis: What does "most consonance" within the entirety of
>the structure of the WTC mean? What is the exact definition of this
>notion?

My methods compile a list of all dyads and attempt to pull them toward
simple ratios (3:2, 5:4, etc.).

>You certainly know the statistical study of my late friend John Barnes
>which indicates an unequal temperament for the WTC and which, due to
>statistics, seems to coincide with my "reconstitution": within its
>error margins.

I have heard that past analyses of the WTC seem to find uneven
distribution of intervals consistent with well-temperaments. My own
analysis does not agree. It is of course possible that my analysis is
wrong!

>Regardless, whatever "most consonance" may signify, can such procedure
>lead to an objective statement that Bach's preferred and applied system
>would have been E.T?

No, certainly not! All I am saying is that we must make our tuning
picks based upon criteria other than "objective". And I am explicitly
applauding historical research into the subject.

>As we are in natural English, (American, if you like), I heartily
>invite you to consult: Kellner, H. A.: J. S. Bach's Well-tempered
>Unequal System for Organs. THE TRACKER, Journal of the Organ Historical
>Society Vol. 40/3, 1996, page 21-27.

I am sorry to say that I don't have good access to scholarly journals,
unless they are on the web. Perhaps you could post a summary?

My own analysis is reported in:

/tuning/topicId_17546.html#17546
/tuning/topicId_17546.html#17562

Thanks for your response!

JdL

🔗ha.kellner@t-online.de

FOR 250 YEARS THE EQUAL TEMPERAMENT THEORY FOR BACH PREVAILED. BOSANQUET, LANGE
AND KELLETAT WERE AMONGST THE FIRST TO CONTEST THE E.T.-HYPOTHESIS. OVER IN THE
U.S.A., OF COURSE, MURRAY BARBOUR. WHEREAS I THINK HE SUFFERED FROM LACK OF
UNEQUAL-TEMPERAMENT-PRACTICE, HIS IMPACT WAS GIGANTIC.

THE TASK I SEE NOW IS PRETTY TOUGH BUT I FACE THE CHALLENGE - AND THANK ALL OF
YOU FOR YOUR CONSTRUCTIVE SUPPORT WHICH INCLUDES POSING YOUR QUESTIONS.

KIND REGARDS,
HERBERT ANTON KELLNER

>
> >
> > Structure of fifths:
> > Nominal Werckmeister III:
> > 8+4 perfect resp. tempered; the latter 696,09 c
> >
> > Bach/Werckmeister "wohltemperirt":
> > 7+5 perfect resp. tempered; the latter 697,28 c
> >
> >
> Why the effort to improve by so little? Isn't splitting a comma into 5 more
> time consuming to evaluate in tuning a keyboard than 4? (There goes the 15
> minutes to tune theory.)
>DO YOU KNOW THE B-MAJOR NETHOD FOR "WOHLTEMPERIRT"? THEREIN, B-D# MOST BEAT 6
TIMES AS RAPIDLY AS B-F#. THIS IS A MATHEMATICAL RESULT FOR DIVIDING THE
PYTH.COM. BY 5.
TO DISCOVER THIS SPECIFIC METHOD TO IMPLEMENT BACH'S SYSTEM, TOOK A PROFESSIONAL
MATHEMATICIAN 3 ABOUT MONTHS AND SUCH A METHOD WAS UNHEARD OF IN THE ENTIRE
HISTORY OF MUSICAL TEMPERAMENT.
>
> > Bach's 5 tempered fifths "wohltemperirt" are
> > BETTER by 1,2 cent than the 4 tempered fifths of Werckmeister III
> >
> > Bach's basic tonal third C-E "wohltemperirt"
> > is by 1,1 cent BETTER than this third in Werckmeister III
> >
> >
>
> While 1.2 cents is perceivable to me, it is not likely to make much of a
> difference in the non-recorded music of real-time J.S. Bach. More
> importantly, where is there any evidence that this is what Bach did? It
> seems more of a modern deduction about what Bach "should have used" than any
> indication that this is indeed historically accurate for the composer in his
> time.
>
> CONCLUSION:
>
> >
> > Anybody accepting Werckmeister III as Bach's tempering system for all 24
> > keys,
> > ACCEPTS that Bach employed an objectively inferior system - as demonstrated
> > under the criteria enounced above.
> >
> >
> Unfortunately, you argument has not been convincing. It's hard enough for
> moderns to accept that Werckmeister promoted a non-12 equal in his first
> description of a 12 majory/minor key circle. Since there is a prevalence of
> Werckmeister's chromatic in central Germany, Bach's distinctiveness would
> require serious evidence to the contrary. I await more.
>
>
> > Or else, what OTHER rational and more appropriate criteria should be
> > applied,
> > for assessing two irregular baroque tuning systems?? (Their fifths being
> > structured 7+5, respectively 8+4).
> >
> If you have evidence of 7+5 tuning for Bach I'd love to hear of it.
> Incidentally, Werckmeister never claimed to have invented "Werckmeister III"
> or "chromatic Werckmeister." He described it as already in practice.
>
>
>
> > REMARKS:
> > There is no sufficient reason that in Werckmeister III the third C-E is
> > bridged
> > by 3+1 fifths, tempered, respectively perfect. Rather than by 4 EQUAL
> > fifths
> > amongst themselves, as in Bach/"wohltemperirt".
> >
>
> Actually there is, since Werckmeister described his temperament as a
> progressive result of Praetorious's quarter-comma meantone. Werckmeister
> keeps the development clear.
>
>
> > To the honor of Andreas Werckmeister it must be stressed that he was
> already
> > 1691 in possession - and the inventor - of the System "Bach/wohltemperirt",
> > as
> > his treatise "Musicalische Temperatur" shows!!
> >
> By 1681, Werckmeister had begun spreading his ideas throughout out central
> Germany. I do not quite understand why you are saying Werckmeister was an
> "inventor" of something called Bach/wolhtemperirt. And yes, I have read the
> 1691 treatise in translation.

I HAVE READ PRCTICALLY ALL THE WERCKMEISZTER WRITINGS, COMING FROM THE TURN OF
THE 17TH TO THE 18TH CENTURY.
HE USED UPON HIS GERMAN TEXT THE GEMATRIA AND THE TRANSLATION OF COURSE,
DESTROYS THESE ENCODINGS. REFERENCE FROM MY WEBSITE:

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
>
>HAK JULY 2, 2001
>
> > TO BE CONTINUED
> > Herbert Anton Kellner
> >
> >
> >
>
> Looking forward to further understanding of your position.
>
> Best, Johnny Reinhard
>
>

🔗ha.kellner@t-online.de

JoJoBuBu@aol.com schrieb:
> In a message dated 7/1/2001 3:47:56 PM Eastern Daylight Time,
> ha.kellner@t-online.de writes:
> >
> >
> > Dear Mr. John A. deLaubenfels,
> > For your interesting statements you sent, I should like to thank you
> > cordially. Let me remark beforehand that you that you do have, rationally,
> > intuitively, or else, emotionally, the appropriate attitude towards the
> > "problem" at hand: You speak, as concerns your result, about DEPRESSINGLY
> > close
> > to 12-tET.
> >
> > Concerning the CRITERIA I took recourse to for assessment:
> >
> > 1001$ are more than 1000$ - nobody would contest that.
> >
> > An enlarged major third smaller by 1 cent than another one, is BETTER.
> >
>
> Just a thought on this matter.
>
> Suppose that for the sake of argument your results showed a major third
> smaller by 0.00001 cents. By your criteria one would think this to be better,
> but in reality this this is inaudible to most, and very probably all,
> listeners ears. Even with 1 cent it is at least debatable whether that
> interval is hearable to anyone ... let alone in a real musical performance.

