Introduction
-
Bach's musical temperament comprises within its circle 5 well tempered
fifths and 7 perfect fifths. This system "wohltemperirt" of J. S.
Bach - the composer's own authentic spelling - is the musical tuning for
Das
wohltemperirte Clavier.
-
The well-tempered fifths are (practically) all equal, being reduced by
1/5 Pythagorean comma, i. e. by about 4.7 cent. It is essential to note
that Bach's system "wohltemperirt" admits all 24 keys, major and minor.
Towards this aim, equal temperament is by no means a prerequisite. Through
several centuries the wrong opinion prevailed, Bach's system were equal
temperament! This system "wohltemperirt" of J. S. Bach - the composer's
own authentic spelling - is the musical tuning for Das wohltemperirte
Clavier.
The method Kirnberger III represents a different unequal baroque
temperament, also accessing all 24 keys. Therein, seven fifths are perfect.
Four of its tempered fifths are reduced by 1/4 syntonic comma, producing
a pure third c-e, which obviously allows an easy check upon laying
the bearings. However, this pure third causes a severe flaw: the keys with
sharps (such as e-major - only 4 sharps!) become pretty harsh and this
fact renders even Kirnberger III, his least unfavourable temperament musically
unsuitable. Better systems do exist!
On the contrary, within Bach's well-tempered system, the third c-e
is not perfect, but rather widened by 2.8 cent and thus e - g-sharp
in particular, is improved. But evidently, a check utilizing a pure third
c-e
will no longer be possible.
The procedure of tuning a harpsichord
"wohltemperirt", laying the bearings, will remain essentially within the
octave below middle c. This range is exceeded downwards only by
the two tones B-flat as well as B - a semitone higher.
Upwards, the range extends no higher than e above middle c.
The peculiar and specific method to proceed is as follows: One descends
from middle c down to g-flat by perfect fifths (thus: c,
f, b-flat, e-flat, a-flat, d-flat, g-flat) with jumps of an octave
upwards wherever necessary in order not to trespass the range. From the
last note
g-flat attained, (enharmonically f-sharp) one tunes
downwards a fifth initially perfect - no beats - towards B. Thus,
we are in B-major now. Then B is pulled up slightly and the fifth
will start beating. At this point it is essential that e-flat is
already available, as it has been reached on the way downward tuning the
chain of perfect fifths. Now listen to the third B - e-flat formed,
henceforth better to be called B - d-sharp as we are in B-major.
Initially, as long as this fifth downward f-sharp - B is perfect,
the third B - d-sharp will be Pythagorean and hence beating violently.
But by pulling up the B, the beats of the intervening third B
- d-sharp within the triad will relax - slowing down. As soon it beats
rhythmically SIX TIMES faster than its fifth, the B is correctly
tempered. (The higher one pulled up the B, the faster the fifth
beats, whereas the beats of the intervening major third B - d-sharp
slow down). This is the specific method for tuning the system, derived
mathematically, known and utilized by J. S. Bach, as my publications -
see below - prove.
Once the proportion between the beats of (B - d-sharp) : (B
- f-sharp) = 6 : 1 is thus accomplished, the B is correctly
tempered. As mathematics show, this procedure divides the Pythagorean comma
by 5. By that value of 4.7 cent the fifth B - f-sharp is reduced!
Thereafter, from this B attained, one tunes a perfect fourth
upwards towards e, generated most conveniently by tuning a perfect
octave upwards, (placed a semitone below middle c), followed by
a perfect fifth down to e. This resulting third c-e must
ever so slightly beat upwards, being enlarged by only 2.8 cent!
Finally, one fits 4 fifths of equal size into c-e, subdividing
this third, exactly as would have to be done for the third c-e of
Kirnberger III; only that the latter one is pure, in contradistinction
to the enlarged basic third within Bach's system.
This subdivision of the third c-e into 4 equal parts should,
of course, be checked:
* The first pair of fifths consisting of c-g and d-a must
beat at about the same rate; under no circumstances the fifth c-g
may beat more rapidly than d-a.
* The second pair of fifths consisting of g-d and a-e
must beat at about the same rate; under no circumstances the fifth g-d
may beat more rapidly than a-e.
* As regards the relation between these two pairs of fifths, g-d
must beat 1.5-times more rapidly than c-g. That is to say, during
the time c-g beats twice, g-d completes three beats.
