P12 in Tunelab Pro / P12 theoretical basics

[Bernhard Stopper]

Not exactly, since you should take  440 * 3,somewhat^(1/19) (because of
inharmonicity of the 3rd harmonic what is not negligible at this point). But
i think we can say that the aural integration of the octave spectrum fits
much better to inharmonicity in P12 than with standard-ET octaves. (The
"acoustic octave" effect.)

friendly wishes.

Bernhard

----- Original Message ----- 
From: "Richard Brekne" <Richard.Brekne@grieg.uib.no>
To: "Pianotech" <pianotech@ptg.org>
Sent: Thursday, June 03, 2004 12:44 AM
Subject: Re: P12 in Tunelab Pro / P12 theoretical basics


> Hi Bernhard
>
> Hi again Bernhard
>
> So lets put this into some more perspective... this is getting really
> interesting by the way... my attention is really caught !
>
> The pythagorean comma of roughly 23.46 is usually divided into 12 parts
> gives 1.955 cents needed to yeild a pure 2:1 octave. If we divide the
> comma into 19 parts then we have as you say roughly 1.235 cents needed
> to end up with a pure 3:1 12th
>
> Translated into our familiar root of twelve and root of nineteen and
> useing 440 as an example... :
>
> 440 * 2^(1/12) twelve times will yeild 880, but 440 * 3^(1/19) twelve
> times will yeild 880.6278 for the 2:1 relationship. Adding abou 0.63 bps
> before inharmonicity is considered to the A3-A4 octave.  Now.... if you
> tune an A3-A4 octave to a pure 6:3 octave type... then the 2:1 type gets
> stretched by a very comparable amount.... i.e. close to 0.63 bps for the
2:1
>
> One of my thoughts all along about the P12th that I do is that it more
> or less automatically takes consideration to inharmoncity.
>
> Just a thought
>
> RicB
>
>
>
>
>
>
> Bernhard Stopper wrote:
>
> > some additional thoughts....
> >
> > *Twelve fifths - pythagorean comma = seven octaves*
> > mathematically *(3/(2^(pc/12)))^12 = (2)^7*
> > is *traditional equal temperament*
> >
> > *Twelve fifths= seven octaves  +  pythagorean comma*
> > mathematically *(3/2)^12 = (2)^(pc/7)^7*
> > is *equal pure 5th (Cordier)*
> >
> > *Twelve 12ths = 19 Octaves + pythagorean comma*
> > mathematically *3^12 = 2^(pc/19)^19*
> > *or*
> > *3^12 = 3^(12/19)^19*
> > *is equal pure 12th *(Stopper)* *
> > *or "acoustic octave transformed pythagorean"*
> >
> >
> > the list of the "pythagorean" given below is not complete, it can be
> > extended to any interval combination of 3 and 2 ratio of the keyboard
> >
> > regards,
> >
> > Bernhard Stopper
>
>
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