I agree, Rainer, I think it's accepted there's no _perfect_ way to introduce meantone or its variants to lute. The trade off of 2 or 3 cents (which I will barely --if at all-- notice) in a 5th or octave for 10 cents for nicer thirds is acceptable to me. - nb. If one is using gut strings all bets are off after a week or so anyway, esp'ly anything above the 5th fret. This related to the age of the gut, weather, finger oils, overstrumming/plucking/fingering wear. - nb2. The act of pressing or over pressing a digit behind a fret will also affect that tuning. - nb3. The attempted meantone fretting of fixed fret instruments like citterns and orpharions shows that it was given at the very least lip service and, at most, worth spending money on and a real pursuit at the time. To my ear, a gut strung instrument that stays in perfect equal temperament for any length of time is equally a chimera. ... unless you're wealthy. This is my 2 cents and you may continue with your mathematics now. Sean
On Sun, May 13, 2018 at 12:30 PM, Rainer <[1]rads.bera_g...@t-online.de> wrote: A clarification: Suppose you want to place the frets on a lute so that: 1) All unisons are pure. That means the f on the second course has the same frequency as the open 1. course, the g on the second string has the same pitch as the b on the 1. course, ... over all courses. 2) All octaves are pure. The c on the fourth course has half the frequency of the first course, .... Then you have no choice left except equal temperament. The reason is that lutes cannot be tuned in any meantone temperament. You always have to accept false unisons or octaves. Rainer PS I use 1/6 comma meantone. However, I have no Baroque lutes nor have I ever tried accord nouveaux... On 13.05.2018 21:04, Ralf Mattes wrote: Am Sonntag, 13. Mai 2018 20:43 CEST, Ron Andrico <[2]praelu...@hotmail.com> schrieb: Ralf and Rainer, I believe you are in agreement. Octaves, fourths and fifths are pure and other intervals are an approximation. No, that's not what I (or Rainer) said. And it's wrong: Octaves are alway pure in all (western) tuning systems. Fifth/forth can be pure but then will be incomensurable with octaves (i.e. you can't stack fifth/forth and ever end up with pure octaves. As a matter of fact no pure intervals are comensurable. That IS a mathematic fact). Cheers, RalfD To get on or off this list see list information at [3]http://www.cs.dartmouth.edu/~wbc/lute-admin/index.html -- References 1. mailto:rads.bera_g...@t-online.de 2. mailto:praelu...@hotmail.com 3. http://www.cs.dartmouth.edu/~wbc/lute-admin/index.html