BUT THE TWO DIFFERENCES I CONSIDERED FOR MY CRITERIA ARE 1.2 AND 1.1 CENT,
RESPECTIVELY. AND NOT 0.00001 CENT!
LET'S TURN THE MATTER AROUND; COULD ONE JUSTIFY THAT "WOHLTEMPERIRT" OVERALL
WERE A PRIORI WORSE THAN W III, DESPITE THAT FACT THAT THESE TWO CHARACTERISIC
VALUES PICKED OUT FOR ASSESSMENT ARE, IN FACT, "BETTER" IN THE ORDER OF 1 CENT
THAN FOR W III.
THE BASIC TONAL THIRD IN BOTH CASES CO-INCIDES TO THE ORDER OF 1 CENT. BUT
THERE IS ANOTHER SNAG IN W III: IF WITHIN THIS THIRD OF WERCKMEISTER THE 3 FIFHS
C-G-D-A ARE TEMPERED AND FOLLOWED BY A-E perfect TO SPAN THS THIRD C-E, THEM
THIS PERFCET FIFTH SQUEEZES THE THREE TEMPERED FIFTHS TO BE MORE TEMPEREED THAT
UNAVOIDABLE. rEMEDCY: EQUALIZE ALL FOUR THIRDS IN NO MATTER WHAT EXACT SIZE OF
C-E. IN THIS SENSE, W III IS PROBABLY, OBJECTIVELY, INFERIOR TO
"WOHLTEMPERIRT".

>
> Even if perhaps a few individuals could hear this extremely tiny difference
> it would be the extreme minority of individuals with this ability ... if any
> individuals. This is not even to mention whether by ear somone tuning these
> instruments really could have made the very fine adjustments and been able to
> tell the difference.
>
> I suppose it is "possible" that 0.00000001 percent of the population could
> have heard the difference, heard that these intervals were better by your
> criteria, and successfully tuned the instrument according to this "better"
> criteria. I suggest, however, that when intervals are as small as the
> differences are here that it becomes less and less important to make
> distinctions about one system or interval, or etc, as being better at all.
>
> To sum up for an interval to be a better interval three criteria must be met
> as oppossed to just your one criteria.
>
> The FIRS is that you must show via some logical method that the interval is
> better, and these one cent differences you speak of seem to do this fine(in
> this isolated instance).
ONE MIGHT SUPPOSE, AN INTERVAL THAT IS CLOSER TO ITS CORRESPONDING "PERFECT"
VALUE OR SIZE IS "BETTER"!?
>
> The SECOND point however is that these differences must be audible to human
> beings. In this case I think it is a fair argument to say that "if" these
> differences are audible, that only the extreme minority can hear them, if
> anyone at all. If the differences between these intervals are not hearable
> then the tuning system is not better or worse but exactly the same to the
> listener. (In this context listener includes the person tuning the
> instrument, the composer, and the audience).
ANY PROFICIENT TUNER TEMPERING VIA BEATS WILL DESCERN MUCH MORE SHARPLY SLIGHT
DIFFERENCES BETWEEN, SAY, "WOHLTEMPERIRT" AND W III THAT A NAIVE LISTENER.
BUT CAN'T ONE EXPECT THAT A "BETTER" SYSTEM IN MY SENSE WILL GIVE A "BETTER
IMPRESSION" - REGARDLESS IF THE LISTENER BECOMES RATIONALLY AWARE OF THAT OR
NOT?? THIS THOUGHT NATURALLY LEADS TO YOUR LAST AND INTERESTING POINT YOU MAKE!
>
> LASTLY it must be possible to tune instruments correctly to this very small
> interval distance "by ear." Certainly electronic instruments could do this
> interval correctly, but to tune live instruments like this by ear would be
> difficult, especially if criteria two is not met and the interval differences
> are not possible to hear or only possible to be heard by the extreme minority
> of people.
>
> This thread has been very interesting.
>
> Cheers,
> Andy
>KIND REGARDS,
HERBERT ANTON k.
>
>

🔗Paul Erlich <paul@stretch-music.com>

🔗JoJoBuBu@aol.com

>BUT THE TWO DIFFERENCES I CONSIDERED FOR MY CRITERIA >ARE 1.2 AND 1.1 CENT,
>RESPECTIVELY. AND NOT 0.00001 CENT!
>LET'S TURN THE MATTER AROUND; COULD ONE JUSTIFY >THAT "WOHLTEMPERIRT" OVERALL
>WERE A PRIORI WORSE THAN W III, DESPITE THAT FACT THAT >THESE TWO
>CHARACTERISIC
>VALUES PICKED OUT FOR ASSESSMENT ARE, IN >FACT, "BETTER" >IN THE ORDER OF 1
>CENT
>THAN FOR W III.
>THE BASIC TONAL THIRD IN BOTH CASES CO-INCIDES TO THE >ORDER OF 1 CENT. BUT
>THERE IS ANOTHER SNAG IN W III: IF WITHIN THIS THIRD OF >WERCKMEISTER THE 3
>FIFHS
>C-G-D-A ARE TEMPERED AND FOLLOWED BY A-E perfect TO >SPAN THS THIRD C-E, THEM
>THIS PERFCET FIFTH SQUEEZES THE THREE TEMPERED FIFTHS >TO BE MORE TEMPEREED
>THAT
>UNAVOIDABLE. rEMEDCY: EQUALIZE ALL FOUR THIRDS IN NO >MATTER WHAT EXACT SIZE
>OF
>C-E. IN THIS SENSE, W III IS PROBABLY, OBJECTIVELY, >INFERIOR TO
>"WOHLTEMPERIRT".

If one was objectively trying to say that one tuning system is "better" than another then the difference between the two systems must be audible. The jury is not out on whether these 1.2 or what not cent differences are audible.

I'm not saying either system is better - perhaps you misunderstood me on this part - I'm not saying your wrong. I am saying that if your claim asserts "WOHLTEMPERIRT" is better yet the difference between the two systems is objectively inaudible, or audible only to one out of a million people in extremely isolated circumstances, then its really a pointless discussion to say EITHER is better.

I question whether these differences you state are physically audible except in extremely isolated circumstances. My suggestion then would be to show that this is not only objectively good from a scholarly mathematic approach but also that this has some value in the real world. If people, meaning all people tuner composer listener etc, cant hear the difference, except perhaps the extreme minority in extremely isolated circumstances, what then is the point of differentiating between either system?

Cheers,
Andy

🔗John A. deLaubenfels <jdl@adaptune.com>

🔗ha.kellner@t-online.de

Dear Andy,

Thanks to some friendly advice I cease SCREAMING.

We may look now at the following aspect:
Was Bach's system for his WTC the "nominal Werckmeister", or else,
"Wohltemperirt" (that was invented by the very same Werckmeister and, having
its 7 perfect and 5 tempered fifths encoded into his treatise "Musicalische
Temperatur", page 75 - in words seven&five - of chapter 28 (=secundus numerus
perfectus, in terms of Werckmeister's Baroque mathematics).
Musicological papers already published concerning this matter:

Primary reference:
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.

