P12 in Tunelab Pro / P12 theoretical basics

[Richard Brekne]

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