Yahoo Tuning Groups Ultimate Backup tuning Bach structured WTC by number-set of "wohltemperirt"

Dear members,

To the mathematically well-defined system "wohltemperirt" (See my site below)

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

belongs a set of numbers that Bach employed in order to structure his music
under various aspects and by various methods.

This is a philosophically and even spiritually gratifying principle to proceed:
the numbers belonging to harmony are the same that structure the
architecture. This brings about and assures a grandiose unification for the
result of the compositional effort.

Which typical numbers do belong to "wohltemperirt"?

The first interval that can be tempered is the fifth - evidently not the octave
on the harpsichord. Therefore, the tempering of the fifth, i.e. those that are
tempered - is a characteristic quantity. The 21st century would express this
parameter as 4.7 cent. By the way, the cent is a measuring unit extremely
appropriate for and invented because of Equal Temperament; tailored to E.T.

But today, as well as in Werckmeister's and Bach's times, the pure intervals
were expressed as rationes superparticulares, the superparticular ratios,
(N+1)/N: octave 2/1, Fifth 3/2, etc., 4/3, 5/4, 6/5, etc. ...
Werckmeister also mentioned some temperings expressed as superparticular ratios.

Thus, it turns out that the fifths of "Bach/wohltemperirt" are tempered by the
superparticular ratio of 369, being 370/369. This fraction follows as the first
approximant via continued fractions to the fifth, amounting to
1,495953506243...
Provided the fifth has this value, its corresponding third (from these tempered
fifths c-g-d-a-e) and the fifth itself in the central C-major triad beat at
the UNISON.

The Four Duets measure 369 bars, etc, see:

Kellner, H.A.: How Bach quantified his well-tempered tuning within the Four
Duets. English Harpsichord Magazine, Vol. 4, No. 2, 1986(87), page 21-27

Idem: Barocke Akustik und Numerologie in den Vier Duetten: Bachs "Musicalische
Temperatur". In "Bericht ï¿½ber den Internationalen Musikwissenschaftlichen
Kongreï¿½ Stuttgart 1985", Hg. Dietrich Berke und Dorothea Hanemann, Kassel 1987,
Seite 439-449

*******************************************************************
It is to be stressed that the specific single characteristic number for
"wohltemperirt" is 369.
*******************************************************************

Other numbers pertaining to this system, the central C-major triad of which has
its third C-E beating at the same rate as the fifth C-G, derive from the
idea of the trias harmonica perfecta and the concept of the perfection of the
baroque UNITAS =1, TRINITAS = 3. (Rolf Dammann, Der Musikbegriff im Deutschen
Barock, Laaber 1994).

Thus, 3 itself, its square 3*3=9, its cube, 3*3*3=27, and the double and triple
juxtapositions 13, 31, 131, 313 are numerological expressions pertaining to
"wohltemperirt".

Duetto II, Clavierï¿½bung III, 149 bars, is structured 37+75+37 bars.
37 ist structured 9+19+9 bars.
75 is structured 31+13+31 bars - a tri-unitary making up of the numbers of
fifths in the system "wohltemperirt":

The respective numbers of fifths, perfect and tempered are in Bach's system, as
I call it, "Werckmeister/Bach/wohltemperirt", are 7+5. Therefore, the numbers
5, 7, and their dual and triple juxtapositions 57, 75, 577 characterize -
numerologically - the system "wohltemperirt".

The respective numbers of fifths, perfect and tempered, in Werckmeister III
are, in contradistinction, 8+4. Werckmeister divides the Pythagorean Comma by 4.

But it is essential that the single parameter of tempering the "nominal"
Werckmeister III fifth is 295. This yield the value of this "Werckmeister-fifth"
as 1,5/(295/294).

It is to be stressed, that the SINGLE characteristic and specific parameters are

for "Bach/wohltemperirt" 369 and for
Werckmeister III "nominal" 294.