Secondary references:
Is there an enigma in Werckmeister's "Musicalische Temperatur"? English
Harpsichord Magazine, Vol. 3, No. 7, 1984, page 134-136.
One typographical enigma in Werckmeister, "Musicalische Temperatur". English
Harpsichord Magazine, Vol. 3, No. 8, 1985, page 146-151.
Did Werckmeister already know the tuning of J.S. Bach for the "48"? English
Harpsichord Magazine, Vol. 4, No. 1,1985, page 7-11.

What could have been J. S.Bach's own and personal criteria to prefer
"wohltemperirt" to Werckmeister III?

In my opinion, above all the musico-theological significance of the C-major
triad in "wohltemperirt". Not only is any major triad (trias harmonica perfecta)
a musical tri-unitary symbol. But by virtue of the beat ratio third to
fifth being 1:1, the "unitarian" aspect is enhanced.

(The baroque mentality, trias harmonica perfecta and the perfection of the
unitas, are expounded to great depth in the extraordinary book of Rolf Dammann,
"Der Musikbegriff im Deutschen Barock", Laaber Verlag, 3rd edition about 1994.)

If JSB put his emphasis on the musico-theological features of "wohltemperirt",
any mathematical or physical quantities better or worse by about 1 cent could
not have mattered to him! But these parameters are better in "wohltemperirt" and
not worse. Except, one has the opposite personal opinion.

Again technologoically speaking, should one now better prefer today, to perform
WTC in "wohltemperirt" or in W III, even regardless of what was JSB's true
system?

Kind regards,

Herbert A. Kellner

Could it have escaped to JSB that the basic third C-E, of W III, filled up by
3+1 fifths tempered resp. perfect, has 3 fifths being stronger tempered than
necessary. They are, in fact, squeezed together within that third by the last,
perfect fifth A-E.

Could it have escaped to JSB that

JoJoBuBu@aol.com schrieb:
> >BUT THE TWO DIFFERENCES I CONSIDERED FOR MY CRITERIA >ARE 1.2 AND 1.1 CENT,
> >RESPECTIVELY. AND NOT 0.00001 CENT!
> >LET'S TURN THE MATTER AROUND; COULD ONE JUSTIFY >THAT "WOHLTEMPERIRT"
> OVERALL
> >WERE A PRIORI WORSE THAN W III, DESPITE THAT FACT THAT >THESE TWO
> >CHARACTERISIC
> >VALUES PICKED OUT FOR ASSESSMENT ARE, IN >FACT, "BETTER" >IN THE ORDER OF 1
> >CENT
> >THAN FOR W III.
> >THE BASIC TONAL THIRD IN BOTH CASES CO-INCIDES TO THE >ORDER OF 1 CENT. BUT
> >THERE IS ANOTHER SNAG IN W III: IF WITHIN THIS THIRD OF >WERCKMEISTER THE 3
> >FIFHS
> >C-G-D-A ARE TEMPERED AND FOLLOWED BY A-E perfect TO >SPAN THS THIRD C-E,
> THEM
> >THIS PERFCET FIFTH SQUEEZES THE THREE TEMPERED FIFTHS >TO BE MORE TEMPEREED
> >THAT
> >UNAVOIDABLE. rEMEDCY: EQUALIZE ALL FOUR THIRDS IN NO >MATTER WHAT EXACT SIZE
> >OF
> >C-E. IN THIS SENSE, W III IS PROBABLY, OBJECTIVELY, >INFERIOR TO
> >"WOHLTEMPERIRT".
>
> If one was objectively trying to say that one tuning system is "better" than
> another then the difference between the two systems must be audible. The
> jury is not out on whether these 1.2 or what not cent differences are
> audible.
>
> I'm not saying either system is better - perhaps you misunderstood me on this
> part - I'm not saying your wrong. I am saying that if your claim asserts
> "WOHLTEMPERIRT" is better yet the difference between the two systems is
> objectively inaudible, or audible only to one out of a million people in
> extremely isolated circumstances, then its really a pointless discussion to
> say EITHER is better.
>
> I question whether these differences you state are physically audible except
> in extremely isolated circumstances. My suggestion then would be to show
> that this is not only objectively good from a scholarly mathematic approach
> but also that this has some value in the real world. If people, meaning all
> people tuner composer listener etc, cant hear the difference, except perhaps
> the extreme minority in extremely isolated circumstances, what then is the
> point of differentiating between either system?
>
> Cheers,
> Andy
>
>
>
>
>
> You do not need web access to participate. You may subscribe through
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>

🔗JoJoBuBu@aol.com

In a message dated 7/4/2001 12:29:45 AM Eastern Daylight Time,
ha.kellner@t-online.de writes:

> Dear Andy,
>
> Thanks to some friendly advice I cease SCREAMING.
>
> We may look now at the following aspect:
> Was Bach's system for his WTC the "nominal Werckmeister", or else,
> "Wohltemperirt" (that was invented by the very same Werckmeister and,
> having
> its 7 perfect and 5 tempered fifths encoded into his treatise "Musicalische
> Temperatur", page 75 - in words seven&five - of chapter 28 (=secundus
> numerus
> perfectus, in terms of Werckmeister's Baroque mathematics).
> Musicological papers already published concerning this matter:
>
> Primary reference:
> Kellner, H.A.: A propos d'une réimpression de la "Musicalische Temperatur"
> (1691) de Werckmeister. Revue de Musicologie Vol. 71, 1985, page 184-187.
>
> Secondary references:
> Is there an enigma in Werckmeister's "Musicalische Temperatur"? English
> Harpsichord Magazine, Vol. 3, No. 7, 1984, page 134-136.
> One typographical enigma in Werckmeister, "Musicalische Temperatur".
> English
> Harpsichord Magazine, Vol. 3, No. 8, 1985, page 146-151.
> Did Werckmeister already know the tuning of J.S. Bach for the "48"? English
> Harpsichord Magazine, Vol. 4, No. 1,1985, page 7-11.
>
>
> What could have been J. S.Bach's own and personal criteria to prefer
> "wohltemperirt" to Werckmeister III?
>
> In my opinion, above all the musico-theological significance of the C-major
> triad in "wohltemperirt". Not only is any major triad (trias harmonica
> perfecta)
> a musical tri-unitary symbol. But by virtue of the beat ratio third to
> fifth being 1:1, the "unitarian" aspect is enhanced.
>
> (The baroque mentality, trias harmonica perfecta and the perfection of the
> unitas, are expounded to great depth in the extraordinary book of Rolf
> Dammann,
> "Der Musikbegriff im Deutschen Barock", Laaber Verlag, 3rd edition about
> 1994.)
>
> If JSB put his emphasis on the musico-theological features of
> "wohltemperirt",
> any mathematical or physical quantities better or worse by about 1 cent
> could
> not have mattered to him! But these parameters are better in
> "wohltemperirt" and
> not worse. Except, one has the opposite personal opinion.
>
> Again technologoically speaking, should one now better prefer today, to
> perform
> WTC in "wohltemperirt" or in W III, even regardless of what was JSB's true
> system?
>
> Kind regards,
>
> Herbert A. Kellner
>
> Could it have escaped to JSB that the basic third C-E, of W III, filled up
> by
> 3+1 fifths tempered resp. perfect, has 3 fifths being stronger tempered
> than
> necessary. They are, in fact, squeezed together within that third by the
> last,
> perfect fifth A-E.
>
>

Well I understand your point surely, but I still dont think you answered the
question I was attempting to pose which is, "If the margin of error is so
close that the difference between the two systems might be inaudible to most,
if not all, ears why does it matter to determine which system is better?"

I see your sources and such, which is great, but I dont see how your answer
relates to the question I have been attempting to ask.

Could you enlighten as to your point of view on this?