* Under no circumstances the fifth c-g may beat more rapidly
than
g-d.
Detailed bearing plan
Checking for the beats in Bach's
system "wohltemperirt":
* Basic checks:
In the basic triad of c-major c-e-g, the third c-e beats
at the same rate as its fifth c-g.
(Strike the entire fundamental triad c-e-g and listen to its
sonority: this is the triad closest to purity one can attain within a balanced
system for all 24 keys).
The fifth c-g being just a semitone higher than B - f-sharp,
beats at virtually the same rate.
* Further checks:
The third d - f-sharp beats 3-times as fast as the basic third
c-e.
The thirds e-flat - g and e - g-sharp being of equal
size, must beat at the same rate.
Under no circumstances the third e-flat - g must beat more rapidly
than e - g-sharp.
The thirds f-a and g-b being of equal size, must beat
at virtually the same rate.
But under no circumstances the third f-a must beat more rapidly
than g-b.
Spezifikation der Wohltemperirung
Werckmeister (1691) / Bach (1722)
Das wohltemperirte System ist spezifiziert durch dessen fundamentalen C-Dur
Dreiklang, dessen geschärfte Terz c-e gleich schnell mit der
verkleinerten wohltemperirten Quint c-g zusammenschwebt - in optimaler
gegenseitiger Anpassung. Die zweite Oktave der Terz besteht aus vier derartigen
wohltemperirten Quinten c-g-d-a-e. Die Quint
e-h ist rein.
Ab c steigen sechs reine Quinten bis ges (fis) ab;
beim praktischen Cembalostimmen mit den erforderlichen Oktavversetzungen.
Die chromatische Skala der Wohltemperirung lautet der Reihe nach für
die aufsteigende Leiter ab c in cent:
0; 90,2; 194,6; 294,1; 389,1; 498,0; 588,3; 697.3; 792,2; 891,8;
996,1; 1091,1; 1200,0.
Andreas Werckmeister hat dieses System erfunden, wie meine Publikationen
beweisen. Es wurde am 21.12.1975 rekonstituiert - siehe Patent.
Specification of Werckmeister (1691) / Bach (1722)
wohltemperirt
This welltempered system is specified via the fundamental C-major
triad, the sharpened third c-e of which beats at the same rate as
the flattened welltempered fifth c-g in optimum mutual adaptation.
The second octave of the third is made up by four such welltempered fifths
c-g-d-a-e.
The fifth e-b is perfect. From c descend six perfect
fifths until g-flat (f-sharp) is reached, including octave
transpositions where necessary, upon tuning a harpsichord. The chromatic
scale wohltemperirt, ascending successively from c, reads
in cent:
0,0; 90,2; 194,6; 294,1; 389,1; 498,0; 588,3; 697,3; 792,2;
891,8; 996,1; 1091,1; 1200,0.
The inventor of this system was Andreas Werckmeister, as my publications
show. It was reconstituted on 21.12.1975 - see the patent.
Tuning instructions:
Herbert Anton Kellner, Wie stimme ich selbst mein Cembalo? Schriftenreihe
Das Musikinstrument, Heft 19, Verlag Das
Musikinstrument, E. Bochinsky, Frankfurt / Main, 31986, ISBN 3-923639-68-6;
In English: Herbert Anton Kellner, The Tuning of my Harpsichord. Schriftenreihe
18, Das Musikinstrument, Frankfurt
1980, ISBN 3-920-112-78-4;
En Français: Herbert Anton Kellner, J'accorde mon clavecin. Schriftenreihe
31, Das Musikinstrument, 1982, ISBN
3-920-112-89-X;
Japanese translation by Sumi Gunji. ISBN4-88564-170-5 C3037, 1990
Kellner, Herbert
Anton
Born December 25, 1938 in Prague, studied philosophy, physics, mathematics
and astronomy at the University of Vienna. Graduation, Dr. phil., in 1961
with a mathematical thesis. Identified in 1975 the unequal temperament
of Johann Sebastian Bach for Das wohltemperirte Clavier; up to that
time it had been mostly assumed, equal temperament was meant.
Publications include:
The Tuning of my Harpsichord, Schriftenreihe 18, Das Musikinstrument,
E. Bochinsky, Frankfurt/Main 1980. Wie stimme ich selbst mein Cembalo?
Schriftenr. Das Musikinstr. 19, Frkf./M. ³1986.
