re: frugal tuning?
4 nov 2002
arri london wrote:
>> joelncaryn wrote:
>> >did you know that bach generally didn't use well-tempered tuning...
>> no. what did he use?
johann sebastian bach: the well tempered tuning is unequal
technological specification of "wohltemperirt/bach"
dr. herbert anton kellner, the author
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
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.
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)...
piece of cake...
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
* 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
* 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
* 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
under no circumstances the third e-flat - g must beat more rapidly than e -
the thirds f-a and g-b being of equal size, must beat at virtually the same
but under no circumstances the third f-a must beat more rapidly than g-b.
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;
the inventor of this system was andreas werckmeister, as my publications
show. it was reconstituted on 21.12.1975 - see the patent.
peterson autostrobetm 490 features:
c.. pre-programmed temperaments (equal, pure major & minor, mean tone, =
pythagorean, werckmeister iii...
now why is this better than equal-tempering?