*******************************************************************
These numbers are vastly different; the Four Duets measure 369 and not 295 bars.
*******************************************************************

It is of no relevance whatsoever, if the difference between W III and
wohltemperirt cannot be overeard. What matters, is that Bach UTILIZED
"wohltemperirt" and NOT Werckmeister III. Isn't 369 sufficiently different
from 295???

Does one need to be a great mathematician to grasp that 369 is different
from 295???

The specific B-major method achieves tempering the bearings in
the minimal number of NO MORE than 19 steps, (at the same time the number of
closure of the circle of fifths!): 12 fifths and 7 octaves in the opposite
direction assure closure of the circle; 19 intervals altogether).

The B-major tonality in WTC starts at its bar 1913, its prelude ends at
bar 1931. This B-major prelude has 19 bars.

The pieces at the onset of WTC I in C-major and minor measure 131 bars.
The pieces at the onset of WTC I in C#-major and minor measure 313 bars.

Given the B-major method for tempering the fifth B-f# smaller by 1/5 of P, the
Pythagorean Comma, it took a professional mathematician of the 20th century
several weeks to find it.

But looking into the B-major pieces proves that Bach must have been
familiar with this method: he was a learned musician, like Werckmeister.

It was Werckmeister, though, who has invented the system "Werckmeister / Bach /
wohltemperirt":

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 could not shew up to now that THE INVENTOR Werckmeister did
know as well the B-major method. The mathematical background and some
details may be found in:

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

Kind regards to all,

Herbert Anton Kellner

🔗monz <joemonz@yahoo.com>

> From: <ha.kellner@t-online.de>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, July 08, 2001 6:03 AM
> Subject: [tuning] Bach structured WTC by number-set of "wohltemperirt"
>
>

Dear Herr Kellner,

While it seems that you and I are in agreement that there
is a strong possibility that Bach chose his specific tuning
parameters based on numerological (and theological)
considerations... with all due respect, there are a few
contrary opionions I would like to express concerning
your mathematics.

> Thus, it turns out that the fifths of "Bach/wohltemperirt"
> are tempered by the superparticular ratio of 369, being
> 370/369. This fraction follows as the first approximant
> via continued fractions to the fifth, amounting to
> 1,495953506243...

Of course you are stating here as fact that 'the fifths
of "Bach/wohltemperirt" are tempered by the superparticular
ratio of 369, being 370/369', when in reality what you
mean is that 370/369 is a rational approximation of the
actual tempering of each of these particular "5ths" by
1/5 of a Pythagorean comma.

I will follow you in referring to the Pythagorean comma as P.
It has the prime-factor notation (2^-19)*(3^12). The
1/5 equal division of it is ((2^-19)*(3^12))^(1/5).

"P/5" = 1.002713883 = ~4.692002077 cents

Some rational approximations to it:

error from P/5

369/368 = 1.002717391 = ~4.698060004 cents +0.006057927 cent
370/369 = 1.002710027 = ~4.685345347 cents -0.00665673 cent
739/737 = 1.002713704 = ~4.691694061 cents -0.000308016 cent

Thus it can be seen that 739/737 is a much closer rational
approximation to P/5, but it is not superparticular.

But 369/368 *is* superparticular and is a slightly better
rational approximation to P/5 than 370/369.

> But it is essential that the single parameter of
> tempering the "nominal" Werckmeister III fifth is 295.
> This yield the value of this "Werckmeister-fifth"
> as 1,5/(295/294).

error from P/4

295/294 = 1.003401361 = ~5.878559295 cents +0.013556699 cent
296/295 = 1.003389831 = ~5.858665656 cents -0.00633694 cent

As above, the superparticular ratio 296/295 is a better
rational approximation to P/4 than 295/294.

> It is to be stressed, that the SINGLE characteristic
> and specific parameters are
>
> for "Bach/wohltemperirt" 369 and for
> Werckmeister III "nominal" 294.

I think you made a typo here and meant to put "295"
for Werckmeister III.