J'accorde
mon clavecin, Schriftenr. 31, Das Musikinstrument, 1982.
Japanese translation by Sumi Gunji: ISBN4-88564-170-5 C3037, 1990. Numerous
musicological research papers concerning J. S. Bach, such as:
Eine Rekonstruktion
der wohltemperierten Stimmung von Johann Sebastian Bach, Das Musikinstr.
26,
1977, 34-35. In English: A Mathematical Approach Reconstituting J.S.
Bach's Keyboard-Temperament, BACH, The Quarterly Journal of the Riemenschneider
Bach Institute, Berea, Ohio, Editor Elinore Barber, 10/4, October
1979, pp. 2-8, 22. German
patent
DE 25 58 716 C3: Musikinstrumente in fester, optimierter ungleichschwebender
Stimmung für alle 24 Tonarten, claimed 21. Dez. 1975 , granted
14. May 1981. 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. Temperaments for all 24 Keys - A Systems Analysis,
Acustica 52/2, 1982/83, Hirzel Stuttgart, 106-113; Publication of
the lecture delivered July 1980 at the Bruges 6th International Harpsichord
Week. Was Bach a Mathematician? English Harpsichord Magazine and
Early Keyboard Instrument Review (EHM), Editor Edgar Hunt,
2/2,
April 1978, 32-36; Publication of the lecture delivered August 1977 at
the Bruges 5th International Harpsichord Week, 14th International Fortnight
of Music. "Das wohltemperirte Clavier" - Tuning and Musical Structure,
EHM, 2/6 April 1980, 137-140. How Bach quantified his welltempered
tuning within the Four Duets, EHM 4/2, 1986/87, 21-27. Barocke
Akustik und Numerologie in den Vier Duetten: Bachs "Musicalische Temperatur",
Ber. Int. Musikw. Kongr. Stuttgart 1985, Kassel 1987, 439-449. Das wohltemperirte
Clavier - Implications de l'accord inégal pour l'oeuvre et son autographe,
Revue de Musicologie 71, 1985, 184-157. Kepler, Bach and Gauß,
The Celestial Harmony of the Earth's Motion, BACH, Journ. Riemenschn.
Bach Inst. 25/1, 1994, 46-56.
Le tempérament inégal
de Werckmeister/Bach et l'alphabet numérique de Henk Dieben,
RMl. 80/1, 1994, 283-298. J. S. Bach's Well-tempered Unequal
System for Organs, The Tracker, J. Organ Hist. Soc. 40/3, 1996,
21-27. Über die Cembalostimmung für Das Wohltemperirte Clavier,
Michaelsteiner Konferenzberichte
52, "Stimmungen im 17. und 18.
Jahrhundert, Vielfalt oder Konfusion?" Eds. G. Fleischhauer, Monika
Lustig, W. Ruf, F. Zschoch, Michaelstein 1997, 35-44. Stimmungssysteme
des 17. und 18. Jahrhunderts, in "Alte Musik und Musikpädagogik",
Symp., Hochschule für Musik und Darstellende Kunst, Wien, Ed. Hartmut
Krones, Reihe Wiener Schriften zur Stilkunde und Aufführungspraxis,
1;
Böhlau, Wien, Köln, Weimar 1997, 235-265. Baroque-style Organs
well-tempered according to Werckmeister/Bach; Bien tempérer les
orgues de style baroque selon Werckmeister/Bach; Orgeln barocken Stils,
wohltemperirt nach Werckmeister/Bach. ISO Journal N° 4, March 1999,
8-14. Considering the Tempering Tonality B-Major in Part II of the "Well-Tempered
Clavier", BACH, J. Riemenschn. Bach Inst. Vol. 30/1, 10-25.
Göttliche
Unität und mathematische Ordnung - Zahlenalphabet und Gematria von
Andreas Werckmeister bis Joh. Seb. Bach, Österreichische Musik
Zeitschrift Jg. 55, 11/12, 2000, 8-16.
Thanks to Dr. Yo Tomita, Belfast, a comprehensive general bibliography,
"Bach",
serving scholars of all the world, exists:
http://www.music.qub.ac.uk/~tomita/bachbib.html
