[CAUT] The Origins of P12ths tuning.

[Richard Brekne]

Hi Folks

Ok... as promised I hunted down the article that I have said all along 
contains the information that prompted me to create the P-12ths tuning 
on Tunelab in 2000 along with all subsequent discussion on both lists 
prior to Bernhard Stoppers appearance on these lists some years later. 
As it turns out, this article PTG Journal May 1982 by Gary Shulze RTT, 
six years before Stoppers publication,  contains the entire basis for 
the P12th tuning along with the mathematical definition for the a 
Perfect 12ths stretch.

I quote along the accompanying footnote from the article by Shulze and a 
comparison quote from Stoppers own publication defining the comma he 
claims as his.

    "The basis of equal temperament theory is the attempt to make all
    fifths as beatless as possible, while preserving the purity of the
    octave. With the modern piano in equal temperament, the beat rate of
    the fifth does not double with each octave as theory would dictate,
    but remains reasonably pure. In fact, the twetfth and nineteenth are
    no more tempered then are the octave and fifteenth. The "stretch"
    necessitated by inharmonicity seems quite consistent with the
    stretch inherent to the tuning of these intervals pure --- footnote # 2"

That footnote states the following

    "This stretch, disregarding inharmonicity, is exactly equal (in
    cents) to: Log (3/(2^(19/12)) * 1200 / Log 2 for  the perfect 12th
    and Log (6/(2^31/12))*1200/ Log 2 for the perfect ninteenth.

Stoppers quote defining the comma he claims as his from his publication 
in 1988,  six years after the Shulze article

        "now rewriting the pythagorean comma as 3^12/2^19 yields for the
    Stopper comma (sc) (3^12/2^19)^(1/19) or  3^(12/19)/2"

The likenesses in thinking are obvious.

The Shulze article goes on to describe what is the essence of  the 
P-12ths tuning without directly giving it that title. He describes the 
appropriate usage of the 6:1 and 3:1 partials and why they work as they 
do, along with a good deal of other relevant thinking.

There is another article in one of the earlier journals where this 
subject matter is taken up again. It is much shorter column as I recall 
and specifically mentions the 19th root of three in plain text.  I will 
find it as well.

I encourage all of you who are interested in the P-12ths tuning to read 
the article "Influences of Inharmonicity on Aural Tests in Equal 
Temperament", Gary Shulze, RTT Lansing, MI Chapter, PTG Journal May 
1982.  Its good reading and gives some very good answers to many of the 
questions the recent related threads have posed. And I sincerely hope 
the issues surrounding who did what, where, when, and under what 
influences is once and for all put to an honorable rest.

Sincerely
Richard Brekne, RPT