<!doctype html public "-//w3c//dtd html 4.0 transitional//en"> <html> Richard Brekne wrote: <blockquote TYPE=CITE>...I'd like to point out that in a sense the whole point is really that there is no such thing as ET at all. At very best one can equally space only one and the same one partial for every string on the piano. This is illustrated by the fact that a perfectly equal spaceing of say the 4th partial for all 88 notes, is no guarantee that any other partial will be equally spaced. In fact it is more like a guarantee that all other partials will be unevenly spaced.</blockquote> Richard, Of course you are right. Just Intonation Monophonicist (chordist...) Harry Partch held the 12 tone keyboard, and implicitly the piano, accountable for a general out-of-tunedness, but especially with certain Acrosonics we know this is not the case: even despite a good scale (not present on said instrument), string drawing and bridge notching play out their roles even before inharmonicity factors in. However, a body of research has been done on tempering inharmonic sounds - noteably by Bill Sethares. Tempering itself implies maximizing some (or more than one) consonance, be it 3rd, fifth or octave normally, stretched or pure (actually, Partch seemed not to believe in partials effect on tuning...) - in an instrument with inharmonic partials the nth root of two is compromised already (if speaking of equal temperaments); an instrument with inconsistent inharmonicity asks judgement as far as how it is to be tempered, down to its smallest intervals (and so why any form of alternate temperament should not be unfamiliar to a tuner). <blockquote TYPE=CITE> <pre>I have begun to think less of different tuning styles and stretches as right or wrong.. rather as one or another set of priorities from the tuner</pre> </blockquote> Well put indeed. A temperament that maximizes consonance in one integer ratio (like [approximate] 3/2 vs. 5/4) is no less an accomplishment than one that attempts maximizing 2 or 3. But our stricter sense of temperament has appropriateness in mind as well, a correctness that only should be decided with the intended music. At least this ought to be as consistent as possible throughout the compass. Clark (thinks "why stop at 12?")</html>