Huh?? was RE: pRCT got ears again...

[Kent Swafford]

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.