Actually, Dowland's tuning is quite sensible. The rule of 18 sets a lute (in theory, neglecting string stretch caused by sideways displacement at the finger and fret) to very-nearly equal temperament. Apparently, the stretch added by actually fingering the strings brings it very very close.
This isn't too hard to understand if you look at the math. ET12 is based on each higher semitone frequency being in a ratio of <the 12th root of 2> to the previous frequency. From A=440, this is about 440*1.0594631 =466.163761 (using 20 digits for the multiplication and throwing away the seventh decimal on: even still, with 7 decimals, the answer to six decimals is .000003 higher!) The rule of 18 uses a ratio of 1+1/18 =.0555555<keep repeating 5's forever>, and the result of multiplying this times 440 is 464.44444<repeating 4's>. The percentage of error here is on the order of -.36%. It is not hard to raise the pitch of a string by 1/300th of its current frequency with sidestretching. The error is cumulative, because each subsequent fret is set to (1+1/18)^fret#, but the string is not parallel to the frets (unless you are very lucky) which probably adds stretch, but also the side-stretch caused frequency increase decreases as you approach the middle of the string, which probably nullifies the effect, so the highest frets will be off by as much as 7*-.36, or about 2.4%. This is worth a bit of fudging, but the fifths are already flat from pure, and this amount of difference may not be significant. The rule-of-18 ratio for the fifth is 1.45005 (rounded to five decimals) while the ET12 fifth ratio is 1.49831 (again, to rounded to five decimals.) The just intonated fifth is 1/5. This makes rule-of-18 fifths over A=220, before figuring string stretch modifications, 321.211Hz, ET12 is 329.62Hz, and just is 330Hz. Does anyone have good data on frequency deviation caused by string stretch on a Dowland-era 8-course (for instance) lute? As I have mentioned before, at least to my ear, Dowland's and Holborne's music sounds very good with rule-of-18 fret placement, even without modification. (Then again, the lutes I have been able to play have all had the action a bit too high.) It doesn't seem to work so well for other composers, though. Ray On Fri, Oct 3, 2008 at 5:19 AM, Andrew Gibbs <[EMAIL PROTECTED]> wrote: > The LSA file works on my mac - but I found those historical > instructions for placing frets weren't really that useful - I think > some are inaccurate (Dowland's doesn't seem to make much sense). And > there's no instructions on how to tune the open strings - so if you > don't know what temperament you're aiming for, it makes the process a > bit circular - tweak the frets, tweak the open strings, tweak the > frets etc > > My basic understanding is that 16th c lute players would have used > some kind of meantone - ranging from 1/4 to 1/8 - but of course > wouldn't have used that terminology as the fine maths weren't worked > out until later when meantone was on the point of becoming obsolete. > This is my basic understanding - I'm sure someone on this list will > know better/more... > > Andrew > > > On 3 Oct 2008, at 07:22, Omer katzir wrote: > >> Thanks andrew, but i have a problem with LSA file, i see every >> thing as 0, beside the english... >> i used both office 2008 and Neoofice, and it showed me the same >> thing. might be becuase I'm a mac user... > > > -- > > To get on or off this list see list information at > http://www.cs.dartmouth.edu/~wbc/lute-admin/index.html >
