Hi, Kent. Thanks for taking the time to spell this out. You were thinking more globally, while I was thinking more locally. You and John are both correct in the sense in which you were thinking. Point acknowledged. -Mark Schecter 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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