Bach's temperament, 1981
ABSTRACT
In April 1979 Early Music 7/2 236-249 published John Barnes's seminal
article “Bach's keyboard temperament: Internal evidence from the WellTempered Clavier”. Using statistical tools Barnes demonstrated the
inadequacy of Werckmeister III, Kelletat and Kellner temperaments and
proposed a new temperament that provides a very good fitting for the
data.
In January 1980 EM 8/1 129 published a letter by C. Di Veroli showing
that the case for Barnes's temperament was even stronger, and also that
Vallotti's temperament ranked as 2nd best.
In January 1981 EM 9/1 141 published a letter by Herbert A. Kellner
where he declared his belief in the superiority of his proposal and
publications, and challenged C. Di Veroli to prove otherwise. Even
before this letter was published, Early Music had already sent a copy to
C. Di Veroli asking him to write a definitive paper.
In April 1981 EM 9/2 219-221 published C. Di Veroli's article: it
demonstrates the optimality of Barnes's temperament compared with
Werckmeister III, Vallotti and Kellner. It also proves that Kellner's work
on his 'Bach' temperament, and also his refusal to accept Barnes's and
Di Veroli's conclusions, are both fundamentally flawed.
The two letters and the final article are reproduced here.
EARLY MUSIC
VOL 9 NO 2 APRIL 1981
Bach’s temperament
Claudio Di Véroli
Dr Kellner’s letter (EM 9/1, p.141) raises a
number of interesting points. As is by now
widely known, Dr Kellner’s writings on his
‘Bach’ temperament are based on numerology
(the study of coincidences among integer
numbers). Numerology has shown that Bach was
most skilful at writing his music with hidden
numerical allusions. But to start with
coincidences and try to infer ‘facts’ is a
completely different matter. For instance, I was
born in 1946, where 6-4 = 2 (numbers of
keyboards on my harpsichord) , 1+6 = 7 (degrees
of the diatonic scale), 9-4 = 4+1 = 5 (number of
octaves on my harpsichord), 1+9 = 10 (number
of fingers I use to play my harpsichord) and
9-6 = 3 (number of stops on my harpsichord).
Further, the name BACH (represented by
numerology as 2138) can also be obtained from
1946: the 1 is already there, 2 = 6-4, 3 = 9-6 and
8 = 9-1. This is neither magical coincidence nor
dazzling numerical virtuosity on my part, but just
a trivial amusement based on an elementary
mathematical fact: starting from a set of integer
numbers, almost any other set of small integers
can easily be obtained by simple arithmetical
operations. The above coincidences are also
produced by the years 1947, 1948, 1951, 1952,
1953 etc. This shows why numerology is not a
valid scientific method: simply anything can be
concocted with it.
Back to Dr Kellner’s writings. There is no
need to refer to his article ‘Das ungleichstufige
. . .’: for the benefit of his English readers he has
published two articles on his ‘Bach’
temperament in the English Harpsichord
Magazine (ii/2 (1978), pp.32-6; ii/6 (1980),
pp.137-40). The first article consists of three
pages of numerology plus a Conclusion: ‘. . .
However, as regards the authenticity of Bach’s
EARLY MUSIC APRIL 1981
219
tuning, and the rigour inherent in the methods of
musicological research, all of what has been said
here provides of course no proof whatsoever . . .
nothing more than some rather obvious
mathematical allusions . . .’. Oh surprise! He
knows numerology is not a valid research
method! But the most incredible assertion comes
a few lines afterwards: ‘. . . it appears unlikely
that anything different could be drawn up to
perform, with 12 fixed pitches per octave,
keyboard music through all 24 tonalities.’ Does
Dr Kellner really not know that descriptions of
circular temperaments were already printed in
1511? What about the scores of circular
temperaments by Werckmeister, Neidhardt et.
al.? Where is then the ‘uniqueness’ claimed by
Dr Kellner in his letter?
