Hey, Aural Gurus ...

[Bernhard Stopper]

William, Ric, Alan,

the main thing behind the P12 tuning is to take the pythagorean comma from 
the fifths/twelfths side to the octave side. It is a complete different 
thing relating tuning theory. It makes no sense to me to prefer any partials 
pair in the octave, since a pure octave is not what we want if we decide to 
break down a perfect twelfth into 19 steps. Any other thoughts into this 
direction is to try the quadrature of a circle. Although it is highly 
arbitrarily to choose what partial pair. And if you choose any pair, this 
must not be the state where the sum of the beats is minimal. And the state 
where the sum of the beats is minimal, that is the pure state for me. And in 
that case the quadrature of the circle begins again. Pure twelfth tuning 
means a not pure octave in no way.

Read my next post about my new P12 approach called "OnlyPure"

regards,

Bernhard

----- Original Message ----- 
From: "William Ballard" <yardbird@vermontel.net>
To: "Pianotech" <pianotech@ptg.org>
Sent: Friday, April 08, 2005 6:35 AM
Subject: RE: Hey, Aural Gurus ...


> 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
> +++++++++++++++++++++
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