Hey, Aural Gurus ...

[William Ballard]

At 10:39 PM -0500 4/5/05, Alan wrote:
>I'm still curious about the P12th D3-A4 bit, though. Is the logic correct or
>would different pianos require a, shall we say, less-than-perfect 12th?

Just kind of poking my head through the window here and admittedly 
not reading all of the posts in this thread, but.....

Are you asking for an aural temperament in which the octaves have 
been stretched sufficiently that 12ths occurring within the 
temperament compass (yes a two-octave compass) will be pure. That's 
how Ric and the other-'cross-the-ponder-er were laying out octaves in 
general, Ric using an ETD and t'other stating that the mathematical 
principle was there. That was back in September.

I've been laying out my octaves aurally using P12ths for years, and 
as a general rule, if you tune C5 to F3 a pure 12th, the octave C4 to 
C5 will be wide at the 2:1 and 4:2 and narrow at the 6:3. The 2:1 and 
4:2 will be wide because (and oh yes, if) the F3-C4 5th is narrow 
(both at its 3:2 and 6:4). There is no direct interval test to offer 
the same proof about 6:3 level of the C4-C5 octave. But in most good 
scales the 3:1 falls in between the 2:4 and the 6:3. And if your ear 
likes to hear "motionless octaves", the 3:1 is a good balance.

(I call the 3:1 an octave, because it walks like an octave and quacks 
like one. And I use it like one.)

So last September, I came up with a way of establishing a temperament 
octave (A4-A3), stretched on the basis of a pure D3-A4 12th. All I 
can remember at this point was that it was easier than I thought. In 
fact, once I had set D3 pure to A4, the next step may have been to 
locate A3 dead in the middle between those two, ie., such that the 
3:2 between D3-A3 was identical in beat rate to the 2:1 between A3-A4.

My temperament is the classic 3ds&6ths pattern, so with the A#-A4 
octave set, the other three "rungs of the ladder" (C#4 and F4) fill 
in easily, and the F3-F4 and C#3-C#4 get a stretch similar to the 
A3-A4 octave. (Yes, the beat rate of that stretch should slow down, 
as you walk down the scale in frequency. But if you duplicate the 2:1 
stretch of those next two octaves from the first one, you be starting 
from something which you know is wide of the mark. You may even be 
able to hear this extra width in the F3-A3 and C#3-F3 3ds, which 
don't slow down as must as they should. The proper width in cents 
lies somewhere just narrow of these octaves.

When the C#3 is set, your next ladder begins with a C#3-G#4 P12th, 
and the only significant difference between this and the usual 
3ds&6ths pattern is that your ladders are moving downwards instead of 
upwards. (I seem to have remembered it pretty well, although I may 
have a sound file of this temperament recorded that afternoon on my 
laptop, tucked away on some hard drive.)

So Alan, was that what you were looking for, an aural temperament 
built on octaves stretched to the point that 12ths contained within 
the temperament would be pure? Personally I think such a temperament 
is more reliably calculated by an ETD. This aural pattern makes the 
assumption that the properly stretched A3-A4 octave lying inside the 
D3-A4 P12th is one where the D3-A3 5th and the A3-A4 octave which 
make up that 12th beat equally, ie., they "split the difference". 
Admittedly an arbitrary assumption. Also the business of keeping the 
width in ¢ of the F3-F4 and C#3-C#4 octaves equal to that of the 
A3-A4 octave is hardly scientific.

That's why I say that an ETD can do this better than an aural tuner. 
But that doesn't bother me because on most piano, I happy with a 6:3 
octave from C3-C4 on down, and in a piano on which that's too wild, 
nothing else will sound much better.

Back to sleep now...

Bill Ballard RPT
NH Chapter, P.T.G.

"Can you check out this middle C?. It "whangs' - (or twangs?)
     Thanks so much, Ginger"
     ...........Service Request
+++++++++++++++++++++