What I find most irritating is the fact that Dr
Kellner has obviously not bothered to read Mr
Barnes’s article (EM 7/2, pp.236-49), where it is
scientifically proved that Dr Kellner’s ‘Bach’
temperament is not the best suited for Bach’s
WTC. In view of these hard facts, I thank Dr
Kellner for his kind invitation to disprove his
claim, but I cannot possibly accept: it has already
been done by others. Neither can I understand
why Dr Kellner should ‘improve his proposal’.
Since Mr Barnes has already improved it, and
proved the improvement, from now on one
should try to improve on Barnes’s proposal
instead.
The above has not much to do with my letter,
where I said that Vallotti’s temperament was
better suited to the WTC than Dr Kellner’s.
EARLY MUSIC APRIL 1981
220
Hence, I may be expected to prove this assertion.
I will do it briefly, following Barnes’s criterion,
based on what was obviously Bach’s aim: to use
the dissonant intervals less frequently than the
consonant
ones,
in
agreement
with
Werckmeister’s ideas (1691). However, if
frequency and prominence of major 3rds is to be
inversely related to deviation (mistuning), then
the proper fit is not given by a direct proportion
curve (a straight line, as used by Barnes) but by
an inverse proportion curve (an hyperbola or,
more generally, a power curve). Further, I have
preferred to plot frequency against dissonance,
so that deviations with respect to the ‘best-fitting
power curve’ can be measured in terms of tuning
accuracy, rather than frequency of occurrence.
As a measure of goodness of fit I use the mean
absolute deviation (MAD), i.e. the average
vertical distance from each point to the curve.
Table 1 shows the plots produced by the
temperaments of Werckmeister, Vallotti, Kellner
and Barnes. We should also consider the
mistuning of the 5ths, which in Werckmeister is
very noticeable. Let us then compare the four
temperaments by plotting together both their
MAD and their WFD (worst-5th deviation), as
shown in table 2:
Conclusion: J. S. Bach is much more likely to
have used a temperament like Barnes’s or
Vallotti’s than one like Werckmeister’s or
Kellner’s. I must admit however that the
differences are hardly noticeable in performance,
except for Werckmeister’s very obviously flat
5ths.
EARLY MUSIC APRIL 1981
221
In April 1979 Early Music 7/2 236-249 published John Barnes's seminal
article “Bach's keyboard temperament: Internal evidence from the WellTempered Clavier”. Using statistical tools Barnes demonstrated the
inadequacy of Werckmeister III, Kelletat and Kellner temperaments and
proposed a new temperament that provides a very good fitting for the
data.
In January 1980 EM 8/1 129 published a letter by C. Di Veroli showing
that the case for Barnes's temperament was even stronger, and also that
Vallotti's temperament ranked as 2nd best.
In January 1981 EM 9/1 141 published a letter by Herbert A. Kellner
where he declared his belief in the superiority of his proposal and
publications, and challenged C. Di Veroli to prove otherwise. Even
before this letter was published, Early Music had already sent a copy to
C. Di Veroli asking him to write a definitive paper.
In April 1981 EM 9/2 219-221 published C. Di Veroli's article: it
demonstrates the optimality of Barnes's temperament compared with
Werckmeister III, Vallotti and Kellner. It also proves that Kellner's work
on his 'Bach' temperament, and also his refusal to accept Barnes's and
Di Veroli's conclusions, are both fundamentally flawed.
The two letters and the final article are reproduced here.
EARLY MUSIC
VOL 9 NO 2 APRIL 1981
Bach’s temperament
Claudio Di Véroli
Dr Kellner’s letter (EM 9/1, p.141) raises a
number of interesting points. As is by now
widely known, Dr Kellner’s writings on his
‘Bach’ temperament are based on numerology
(the study of coincidences among integer
numbers). Numerology has shown that Bach was
most skilful at writing his music with hidden
numerical allusions. But to start with
coincidences and try to infer ‘facts’ is a
completely different matter. For instance, I was
born in 1946, where 6-4 = 2 (numbers of
keyboards on my harpsichord) , 1+6 = 7 (degrees
of the diatonic scale), 9-4 = 4+1 = 5 (number of
octaves on my harpsichord), 1+9 = 10 (number
of fingers I use to play my harpsichord) and
9-6 = 3 (number of stops on my harpsichord).
