<html> Hello, Bradley. I'm glad to see that great minds are (finally) working alike. As you were writing this post, I was hauling out Jorgensen's tuning book to go over just intervals. I took "just" to mean simple ratios, which expressed themselves as unbeating intervals, but found I was very rusty on the subject of just thirds. Wouldn't it be cute if I were simply wrong, and just thirds and sixths DID beat, and they WERE used in string playing? Well, the glossary was helpful (page 773). "Some commonly known just intervals are the ratios 1:2 octave of 1200 cents, 2:3 fifth of 702 cents, 3:4 fourth of 498 cents, 4:5 major third of 386 cents, 5:6 minor third of 316 cents, and 6:7 small minor third of 267 cents." From this one can see that octaves, fifths, and fourths are extremely close to the tempered intervals, and therefore a string player can play them "just" and still agree quite well with a piano. But the just major third -- 14 cents narrow; the minor third -- 16 cents wide; and the "other" minor third, a useless 33 cents narrow; how to reconcile them? I really don't think that a string player will be willing to bend everything that far out of shape, just to get a non-beating interval, when they are vibrating and usually playing the notes in succession instead of simultaneously most of the time, anyway. I also found page 18, where very early Pythagorean temperaments were discussed. They were derived by tuning octaves, fourths, and fifths just, which yielded wide Pythagorean major thirds. Jorgensen talked about a tempering system used by Boulliau, for fretted instruments. It was a Pythagorean temperament, and gave major thirds of differing sizes, listed in Table 7-2, on page 20. It dates from the 14th century. Jorgensen says about it: "As a just intonation it was the most amazing system in history because one could modulate through all twenty-four major and minor keys in it without encountering wolf intervals, though there were two very harsh fifths. Musically, it served as a substitute for equal temperament. By comparison with equal temperament, it was extremely easy to calculate on paper and transfer to the frets of lutes and viols." It sounds to me like this system may be about as just as a fixed pitch instrument can manage, but the major thirds, as given on page 20, seem to come in three flavors; 6.77 cents wide from just; 12.78 cents wide from just; and 21.5 cents wide from just. I did check out Hilary Hahn, by the way. I went to Amazon and listened to sound clips of her playing Bach.  Good playing. Thank you. From what you were saying, I expected playing with very little vibrato, and strangely narrow (from tempered) major thirds, and wide minor thirds, but I didn't find them. Plenty of vibrato, and I thought she showed a strong bias toward wide major thirds, and toward sharping as she reached for major thirds and sixths, sort of an aggressive reaching for them, which I found very pleasing. Also small semitones when resolving leading tones. And a tendency to sharp the upper register a little, but not to excess. Very tasteful, really. Maybe you are hearing something different, and it would be interesting to find out what it is. Here's the link to get the sound clips. The most telling one for interval size I found to be number five, the Minuet II from the third partita (in E.) <a href="http://www.amazon.com/exec/obidos/ASIN/B0000029UH/qid=1020641792/sr=2-2/ref=sr_2_2/002-2730038-9330451" eudora="autourl">http://www.amazon.com/exec/obidos/ASIN/B0000029UH/qid%3D1020641792/sr%3D2-2/ref%3Dsr%5F2%5F2/002-2730038-9330451</a> A score would have helped, but I went to a piano and found out the note names. The same double-stopped line occurs again and again, which helps to hear it. The lower line is moving, against the stationary upper note (B.) The lower line is G# - A - G# - F# - E - F# - G# - A Then it moves lower, so that E - D# - E can be easily heard. The G# and D# she plays very high, and she resolves them by a very small semitone, smaller than tempered by a good bit. Her G# - B doublestop is a very pleasing (to me) quite narrow minor third. It is smaller than tempered by a good lick; but it is smaller than just by a humongous amount. Now, what were we arguing about, again? Regards, Susan Kline At 05:59 PM 5/5/2002 -0700, you wrote: <blockquote type=cite class=cite cite>I am trying to work on a good definition for just intervals, but I am kind of stuck . . .   I have all the just intervals worked out (i.e. smallest ratios): P5  3/2 P4  4/3 M6  5/3 M3  5/4 m7  7/4 m3  6/5 TT  7/5 m6 8/5 M7 9/5 M2  7/6 m2  8/7   This takes care of the simple ratio aspect, but it doesn't really address the psychoacoustical aspect of beating. You only get beating if you have overtones (sine tones will not create beats even if they are set out-of-tune; you will only hear very, very slight flutterings). Since overtones on the piano grow sharp because of inharmonicity, it's possible to set an interval just (non-beating) but then the fundamental ratios are slightly expanded.   Does anyone have any suggestions?   Bradley M. Snook</blockquote></html>