KELLetat and KELLner intersted in Bach's well tempered system

Dera Dave Keenan,

Thank you for writing to me, wondering about KELL/KELL and asking for
clarification!!

Whilst I had correspondence with Prof. Kelletat some decades ago, it
is, though strangely enough, mere co-incidence that both the names
starting with KELL apparently lead to a predisposition dealing with
musical temperament.

But the system of Bach, though approximately having its fifths
reduced by 1/5 P, is DIFFERENT from that!!

The specification as given in my webpage, canNOT be reconciled exactly and
to full accuracy with a system based upon 1/5 P. attention!!!:

Specification of Werckmeister (1691) / Bach (1722) wohltemperirt

see: http://ha.kellner.bei.t-online.de as follows:

"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, (roughly rounded to 0.1 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 at Xmas 1975, on 21.12.1975)).

If you now compare with a temperament of the same type, but derived
from 1/5 P, then you would notice its rounded cent-values, to the
accuracy of 0.1 cent would agree in all cases, save one.

For the sake of precision I will now state the true well-tempered fifth
somewhat more accurately:

Qw = 1,4959535062432300212473083995637961...
I must confess that I am less keen on the fifth belonging to P/5:
Q5 = 1,49594019392827

These two fifths are different.

The basic third Tw of "wohltemperirt", i.e. C-E follows readily from Qw via
Tw = (Qw^4)/4

The entire (truly theoretical) scale well-tempered, accurate to about 14 digits
- in view of the known structure of fifths - can be calculated easily. Should
there be any interest manifested for this table, I would publish it with
pleasure for our tuning-group, together with correspondingly accurate
cent-values. Of no practical value, though.

Kind regards,

Herbert Anton Kellner

Dave Keenan schrieb:
> Dr Kellner's Bach tuning appears (undated) in the Scala archive as:
>
> ! kellner.scl
> !
> Herbert Anton Kellner's Bach tuning. 5 1/5 Pyth. comma and 7 pure
> fifths
> 12
> !
> 90.22500
> 194.52600
> 294.13500
> 389.05200
> 498.04500
> 588.27000
> 697.26300
> 792.18000
> 891.78900
> 996.09000
> 1091.00700
> 2/1
>
> Curiously, the following also appears:
>
> ! kelletat.scl
> !
> Herbert Kelletat's Bach-tuning (1967)
>
> 12
> !
> 90.00000
> 196.00000
> 294.00000
> 386.90000
> 498.00000
> 588.00000
> 700.00000
> 792.00000
> 892.00000
> 996.00000
> 1086.00000
> 2/1
>
> The astounding similarity in the author's names, both proposing a
> tuning for Bach's well-temperament, has me wondering whether the
> second is in fact an early proposal by Dr Kellner?

Contrary to several historic and contemporary scholars, I am unable to
propose more than BUT ONE SOLE musical temperament for all 24 keys!

In this context, one might look into:

Kellner, H.A.: Temperaments for all 24 Keys - A Systems Analysis. Acustica, Vol.
52/2, 1982/83. S. Hirzel Verlag Stuttgart. Seite 106-113. Publication of the
lecture delivered July 1980 at the Bruges 6th International Harpsichord Week

Incidentally, Kelletat usually starts off with a pure third C-C. But documents
report that JSB has instructed Kirnberger to tune ALL major thirds larger than
pure; identical with Werckmeister's prescriptions. Therefore, Kirnberger III and
the like, with (a) pure fifth, were not necesarily good candidates for Bach's
"wohltemperirt" system.

Best regards,
Herbert-Anton Kellner

>
> -- Dave Keenan
>
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