calculation of 369 etc

Dear Johnny,

Thanks for your valuable contribution of arguments.

To your further exciting message I will be pleased to respond later!
I feel its best to focus specifically on

> "However, while I have read all your posted materials carefully, I
> still DO NOT understand how your achieved 369?"
****?******

I have prepared below something for you and adjoined some further
material.

Let's concede to Werckmeister he knew the appoximation P=74/73.
Thus,
74 5*74 370 370*369*368*367*366
___= ____ = ___ = ___________________
73 5*73 365 369*368*367*366*365

Remains 370/369 as an approximation they divided a comma
in Baroque elementary maths, e.g. Sorge.

370 is the ratio superparticularis of 369.
But why not take, e.g. 369/368?

Answer: try it out to close the circle by reducing 5 fifths
in turn by SUP.RAT. of 369, 368, 367, etc.

You will find that the FIRST approximation of how to reduce the fifth
by 1/5 of P is 370/369; SUP.RAT. of 369. (The Four Duets: 369 bars).

In fact, trying out 371/370, yields a still better approximation
of 1/5 P in form of a SUP.RAT. By the way, at that time the first
occurrence in German Maths of continued fractions expansions
(Schwenter) appeared. See:
Kellner, H.A.: Le temp�rament in�gal de Werckmeister/Bach et l'alphabet
num�rique de Henk Dieben. Revue de Musicologie Vol. 80/2, 1994, 283-298,
in particular p.298.

*
1 Werckmeister knows that by tempering a perfect fifth smaller
by 370/369, reduces it by 1/5 Pythagorean comma. Check the
calculation above if you don't believe. Then Werckmeister must -
notionally and mathematically - been even more thoroughly familiar
with the distinction Pythagoeran/Syntonic comma. The accuracy
of 370/369 is astounding. Sorge has published this efficient and
simple method of how to calculate, in one of his treatises.
(pertaining to "Baroque elementary mathematics").

2 The Tables Werckmeister presents about the deviations of the thirds
in the various systems, (as depending on the fifths' temperament),
from pure, are neither in units of Syntonic, nor in Pythagorean-comma
fractions!!
Many writers have misunderstood that essential fact, accepted without
any objection by uncritical readers; probably non-mathematicians.

In "wohltemprirt", as an example and illustration for clarification,
the third C-E is enlarged by about 2.8 cent. (NOT equal to P/5)

The fifths are reduced by about 1/5 P. Werckmeister would have written

(see Facsimile, Musicalische Temperatur, p. 78ff.,)

in his book in this case: the third is enlarged by 1/5 comma!!
Try to verify that by yourself in detail. By COMMA Werckmeister meant
in this context just any MICRO-INTERVAL!!
There exits no neutral comma with Werckmeister, but he utilizes
MICRO-INTERVALS of various sizes, whereever necessary.

*
In one of his later books Werckmeister mentions E.T. and
he associates it with Christian principles: humiliation
(reduced fifths!) and equality (facing God?).
It is known that organ builders, following mean-tone, stuck to mean-tone
and were reluctant to leave it and go for E.T.
Werckmeister would not have missed to recognize the
shortcomings of E.T. for organs.
And he could not refrain from saying somewhat (translated from memory) :
And yet, I hold it that the diatonic keys should remain somewhat
purer than the remote ones. This, IMO, is not really enthusiasm for E.T.!
*
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, 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 �ber den Internationalen Musikwissenschaftlichen
Kongre� 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, 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

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
"nominal" divides the Pythagorean Comma by 4.

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).

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

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

*******************************************************************
Vastly different numbers ; the Four Duets measure 369 and not 295 bars.
*******************************************************************

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 r�impression de la "Musicalische Temperatur"
(1691) de Werckmeister. Revue de Musicologie Vol. 71, 1985, page 184-187.

I could not prove 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 f�r Helmut Walcha, Hg. W. Dehnhard
und G. Ritter. Evang. Presseverband in Hessen und Nassau,
Frankfurt/Main 1978. Seite 75-91

Kind regards,

Herbert Anton