Andy

🔗ha.kellner@t-online.de

Dear Andy,

Better or worse tuning systems? Superiour or inferior?

The assessment of a tuning system does NOT depend on whether
differences in the intervals are audible, or not. In the eventuality, that
these differences in listening to
Das wohltemperirte Clavier
* are audible, then the system that sounds better, will obviously be BETTER,
will
have to be designated as the better system.

In case the difference in listening to
Das wohltemperirte Clavier
* is only marginally audible, or inaudible at all, then that system being
mathematically, physically, or else, as judged from a competent
engineering-point of view superior, WILL HAVE TO BE CONSIDERED as BETTER.
We have no other choice!
Otherwise, the opposite alternative is not really to my taste and comprehension.

Kind regards,

Herbert Anton Kellner

JoJoBuBu@aol.com schrieb:
> In a message dated 7/4/2001 12:29:45 AM Eastern Daylight Time,
> ha.kellner@t-online.de writes:
>
>
> > Dear Andy,
> >
> > Thanks to some friendly advice I cease SCREAMING.
> >
> > We may look now at the following aspect:
> > Was Bach's system for his WTC the "nominal Werckmeister", or else,
> > "Wohltemperirt" (that was invented by the very same Werckmeister and,
> > having
> > its 7 perfect and 5 tempered fifths encoded into his treatise "Musicalische
> > Temperatur", page 75 - in words seven&five - of chapter 28 (=secundus
> > numerus
> > perfectus, in terms of Werckmeister's Baroque mathematics).
> > Musicological papers already published concerning this matter:
> >
> > Primary reference:
> > 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.
> >
> > Secondary references:
> > Is there an enigma in Werckmeister's "Musicalische Temperatur"? English
> > Harpsichord Magazine, Vol. 3, No. 7, 1984, page 134-136.
> > One typographical enigma in Werckmeister, "Musicalische Temperatur".
> > English
> > Harpsichord Magazine, Vol. 3, No. 8, 1985, page 146-151.
> > Did Werckmeister already know the tuning of J.S. Bach for the "48"? English
> > Harpsichord Magazine, Vol. 4, No. 1,1985, page 7-11.
> >
> >
> > What could have been J. S.Bach's own and personal criteria to prefer
> > "wohltemperirt" to Werckmeister III?
> >
> > In my opinion, above all the musico-theological significance of the C-major
> > triad in "wohltemperirt". Not only is any major triad (trias harmonica
> > perfecta)
> > a musical tri-unitary symbol. But by virtue of the beat ratio third to
> > fifth being 1:1, the "unitarian" aspect is enhanced.
> >
> > (The baroque mentality, trias harmonica perfecta and the perfection of the
> > unitas, are expounded to great depth in the extraordinary book of Rolf
> > Dammann,
> > "Der Musikbegriff im Deutschen Barock", Laaber Verlag, 3rd edition about
> > 1994.)
> >
> > If JSB put his emphasis on the musico-theological features of
> > "wohltemperirt",
> > any mathematical or physical quantities better or worse by about 1 cent
> > could
> > not have mattered to him! But these parameters are better in
> > "wohltemperirt" and
> > not worse. Except, one has the opposite personal opinion.
> >
> > Again technologoically speaking, should one now better prefer today, to
> > perform
> > WTC in "wohltemperirt" or in W III, even regardless of what was JSB's true
> > system?
> >
> > Kind regards,
> >
> > Herbert A. Kellner
> >
> > Could it have escaped to JSB that the basic third C-E, of W III, filled up
> > by
> > 3+1 fifths tempered resp. perfect, has 3 fifths being stronger tempered
> > than
> > necessary. They are, in fact, squeezed together within that third by the
> > last,
> > perfect fifth A-E.
> >
> >
>
> Well I understand your point surely, but I still dont think you answered the
> question I was attempting to pose which is, "If the margin of error is so
> close that the difference between the two systems might be inaudible to most,
> if not all, ears why does it matter to determine which system is better?"
>
> I see your sources and such, which is great, but I dont see how your answer
> relates to the question I have been attempting to ask.
>
> Could you enlighten as to your point of view on this?
>
> Andy
>

🔗JoJoBuBu@aol.com

In a message dated 7/5/2001 8:57:52 AM Eastern Daylight Time,
ha.kellner@t-online.de writes:

> Dear Andy,
>
> Better or worse tuning systems? Superiour or inferior?
>
> The assessment of a tuning system does NOT depend on whether
> differences in the intervals are audible, or not. In the eventuality, that
> these differences in listening to
> Das wohltemperirte Clavier
> * are audible, then the system that sounds better, will obviously be
> BETTER,
> will
> have to be designated as the better system.
>
> In case the difference in listening to
> Das wohltemperirte Clavier
> * is only marginally audible, or inaudible at all, then that system being
> mathematically, physically, or else, as judged from a competent
> engineering-point of view superior, WILL HAVE TO BE CONSIDERED as BETTER.
> We have no other choice!
> Otherwise, the opposite alternative is not really to my taste and
> comprehension.
>
> Kind regards,
>
>

That was my point. I think its useless to say either system is better because
the difference is inaudible. That is the alternative and other choice. When
we are discussing systems of music if the system is not audible, or in this
case an audible difference between systems, then the theory is useless. It
would be vary analogous to writing songs above 25,000 Hz. One can make all
the theory in the world about writing songs above 25,000 Hz but no one will
ever hear what that person is talking about because its not possible ... I
see this as analagous to the present situation.

From an engineering point of view??? ... Engineers realize that some things
are mathematically insignificant. In music if theory discusses inaudible
things it IS mathematically insignificant. The question here is really,

"Where does music theory/tuning theory stop being useful?"

Music theory/tuning theory stops being usefull when humans can no longer tell
the difference between systems as has happened here.

To say it doesn't matter whether the difference is audible is not correct. We
are talking about MUSIC.

Cheers,

Andy

🔗ha.kellner@t-online.de

Dear Andy,

Tanks for your reply with your salient arguments. We are converging.

1) Do you think Bach himself would have preferred W III or "wohltemperirt"?
if one or the other, for what possible reasons??

2) Are ancient tunings easier to reconstitute: for organs or harpsichords???

Kind regards,

Herbert Anton Kellner; cont'd, see below

JoJoBuBu@aol.com schrieb:
> In a message dated 7/5/2001 8:57:52 AM Eastern Daylight Time,
> ha.kellner@t-online.de writes:
>
>
> > Dear Andy,
> >
> > Better or worse tuning systems? Superiour or inferior?
> >
> > The assessment of a tuning system does NOT depend on whether
> > differences in the intervals are audible, or not. In the eventuality, that
> > these differences in listening to
> > Das wohltemperirte Clavier
> > * are audible, then the system that sounds better, will obviously be
> > BETTER,
> > will
> > have to be designated as the better system.
> >
> > In case the difference in listening to
> > Das wohltemperirte Clavier
> > * is only marginally audible, or inaudible at all, then that system being
> > mathematically, physically, or else, as judged from a competent
> > engineering-point of view superior, WILL HAVE TO BE CONSIDERED as BETTER.
> > We have no other choice!
> > Otherwise, the opposite alternative is not really to my taste and
> > comprehension.
> >
> > Kind regards,
> >
> >
>
> That was my point. I think its useless to say either system is better because
> the difference is inaudible. That is the alternative and other choice. When
> we are discussing systems of music if the system is not audible, or in this
> case an audible difference between systems, then the theory is useless.
*********************
> It would be vary analogous to writing songs above 25,000 Hz. One can make all
> the theory in the world about writing songs above 25,000 Hz but no one will
> ever hear what that person is talking about because its not possible ... I
> see this as analagous to the present situation.
>
To every musical temperament, provided it is mathematically defined, or in
engineering terms, belongs a set of numbers, of course.

