Technical Library

TEMPERAMENTS XIII: Kellner

Kellner 3K gif There has been much debate over the years about exactly what sort of temperament might have been Bach’s intention for those two immortal volumes of Das wohltemperirte Clavier, let alone his other works. Bach himself had ample opportunity to include any of his thoughts on tunings in the various prefaces to several musical works published in his lifetime, but for me the fact remains he chose not to. (But see the recent Bach/Lehman temperament researched by Dr Bradley Lehman.)

One temperament which has, however, found some favor with European musicians especially, was ratified by the late Dr Herbert Anton Kellner in 1975, and published in 1980. His little book The tuning of my harpsichord is available in several languages including Japanese.

While Dr Kellner’s thesis goes to great lengths to prove that this is indeed the Bach temperament, citing the arrangement of gems on Bach’s signet ring and other numeralogical “proofs” to establish that there absolutely must be 5 tempered fifths and 7 perfect fifths, you will, I trust, permit me to remain a little scientifically detached, or even somewhat skeptical. Let’s agree, though, that it is a wonderful temperament.

Here’s how you go about setting Kellner:

1. Tune your c'' to the tuning fork, and tune down a perfect octave to middle c'. Tune down another octave to tenor c as well, because you are going to be mostly working in this octave below middle c' for this temperament.

2. Tune all the fifths from the flat side of C around the circle of keys absolutely perfect. Unlike Kirnberger, where for accuracy I asked you to stop half-way at the E^b, and begin again around the sharp side of E in the opposite direction, this time continue in the one direction from C through F^#. Keep all these fifths beatless.You know what a real perfect fifth sounds like by now, I hope.

3. Temporarily tune your tenor B a perfect fifth below f^#. Now you are ready for the crux of this Kellner temperament. Play the B Major triad, and observe its harshness. Afterall, its major third is Pythagorean, ie determined by four perfect fifths. (You should know this—you’ve just tuned them!) Slowly raise the tenor B until the third B–d^# beats six times are fast as the fifth B–f^#. To do this, you will be continually checking the third against the fifth until satisfied. Don’t ask me how many beats per second, just do it! When you have it right, the fifth B–f^# measures exactly a fifth comma narrow.

4. Tune up the octave to b, and down a perfect fifth to determine e. This will leave a slightly wide third from c–e.

5. The four remaining fifths around the circle between C and E must now be made equally narrow and rough, but they are only one fifth of a comma narrow, not quite as rapidly beating as your quarter comma fifths of Kirnberger. Remember you have lost the perfect third between c-e. In fact, when you are done, the C Major triad is useful as a check: The third c–e should beat at the same speed at c–g. As always, if you are tuning intervals which are theoretically the same size, the beat speeds increase as the pitch ascends. Therefore, your B–f^# beats slightly slower than c–g, which itself is slightly slower than d–a. And of course, your e–b doesn’t beat at all, because it must be perfect in Kellner.

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