Well Temperaments (with tempered octaves)

[Billbrpt@AOL.COM]

In a message dated 3/21/99 2:12:42 AM Central Standard Time,
pianotoo@imap2.asu.edu writes:

<< Hi Jerry:
 
 You asked for a plan of attack in tuning the octave in Historical 
 Temperaments. You still can use the M3rd-10th-17th tests. They just won't
 progress evenly as in equal temperament. Bill Bremmer takes advantage of 
 the opportunity to spread out the 5ths which are narrowed more than 
 usual in Equal Temperament. Perhaps he will jump in a give some suggestions
 about how to go about this.
 
 Jim Coleman, Sr.
  >>

I can give you the same idea I came up with for tuning octaves in ET.  To me,
it seems the most natural, the easiest (even *mindless*) way to tune octaves
yet I know of no one else who does it this way.  As I progress outwards from
the temperament octave, I try to "pull the tempering out" of the 5th as much
as I can without making an obvious beat in the octave and being mindful of how
fast I make the 4th beat.  Once into the 5th octave, the 4ths cease to be of
concern.

Once I have 2 full octaves tuned, I compare the double octave and the octave
and a 5th and make an exact compromise between the two.  Example:  When I
reach F5, I play F3 and F5 together, tune that double octave slightly wide,
then compare it with A#3 (Bb3)-F5, the octave and a 5th.  I try to make both
intervals "apparently" in tune (Equal Beating).  I continue all the way to C8
with this pattern and do the reverse in the bass.

To play and sustain a double octave or an octave and a 5th, you need both
hands and the sostenuto pedal.  (I am the *only* person in the *world* I know
of who uses the sostenuto pedal to tune the piano but I don't know how I could
do it any other way).  You strike the keys, press the sostenuto pedal and tune
your wide intervals.  If the piano has no sostenuto pedal, you can use the
damper pedal in the same manner that you would the sostenuto, that is, press
it *after* striking the keys, there will just be a little more extraneous
sound than with the notes you are tuning being completely isolated.

In a vertical, I have the strip mute under the dampers in the treble section
and therefore do not need a pedal.  Once you have tuned to the point in the
treble where there are no more dampers, you don't need the pedal any more.

If you are tuning ET, this will produce perfectly evenly ascending 10ths and
17ths all the way to the top and bottom.  If you are tuning a Well
Temperament, the compromise made between a double octave and an octave and 5th
will be a little more difficult when the 5th in question is tempered.  It may
mean that your double octave will have a slight beat.  However, when tuning a
double octave against a pure 5th, reconciling the two will be easy and the
double octave will not have much stretch to it at all.  This will naturally
mean that your octaves will vary in size up and down the scale.  This is why
they are called "tempered octaves".  No FAC program with HT correction figures
can duplicate this kind of compromise.

I described this procedure in the Journal article that featured mine and two
other responses to Jim Coleman's article about ET with pure 5ths.  (I don't
remember what that date was).  I also describe it in my aural instructions for
tuning the Equal Beating Victorian Temperament:

<< When expanding the outer octaves, try to reconcile the double octave
    and the octave and a 5th (12th) so that the double octave and the 12th
    beat exactly the same. (A very slight beat, almost inaudible). This
    will naturally result in octaves which vary slightly in size up and
    down the keyboard. These will quite effectively be "tempered
    octaves".>>

When tuning the 1/7 Comma Meantone, I do the very same thing with one
exception for the key of Ab, the "wolf" key.  Since the Ab is the flattest
note of the scale, I stretch the Ab octaves all the way to the top as much as
I reasonably can.  In the bass, I tune the Ab octaves as narrowly as I can.
For the Eb's, I tune 2:1 octaves all the way to the top and stretch them in
the bass as much as I can.  This "reconciles" the dissonance of the "wolf" Ab-
Eb interval so that it is not so apparent in the outer octaves.  Again, this
type of octave manipulation cannot be duplicated with an FAC program.

Bill Bremmer RPT
Madison, Wisconsin