<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> <HTML><HEAD> <META http-equiv=Content-Type content="text/html; charset=iso-8859-1"> <META content="MSHTML 6.00.2715.400" name=GENERATOR> <STYLE></STYLE> </HEAD> <BODY bgColor=#ffffff> <DIV><FONT face=Arial size=2>I am trying to work on a good definition for just intervals, but I am kind of stuck . . . </FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>I have all the just intervals worked out (i.e. smallest ratios):</FONT></DIV> <DIV><FONT face=Arial size=2>P5  3/2</FONT></DIV> <DIV><FONT face=Arial size=2>P4  4/3</FONT></DIV> <DIV><FONT face=Arial size=2>M6  5/3</FONT></DIV> <DIV><FONT face=Arial size=2>M3  5/4</FONT></DIV> <DIV><FONT face=Arial size=2>m7  7/4</FONT></DIV> <DIV><FONT face=Arial size=2>m3  6/5</FONT></DIV> <DIV><FONT face=Arial size=2>TT  7/5</FONT></DIV> <DIV><FONT face=Arial size=2>m6 8/5</FONT></DIV> <DIV><FONT face=Arial size=2>M7 9/5</FONT></DIV> <DIV><FONT face=Arial size=2>M2  7/6</FONT></DIV> <DIV><FONT face=Arial size=2>m2  8/7</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>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.</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>Does anyone have any suggestions?</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>Bradley M. Snook</FONT></DIV></BODY></HTML>