Andy Stefik has argued that the difference between
Bach/wohltemperirt and Werckmeister III is inaudible
and thus insignificant, and you responded (and I agreed)
that Bach would have made his numerical choices based
on numerological and not acoustical considerations.

I would suggest that the fact that these differences
are inaudible may indeed have something to do with the
particular choices of numbers made by Bach, as there
are better choices available which, according to your
analysis, he chose not to use.

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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🔗ha.kellner@t-online.de

monz schrieb:
>
Dearn Monz, Thanks very much for your stimulating, exciting and corrective
observations!

> > From: <ha.kellner@t-online.de>
> > To: <tuning@yahoogroups.com>
> > Sent: Sunday, July 08, 2001 6:03 AM
> > Subject: [tuning] Bach structured WTC by number-set of "wohltemperirt"
> >
> >
>
> Dear Herr Kellner,
>
>
> While it seems that you and I are in agreement that there
> is a strong possibility that Bach chose his specific tuning
> parameters based on numerological (and theological)
> considerations... with all due respect, there are a few
> contrary opionions I would like to express concerning
> your mathematics.
>
>
> > Thus, it turns out that the fifths of "Bach/wohltemperirt"
> > are tempered by the superparticular ratio of 369, being
> > 370/369. This fraction follows as the first approximant
> > via continued fractions to the fifth, amounting to
> > 1,495953506243...
>
To avoid misunderstandings: the exact fifth
"w-t" is 1,495953506243...
and I'd better formulated here:
Thus, it turns out that the fifths of "Bach/wohltemperirt"
are tempered
TO A FIRST APPROXIMATION VIA CONTINUED FRACTIONS
by the superparticular ratio of 369, being
370/369. This fraction follows as the first approximant
via continued fractions to the
EXACT "WOHLTEMPERIRTE" fifth, amounting to
1,495953506243...

>
> Of course you are stating here as fact that 'the fifths
> of "Bach/wohltemperirt" are tempered by the superparticular
> ratio of 369, being 370/369', when in reality what you
> mean is that 370/369 is a rational approximation of the
> actual tempering of each of these particular "5ths" by
> 1/5 of a Pythagorean comma.
***********
***********

NO, I don't mean that. I am speaking about the approximation to the
"welltempered" Bach-fifth which is the one beating at the same rate as the
third C-E made up by 4 such fifths (C-G-D-A-E, upto 2 octaves). The fifth "w-t"
is to all PRACTICAL intents and purposes equal to the fifth reduced by 1/5 P.
But this equality does not NOT hold mathematically exactly.

I often hear "reproaches" that upon listening, W III and "wohltemperirt" cannot
be discerned. Here I my self am inclined to say, in PRACTICE, these two fifths
should not be discerned "w-t" and the fifth reduced by 1/5 P.
>
>
> I will follow you in referring to the Pythagorean comma as P.
> It has the prime-factor notation (2^-19)*(3^12). The
> 1/5 equal division of it is ((2^-19)*(3^12))^(1/5).
>
As I said above, pursuing the theory in EXACT terms, the Pythagorean Comma P is
entirely irrelevant in this context!

> "P/5" = 1.002713883 = ~4.692002077 cents
>
>
> Some rational approximations to it:
>
> error from P/5
>
> 369/368 = 1.002717391 = ~4.698060004 cents +0.006057927 cent
> 370/369 = 1.002710027 = ~4.685345347 cents -0.00665673 cent
> 739/737 = 1.002713704 = ~4.691694061 cents -0.000308016 cent
>
>
> Thus it can be seen that 739/737 is a much closer rational
> approximation to P/5, but it is not superparticular.
>
I have values in agreement, like this:
1 / 1
369 / 368
739 / 737
7020 / 7001
7759 / 7738
38056 / 37953
45815 / 45691
83871 / 83644
297428 / 296623
4842719 / 4829612

> But 369/368 *is* superparticular and is a slightly better
> rational approximation to P/5 than 370/369.
>P/5 and its approximations are irrelevant. The relevant approximations to the
Bach fifth, 1,495953506243. ...

are:

1 / 1
370 / 369
371 / 370
1112 / 1109
4819 / 4806
25207 / 25139
30026 / 29945
55233 / 55084
140492 / 140113
1600645 / 1596327
3341782 / 3332767
4942427 / 4929094

The FIRST superparticular approximation stems from 369.

