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 --------------------------------------------------------------- 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
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