P12 in Tunelab Pro / P12 theoretical basics

[Richard Brekne]

Bernhard Stopper wrote:

> Ric,
> I agree that the tuning itself was tendencially practiced more or less 
> by good tuners already long time ago.
>  
> What i claim as new is mainly the theory behind the P12 tuning. 
> Especially the transformation of the standard  12-5th circle that has 
> to be closed with 7 octaves into a 12-12th circle that is closed with 
> 19 octaves. And the direct transformation of the pythagorean tuning 
> into equal P12 by replacing simply the mathmatical 2/1 octave 
> ratio with the "acoustic octave" (later explained).
> (The effect of inharmonicity can be divided out at this point, it is 
> added later as instrument immanent factor that stretches all ratios 
> according to the instrument inharmonicity curve)
>  
> For those who are not familiar with any maths the traditional fifth 
> circle can be expressed in words as
>  
> Twelve fifths = seven octaves + pythagorean comma
>  
> Mathematically this can be written as ( 3/2 for 5th, pc for 
> pythagorean comma and 2 for octave ratio):
>  
> (3/2)^12 = (2)^7 + pc
>  
> Now comes the transformation trick:
>  
> Dividing out the twelve fifths give:
>  
> (3^12) / (2^12) = 2^7 + pc
>  

Bernhard..

Something doesnt add up here... perhaps just a notation confusion... but

(2)^7 + pc is the exact same thing as 2^7 + pc

so essentially the above two statements mean that

(3/2)^12 = (3^12) / (2^12) 

which clearly cannot be.

Please correct and then repost. 

Otherwise... I just have to say my own rendering was much more seat of 
the pants as it were.  I simply noticed that we tune using very  close 
to perfect 12ths in the tenor/treble/diskant registers as a rule... so I 
tried it out and used the previously mentioned procedure with Tune Lab 
97 to help assure a truly P 12th interval everywhere, and to divide the 
<<temperament 12th>> into its associated curve.  I wanted at the time to 
see if the idea would sound good.  It did, and viola... the P 12ths 
tuning was <<born>> for me... which I related to others a couple years ago.

I never did get around to trying to figure a mathematical justification 
for it... as clearly it was not really needed to understand it sounded 
good, and to justify its usage based on that.

I'm leaving for Japan in a couple days, so its doubtful I will be able 
to continue this thread much more beyond tomorrow evening, and I really 
had wanted to browse through my old journals and look for that reference 
I mentioned yesterday.  Seemed to me that somebody stateside wrote 
something rather along the lines you sketch out in the late 70's... tho 
I wont make any absolute claim in this regard until I re-find the 
article.  Regardless of the time, I remember smiling to myself about it 
as it provided ready made justification that my own stumbling onto the 
idea was a perfectly feasible way of tuning. Your own work provides even 
greater support... or perhaps better said the other way around.... the 
fact that my own experiences from a purely empirical direction bear out 
the tuning goes to support the claims you made some 15 years ago.

Cheers
RicB