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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?
 
>http://ha.kellner.bei.t-online.de/

hmmm: 

  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
  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.

  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
  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.

  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.

  http://www.music.qub.ac.uk/~tomita/bachbib.html

  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?

nick




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