<!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> <DIV>Robin, thanks for everything that you wrote; it was very interesting reading.</DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV>I think that you are right when you point out that a 9/8 just M2 is closer to EQT than 7/6. But EQT doesn't really have anything to do with the justness of intervals, right? A simple just interval is a simple just interval, whether or not it is even included in our scale. They are not used on the piano because we favor intervals like the P5 and M3 over intervals like the M2 and m2.</DIV> <DIV> </DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>><EM> The ratio of an interval, if smaller than an octave, taken ascending,  when multiplied by the ratio of the interval that is the musical  inversion of the first, also taken ascending,  must equal two.</EM></FONT></DIV> <DIV>Hummm . . . I don't think that this is an applicable assumption. What you say turns out to be true, but does that really factor in to what constitutes the justness on an interval? Besides, all the intervals that I put down are also true for these parameters.</DIV> <DIV> </DIV> <DIV> </DIV> <DIV><FONT face=Arial size=2><EM>. . . the fundamental nature of just intervals; they are the paradigm of the tempered intervals we are accustomed to</EM>.  </FONT></DIV> <DIV><FONT>I don't think that I understand this statement clearly. But I did not mean to imply that tempered intervals can start to sound 'just' simply because we are accustomed to them.</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2><EM>> Listening to choral music, great orchestral performances and string music and, in particular, string quartets, I don't sense the unpleasant aspect of the tempered values; it is obvious to me that their tuning of harmonic values, if not completely just,  is substantially closer than tempered values.</EM></FONT></DIV> <DIV><FONT face=Arial size=2><FONT face="Times New Roman" size=3>I have been arguing about this with Susan for a while, and my opinion sides with that statement exactly. The only thing that I would add is that we also temper intervals occasionally to help increase tension (and also just play out-of-tune), but what we are shooting for is just.</FONT></FONT></DIV> <DIV> </DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2><EM>>   I believe that people in general are conditioned on an unconscious level by the vast amount of music they hear, the great bulk of which is in equal temperament, so that, when asked to make sounds or sing a pitch level, the sounds so generated will be centered around the  frequencies generated by Equal Temperament at A-440 to a significant degree such that this cannot possibly be random. </EM></FONT></DIV> <DIV><FONT face=Arial size=2><FONT face="Times New Roman" size=3>That is a good thesis statement, but it is not true. Go to any very high level music school/conservatory and listen to some an ear training class. Even though the musicians are very highly trained on their specific instrument, I have never heard anyone even come close to centering around EQT. Even some of the pianist that have perfect pitch have a difficult time.</FONT> <FONT face="Times New Roman" size=3>I am not saying that it is impossible, but I am saying that it does not happen without specific training. There are some string players that specialize in duo work; they must train very hard to approximate EQT.</FONT></FONT></DIV> <DIV><FONT face=Arial size=2></FONT><FONT face=Arial size=2> </DIV> <DIV> </DIV> <DIV><BR><EM>>      As people have an unconscious, conditioned,  memory of pitch, they similarly have a conditioned sense of tempered intervals, particularly thirds and sixths . . .</EM></DIV> <DIV><FONT face="Times New Roman" size=3>I disagree (I think, if I understand correctly): if you really think that people have a good sense on tempered intervals, then you should have no problem setting an EQT  temperament octave using only tempered thirds or sixths.</FONT></DIV> <DIV><FONT face="Times New Roman" size=3></FONT> </DIV> <DIV><FONT face="Times New Roman" size=3></FONT> </DIV> <DIV><FONT face="Times New Roman" size=3>You have many interesting points (I think), but I am not sure if I understood half of them: maybe shorter sentences would help?</FONT></DIV> <DIV><FONT face="Times New Roman" size=3></FONT> </DIV> <DIV><FONT face="Times New Roman" size=3></FONT> </DIV> <DIV><FONT face="Times New Roman" size=3>Bradley M. Snook</FONT></DIV> <DIV> </DIV> <DIV></FONT> </DIV></BODY></HTML>