It took me a while, but was able to run through some of the numbers; if you are interested I can give you my results from A4-C7. The difference between 3:1 and 2:1 is not that much, but it does add a small amount of "extra" stretch to the overall temperament. Why do you feel that this small amount of stretch past a 2:1 octave is helpful? Using your method with my piano, I get a .190 total beatrate and a +0.3729 cent additional stretch at A4-A5; at C7-C8 I get a 3.819 beatrate with a +1.5476 cent additional stretch. It is an interesting approach, but I'm not sure that it is a good idea to sacrifice the sound of a single octave like that. I personally do add a very small amount of "extra" stretch to my single octaves (past that of a non-beating 2:1). But for me, I feel I get a better result by balancing the sound of a 2:1/4:1 single octave. In addition to closer coinciding overtones, this method also allows for a better sounding double octaves. Bradley M. Snook
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