Kirnberger II

Here is a rewrite of my paragraph on Kirnberger for the Harrison
review... Better??

>Essentially all the perfect fifths (and therefore fourths) of the
tuning are "just" (meaning no beats), with significant tempering of
only one note, the final "A" at the end of the tuning cycle. This
tuning has some extraordinary features: just intonation major triads
on C and G and just intonation minor triads on E and B. It is no
wonder that Lou Harrison is so enamored of this tuning with its
>affinity to just intonation.

So, if I'm understanding the Kirnberger tuning properly, all the
"pain" of fitting just fifths within the octave is concentrated in
only one place, with a rather significant tempering of A natural at
the end of the tuning cycle.

So, in other words, rather than being distributed, the tempering
adjustment is big (1/2 Pythagorean comma), and only in one place...
(??)

>So, if I'm understanding the Kirnberger tuning properly, all the
>"pain" of fitting just fifths within the octave is concentrated in
>only one place, with a rather significant tempering of A natural at
>the end of the tuning cycle.

Or, equivalently, the beginning!

More significant, though, is the four schismatic triads thus produced, which
I would wager figure heavily in this piece.

>So, in other words, rather than being distributed, the tempering
>adjustment is big (1/2 Pythagorean comma), and only in one place...
>(??)

Instead of having one wolf fifth, as normal 12-tone Pythagorean tuning has,
Kirnberger II splits the wolf in half and distributes it among two adjacent
fifths: D-A and A-E. And, by putting these fifths on the "near end" of the
circle rather than the far end, one can take advantage of some lovely
schismatic thirds: C-E, G-B, D-F#, E-G, and B-D.

What is the difference between this and Kirnberger I tuning?

Thanks,
Paolo

Paolo wrote,

>What is the difference between this and Kirnberger I tuning?

Kirnberger I is a just tuning. The most common Kirnberger tuning by far is
Kirnberger III, which is what you can assume if someone just says
"Kirnberger".

In Kirnberger III, each of the fifths C-G, G-D, D-A, and A-E is reduced by
1/4 syntonic comma, making C-E a just major third; E-B is just; the tuning
proceeds flatwards from C in just fourths until Gb is reached. B-Gb is then
only a schisma flat of a just fifth.

-Pablo