Bach's B-major tempering-method "wohltemperirt"
herbert anton kellner
Bach's B-major tempering-method
Datum:
Mon, 17 june 2002
Von:
ha.ke...@t-online.de (herbert anton kellner)
Foren:
rec.music.early, alt.music.j-s-bach
Dear members,
This time, something for mathematically inclined ones!!
Kind regards,
Herbert-Anton Kellner.
You may also look into
http://members.tripod.de/Herbert_Anton_Kellne no "R"!
or into
http://members.aol.com/kellnerha/index.html
CONSIDERING THE TUNING METHOD
OF BACH'S SYSTEM "WOHLTEMPERIRT"
via the B-major triad
In the B-major triad, the following proportion
of beats between third and fifth must hold:
(B1-D#2) : (B1-F#2) = 6 : 1
The purpose of this note is to prove mathematically
that assuring this relation upon tempering a harpsichord,
will produce a fifth B1-f#2 reduced by 1/5 of the
pythagorean comma P with respect to the perfect fifth.
The circle of fifths in Bach/"wohltemperirt" is as follows,
the 5 "t" meaning a well-tempered fifth; the remaining
7 fifths are perfect:
Ab Eb Bb F C t G t D t A t E H t F# C# G#
Let us now consider No. 1, 2, 3, 4 and 5, harmonic partials of C2,
one octave below middle C:
1. 2. 3. 4. 5.
C2 MA3rd E2 mi3rd G2 C3 G3 C4 E4
In J. S. Bach's system „wohltemperirt",
The Major 3rd third C-E beats at the same rate as
The Fifth C-G;
In optimal mutual adaptation.
From C2 ascend 4 equal fifths „wohltemperirt",
C-G-D-A-E to E4, two octaves above the E2 of the
basic well-tempered triad C2-E2-G2.
The fifth Qw "wohltemperirt" results numerically as
Qw = 1,495953506243
The perfect fifth, of course, measures
Q = 1,5 or, as fraction, 3/2
If two close frequencies f1 and f2 sound, (they may be
pure sine-waves) then beats of the frequency f2-f1 will result.
What about the beat frequencies of tempered fifths and
thirds on the harpsichord? Where are the "close frequencies"
that produce beats arising from their small difference?
In fact, these beats arise between the harmonic partials
of the fundamental note and the upper note of the interval.
As an example, see above, the fundamental C on the harpsichord
is accompanied by numerous harmonic partials.
Bach's tuning method is based upon the B-major
triad B-D#-F#. The two crucial intervals involved are
the fifth B-f# and
the third B-D#.
The FIFTH B2-F#3 on the keyboard:
What are the first partials belonging to the
constituents of the fifth B2-F#3 ?
1. 2. 3.
B2 B3 F#4
1. 2.
F#3 F#4
Thus, the third partial of B2 coincides with
the second partial of F#3.
The THIRD B2-D#3 on the keyboard:
What are the partials belonging to the
constituents of the THIRD B2-D#3 ?
1. 2. 3. 4. 5.
B2 B3 F#4 B4 D#5
1. 2. 3. 4.
D#3 D#4 G#4 D#5
Thus, the fifth partial of B2 coincides with
the fourth partial of F#3.
The beats of a tempered fifth designated Qx arise from
the nearly coinciding partials 3 and 2, fundamental
and the fifth itself. Let us assume - without restriction of
generality - 1 as the relative frequency of the fifth's
fundamental. Then the relative beat frequency S(Qx) is:
S(Qx) = 2*Qx - 3
The beats of a tempered third designated Tx arise from
the nearly coinciding partials 5 and 4, fundamental and
the third itself. Let us assume - without restriction of
generality - 1 as the relative frequency of the third's
fundamental. Then the relative beat frequency S(Tx) is:
S(Tx) = 4*Tx - 5 .
The enlarged third Tx beats 6-times as fast
as the reduced fifth Qx :
S(Tx) = - 6*S(Qx). Inserting Tx and Qx:
4*Tx - 5 = -6*(2*Qx - 3), or,
4*Tx-5 = -12*Qx +18, or,
4*Tx = -12*Qx +23
Both unknown intervals Tx and Qx will now be calculated.
Consider the relevant extract of the fifth's circle
in Bach's system "wohltemperirt",
B t F# 0 C# 0 G# 0 D#
From B up to D#. The first fifth is tempered, but the
three following ones are perfect; To arrive
from B at D#, one multiplies:
Qx * 1,5 * 1,5 *1,5
In order to transfer this D# down into the correct octave,
namely as D# of the basic tempering octave D#2, one must
descend by two octaves; factor 4:
Tx = Qx * 1,5 * 1,5 *1,5 / 4. Or,
Tx = Qx *3 *3 *3 /2/2/2/2/2.
