Equal beating meantones (was Re: Some more periodicity buzz tests - with chords! Please listen!)

Hi Jacques.

Thanks for your contribution. Interestingly enough, when I first calculated what you call "Madame", it almost immediately reminded me of 1/4-Pyth.comma meantone. And there's one more thing to say here. When I was classifying these meantones back in 2004-2005, I didn't know about the 18th century temperament of Robert Smith; later, Manuel told me that Smith's meantone actually had equal beat rates for 5/3 and 3/2, which makes a minor second of ~120.10856 cents (recall that meantone was used in England right up to late 1840s). And when you consider that "Nikkad" was one of my absolute favorites during 2002-2003 and its major second is ~120.33021 cents, ... Well, isn't that a good coincidence?

Also, back in 2002, I was not thinking about rationalized versions of these temperaments. And, to be honest, even nowadays, I can imagine rationalizing one chord; but rationalizing the entire chromatic scale? That must be full of terribly large numbers. -- Anyway, at the end of 2004, I was thinking more and more about those "quasi-meantone" tunings where some fifths beat heavily and others only have "gentle" beating. So I thought that I could start with a particular frequency like 440Hz or 438Hz and then tune the fifths and thirds just by counting beats per second, which means that the factors can be rational (is the starting frequency is an integer and the beat frequencies are also integers). Finally, I made some scales this way and three of them eventually made it to Manuel's scale archive -- if you have a copy, you should find them by the names starting with "parizek_qmeb".

Petr