In a message dated 10/26/00 7:35:38 PM Central Daylight Time, rscott@wwnet.net (Robert Scott) writes: << HT enthusiasts: Based on aural tuning directions that Bill Bremmer provided, I have calculated the following offsets for his EBVT temperament: F 3.10 F# -1.28 G 4.50 G# 2.49 A 0.00 A# 1.63 B 0.58 C 5.35 C# 0.98 D 1.81 D# 2.00 E -0.86 or, if you prefer offsets that are more zero-centered: F 1.41 F# -2.97 G 2.81 G# 0.80 A -1.69 A# -0.06 B -1.11 C 3.66 C# -0.71 D 0.12 D# 0.31 E -2.55 In finding these offsets I developed a software tool called the Temperament Designer which can hopefully help to translate aural tuning instructions into offsets from ET for other "historical" temperaments as well. One difficulty in translating aural instructions is that they depend somewhat on inharmonicity. So the computer program that I designed uses an ET tuning file and an inharmonicity file on which the calculation of beats is based. Then it provides the ability to modify the ET tuning according to aural instructions based on interval beats. Once the program was written, it took me about 10 minutes to follow the aural instructions for Bill's EBVT and derive a "virtual tuning". Then the offsets for a temperament octave are written out to a file which can be read into TuneLab, or manually entered into an SAT or RCT. The effect of inharmonicity in this process, while noticeable, is small. Therefore I have every hope that the offsets derived based on the "typical" inharmonicity data that I used will apply equally well to most pianos. I also investigated the effect of starting with two different "ET" tuning curves using different partials. That also did not affect the outcome very much, as long as the ET tuning curve was a reasonably good match to the inharmonicity. >> Thank you so much, Mr. Scott for coming up with these figures. I will try today to see how well they work. Regarding the inharmonicity problem, I discovered something that may be of use to you when I was trying to create a Meantone Temperament by a Direct Interval Programming method. Any Meantone can be thought of as a chain of 11 tempered 5ths, all tempered by the same amount. The theoretical amount was assumed to be not exactly correct but knowing that different pianos have different amounts of inharmonicity, the problem was to determine how much to change the theoretical amount. I came up with the answer that there could only be 3 choices, High, Moderate or Low Inharmonicity. If for example, the tempered 5th of 1/7 Comma Meantone is theoretically -3.07 cents (dropping the -.07, leaving 3), I could drop another .1 cent for a Low, (-2.9 cent 5ths), .2 for a Moderate, (-2.8 cent 5ths) and a .3 (-2.7 cent 5ths) for a High Inharmonicity piano. The difference at the end of the chain of 5ths is significant: 1.1 for Low, 2.2 for Moderate and 3.3 for High. Any more of a difference would bring me into the realm (too close) of another comma, say 1/8 Comma. This difference means that a Steinway (High Inharmonicity) will have a "Wolf" that is 2.2 cents less wide and less dissonant than a Kawai, Baldwin SD-10 or Mason & Hamlin, for example. So, the suggestion is that instead of having just one "typical" Inharmonicity factor, there could be three. That determination could be made by a simple sampling, the way Dr. Sanderson did with the original "Stretch Factor" sample (which is what I use). This choice of three would hopefully reflect the same kinds of differences that aural tuners make naturally. To answer Avery's question, your second set of "Zero-Centered" figures gives the smallest possible deviation from ET. This makes the temperament and tuning the most "compatible" it can be with other instruments tuned to Standard Pitch. If tuned to these figures, a piano played unison for unison with another keyboard tuned in ET will reveal that most notes seem to be "in tune", some even dead on and just a few with a slight and slow beat, not enough to be perceived as being "out of tune" with the instrument tuned in ET. Certainly with orchestral instruments, even the oboe, the EBVT, if tuned with the proper pitch correction, will be compatible with all other instruments and can be used to play any kind of music that ET is called upon to play. Indeed, instrumentalists will often find that the piano sounds more "in tune" this way. The beating, the resonance and the intervals which are really pure will much more likely match the kind of intonation that orchestral musicians try to achieve naturally. My sincerest thanks and regards, Bill Bremmer RPT Madison, Wisconsin
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