Tx = Qx*27/32.
But we had above :
4*Tx = -12*Qx + 23 . Thus, eliminating Tx,
4*Qx*27/32 = -12*Qx + 23
Qx*27/8 = -12*Qx + 23
27Qx = -96Qx + 184
123Qx = 184
The final result, in fraction form, reads:
Qx = 184/123.
Further, as a decimal number:
Qx = 1,4959349593495
For comparison, the exact value above of the
fifth "wohltemperirt"was:
Qw = 1,495953506243
The quotient between Qw and Qx amounts to
1,000012398195, corresponding to about
0,02 cent.
To all practical intents and purposes, due to the
very small value of only 0,02 cent, this can be
considered as an extremely close approximation
to the exact fifth "wohltemperirt".
This concludes the mathematical derivation, proving
Bach's tempering-method via the B-major triad.
Dr. Herbert Anton Kellner.
For published literature, see:
Kellner, H.A.: Das ungleichstufige, wohltemperierte Tonsystem.
In "Bach-stunden", Festschrift für Helmut Walcha,
Hg. W. Dehnhard und G. Ritter. Evang. Presseverband
in Hessen und Nassau, Frankfurt/Main 1978. Seite 75-91
Edgar De Blieck
"herbert anton kellner" <ha.ke...@t-online.de> wrote in message
news:3D0DB53C...@t-online.de...
This is very interesting - I wonder if the clever people over at
Rec.Puzzles would appreciate this one. Perhaps you could cross post it, to
see what they make of it!
EDEB.
tlste...@tpgi.com.au
<Debl...@btopenworld.com> wrote:
> "herbert anton kellner" <ha.ke...@t-online.de> wrote in message
> news:3D0DB53C...@t-online.de...
> > Betreff:
> > Bach's B-major tempering-method
> > Datum:
> > Mon, 17 june 2002
> > Von:
> > ha.ke...@t-online.de (herbert anton kellner)
> > Foren:
> > rec.music.early, alt.music.j-s-bach
> >
> >
> >
> >
> > Dear members,
> > This time, something for mathematically inclined ones!!
> >
> > Kind regards,
> >
> > Herbert-Anton Kellner.
> >
> > You may also look into
> > http://members.tripod.de/Herbert_Anton_Kellne no "R"!
> >
> > or into
> >
> > http://members.aol.com/kellnerha/index.html
<snip>
>
> This is very interesting - I wonder if the clever people over at
> Rec.Puzzles would appreciate this one. Perhaps you could cross post it, to
> see what they make of it!
>
> EDEB.
So I guess we can put you down amongst the "not mathematically inclined"
ones, yes?
--
Cheers!
Terry
(remove the numbers if replying direct)
M. Schulter
> Bach's tuning method is based upon the B-major
> triad B-D#-F#. The two crucial intervals involved are
> the fifth B-f# and
> the third B-D#.
Hello, there and thank you for raising here a question which is indeed a
"musical puzzle": what kind of temperament, or type of temperament, Bach
might have favored for his Well-Tempered Klavier (WTK).
While "well-tempered" is generally agreed to mean a circulating
temperament, so that all pieces could be played without retuning, and more
specifically a 12-note circulating temperament (in contrast to the 31-note
systems of Vicentino in 1555 and Colonna in 1618, for example), this
leaves lots of choices.
Personally, I'm not sure if Bach intended one specific temperament,
although your reading of certain clues to suggest seven pure and five
tempered fifths at 1/5 Pythagorean comma narrow is very interesting.
First, I would say that your proposed temperament is indeed an excellent
and period-appropriate one. With 18th-century well-temperaments, as with
16th-century meantones, it's very likely that just about any conceivable
shade was tuned at some time and place. A 1/5 Pythagorean comma scheme
would produce a "key color" somewhat less extreme that Werckmeister III,
for example, and a bit more pronounced than something like Vallotti-Young.
To your mathematical presentation, I might have one general stylistic
question: why would Bach base his favorite tuning on the major third B-D#,
a remote interval which could be taken as a "modified meantone" adjustment
of the meantone diminished fourth B-Eb?
Have I read this point correctly? If the mathematics of B-D# are simply a
consequence of other adjustments of nearer intervals or triads, then my
question would be misplaced: the nearer adjustments would be the main
motive, and the result for B-D# simply a pleasing result of this.
What follows assumes that B-D# is adjusted for its own sake, a reading or
misreading I would be glad to have corrected, apologizing for any
misunderstanding.
