(original message from Paul Tizzard)
""Date: Sat, 27 Oct 2001 21:17:36 +0200
From: "Paul Tizzard" <ptizzard@mweb.co.za>
Subject: Re: more on this temperament thing
Hi List,
Apologies for picking up on this thread so late, but I would like to know
how to tune some of these Historical Tuning's aurally. On the List I've
only
ever seen cent values for ETD's. Any info, would as always, be appreciated.
Paul Tizzard""
The following from Dr. H.A. Kellner
Check out his websites and you decide whether this is the keyboard tuning
of J.S.Bach.
Paul Bailey
Modesto, CA
============================================
3.4 The well-tempered tuning of J.S. Bach
The temperament underlying the "well-tempered Clavier",i.e. the
"Forty-Eight" and Bach's other compositions for this instrument is
described here. This tuning system which I have reconstructed is based upon
one single well-tempered major triad, the sharpened third of which beats at
the same rate from above, as its flattened fifth beats from below. In the
spirit of the baroque theological and number-mystical music theory a triad
was considered a symbol of the Trinity, or a Tri-Unisonus. The major triad,
in particular, represented the trias harmonica perfecta, having the
proportions 4:5:6.This is closest to the Unitas which concept of symbolized
the perfection in the baroque musico-theological treatises:see
Werckmeister.Dammann. Obviously the triad can be tempered best and in the
most natural way if its constituent intervals of major third and fifth are
mutually adapted to each other by adjusting their beats to occur at the
same rate, at 1:1. the ratio of the "Unitas". The triad of the minor mode
need not be considered here, because its proportion 10:12:15 is less
perfect from the outset and more remote from the unity. Therefore, as a
result of our tempering, a major triad emerges the well tempered fifth of
which beats at the same rate from below, as its major third beats from
above: on the whole, a symbol for the Tri-Unity in the baroque sense.
If we now try to close a circle of 12 fifths, it is found that 5
well-tempered fifths together with 7 pure fifths will yield almost exactly
7 octaves. Therefore, a musical temperament derived from the well-tempered
triad, must contain seven pure and five tempered fifths. This crucial triad
will be placed upon C-major on the keyboard, the center of tonality.
Amongst the five well-tempered fifths, four will make up the slightly sharp
well-tempered third c-e, beating from above. Six of the pure fifths descend
successively from c. the seventh pure fifth to be tuned above e: e-b.such
that the remaining well-tempered fifth will be located on b-F#. This
distribution of fifths ensures at the same time that there will be only one
single well-tempered triad proper, namely on the central key of C-major,
which is in line with the baroque concept of the perfection of the
"Unitas". The thirds of the other triads are sharper and their beats as a
consequence, are more vibrant.
After these specifications, the procedure for laying the bearings will be
described. Firstly, from the middle c, six pure fifths are laid downwards
between c1 and f#o a tritrone below. A seventh , only provisionally pure
fifth descending to b is then tuned:f#-B. This produces initially a
pythagorean major third B-d#0, and the essential step for well-tempering
can now be carried out. It can be done accurately and at once, avoiding
digressions of trial and error. In this step, B must be pulled up,
moderating thereby the pythagorean third, such that it eventually beats six
times at the rate of the well-tempered fifth B-f#0 within this B-major
triad resulting from the appropriate tempering.
(INSERT musical notation of B-F# half notes with B-D# triplet eighth notes)
This method to be applied is derived from a comprehensive theory, which
need not be discussed here.
The procedure may now continue by laying a slightly sharpened, tentative
major third c0-e0. beating slightly from above. If by good luck it has been
tempered correctly, the e0 must prove to be a perfect fourth above B, or if
the octave B-b has already been tuned, e0 must be a pure fifth below. In
any case this check via B or b overrides the e0 estimated initially, which
may need to be readjusted slightly to comply. Thereafter four identical,
well-tempered fifths must be fitted into this slightly enlarged
well-tempered third c0-e0, according to the basic task. Here this
subdivision turns out to be substantially simplified. Already the first
step for the fifth c-g takes advantage of the fact that it beats at the
same rate as the third c-e of the well-tempered C-major triad, and
virtually at the same rate as the B-f#, identically tempered, but lower by
a semitone. This supplementary relation is most useful, as it further
facilitates the basic procedure of partitioning a third which occurred
first in mean-tone. For the present tuning, the next well-tempered fifth
g0-d1 follows, together with an octave-transposition down to d0.The routine
proceeds by tempering d0-a0 and a0-e1, which latter tone, an octave above,
must not be further changed. The division of the third c-e is now checked
and verified by comparing the first pair of fifths c0-g0, d0-a0, with the
second pair comprising g0-d1 and a0-e1. Thereby the fifth G-d must be at
3/2-times the rate as c0-g0, which means that g-d produces three beats
during the time it takes c0-g0 to beat twice. This can be easily
distinguished by striking these two intervals alternately and comparing the
rhythm of their beats. The upper fifth in any pair, lying one tone above
its neighbor, would beat faster by a mere 10%. In practice it is sufficient
to ensure that the lower fifth never beats faster than the upper one .
