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

[Kent Swafford]

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.
>