Kellner w.t. bearing plan.

[Paul N. Bailey]

Arnold, if all this doesn't fall into place instantly, remember how
long it took to learn equal temperament? This will yield much sooner
and easier than that puzzle did, but not instantly, perhaps...

If the contrasts between the 'simple' keys and the 'extreme' keys
are too much for you in this temperament, don't give up.....
think of it this way;

	Equal temperament has 14 cent wide thirds; Kellner's and
many other well temperaments have one or more 21.5 cent thirds,and
thirds as small as pure or 2 cents wide of pure,  which may be more
contrast than your ears are comfortable hearing, at least at first;
but there are "harmonically balanced, tonal temperaments" (another
way of describing well temperaments ) that have
less contrast, say 9 cents to 18 cents, which are still quite musical
and authentic.


If you have nerve enough, you can start simply by making your fifths
slightly narrower (or rougher) among the 'white' or long keys, and
slightly  fatter or smoother in the fifths that have one or two 'black'
or short keys. C-F is an exception; it may be pure...(also, in some
well temperaments, E-B may be pure).

You will know if it's coming out right, generally, because the thirds
in the 'simple' keys will be a little smoother, and the thirds in the
keys with a lot of accidentals will be a bit more busy....

Don't worry too much at first if you are doing it EXACTLY right...
you will get a sense of how it all works, pretty soon.

I do , by the way, have a very high opinion of the Kellner temperament;
there is no one and only best temperament, but Kellner is an especially
good one, and when you start to understand the bearing plan, you will
appreciate that it calls for very little 'estimating', and even if you
are not thoroughly accurate, the overall effect will be right anyway...

Good luck, and "keep us posted"

			Paul Bailey RPT


If a fifth is just (pure) , as a major triad, the minor third beats
three times while the major third beats twice; as a minor triad, the
minor and major thirds beat at the same rate.



H. Kellner proposed  J.S.Bach WTC well temperament
Cents Dev from standard  e.t.:

C=+8.0	C#=-1.7	D=+2.8  D#=+2.2  E=-2.8   F=+6.1   F#=-3.6

G=+5.5  G#=+0.3   A=0   A#=+4.2   B=-0.9

(the following i believe to be from Herbert Kellner, tho I haven't seen the
actual book.... P.B.)


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	Descendin pure fifth.
4. Bb-bb	Transposition by asecnding 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 thrid B-d#0must
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 			perfoming 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..... 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 reconsrtruction of Bach's well-tempered tuning will be
recieved with some secpticism, or may even be rejected by those who know better.
Whereas I see no remedy for the secptics, 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.