Werkemeister III by Direct Interval

[Billbrpt@AOL.COM]

In a message dated 9/3/99 6:25:25 AM Pacific Daylight Time, 
103445.713@compuserve.com (Paul N. Bailey) writes:

<< I think these would be the cents deviation for the Werk. III  in 
 Owen Jorgensen's  little book The Equal Beating Temperaments:
  >>
Hi, Paul,

Glad to see you posting again.  I'd sure like to see your meantone 
temperament showcased at the Convention.

There is a very good way to use the SAT in the Direct Interval Programming 
(DIP) mode to tune this temperament.  Using this method will get you the same 
results as the very best aural tuning.  Using correction figures of an FAC 
program works only as well as that program works for that piano in ET.

You first need to determine which broad category of Inharmonicity the piano 
has, High, Moderate or Low.  You can use the old Stretch Calculator sampling 
method to make this determination.  In the Tune mode, set the SAT to F5, play 
the note and stop the lights.  Press Octave Up to F6 and stop the lights 
again. A reading below 5 would be considered Low, 5 to 5.8 would be Moderate 
and 5.9 or above would be High, as simple as that.

Next, you do the same as you would do to tune a 1/4 Comma Meantone 
Temperament.  Take the value for the Syntonic Comma, which is 21.5 and divide 
it by 4 to get 5.38.  Subtract 2 that is already built in between each 5th 
the SAT has to get 3.38, round that off to 3.4.  This is the theoretical size 
of the 5th to be programmed.  Now you must adjust that size for 
Inharmonicity.  For a Low Inharmonicity piano, subtract .2 to get 3.2, for a 
Moderate, subtract .3 to get 3.1 and for a High Inharmonicity piano, subtract 
.4 to get 3.0.

If you have a Steinway, which has a relatively High Inharmonicity scale, it 
is particularly easy to program in the figures, they are all multiples of the 
number 3.  Starting on A3 reading on the 5th Octave, program in 0.0.  Press 
Shift and Store to store the value for that note. Move up to D4 and adjust 
the SAT to read that note on D5 and program in the figure 3.0. Press Shift 
and Store to store the value for that note. Move back down to G3 which will 
automatically read on G5 and program in the figure 6.0.  Press Shift and 
Store to store the value for that note.  Move up to C4, adjusting the SAT to 
read on C5 and program in the figure 9.0.  Press Shift and Store to store the 
value for that note.  Now tune carefully all of these notes according to the 
program you have established so far.

Now you must tune the G3-B3 3rd.  Theoretically, it would be pure but because 
your octave will be stretched, the 3rd will be just slightly expanded but 
below the threshold of where you would hear a beat, less than 1 cent.  Move 
to the Tune mode and set the SAT to G5.  Play the note G3 and stop the 
lights.  Press Shift and reset so that the SAT now reads 0.0.  For a Low 
Inharmonicity piano, dial in 0.7, for a Moderate, dial 0.8 and for a High, 
such as this Steinway, dial in 0.9.  Now tune the note B3 to that setting.  
Now, cancel out your offset and return to your page and set the SAT to the 
note B3 which will automatically read on B5.  Play the note B3 and stop the 
lights.  Press Shift and Store to store the value for that note.

Now, to get the last of your tempered intervals, subtract your adjusted 1//4 
Syntonic Comma value from the figure you now have for B3.  In the Steinway, 
that value is 3.  Let's say the B3 ended up reading 1.0.  If you subtract 3 
from that you would have     -2.0.  Program that value for the note F#3 which 
will automatically read on F#5. Press Shift and Store to store the value for 
that note.

Now from here on out, you can use Direct Interval Programming to tune the 
remaining pure 4ths and 5ths.  It is very easy.  You have already tuned F#3, 
leave the SAT there and play C#4.  Tune that note to stop the lights.  Now 
set the SAT to C#4 and adjust the SAT so that C#4 will read on C#5.  Play C#4 
and stop the lights.  Press Shift and Store to store the value for that note.

Move the SAT down to G#3 which will automatically read on G#5.  Play the note 
C#4 and stop the lights.  Now tune G#3 to that reading.  Press Shift and 
Store to store the value for that note.  Now continue through the rest of the 
temperament the same way.  You will get D#4 from G#3.  You can get the F3 
from C4 and the A#3 from the F4 after you have tuned you octave or from D#4.  
They will all match up.

With DIP, you can now tune the kind of octaves you really want by selecting 
the coincident partials you want and either tune them perfectly or make 
whichever kind of compromise you want.  You have the choice, you are in 
control.  You are telling the SAT what to do, it does not offer you 
information that may be incorrect.

Good Luck!

Bill Bremmer RPT
Madison, Wisconsin