[pianotech] Excerpts from Gary Shulze mail on P-12ths

[Richard Brekne]

For your edification:

.... "Influences of Inharmonicity on Aural Tests in Equal Temperament". 
It's nice to see that it holds up after all these years. A friend and 
former colleague called me and told me about the on-going discussion. 
This article was reprinted, with a minor correction, in "The PTG Exams: 
A Source Book"in 1993. I am somewhat embarrassed by the pomposity of the 
title - I later retitled it as "Perfecting the Fifth" for the Twin 
Cities PTG chapter newsletter.
 
My entire intent of the article was to propose "...incorporating a 
different, although still compatible, system of temperament: the equally 
tempered P19." And to echo Dr. Sanderson's thinking that Inharmonicity 
isn't a disease, but a blessing. The piano, because of it's 
inharmonicity, satisfies both melodically and harmonically. For a "fixed 
pitch" instrument, this is astounding.
 
I've always been an aural tuner, and (I was under the influence of Owen 
Jorgensen at the time as his teaching assistant) set about to propose 
aural tests for proving P12s & p19s. I threw in the comma footnote just 
for the hell of it. I always suspected that I was the first to propose 
this in writing - but I had already heard and deduced through 
discussions with other tuners that P12s may be the way things actually 
worked.
 
I even did up a program in Fortran (I think it was), doing "paper 
tunings",  using and comparing P12 and P19 temperaments, incorporating 
inharmonicity equations, and comparing the results to a S&S "D' tuning 
measured with an Accu-tuner. This, back in '81 was tedious. I had reams 
of graph paper all over the place. That's when I knew I was on to 
something. But the focus was meant to be always "aural". I've always 
believed (call me a Luddite) that aural tuning is more accurate and 
faster. But I admit prejudice. I do think its more fun. I once wrote 
elsewhere that tuning was a "complex puzzle with no real solution. The 
pieces aren't made to fit. We can only produce the illusion of a 
solution: we are forced to cheat".

Gary Shulze
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I ran across a companion paper I did for an independent study in 
acoustics at Michigan State, dated June 8, 1981.
 
My goodness was it ever ponderous, with an equation taking up half a 
page that I used to compare various tuning ratios under the influence of 
inharmonicity with the readings from actual tuning.
 
One quote from the abstract, which echoes somewhat the PTG article, and 
expands a little also:
 
"...assuming that the best tuning achieves the most nearly coincident 
partials while exhibiting a smooth rate of deviation, it is shown that, 
of any one ratio, a 6:1 tuning comes closest to 'ideal'... This theory 
is also shown to comply well with standard tuning practice, and allows 
for an extra degree of control over  accumulated errors."
 
Anyhow, the gist & thrust of all this was to champion the use of P19 & 
P12, and also, concurrent with using the "equal beating" aural tests 
presented in the article, draw attention to the importance of the minor 
3rd. i.e. striving for a consistent relationship between major & minor 
thirds.

Gary Shulze