🚀 go-pugleaf

Charles:

In article <4u5kjs$9n4@newsbf02.news.aol.com>, frankspr@aol.com
(FrankSpr) writes:

 > Thanks for helping out.  If you could reformulate the calculation
 > once again to double check the number I would feel more
 > comfortable with the number you generated.

Oh, yeah.  it's an easy job.

 > The way you calculated the figure put an automatic stop of the
 > calculation after 1000 digits.  If you could please try using the
 > formula I've given then saving the number to memory and then
 > multiplying the figure by itself 12x you will know how close to 2
 > you are.

 > In other words:
 > axa=a2xa=a3xa=a4xa=a5xa=a6xa=a7xa=a8xa=a9xa=a10xa=a11xa=a12 which
 > should be 2.0 on the nose!

Multiplying a = 1.059463094359295264561....439999091018 using the above
formula you will get:  (* is 999th of decimal)        *

          a12=  1.999999999999999999999....99999999999693725902537.....
                                                      *
The origin of the error is justified as the follows.  When I gave
the 1000 digits number, I cut off after the 1000th digits.  Which
resulted the number "a" a bit smaller than the acurate value of
2^(1/12).  This round off error was E = 0.000000...001352025...
                                                    *
After you multiplied the number "a" twelve times, the error has
grown up and turned out:
          a12 almost=  2 -  (24/a) E = 2 - 0.0000...000306275...   
                                                       *
 > If you are successful in obtaining that number if you would be so
 > kind as to subtract the 12th root of two figure you have by 1.0 to
 > obtain .059...  and dividing the 12th root of 2 by that number I
 > would then have the correct number to calculate fret spacing with.

You are welcome.  I will put the result at the end of this article.
I can guarantee the number is so accurate that you cannot notice
the error even when you make a big guitar as large as this universe :)

 -- hiro

a/(1-a) =

17.81715374510576755349010557614300395112017631805461948734377191138\
504344973412061925242909876135775463985065167556884571538724490277295960\
317491869763577723223582698712727491101158263025786071616640644839938481\
691335935584603745200861428634922313322191880028909958038680210836998913\
402482415542287931703592972142006210933591540140831942381871738115955976\
517170712072458387470960219955930434414207902190784665069001360139996629\
480551237509097121933598995498228535741908009099235445240449375804267399\
671849221715358743979339407625196872106074873445351162800572821860336878\
612690392003825544906284036541547637779543602328811574257540243479719377\
517493156708424949773385290343193827493888245698164338811541196869044372\
207539042889212547832863095235441709368978757716322416502522353185443956\
100205786717352170263220241243629703918087472214985302507392840297554874\
577710765328565146704861359647215164763280966890917576850388686468954817\
65249085877653074385674154506292788420304678090142744161146181955038