Hi RicM I got the feeling that this was more or less what Dr Sanderson was saying.... tho he didnt come right out and say so directly about thirds. He did say right out that inharmonicity changes the theoretical picture relative to 5ths and 4ths... which should also have their contiguous relationships. So I imagine he would agree real quickly with what you point out below. This goes back to what I tried to say to David Andersen about the whole definition of ET being really quite impossible to implement on the piano. Inharmonicity precludes that, and we are left with an ET adjusted for inharmonicity which essentially breaks some of the rules for what an ET is. Cheers RicB Richard Moody wrote: > The beat rates of contiguous 3rds if tuned in ET beat at the ET > ratio, which is NOT 5/4. Take the beat rate of C--E and then > E--G# and you will see the ratio is not the EXACT ratio of 5/4. > Actually the ratio of beat rates of two contiguous 5/4 3rds are > zero because those two thirds 3rds have no beat. Dr Sanderson > must have been giving a generalized explanation rather than a > mathematically correct one. The proof of the ratios of 3rds in > ET is that ratio cubed, (or ^3) equals 2. 1.25992105 cubed > equals 2 > > The ratio of the frequencies of ET 3rds is 1.25992105 , the > ratio of their beat frequencies is 1.25992105 . How can it be > otherwise? -----ric >
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