Without enouncing here the respective sets pertaining to "wohltemperirt" or
else, to Werckmeister III,

is it conceivable that a mathematician-musician like JSB has structured his
music employing and taking recourse to the numbers of such a set.

If we find numbers of the set W III, then W III was his preferred tuning.

If, on the contrary, we discover numbers of the set belonging to "wohltemperirt"
in his music,
then we would have to conclude, as concerns J. S. Bach, "wohltemperirt" must
have been preferable, better, for him. And preferable, because he deems it more
appropriate, or BETTER.

But such aspects are not music-theory, but belong to the realm of
musical practice, analyzing the compositional process.

Kind regardss,
Herbert-Anton

> >From an engineering point of view??? ... Engineers realize that some things
> are mathematically insignificant. In music if theory discusses inaudible
> things it IS mathematically insignificant. The question here is really,
>
> "Where does music theory/tuning theory stop being useful?"
>
> Music theory/tuning theory stops being usefull when humans can no longer tell
> the difference between systems as has happened here.
>
> To say it doesn't matter whether the difference is audible is not correct. We
> are talking about MUSIC.
>

> Cheers,
>
> Andy
>

🔗JoJoBuBu@aol.com

In a message dated 7/5/2001 3:03:52 PM Eastern Daylight Time,
ha.kellner@t-online.de writes:

> Dear Andy,
>
> Tanks for your reply with your salient arguments. We are converging.
>
> 1) Do you think Bach himself would have preferred W III or "wohltemperirt"?
> if one or the other, for what possible reasons??

I dont know. If the numbers are as close as yours seem to indicate between
the two systems, then I dont think it matters what he would prefer.

>
> 2) Are ancient tunings easier to reconstitute: for organs or harpsichords???
>

No matter what instrument is being tuned 1 cent is still extremely small.

> Kind regards,
>
> Herbert Anton Kellner; cont'd, see below
>
> JoJoBuBu@aol.com schrieb:
> > In a message dated 7/5/2001 8:57:52 AM Eastern Daylight Time,
> > ha.kellner@t-online.de writes:
> >
> >
> > > Dear Andy,
> > >
> > > Better or worse tuning systems? Superiour or inferior?
> > >
> > > The assessment of a tuning system does NOT depend on whether
> > > differences in the intervals are audible, or not. In the eventuality,
> that
> > > these differences in listening to
> > > Das wohltemperirte Clavier
> > > * are audible, then the system that sounds better, will obviously be
> > > BETTER,
> > > will
> > > have to be designated as the better system.
> > >
> > > In case the difference in listening to
> > > Das wohltemperirte Clavier
> > > * is only marginally audible, or inaudible at all, then that system
> being
> > > mathematically, physically, or else, as judged from a competent
> > > engineering-point of view superior, WILL HAVE TO BE CONSIDERED as
> BETTER.
> > > We have no other choice!
> > > Otherwise, the opposite alternative is not really to my taste and
> > > comprehension.
> > >
> > > Kind regards,
> > >
> > >
> >
> > That was my point. I think its useless to say either system is better
> because
> > the difference is inaudible. That is the alternative and other choice.
> When
> > we are discussing systems of music if the system is not audible, or in
> this
> > case an audible difference between systems, then the theory is useless.
> *********************
> > It would be vary analogous to writing songs above 25,000 Hz. One can
> make all
> > the theory in the world about writing songs above 25,000 Hz but no one
> will
> > ever hear what that person is talking about because its not possible ...
> I
> > see this as analagous to the present situation.
> >

(you said)

> To every musical temperament, provided it is mathematically defined, or in
> engineering terms, belongs a set of numbers, of course.

Certainly. We can't let the numbers take us in to the realm of insignificance
however.

>
> Without enouncing here the respective sets pertaining to "wohltemperirt" or
> else, to Werckmeister III,
>
> is it conceivable that a mathematician-musician like JSB has structured his
> music employing and taking recourse to the numbers of such a set.

Certainly. I suggest however that when numbers get down to the inaudible then
it makes no difference what the numbers say or what the musician believed he
was preferring. We are discussing an art form which is based on sound. Also
tuning systems, or music compositions, are something that are intended to be
perceived. If the difference between two tuning systems can not be perceived
then it is rather pointless to make distinctions between them. This suggests
to me that decidng exactly what Bach preferred is not important.

I recently read a book on computer game graphics to draw an analogy. The
author suggested that if it looks right, it is right. Why? The mathematics
involved with graphics programming, he says, is important of course but if
your math , within a game for example, describes things inperceivable, then
it is pointless to spend time coding that extra math.

In other words I'm not saying Bach did/did not want this or that tuning
system. I am saying its NOT an important question to ask which system Bach
preferred if and only if the two systems you suggest are so close as to be
not perceivable, or debatably perceivable to one in a trillion people.

For example suppose two thing.

1. Suppose Bach preferred what you think he preferred.

2. Suppose that the difference between the two systems is inaudible.

If both of these things are true then it is still not possible to tell which
system Bach used. Why, because it wasn't possible for Bach to accurately
perceive which system he was using anyway. He might have liked one set of
numbers over another surely, but who cares, he more than likely couldn't
accurately perceive which one he was using ... hence making it an unimportant
question.

Lets take this from another perspective. If for example you said instead,
"What set of numbers did Bach prefer, even though he could not actually
perceive the differences between the two sets of numbers?" Then I would say
that that is a good question. It recognizes that it is unlikely to be known
what he actually used and also recognizes that the two systems are
inperceivably close and should be considered as such. Perhaps this is a
clarification of the word preferred. your question seems to ask what did he
prefer in theory and in practice, or what numbers did he prefer and what did
he actually do.

Where as my question recognizes that all of this is in theory and NOT in
practice. In practice it is very possibly impossible to tell these things
apart by ear (since we are discussing Bach not computers of the present day).
It asks what numbers he would prefer and says, these are just numbers and
have nothing to do with what he actually did.

Cheers,
Andy

>
> If we find numbers of the set W III, then W III was his preferred tuning.
>
> If, on the contrary, we discover numbers of the set belonging to
> "wohltemperirt"
> in his music,
> then we would have to conclude, as concerns J. S. Bach, "wohltemperirt"
> must
> have been preferable, better, for him. And preferable, because he deems it
> more
> appropriate, or BETTER.
>
> But such aspects are not music-theory, but belong to the realm of
> musical practice, analyzing the compositional process.
>
> Kind regardss,
>

🔗Paul Erlich <paul@stretch-music.com>

🔗Afmmjr@aol.com

🔗Paul Erlich <paul@stretch-music.com>

🔗Afmmjr@aol.com

In a message dated 7/5/01 7:49:01 PM Eastern Daylight Time,
paul@stretch-music.com writes:

> --- In tuning@y..., Afmmjr@a... wrote:
> > Since there is ONLY a 1.2 cent difference
> > between your superior suggestion and Werckmeister's only chromatic
> > temperament, it seems based more on numerology than sound.
>
> Again, this is not the case. The fifths F-C, A-E, and E-B are just in
> Werckmeister III but 4-5 cents flat in Kellner's suggestion.
> Meanwhile, B-F# is just in Kellner's suggestion but 5-6 cents flat in
> W III.
> >

Thanks for the correction, Paul.

