<HTML>In a message dated 8/18/02 10:28:42 AM Central Daylight Time, Richard.Brekne@grieg.uib.no (Richard Brekne) writes: <BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">The following is an except from appendix F of the SAT manual. It gives an explanation by Dr Sanderson that shows why David Anderson was "correct" about 4ths having the same slow beat rates, and why I was "correct" in stating that 4ths should have a contiguous relationship as 3rds do. "Two contiguous musical intervals are intervals that touch each other, in other words, share the note in the middle. Tests that use contiguous intervals are easy to learn and use, and tell the tuner explicitly which notes are at fault and what to do to correct them. Contiguous major thirds will beat in the ratio of four to five because the major third itself consists of two notes whose frequencies are in the ratio of four to five. Displacing any interval up the keyboard will speed it up theoretically in the ratio of the frequencies of the two root notes involved.  Therefore two contiguous major thirds should beat in the ratio of four to five, two contiguous minor thirds in the ratio of five to six.Similarly, two contiguous fourths should beat in the ratio of three to four and two contiguous fifths in the ratio of two to three.  However, on the piano this theoretical relationship holds well only for the major and minor  thirds.    The  fourths  and  fifths  are  so strongly  affected  by  inharmonicity  that  these contiguous intervals beat at almost the same speeds" </BLOCKQUOTE> Thanks for posting this, Richard.  It does provide the standard, textbook answer.  It applies to the very neutral sound that is striven for in the PTG Tuning Exam which many people look to as being ideal. But as always, this is not the only satisfactory compromise there can be.  In reality, the most appealing (for most people) stretch in an ET tuning will cause the 5ths to become pure and even expanded somewhere in the 6th octave and beyond, the same is true for the lower part of the 2nd octave and below.  Steve Fairchild RPT demonstrated this at more than one PTG Annual Convention a decade and more ago.  From what I have been reading that you are doing, it is along these lines. Consider this:  The higher the partials the more wildly distorted inharmonicity becomes.  Therefore, trying to "tune" notes in the upper 6th octave through to C8 to make them match in every way the notes below is quite impossible.  One kind of match will surely cause another obvious mismatch.  Virgil Smith RPT's aural tuning approach which uses a trained and experienced perception to create a nice, blended compromise without heavily favoring or singling out any particular set of partials has always worked for him. On the other hand, he knows how to create a *contracted* tuning when he wants to and also seems to know how to make triple octaves be in tune when he wants that.  I learned that from him, in fact, way back in the early 1980's at a Regional Seminar in Northern Illinois.  I used to say that I learned how Rapidly Beating Intervals (RBI) should sound from the late George Defebaugh RPT and how Slowly Beating Intervals (SBI) should sound from Virgil Smith. Since the 4th is the same interval as the 5th, only inverted, it makes sense that if the 5ths are caused to be closer than theoretical to pure, actually pure or even expanded, the corresponding 4ths will become faster as a result.  If, in the mid range, the 5ths are allowed to be more contracted, causing the 3rds to beat more mildly, so will the 4ths beat more slowly. Concentrating on getting nice, even 4ths across the range where they are audible is not a bad idea at all; it will result in a very pleasing and harmonious arrangement.  It however, is not the only workable idea.  In my experience, the 4ths diminish in audibility rather quickly in the middle of the 5th octave, so I no longer pay attention to them after that.  I believe this is because the strength of these coincident partials is somewhat less than others being considered in the same range.  Also, anywhere below the middle of the 2nd octave, the 4th just becomes a growl which is not necessary to ponder; other intervals such as double octaves, octaves and 5ths, 10ths and 17ths are more important. Bill Bremmer RPT Madison, Wisconsin <A HREF="http://www.billbremmer.com/">Click here: -=w w w . b i l l b r e m m e r . c o m =-</A> </HTML>