The second approximation, from 370 and reads 371/370.
But again, 369 = 9*41 (!)
If the entire WTC I+II is counted with 5750 bars, then one may look into the
bars from 368 to 371 of WTC II, in "some sense", in fact, an approximation
like 741/739 turns up here): ref.:

Kellner, H.A.: "Das wohltemperirte Clavier" - Implications de l'accord inï¿½gal
pour l'ï¿½uvre et son autographe. Revue de Musicologie Vol. 71, 1985, page 143-157
>
>
> > But it is essential that the single parameter of
> > tempering the "nominal" Werckmeister III fifth is 295.
> > This yields the value of this "Werckmeister-fifth"
> > as 1,5/(295/294).
>MY MISPRINT; please read here :
as 1,5/(296/295) Thanks!

The superparticular ratio of the LOWER value, 295, is to be taken

> error from P/4
> """"""
(means: deviation from P/4)

> 295/294 = 1.003401361 = ~5.878559295 cents +0.013556699 cent
> 296/295 = 1.003389831 = ~5.858665656 cents -0.00633694 cent
>
> As above, the superparticular ratio 296/295 is a betterï¿½ï¿½ï¿½

(I do not understand, how you do not see agreement of our results here).

> ï¿½ï¿½ï¿½ï¿½rational approximation to P/4 than 295/294.
>
For Werckmeister III "nominal" I find the approximations

1 / 1
295 / 294
296 / 295
887 / 884
6505 / 6483
7392 / 7367
13897 / 13850
35186 / 35067
49083 / 48917
8232047 / 8204206

>
> > It is to be stressed, that the SINGLE characteristic
> > and specific parameters are
> >
> > for "Bach/wohltemperirt" 369 and for
> > Werckmeister III "nominal" 294.
>

(One notes that 369=41*9 - one of the innombrable co-incidences in this domain)

>
> I think you made a typo here and meant to put "295"
> for Werckmeister III. YES; INDEED; DEAR MONZ!!
>
>
> Andy Stefik has argued that the difference between
> Bach/wohltemperirt and Werckmeister III is inaudible
> and thus insignificant, and you responded (and I agreed)
> that Bach would have made his numerical choices based
> on numerological and not acoustical considerations.
>
> I would suggest that the fact that these differences
> are inaudible may indeed have something to do with the
> particular choices of numbers made by Bach, as there
> are better choices available which, according to your
> analysis, he chose not to use.
>
>
Dear Monz, you can certainly easily imagine - notwithstanding there are
excellent recordings of WTC in E.T. - how I would judge a prori recordings of
WTC in E. T. and in Werckmeister III. The step from E.T. as regards hearing is
so gigantic, that the difference - ï¿½ la limite - between "wohltemperirt" and W
III does not really matter.

What interests me besides, however, is the recondite structuring of (virtually
all!!!?) Bach's music via the "number-set".

On the whole, it is pretty useless to let W III "play" versus "wohltemperirt",
because Werckmsietr was any way the inventor of "wohltemperirt", as his
"Musicalische Temperatur", 1691 proves.

But it will take, I'm afraid, as long as it took for the E.T, hypothesis, that
one speaks and applies W III. Even worse, one may pretend that W III was Bach's
system.

For the simple reason, due to the very human nature, it is IMPOSSIBLE that a
contemporary person enters musicology, interdisciplinary, to resolve that
enigma that another mathematician, but musician and genius created in 1722.

HAK

>
> -monz
> http://www.monz.org
> "All roads lead to n^0"
>
>
>
>
>
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🔗jpehrson@rcn.com

🔗ha.kellner@t-online.de

Dear Mr. Joseph Pehrson,

Thank you for your kind message.

