Bach structured WTC by number-set of "wohltemperirt"

[Paul N. Bailey]

This is forwarded from the TUNING list.

The 19 step bearing plan and the cents offset from ET have appeared on this
list
a few times, but could be posted again.

                                Paul Bailey

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Message: 3
   Date: Sun, 8 Jul 2001 15:03:24 +0200
   From: ha.kellner@t-online.de
Subject: 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 uber den Internationalen Musikwissenschaftlichen 
Kongres 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, Clavierubung 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 reimpression 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 fur 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