Alternative Temperaments

[Richard Brekne]

Hi list.

To those of you who are interested in this kind of thing, one Paul
Erlich has recently contacted me and asked me to put him in contact with
some of you. I have an article of his tied to my web page. This may be
viewed at :

 http://home.c2i.net/ric/22ALL.pdf

In addition he asked me to post parts of the e-post he sent me for your
reading pleasure. As I know little of these things, and included his
article simply because it appeared well thought out, and interesting, I
simply present the following for you all to do with as you wish. He
seems like an interesting and very engaged fellow. Tho much of his work
has to do with instruments requiring other then 12 tones to the octave,
you may find some of it relative in some sence or another. Hopeing this
is of use to some of you.

Richard Brekne
I.C.P.T.G.  N.P.T.F.
Bergen, Norway

Richard,

Thank you for including a link to my paper Tuning, Tonality, and
Twenty-Two-Tote Temperament! How did you find out about it? What do you
think of it? I should let you know that an html version of the paper
(currently with some problems, hopefully to be revised soon) is
available,
and both versions are linked to from
http://www-math.cudenver.edu/~jstarret/Erlich.html.

Keep up the good work!

-Paul


Hi Richard!

Although it's true that 22-tone equal temperament can't be fully
acheived on
an acoustic piano, there is a 12-out-of-22 subset which I call the
"hexachordal dodecatonic scale" which, as I mention in the paper, falls
quite naturally on the standard piano keyboard, and allows one to
experiment
with many of the ideas in my paper, including the decatonic scales, of
which
it contains 3.

In this tuning, each "semitone" on the keyboard is 2/22 octave, except
E-F
and B-C, which are 1/22 octave. This scheme has many wonderful
properties.
For instance, the chords

A C# E G
Ab C Eb F#
E G# B D
Eb G Bb C#
D F# A C

are tuned to a good approximation of a 4:5:6:7, and

A C Eb G
Bb C# E G#
D F Ab C
Eb F# A C#
E G Bb D

are tuned to a good approximation of a 1/7:1/6:1/5:1/4.

This may be of interest to anyone who's used meantone temperament (my
piano
is tuned in meantone) to experiment with 7-limit harmony. In meantone,
only
two good 4:5:6:7s,

Bb D F G#
Eb G Bb C#

and two good 1/7:1/6:1/5:1/4s,

Eb F# A C#
Bb C# E G#

exist, though their tuning in 1/4-comma meantone is better than those of

22-equal.

I could go on and on . . .

It is heartening to hear that someone is studying tuning in as much
depth as
you. It is something I'm deeply interested in. I started with piano but
now
play more guitar, and I have a 22-tone one that I play once in a while.
I
would love to see a 22-tone piano constructed -- it would be a great
help
for the advancement of music.

-Paul


For your thoughts tonight: for 5-limit harmony, I derived that
7/26-comma
meantone is the best meantone temperament (the size of the fifth is in
footnote 49 in my paper) using an equal-weighted RMS criterion. The only

historical account I know of someone using this criterion before me is
Wesley Woolhouse; however he did not carry out an exact calculation and
simply found that the result was close to 50-tone equal temperament,
which
he then decided to approximate with 19-tone equal temperament . . .
Other
meantones "derived" throughout history include 5/18-comma meantone
(Robert
Smith), 3/14-comma meantone (?), 2/7-comma meantone (the first,
Zarlino),
etc.

-Paul