My research-work into Bach's tuning ran as follows:

1965: I learned harpsichord tuning, mean-tone, Kirnberger, etc. ...
1975, December: I established the system "wohltemperirt" 10 yrs later.
1976, Spring: I found the method to deduct 1/5 of the Pythagorean
Comma from the fifth B-f#

Thereafter, I looked into the B-major pieces ( of the "tempering tonality")
and noted that Bach had included many allusions to his tuning therein.

I also discovered thanks to the facsimile of Werckmeister's "Musicalische
Temperatur", that at its date, year 1691, Werckmeister must have already
been familiar with "Bach's" system "wohltemperirt".

Thereafter I noted that Bach has structured many compositions taking
recourse to numbers belonging to the system "wohltemperirt"; to notions
belonging to the Musikbegriff im Barock (the notion of music in German Baroque
epoch, such as related to the perfection of the baroque UNITAS, the unity. This
is a musico-theological notion important in Werckmeister's treatises).

Continuing such research into Bach's entire oeuvre, might be occupying
future researchers for many years to come; I had to limit myself to few
of the compositions, such as the Wohltemperirtes Clavier, Four Duets from
Clavierï¿½bung III, etc.

The details concerning these statements about my results can be found in my
articles that are listed in my website:

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

Best regards,
Herbert Anton Kellner

jpehrson@rcn.com schrieb:
> --- In tuning@y..., ha.kellner@t... wrote:
>
> /tuning/topicId_26100.html#26100
>
> If all of this is true, it's really fascinating to think that Bach
> structured the length and number of measures of his music according
> to an overriding tuning proportion and principle!
>
> I wonder in how many pieces he used these kinds of "number games..."
>
> It certainly seems, given his attention to detail and interest in
> other musical puzzles that this could be possible!
>
> I have *never* heard of this before!
>
> Thank you so much for the information!
>
> _______ ______ ________
> Joseph Pehrson
>
>
>
> You do not need web access to participate. You may subscribe through
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>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗jpehrson@rcn.com

🔗ha.kellner@t-online.de

Dear Mr. Joseph Pehrson,

My texts to our groups are tiny extracts from my published papers. My most
appropriate text for Anglo-American readers is, in my opinion,

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

This is a general paper, indication the "philosophy" and general approach
attempting and striving to reveal the terribly rational means to compose, of
this miraculous "mathematician-musician". Other papers deal with detailed
analyses of how J S B laid out and structured (mathematically/ numerologically)
his musical works. I take a bold mathematical and congenial approach for this
sort of my work. I have been thinking about composing music from my age of 3 or
4 years, but never done anything myself - onlyo bserved what others do. After
all my maths, inparticular for J.S.B. this discipline is apt for HIM.
I like best to write in English - as perhaps my list of publications indicate I
present in my site

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

Kind regards,

Herbert-Anton K

jpehrson@rcn.com schrieb:
> --- In tuning@y..., ha.kellner@t... wrote:
>
> /tuning/topicId_26100.html#26142
>
> >
> >
> > The details concerning these statements about my results can be
> found in my
> > articles that are listed in my website:
> >
> > http://ha.kellner.bei.t-online.de
> >
> > Best regards,
> > Herbert Anton Kellner
> >
>
> Hello Herbert Kellner!
>
> Thank you so much for your continuing explanation of your discovery
> of Bach's tuning. I can see on your page the subtle distinction
> between that tuning and Werckmeister. Did Bach also use
> Werckmeister, and this particular "wohltemperirt" tuning as well??
>
> In any case, I didn't see on your page any references to the number
> of measures of Bach's pieces as determined by numerical details of
> his tuning. I guess the only place you have posted this material so
> far is the Tuning List. Am I correct in that, or am I missing
> something.
>
> Thank you so very much for this fascinating discussion...
>
> ________ _______ ______
> Joseph Pehrson
>
>
>
> You do not need web access to participate. You may subscribe through
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>