> > In preparing a September 29th performance of Brandenburg, I find
> great
> > difficulty in adopting your suggestion as Bach's preference.
> Playing
> > recorders in Werckmeister III makes for a very sharp distinction
> between
> > intervals. It uses only the pedestrian 1/4 comma flat fifth and
> the pure
> > perfect fifth. Anything dividing into fifths is monochord
> activity, and Bach
> > did not tune this way.
>
> I don't understand what you mean by "dividing into fifths". You mean
> the 1/5 comma, as opposed to 1/4 comma, temperament of the non-just
> fifths? In what way is one of these "monochord activity" and the
> other not?
>
Breaking the comma into half, and then into half again is easier to navigate
that dividing into whole fifths. Bach could tune in 15 minutes, and without
a monochord.

> Note: I'm not agreeing with Kellner's suggestion, only trying to keep
> the discussion logical and concrete.
>
>
>

Only an exact tuning for a keyboard could hold such precision. The other
instruments that play have to be organized differently. Werckmeister III
sits beautifully and add much (IMO) to the melodies that are featured. Each
of the quirks in each of its keys reflect keys that have been used already in
Western music history.

Have I missed the explanation for the belief that Bach preferred 7 and 5?
This seems to be the most important point.

Johnny Reinhard (practicing recorder feverishly)

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., Afmmjr@a... wrote:
> In a message dated 7/5/01 7:49:01 PM Eastern Daylight Time,
> paul@s... writes:
>
>
> > --- In tuning@y..., Afmmjr@a... wrote:
> > > Since there is ONLY a 1.2 cent difference
> > > between your superior suggestion and Werckmeister's only
chromatic
> > > temperament, it seems based more on numerology than sound.
> >
> > Again, this is not the case. The fifths F-C, A-E, and E-B are just
in
> > Werckmeister III but 4-5 cents flat in Kellner's suggestion.
> > Meanwhile, B-F# is just in Kellner's suggestion but 5-6 cents flat
in
> > W III.
> > >
>
>
> Thanks for the correction, Paul.
>
> > > In preparing a September 29th performance of Brandenburg, I find
> > great
> > > difficulty in adopting your suggestion as Bach's preference.
> > Playing
> > > recorders in Werckmeister III makes for a very sharp distinction
> > between
> > > intervals. It uses only the pedestrian 1/4 comma flat fifth and
> > the pure
> > > perfect fifth. Anything dividing into fifths is monochord
> > activity, and Bach
> > > did not tune this way.
> >
> > I don't understand what you mean by "dividing into fifths". You
mean
> > the 1/5 comma, as opposed to 1/4 comma, temperament of the
non-just
> > fifths? In what way is one of these "monochord activity" and the
> > other not?
> >
> Breaking the comma into half, and then into half again is easier to
navigate
> that dividing into whole fifths.

Please explain how, specifically in the process of tuning Werckmeister
III, one encounters a full comma, and then what it means, in practical
terms, to break it in half, and in half again. For tuning 1/4-comma
meantone, this might apply, but it seems to me that in Werckmeister,
there isn't even a single just major third, so how does one arrange
this fancy footwork? Does one first tune E to C, tune the fifths
in-between as in 1/4-comma meantone, and then _retune_ E to be just
with A?

🔗monz <joemonz@yahoo.com>

🔗ha.kellner@t-online.de

In fact, in the B-major triad B-d#-f# the crucial tempering is effected; please
just look into the detailed bearings- method I describe in my site:

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

Kind regards
#
Herbert-Anton

Kind regards,

Herbert-Anton
>
> Thanks for the correction, Paul.
>
> > > In preparing a September 29th performance of Brandenburg, I find
> > great
> > > difficulty in adopting your suggestion as Bach's preference.
> > Playing
> > > recorders in Werckmeister III makes for a very sharp distinction
> > between
> > > intervals. It uses only the pedestrian 1/4 comma flat fifth and
> > the pure
> > > perfect fifth. Anything dividing into fifths is monochord
> > activity, and Bach
> > > did not tune this way.
> >
> > I don't understand what you mean by "dividing into fifths". You mean
> > the 1/5 comma, as opposed to 1/4 comma, temperament of the non-just
> > fifths? In what way is one of these "monochord activity" and the
> > other not?
> >
> Breaking the comma into half, and then into half again is easier to navigate
> that dividing into whole fifths. Bach could tune in 15 minutes, and without
> a monochord.
>
>
>
> > Note: I'm not agreeing with Kellner's suggestion, only trying to keep
> > the discussion logical and concrete.
> >
> >
> >
>
> Only an exact tuning for a keyboard could hold such precision. The other
> instruments that play have to be organized differently. Werckmeister III
> sits beautifully and add much (IMO) to the melodies that are featured. Each
> of the quirks in each of its keys reflect keys that have been used already in
> Western music history.
>
> Have I missed the explanation for the belief that Bach preferred 7 and 5?
> This seems to be the most important point.
>
> Johnny Reinhard (practicing recorder feverishly)
>

🔗Afmmjr@aol.com

🔗JoJoBuBu@aol.com

🔗ha.kellner@t-online.de

Paul Erlich schrieb:
> --- In tuning@y..., Afmmjr@a... wrote:
> > Since there is ONLY a 1.2 cent difference
> > between your superior suggestion and Werckmeister's only chromatic
> > temperament, it seems based more on numerology than sound.
>
> Again, this is not the case. The fifths F-C, A-E, and E-B are just in
> Werckmeister III but 4-5 cents flat in Kellner's suggestion.
> Meanwhile, B-F# is just in Kellner's suggestion but 5-6 cents flat in
> W III.
> >
In "woltemperirt", the fifth is flat by P/5, P being the pythagorean comma of
23.5 cent. Thus, 4.7 cent.

> > In preparing a September 29th performance of Brandenburg, I find
> great
> > difficulty in adopting your suggestion as Bach's preference.
> Playing
> > recorders in Werckmeister III makes for a very sharp distinction
> between
> > intervals. It uses only the pedestrian 1/4 comma flat fifth and
> the pure
> > perfect fifth. Anything dividing into fifths is monochord
> activity, and Bach
> > did not tune this way.
>
The monochod Werckmeister mentions and quantifies its dimension is not meant
to be a tool for tempering harpsichords, but rather, just a didactic gadget to
make and illustrate his points for his readers. Any harpsichord tuned by ear it
"tuned in itself" and not viamonochrds.

> I don't understand what you mean by "dividing into fifths". You mean
> the 1/5 comma, as opposed to 1/4 comma, temperament of the non-just
> fifths? In what way is one of these "monochord activity" and the
> other not?
>
> Note: I'm not agreeing with Kellner's suggestion, only trying to keep
> the discussion logical and concrete.
>
>
> You do not need web access to participate. You may subscribe through
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🔗ha.kellner@t-online.de

Dear Johnny Reinhard,

The solution for Werckmeister's preferred system, out of his Musicalische
Temperatur, is published in:
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.

I might note at this point already:
The pure or perfect intervals are the "superparticular ratios" (N+1)/N, of the
integers, 1,2,3, etc. Thus, Octave 2/1, fifth 3/2, fourth 4/3, major third
5/4, etc., etc.

Werckmeister mentions a temperament of 1001/1000, or so; it does not matter. But
his op. cit indicates clearly the number 369. Temper by its ratio
superparticularis, i. e. by 370/369 = B, the perfect fifth, then you get the
"wohltemperirt"/Bach fifth!!!

