Kirnberger temperament

Hello Ken,

I hope you don't mind, but I decided to send a copy of this
response to the Tuning List as well as to you directly.

> From: "Ken Miller" <mmusic1@mindspring.com>
> To: <monz@juno.com>
> Date: Tue, 27 Nov 2001 10:23:59 -0700
> Subject: microtonal dictionary
>
> I was looking for the definitions for several particular
> tunings and wasn't able to find them on your site. Your site
> does have a lot of information I haven't even dreamed of before.
> In particular, I was looking for the definitions of these
> tunings, Kirnberger and werkmeister. Are thes familiar?
> Are they called something else on your site? I found these
> tunings available on several digital keyboards and would
> like to experiment with them. Thanks for your input.

Thanks for the kind words about my work... glad you find
it useful.

I refer to the Kirnberger and Werckmeister as "well-temperaments".
There's an entry for that in the dictionary, but unfortunately,
I haven't devoted a lot of time to either of these theorists yet.

Please note that what I define as "well-temperament" is also
known to other theorists as "irregular" or "circulating" temperament.
All three names are commonly accepted today.

Actually, it's really interesting to me that you write this to
me at this particular time, because while I studied Werckmeister
a bit about two years ago (and do hope to eventually someday do
more research on his tunings), I have just in the last two weeks
been studying the temperament known as "Kirnberger III",
because I've been learning Beethoven's "Moonlight Sonata" and
believe that Kirnberger III may have been a typical tuning on
Beethoven's piano at the time he wrote the sonata.

So I've actually made a MIDI-file of the beginning of Beethoven's
piece in Kirnberger III tuning, and of course, have constructed
lattice diagrams with which to analyze it.

One reason I chose Kirnberger III as typical is because it is
so conceptually simple, and therefore quite easy to tune by ear
as well. Here's how it works:

Our reference will be "C", and the set of pitches tuned will
be given the letter-names Db to F#. The "major 3rd" C:E is tuned
to the "pure" 5:4 ratio. The "5ths" and "4ths" C:G:D:A:E are
all tuned equally slightly narrow (i.e., 1/4-syntonic-comma)
to match the 5:4 already tuned between C:E. Then all other
"4ths/5ths" are tuned as 3:2 ratios in two chains: one going
thru the "cycle of 4ths" from C, and the other thru the
"cycle of 5ths" from E.

Thus, my lattice for it looks like:

Db
/
/
Ab
/
/
Eb
/
/
Bb
/
/
F
/
/
C _
G _
D _
A _
E
/
/
B
/
/
F#

It just so happens that the "wolf 5th" F#:Db is
~700.0012801 cents, not all that much of a "wolf":

note prime-factor ratio ~cents

Db 3^-5 256:243 90.22499567
F# 3^2 * 5^1 45:32 590.2237156

I have tables of interval sizes, implied proportions, etc.,
and will eventually be putting it all into a webpage, along
with my audio files of Beethoven's piece.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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