When implementing this tuning, the above supplementary checks to the basic
task contribute to an optimal accuracy, and together with the unique step
to generate B, at a total number of not less than seven pure fifths, allows
an astounding precision to be attained. In fact, should the count of 6 for
the ratio between third and fifth for the beat rate within B-d#-f# be
missed and 5 or 7 effected instead, this would imply not more than 0.6
cents error for the B thus established! Therefore Bach's tuning can be laid
easily, quickly and accurately. For the sake of clarity, the 19 steps of
the entire procedure are tabulated below:
1. c1-c0 Descending octave from middle c.
2. c1-f0 Descending pure fifth from middle c. Verify pure fourth
c0-f0.
3. f0-Bb Descending pure fifth.
4. Bb-bb Transposition by ascending octave. Verify pure fourth
f0-bb0.
5. bb0-eb0 Descending pure fifth. Verify pure fourth Bb-eb0.
6. eb0-eb1 Transposition by ascending octave. Verify pure fourth
bb0-eb1.
7. eb1-ab0 Descending pure fifth. Verify pure fourth eb0-ab0.
8. g#0-c#0 Descending pure fifth.
9. c#0-c#1 Transposition by ascending octave. Verify pure fourth
g#0-c#1.
10. c#1-f#0 Descending pure fifth, Verify pure fourth c#0-f#0.
11. f#0-B Well-tempered fifth. Within the B-major triad the third
B-d#0 must beat six times .faster than the fifth B-f#0.
12. B-b0 Transposition by ascending octave.
13. b0-e0 Descending pure fifth. Verify pure fourth B-e0.
14. e0-e1 Transposition by ascending octave. Verify pure fourth
b0-e1.
15. c1-g0 Well-tempered fifth. flattened. must beat from below at the
same rate as the third c0-e0 beats within the
C-major triad. Beats slightly faster- virtually at the
same pace- as the well-tempered fifth B-f#0 a semitone
below.
16. g0-d1 Well-tempered fifth. Flattened to beat 3/2-times faster
than c0-g0.
17. d1-d0 Transposition by descending octave.
18. d0-a0 Well-tempered fifth. Beats hardly faster-virtually at the
same rate- as c0-g0 a whole tone
below.
19. a0-e1 Check:This must be a well tempered fifth, e1 must not be
changed. Compare beat- rate with g0-d1 a whole tone
below.
Play c-e-g and listen to this well-tempered C-major triad
which opens the door to performing
music in all 24 keys, both major and minor.
This tuning for Das wohltemperirte Clavier by Johann Sebastian Bach
complies with all the principles of Werckmeister discussed at length in the
preceding paragraph. There are some pure fifths, some tempered ones, and
all the major thirds are sharpened. The unique best major third c-e indeed
beats very slightly from above. However, due to the ear's capability of
identification by tolerance, this third could still pass as a virtually
pure interval. Its offset from purity is less than the graduation between
the scaled other thirds within this temperament. There are four steps of
the five values up to the pythagorean interval.
It may well be that this reconstruction of Bach's well-tempered tuning will
be received with some scepticism, or may even be rejected by those who know
better. Whereas I see no remedy for the sceptics, I invite others to
substantiate that indeed Johann Sebastian Bach's keyboard temperament has
been described here. Such evidence- and how could it be otherwise- will be
rooted in this master's music, and should not be too difficult to
establish.
( cents deviation from modern e.t.)
C=+8 B=-0.9 A#=+4.2 A=0.0 G#=+0.3 G=+5.5
F#=-3.6 F=+6.1
E=-2.8 D#=+2.2 D=+2.8 C#=-1.7
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