My first reaction is that if Bach were basing his tuning on an ideal
mathematical realization of the _trias harmonica_ in a tempered setting,
wouldn't he focus on something like C-E-G (the natural triad in Ionian,
rated as the first and leading mode by Lippius and Bernhard in the earlier
German tradition), or possibly F-A-C, etc.?
It is true that there is an earlier German tradition of focusing on
certain remote intervals: thus Arnold Schlick (1511) describes how to
adjust Ab/G# so that it forms a reasonably acceptable major third Ab-C,
and a marginally acceptable E-G# for use in certain ornamented
cadences. This tuning, by the way, like the apparent Ab-C# meantone scheme
described by Ramos in 1482 (not to be confused with his just monochord
scheme), reflects an outlook in which G# was not regarded as a basic and
vital alteration -- unlike the situation by around 1520-1530, when this
accidental was essential, for example, to produce a major third above the
final of E Phrygian.
However, while Schlick was pragmatically seeking to maximize the range of
available cadences while also accommodating his preference for Ab, I'm
tempted to ask why a mathematically precise arrangement for the _trias
harmonica_ would one of its most remote and "least perfect" instances by
the usual standards of a period well-temperament (in which the nearer keys
have those thirds closest to the ideal ratios in this style of 5:4 and
6:5).
Certainly the _trias harmonica_ is a basic element of German music theory
in this era: Fux, for example, embraces it in _Gradus ad Parnassus_
(1725). The question I have is why B-D#-F# as a criterion for Bach's
tuning -- unless, as I say above, the tuning of B-D# is a mathematical
result of adjustment nearer thirds.
Most appreciatively,
Margo Schulter
msch...@value.net
Edgar De Blieck
> > > Dear members,
> > > This time, something for mathematically inclined ones!!
> > >
> > > Kind regards,
> > >
> > > Herbert-Anton Kellner.
> > >
> > > You may also look into
> > > http://members.tripod.de/Herbert_Anton_Kellne no "R"!
> > >
> > > or into
> > >
> > > http://members.aol.com/kellnerha/index.html
>
> <snip>
>
> >
> > This is very interesting - I wonder if the clever people over at
> > Rec.Puzzles would appreciate this one. Perhaps you could cross post it,
to
> > see what they make of it!
> >
> > EDEB.
>
> So I guess we can put you down amongst the "not mathematically inclined"
> ones, yes?
>
Not at all! - I just think that they would get a blast out of it over there,
and didn't want to cross post it myself, in case this was considered rude.
Personally I think it's fascinating! No sarcasm intended.
EDEB.
Max Schmeder
Dear Bach group,
This is Margo Schulter, one of the gems of the internet and prize of
rec.music.early. She's helped me and countless others on that
newsgroup. Let's see if we can't sweet-talk her into visiting us
sometimes. :) -Max
herbert anton kellner
I realize that I better ought to put my notes concerning the
tempering-method for "Werckmeister/Bach/wohltemperirt" into
context!
The sequence of points to be tackled below will be as follows:
I. Two derivations of the distribution of perfect and tempered
fifths
II. Laying the bearings - descending by perfect fifths from
middle C
III. Repercussions of the tempering method upon Bach's settings
composed in the "tempering tonality" B-major
IV. An incompetent individual and his wrong beat-calculations
web-page
I. Two derivations of the distribution of perfect and tempered fifths
A first publication to "reconstruct" the system "wohltemperirt" of
Werckmeister/Bach started off with the musico-theological notions
of the perfection of the unitas and the tri-unitary symbolism of the
triad, as enounced by Werckmeister in all his treatises; see also the
brilliant book by Rolf Dammann, Der Musikbegriff im Deutschen
Barock, Laaber Verlag, 3rd edition.
Thus, FIRST derivation:
Kellner, Herbert Anton: A Mathematical Approach Reconstituting
J.S. Bach's Keyboard-Temperament. BACH, The Quarterly Journal
of the Riemenschneider Bach Institute, Berea, Ohio. Vol. 10/4,
October 1979, page 2-8 and 22.
Recent Reprint: BACH, The Journal of the Riemenschneider Bach
Institute, Berea, Ohio. Vol. 30/1, Spring – Summer 1999, page 1-9
A second publication to "reconstruct" the system "wohltemperirt" of
Werckmeister/Bach I based upon the method of the "systems
approach" employed e. g. in planning and developing advanced
aerospace projects such as satellite systems.
Thus, SECOND derivation:
Kellner, H.A.: Temperaments for all 24 Keys - A Systems Analysis.