And 1.5^8*(1.5/B)^4 approximates 128 to sufficiently high degree of accuracy.
The result/128 = 1.0027.

For studying baroque musical mathematics, it is indispensable to go into
baroque music theory, nomenclature and notions, and into baroque "elementary
mathematics".

Kind regards,

Herbert-Anton K.

Afmmjr@aol.com schrieb:
> In a message dated 7/5/01 4:25:32 PM Eastern Daylight Time, JoJoBuBu@aol.com
> writes:
>
>
> > This suggests
> > to me that decidng exactly what Bach preferred is not important.
> >
> >
>
> Herr Kelner, please, for the American audience especially, explain how it is
> that Bach chose 7 + 5 rather than 8 + 4. Arcane interpretation of a
> pre-Masonic-like burying of information makes no sense to me for
> Werckmeister.
>
> Werckmeister's book had been available a full 10 years before it was
> published in Quedlinberg in 1691. Since there is ONLY a 1.2 cent difference
> between your superior suggestion and Werckmeister's only chromatic
> temperament, it seems based more on numerology than sound.
>
> In preparing a September 29th performance of Brandenburg, I find great
> difficulty in adopting your suggestion as Bach's preference. Playing
> recorders in Werckmeister III makes for a very sharp distinction between
> intervals. It uses only the pedestrian 1/4 comma flat fifth and the pure
> perfect fifth. Anything dividing into fifths is monochord activity, and Bach
> did not tune this way.
>
> Some time soon I hope to check out the Gematria in my Rasch edition of
> Musicalische Temperatur and read with great interest what inspired you to
> interpret Bach in terms of 7 and 5. Werckmeister stood by his Musicalische
> Temperatur throughout his entire life. There was no recanting. But you say
> there is a hint buried within?
>
> Best, Johnny Reinhard
>

🔗ha.kellner@t-online.de

Paul Erlich schrieb:
> --- In tuning@y..., Afmmjr@a... wrote:
> > In a message dated 7/5/01 7:49:01 PM Eastern Daylight Time,
> > paul@s... writes:
> >
> >
> > > --- In tuning@y..., Afmmjr@a... wrote:
> > > > Since there is ONLY a 1.2 cent difference
> > > > between your superior suggestion and Werckmeister's only
> chromatic
> > > > temperament, it seems based more on numerology than sound.
> > >
> > > Again, this is not the case. The fifths F-C, A-E, and E-B are just
> in
> > > Werckmeister III but 4-5 cents flat in Kellner's suggestion.
> > > Meanwhile, B-F# is just in Kellner's suggestion but 5-6 cents flat
> in
> > > W III.
> > > >
> >
> >
> > Thanks for the correction, Paul.
> >
> > > > In preparing a September 29th performance of Brandenburg, I find
> > > great
> > > > difficulty in adopting your suggestion as Bach's preference.
> > > Playing
> > > > recorders in Werckmeister III makes for a very sharp distinction
> > > between
> > > > intervals. It uses only the pedestrian 1/4 comma flat fifth and
> > > the pure
> > > > perfect fifth. Anything dividing into fifths is monochord
> > > activity, and Bach
> > > > did not tune this way.
> > >
> > > I don't understand what you mean by "dividing into fifths". You
> mean
> > > the 1/5 comma, as opposed to 1/4 comma, temperament of the
> non-just
> > > fifths? In what way is one of these "monochord activity" and the
> > > other not?
> > >
> > Breaking the comma into half, and then into half again is easier to
> navigate
> > that dividing into whole fifths.
>
> Please explain how, specifically in the process of tuning Werckmeister
> III, one encounters a full comma, and then what it means, in practical
> terms, to break it in half, and in half again. For tuning 1/4-comma
> meantone, this might apply, but it seems to me that in Werckmeister,
> there isn't even a single just major third, so how does one arrange
> this fancy footwork? Does one first tune E to C, tune the fifths
> in-between as in 1/4-comma meantone, and then _retune_ E to be just
> with A?

The expression of well tempered tunings, mathematically, is done via fractions
of comma, e.g. 1/4 S where S means the syntonic Comma for (classical) or
Praetorian meantone.

For Bach"wohltemperirt", it is the fraction P/5, P being the Pythagorean comma.

The tempering of Werckmeister III was formerly considered to be a tricky
business, just for the reaqson you mention: C-E is NOT a perfect third. However,
I published a simple tempering method in my booklet (quoted in my website:

http://ha.kellner.bei.t-online.de ,

but now probably out of print):

Kellner, H.A.: The Tuning of my Harpsichord. Schriftenreihe Heft 18. Verlag Das
Musikinstrument, E. Bochinsky, Frankfurt/M 1980. ISBN 3-920-112-78-4
0-933224-35-4?

"The Tuning of My Harpsichord" was reviewed by the renowned specialist for
temeprament: Douglas Leedy, American Musical Instrument Society, AMIS, 7 (1981):
121-29. To my surprise, he mentioned that my method proposed had been unkonwn
earlier.

How to proceed, then, if C-E is not pure and we want to divide the Pythagorean
Comma by 5??

Tune from C downward a chain of EIGHT perfect fifths:

C-F-Bb-Eb-Ab-Db-Gb-Cb-Fb.

C-Fb, enharmonically C-E, will be a third smaller than pure, by 1.9537 cent.
Divide, this "third", as in meantone, where the third is pure, into 4 equal
parts (for the procedure of dividing a third into four fifths, you may look into
my webpage).
Unce, this interpolation C-G-D-A-E into C-E is acomplished, these fifths are
reduced by a fourt of one pythagorean comma., P/4.

But as in nominal W III a-e must be perfct, you tune a-e this way, i.e perfect
and this will shift the last tempered fifth up onto E-B.

Voilï¿½

Herbert Anton Kellner

P. S. For someone who needs not tune harpsichoirds regularly, this procedure is,
IMPO entirely irrelevant.

>
>
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🔗ha.kellner@t-online.de

Afmmjr@aol.com schrieb:
> In a message dated 7/5/01 7:49:01 PM Eastern Daylight Time,
> paul@stretch-music.com writes:
>
>
> > --- In tuning@y..., Afmmjr@a... wrote:
> > > Since there is ONLY a 1.2 cent difference
> > > between your superior suggestion and Werckmeister's only chromatic
> > > temperament, it seems based more on numerology than sound.
> >
> > Again, this is not the case. The fifths F-C, A-E, and E-B are just in
> > Werckmeister III but 4-5 cents flat in Kellner's suggestion.
> > Meanwhile, B-F# is just in Kellner's suggestion but 5-6 cents flat in
> > W III.
> > >
>Again, please note B-f# is the tempered fifth Nï¿½ 5, the 4 others filling the
major third C-: c-g-d-a-e.
Herbert Anton
>
> Thanks for the correction, Paul.
>
> > > In preparing a September 29th performance of Brandenburg, I find
> > great
> > > difficulty in adopting your suggestion as Bach's preference.
> > Playing
> > > recorders in Werckmeister III makes for a very sharp distinction
> > between
> > > intervals. It uses only the pedestrian 1/4 comma flat fifth and
> > the pure
> > > perfect fifth. Anything dividing into fifths is monochord
> > activity, and Bach
> > > did not tune this way.
> >
> > I don't understand what you mean by "dividing into fifths". You mean
> > the 1/5 comma, as opposed to 1/4 comma, temperament of the non-just
> > fifths? In what way is one of these "monochord activity" and the
> > other not?
> >
> Breaking the comma into half, and then into half again is easier to navigate
> that dividing into whole fifths. Bach could tune in 15 minutes, and without
> a monochord.

This he did in about 15 minutes via the B-major triad - see my website for
details. This method renders, turns out first of all, the fifth B-f# tempered
smaller by P/5, whilst P is the pythagorean comma. See:

Kellner, H.A.: Das ungleichstufige, wohltemperierte Tonsystem. In
"Bach-stunden", Festschrift fï¿½r Helmut Walcha, Hg. W. Dehnhard und G. Ritter.
Evang. Presseverband in Hessen und Nassau, Frankfurt/Main 1978. Seite 75-91

Bach's practice of the B-major method is borne aout by close inspection and
analysis of the B-major pieces of the WTC I; how could it be otherwise?
See, e.g.:
Kellner, H.A.: Was Bach a Mathematician? English Harpsichord Magazine and Early
Keyboard Instrument Review. Editor Edgar Hunt. Vol. 2, No. 2, April 1978, page
32-36. Publication of the lecture delivered August 1977 at the Bruges 5th
International Harpsichord Week, 14th International Fortnight of Music
>
>
>
> > Note: I'm not agreeing with Kellner's suggestion, only trying to keep
> > the discussion logical and concrete.
> >
> >
> >
>
> Only an exact tuning for a keyboard could hold such precision. The other
> instruments that play have to be organized differently. Werckmeister III
> sits beautifully and add much (IMO) to the melodies that are featured. Each
> of the quirks in each of its keys reflect keys that have been used already in
> Western music history.
>
> Have I missed the explanation for the belief that Bach preferred 7 and 5?
> This seems to be the most important point.

Although it is an extreme proposal or suggestion, studying all my publications
quoted in the website for the benefit of thoise interested, will show that there
is amusing and rational mathematics, that can be understood. Otherwise, one may
prefer to speak about "belief".
Kind regards,
Herbert Anton.

>
> Johnny Reinhard (practicing recorder feverishly)
>
As a recorder has no strings, the pitches of this instrument are flexible can
fit indiscriminately to "Bach/wohltemperirt", as well as to Werckmeister III.

HAKellner

🔗ha.kellner@t-online.de

monz schrieb:
>
> > From: <JoJoBuBu@aol.com>
> > To: <tuning@yahoogroups.com>
> > Sent: Thursday, July 05, 2001 1:23 PM
> > Subject: Re: [tuning] Re: Inferior tuning system for J. S. Bach??
> >
> >
> > In a message dated 7/5/2001 3:03:52 PM Eastern Daylight Time,
> > ha.kellner@t-online.de writes:
> >
> > >
> > > 1) Do you think Bach himself would have preferred W III or
> "wohltemperirt"?
> > > if one or the other, for what possible reasons??
> >
> >
> > I dont know. If the numbers are as close as yours seem to indicate between
> > the two systems, then I dont think it matters what he would prefer.

The question is, are there, perhaps, other characteristic numbers, not
necessarily related to intervals/(hearing-perception) that specify and allow
more clearly the discerning the two otherwise practically identical systems.

My terminus or notion of INFeRIOR tuning system was, after all, a practical,
thought- and discussion stimulating provokation - wasn't it?
>
>
> Ahh... but musicologists have found that Bach and Schoenberg (among
> others) were, like many of us microtonalists, numerologists, or at
> least they flirted with numerology. So Bach might have numerological
> reasons

***Dear monz, Sensational:
***Nothing is as CORRECT as your statement here!! I should only like to add,
***that there were, furthermore, THEOLOGICAL reasons for JSB - this great
***Lutheran.

for choosing a certain tuning over another *even if the
> differences were not audible*. I certainly wouldn't rule that out
> as a factor in how he made his tuning choices. I'm quite certain
> that Bach thought rather deeply about the subject of tuning,
> but would prefer to read Herr Kellner's articles before offering
> any further opinions.

It is a desperate situation for me, having written through 25 years by now so
much: what should I - for efficiency - and for saving your time, recommend to
you?
My article in The Tracker would, perhaps, be my first choice, also, this should
be easily avilable in libraries and muscal circles in the US.

Kellner, H.A.: J. S. Bach's Well-tempered Unequal System for Organs. THE
TRACKER, Journal of the Organ Historical Society Vol. 40/3, 1996, page 21-27

Also available, the first, basic paper, I rewrote in English::

Kellner, H.A.: A Mathematical Approach Reconstituting J.S. Bach's
Keyboard-Temperament. BACH, The Quarterly Journal of the Riemenschneider Bach
Institute, Berea, Ohio. Editor Elinore Barber. Vol. 10/4, October 1979, page 2-8
and 22.
>

Yes, from 1979 (!!), but already reprinted 1999 in my lifetime to refresh
memories or encourage novices:

in: BACH, The Journal of the Riemenschneider Bach Institute, Berea, Ohio. Editor
Melvin Unger. Vol. 30/1, Spring - Summer 1999

> -monz
> http://www.monz.org
> "All roads lead to n^0" and in the extreme, to 0^0
>
>
>
>
>
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🔗monz <joemonz@yahoo.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗ha.kellner@t-online.de

Dear Andy, dear Monz,

A recent case of serious numerology of a composer was Scriabin. In fact, he
devised structures, (bar-wise) for his compositions under and from
numerological principles. This led him (musically) to the sore problem that in
some cases he could not brige easily the gaps (empty bars) satisfactorily by
music to be apropriately inserted. He insisted, however, on his preconceived
structures. Due to these gaps produced here and there, termination of his
compositional work was sometimes delayed, by weeks, perhaps by months.

JSB, as I observed, also laid out his pieces bar-wise. For the WTC I, for
exemple, in terms of the length of pieces, praeludes and fugues, 2088 bars
without the repetion AABB in the last piece, exists the magic rectangle
discovered by the Durchman Henk Dieben; ref:

Kellner, H.A.: Le tempï¿½rament inï¿½gal de Werckmeister/Bach et l'alphabet
numï¿½rique de Henk Dieben. Revue de Musicologie Vol. 80/2, 1994, 283-298

(For Skriabin, Manfred Kelkel, musicologist scholar and composer, is a
distinguished authority)

Rolf Mï¿½ser, musician and interested and skilled in chess (!!), discocvered
further extraordinary magic rectangles and squares in WTC I and WTC II:

Mï¿½ser, Rolf
Bach und die drei Temporï¿½tsel
Das wohltemperirte Clavier gibt Bachs Tempoverschlï¿½sselung
und weitereGeheimnisse preis.
Basler Studien zur Musik in Theorie und Praxis, Bd. 2
Musik-Akademie der Stadt Basel
Peter Lang AG, Bern 2000 ISBN 3-906764-77-X 497 p.,

Kind regards,
Herbert-Anton

JoJoBuBu@aol.com schrieb:
> In a message dated 7/6/2001 2:52:48 AM Eastern Daylight Time,
> joemonz@yahoo.com writes:
>
>
> > Ahh... but musicologists have found that Bach and Schoenberg (among
> > others) were, like many of us microtonalists, numerologists, or at
> > least they flirted with numerology. So Bach might have numerological
> > reasons for choosing a certain tuning over another *even if the
> > differences were not audible*. I certainly wouldn't rule that out
> > as a factor in how he made his tuning choices. I'm quite certain
> > that Bach thought rather deeply about the subject of tuning,
> > but would prefer to read Herr Kellner's articles before offering
> > any further opinions.
> >
> >
> > -monz
> > http://www.monz.org
> >
>
> I didn't. In fact I said that specifically below the spot where you quoted
> from.
>
> Andy
>