Further, the name BACH (represented by
numerology as 2138) can also be obtained from
1946: the 1 is already there, 2 = 6-4, 3 = 9-6 and
8 = 9-1. This is neither magical coincidence nor
dazzling numerical virtuosity on my part, but just
a trivial amusement based on an elementary
mathematical fact: starting from a set of integer
numbers, almost any other set of small integers
can easily be obtained by simple arithmetical
operations. The above coincidences are also
produced by the years 1947, 1948, 1951, 1952,
1953 etc. This shows why numerology is not a
valid scientific method: simply anything can be
concocted with it.
Back to Dr Kellner’s writings. There is no
need to refer to his article ‘Das ungleichstufige
. . .’: for the benefit of his English readers he has
published two articles on his ‘Bach’
temperament in the English Harpsichord
Magazine (ii/2 (1978), pp.32-6; ii/6 (1980),
pp.137-40). The first article consists of three
pages of numerology plus a Conclusion: ‘. . .
However, as regards the authenticity of Bach’s
EARLY MUSIC APRIL 1981
219
tuning, and the rigour inherent in the methods of
musicological research, all of what has been said
here provides of course no proof whatsoever . . .
nothing more than some rather obvious
mathematical allusions . . .’. Oh surprise! He
knows numerology is not a valid research
method! But the most incredible assertion comes
a few lines afterwards: ‘. . . it appears unlikely
that anything different could be drawn up to
perform, with 12 fixed pitches per octave,
keyboard music through all 24 tonalities.’ Does
Dr Kellner really not know that descriptions of
circular temperaments were already printed in
1511? What about the scores of circular
temperaments by Werckmeister, Neidhardt et.
al.? Where is then the ‘uniqueness’ claimed by
Dr Kellner in his letter?
What I find most irritating is the fact that Dr
Kellner has obviously not bothered to read Mr
Barnes’s article (EM 7/2, pp.236-49), where it is
scientifically proved that Dr Kellner’s ‘Bach’
temperament is not the best suited for Bach’s
WTC. In view of these hard facts, I thank Dr
Kellner for his kind invitation to disprove his
claim, but I cannot possibly accept: it has already
been done by others. Neither can I understand
why Dr Kellner should ‘improve his proposal’.
Since Mr Barnes has already improved it, and
proved the improvement, from now on one
should try to improve on Barnes’s proposal
instead.
The above has not much to do with my letter,
where I said that Vallotti’s temperament was
better suited to the WTC than Dr Kellner’s.
EARLY MUSIC APRIL 1981
220
Hence, I may be expected to prove this assertion.
I will do it briefly, following Barnes’s criterion,
based on what was obviously Bach’s aim: to use
the dissonant intervals less frequently than the
consonant
ones,
in
agreement
with
Werckmeister’s ideas (1691). However, if
frequency and prominence of major 3rds is to be
inversely related to deviation (mistuning), then
the proper fit is not given by a direct proportion
curve (a straight line, as used by Barnes) but by
an inverse proportion curve (an hyperbola or,
more generally, a power curve). Further, I have
preferred to plot frequency against dissonance,
so that deviations with respect to the ‘best-fitting
power curve’ can be measured in terms of tuning
accuracy, rather than frequency of occurrence.
As a measure of goodness of fit I use the mean
absolute deviation (MAD), i.e. the average
vertical distance from each point to the curve.
Table 1 shows the plots produced by the
temperaments of Werckmeister, Vallotti, Kellner
and Barnes. We should also consider the
mistuning of the 5ths, which in Werckmeister is
very noticeable. Let us then compare the four
temperaments by plotting together both their
MAD and their WFD (worst-5th deviation), as
shown in table 2:
Conclusion: J. S. Bach is much more likely to
have used a temperament like Barnes’s or
Vallotti’s than one like Werckmeister’s or
Kellner’s. I must admit however that the
differences are hardly noticeable in performance,
except for Werckmeister’s very obviously flat
5ths.
EARLY MUSIC APRIL 1981
221