<HTML>In a message dated 11/28/02 7:08:11 AM Pacific Standard Time, Tvak@aol.com writes: <BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">Here are the offsets from that harmonium temperament:</BLOCKQUOTE> 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: <BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">        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 </BLOCKQUOTE> 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</HTML>