Acustica, Vol. 52/2, 1982/83. S. Hirzel Verlag Stuttgart. p. 106-113.
Publication of the lecture delivered July 1980 at the Bruges 6th
International Harpsichord Week
Both approaches - independent from each other - yield a system
having 7 perfect and 5 tempered fifths. Thus, a division of the
Pythagorean comma by 5.
In detail:
There are 4 tempered fifths C-G-D-A-E
There is one "tempering fifth" B-F#
The remaining 7 fifths are perfect.
II. Laying the bearings - descending by perfect fifths from middle C
Upon laying the bearings, one tunes downwards the 6 perfect fifths
C-F-B-flat-E-flat-A-flat-D-flat-G-flat.
Of course, octave transpositions upwards intervene wherever
necessary.
At this point, one may tune a seventh perfect fifth down towards the
B that is a semitone below the C one octave below the middle C.
At this place, the B-major triad can now be considered: B-E-flat
(D#) - F#.
It is essential that in the course of laying the chain of descending
fifths the third D# IS AVAILABLE within the B-major triad.
The Pythagorean comma must still be divided by 5 and subtracted
from a perfect fifth.
As concerns this problem, any mathematician, such as
Johann Sebastian Bach, or
Herbert Anton Kellner,
will discover that for this purpose of dividing the comma by 5 and
subtracting it from the perfect fifth B-F#, the following proportion of
beats within the triad of the tempering-tonality must be attained:
(B-D#) : (B-F#) = 6 : 1.
In practice, as long as the provisional fifth G-flat-B is perfect, the
Pythagorean third B-D# will beat violently. But pulling up the B
slightly and carefully, the fifth B-f# will start beating but at the same
time the violent beats of B-D# will be moderated.
The higher one pulls up the B, the better becomes the third B-D# and
the worse, the fifth B-F# gets out of tune.
See: http://ha.kellner.bei.t-online.de (Acoustics, mathematics)
The B-major triad furnishes this way for "well-tempering" its own metronome.
Such a
tempering-method was unheard of in the entire history of musical tempering
prior to
my publication:
Kellner, H. A.: Das ungleichstufige, wohltemperierte Tonsystem. In
"Bach-stunden",
Festschrift für Helmut Walcha, Hg. W. Dehnhard und G. Ritter. Evang.
Presseverband in Hessen und Nassau, Frankfurt/Main 1978. p. 75-91
My latter publication derives and proves in great detail this procedure.
Furthermore,
and very essential, it furnishes an error analysis via differential calculus
of the
method. In fact, the method proves to be pretty INSENSITIVE whether you happen
to
take the proportion 1 : 5, or 1 : 7 due to inaccuracies in tempering the
nominal and
correct proportion 1 : 6.
At this opportunity I invite all mathematicians to devise a better and more
accurate
tempering method. In "closer(?)" triads, but HOW??
(It is worth noting that once the correct B - about 1 octave below middle C -
is
attained, the third C-E follows from a perfect fourth B-E. This third has to
be
partitioned into the remaining 4 well-tempered fifths).
III. Repercussions of the tempering method upon Bach's settings composed in
the
"tempering tonality" B-major
Within Bach's Well Tempered Klavier, looking into the pieces in the "tempering
tonality" show clearly that Bach must have known that specific tempering
procedure.
Such aspects I have first published as early as 1977 but obviously, read by no
scholar
active at present:
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
Kellner, H. A.: Das wohltemperirte Clavier - Tuning and Musical Structure.
English
Harpsichord Magazine Vol. 2, No. 6, April 1980, page 137-140
Other aspects and methods concerning "proofs of authenticity" are easily
accessible in
my recent paper:
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
IV. An incompetent individual and his web-page with wrong beat-calculations
From time to time I have been receiving messages pointing out to me that there
exists
a web-page presenting a wrong calculation of the tempering method in B-major.
Worse, it claims, my own calculations are wrong. That's somehow annoying.
But why should I mind if someone chooses making a fool of himself in public. I
insist
that anybody should have the liberty of publishing his idiocies in the web.
Imagine
that this individual has COPYRIGHTED his rubbish! His self-confidence is
inversely
related to his competence and obviously, he will never be able to identify the
snag in
his "ingenious" calculations. It may be a psychiatrist in Berlin? Who can name
him?
P. S.:
Within C-E-G, The third C-E will beat at the same rate as the fifth C-G!
At the perfection of the UNITAS.
I do hope all that contributes to resolving your interesting queries!
Thanking once more to Prof. Margo Schulter for her substantial, critical and
constructive text, I remain,
with kind regards,
Herbert Anton Kellner
"M. Schulter" schrieb: