Sorry for the blip in the following incorrect sentence: "The more useful note to look at might be A#3. In checking the 4:2 F3- F4 octave, the difference in beat rate between the F3-A#3 4th and the A#3-F4 5th _is_ the beat rate of the 4:2 octave. Any change in the tuning of A#3 will move the beat rates of the 4th and 5th in opposite directions." What I meant to discuss was that A#3 was a check note for the F3-F4 octave, and thus any movement of the A#3 would not change the _difference_ between the beat rates of the F3-A#3 4th and the A#3-F4 5th. This means A#3 is a great check note for the F3-F4 octave, but there is still only one place to put the A#3 correctly and maintain equal temperament. The point, as poorly as I made it, is still that you can't slow down both the 4ths and 5ths and maintain equal temperament. Kent On May 4, 2006, at 6:15 AM, Kent Swafford wrote: > > On May 4, 2006, at 12:09 AM, Mark Schecter wrote: > >> Hi, John. >> >> Sorry, but that's not correct. > > No, John was correct. If you assume that equal temperament is being > tuned, then there is indeed an inverse relationship between the > beat rates of the fourths and fifths. That is, if you widen the > octave within which equal temperament is tuned, then all the fifths > get wider (slower beat rate) and all the fourths get wider (faster > beat rate). > >> If the fifth above the bottom note of a 4:2 octave is made less >> contracted, approaching just, the fourth below the top note will >> also get slower, as it contracts from its expanded state toward just. > > The beat rate relationship that you cite is correct, but you > describe raising the C4 within an F3-F4 octave, which you cannot do > and still maintain equal temperament. If the F3-F4 octave is > properly tuned, then one could tune the C4 so that both the F3-C4 > fifth and the C4-F4 4th are just, but that is irrelevant to equal > temperament. > > The more useful note to look at might be A#3. In checking the 4:2 > F3-F4 octave, the difference in beat rate between the F3-A#3 4th > and the A#3-F4 5th _is_ the beat rate of the 4:2 octave. Any change > in the tuning of A#3 will move the beat rates of the 4th and 5th in > opposite directions. > > Assume an accurately tuned temperament octave; if you raise the F4 > and widen the F3-F4 octave, then you would also have to raise > _both_ the A#3 and the C4, widening _all_ of the intervals within > the octave. Widening all the intervals would speed up the 4ths but > slow down the 5ths. > > You can't slow down both the 4ths _and_ 5ths and still maintain > equal temperament. > > HTH. 8^) > > > Kent > > > > >> For example, if the octave is F3-F4, and the fifth above F3 is C4, >> the act of lowering C4 to contract the fifth, expands the fourth >> C4-F4. Contrariwise, if you then raise C4 to slow the fifth F3-C4, >> so doing also contracts the expanded fourth C4-F4 toward just, or >> beatless. It's easier to picture than to say. HTH. >> >> -Mark >> >> John M. Formsma wrote: >>> How do you get 4ths and 5ths to both be slower? In equal >>> temperament, a >>> slower 5th means a faster 4th. >
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