Harmonium temperament-New figures offered

[SidewaysWell1713@aol.com]

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In a message dated 11/28/02 7:08:11 AM Pacific Standard Time, Tvak@aol.com 
writes:


> Here are the offsets from that harmonium temperament:

To the right, I have placed the figure I suggest would make the piano sound 
better yet still be compatible and supportive of the harmonium's sound:

>         A :     0          1.0
>         G# :    -3.7     -1.5
>         G :     0           1.0
>         F# :    -1.7      -0.5
>         F :     -2.2       -0.5
>         E :     -1.1       -1.0
>         D# :    -3         -2.0
>         D :     -3          -1.0
>         C# :    0.5        -0.5
>         C :     -5           -1.0
>         B :         -3.2    -2.0 
>         A# :   -4.1        -2.0
> 
> 


This will result in what is known as a Quasi Equal Temperament (quasi means 
almost).  Many 19th Century tuners actually tuned something along these 
lines.  It does not meet the rules or requirement of a Well Tempered Tuning 
but the time to let go of that theory of correct harmonic balance is 
precisely the the instance when you would consider a Quasi Equal Temperament.

When you play these 3rds, they won't sound too bad at all from the viewpoint 
of ET and neither will the 4ths and 5ths.  Careful picking through the 
temperament however, would reveal the unevenness.  It would basically sound 
like ET with a few obvious but small errors.  The harrmony when music is 
played will all sound good but will not exactly fit the logic of smooth 
progression along the cycle of 5ths.  One of Jason Kanter's graphs would 
reveal an irregular, wave type pattern instead of the bell shaped curve.  In 
this circumstance, that is not at all important.

The way I would do this aurally is to actually tune the F3-F4 octave unison 
by unison to the harmonium the best possible.  Now, listen to what that 
creates and try to smooth it out by adjusting the most extreme sounding 
intervals but also by changing it the least amount possible.

Your ETD makes it much easier to mathematically analyse but it also has no 
mind of its own.  It can easily make you tune something which is *exactly 
wrong*.  Not that it matters because there will be no ET in this 
circumstance, the numbers I provided you end up cutting the average amount 
that a note is off of ET from -2.2 to -0.8.  The average pitch of the new set 
of numbers is 1.3 cents sharp of the harmonium.  I bet it would be perceived 
as "matching" perfectly.  The range in sizes of 3rds is from 11 to 17 cents, 
all of which are quite tolerable sounding.  The Eb-G 3rd is the largest but 
where it is in the scale makes that not seem so bad.  It is contiguous to the 
smallest interval, the G-B which will actually make both of them seem to beat 
more alike, not be so different from each other.  As it turns out, the 
average width of 3rds with my proposed figures is the same as ET, 14 cents.

Here are the sizes of 3rds of the harmonium tuning and the proposed piano 
tuning:

Harmonium     Piano

C     17.9         14.0
C#    16.7         14.0
D      15.3         14.5
D#    17.0         17.0    (no change)
E      16.6         13.5
F      16.2         15.5
F#    16.4          12.5   
G      17.2         11.0   (largest change and improvement)
G#    15.3         14.5
A      14.5         12.5
A#     15.1        15.0
B       14.2        14.0

Actually, the harmonium does not look all that bad from this point of view.  
The 3rds are not even and they are all a little more sour than in ET.  The 
least used and nearly the most remote key, B sounds the sweetest and the most 
used, the expected "home" key of C sounds the worst.  Believe me, I've heard 
plenty of temperaments that match that description, mainly the attempts to 
tune the elusive ET with pure 5ths.  When you only know about ET and have 
rejected completely the idea that anything else exists, this discrepancy is 
imperceptible to you, it sounds "equal enough".

I have often read in the past where reconciling the two instruments in such a 
circumstance was considered an impossibility.  The advice was always to 
simply tune the piano to the A on the harmonium (or organ or whatever) and 
that would provide the best match possible.  But with just a little careful 
study on my part, I can provide you with a piano which sounds good on its own 
and does also seem to match the harmonium.  I hope it works out that way and 
that the musicians are truly impressed with what you could do.

Bill Bremmer RPT

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