Keenan Tuning

Paul Erlich has kindly referred to "Keenan tuning" a couple of times recently, as the only known member of a unique class. I believe this is the class of tunings that obtain a benefit, in terms of increased quantity and accuracy of 9-limit harmonies, from having two different sizes of usable fifth (giving 3 sizes of scale step).

It isn't a well-temperament because it has two wolves. It turns out that there is a previously unsuspected local error-minimum on the continuum between 9-limit Just scales and tempered scales having two-chains of near-22-tET-sized fifths a half octave apart (Graham Breed's "diaskismic scales").

I've finally put some stuff about it on my website. http://dkeenan.com/Music/KeenanTuning.htm

It includes a pretty colored chart showing the minimum in the error surface and you can download the spreadsheet that generated it, in case you want to play with different error weights or whatever.

Regards,
-- Dave Keenan
http://dkeenan.com