Talk:Logarithmic approximants: Difference between revisions

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= ARCHIVED WIKISPACES DISCUSSION BELOW =
{{WSArchiveLink}}
'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''


<span style="color:#800000">'''PLEASE MAKE ANY NEW COMMENTS <u>ABOVE</u> THIS SECTION.'''</span> Anything below here is for archival purposes only.
== Cleanup and notes ==


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I cleaned up a lot of the formatting on this page. It is in a much better state now.


== Pade(2,1) ==
There's some genuinely cool stuff buried in here, and I haven't really taken the time to understand all the details.
Have you taken a look at the Pade(2,1) approximant?  I compute it as $\frac{1}{4} \frac{(r-1)(r+5)}{2r+1}$. Calculations show it beats the quadratic approximant on the interval $[1,4]$ at the endpoints of which the two approximants agree. (The two are nearly identical on that interval, with a max difference of $\approx .006$.)  The approximants are fairly simple (generally simpler than the Pade(1,2) approximants).  E.g. for r = 2/1 we have the approximant 7/5 (ignoring the constant 1/4).  A first computation shows that this approximation spots 34EDO by comparing the approximants for 5/4 and 3/2:  25/56 and 13/16 resp.  We don't get equality, but minimizing the difference m*(13/16)-n*(25/56) gives m=11, n=20 with 11*(13/16) - 20*(25/56) = 1/112, a unit fraction with denominator the LCM of the approximant denominators.
I believe this material can be presented in much better ways. Not sure exactly how to go about this. A good start would maybe be to split off approximants into a more encyclopedic article rather than this essay format. The section on argent tuning is also of high interest, since it relates to what we now call the [[argent comma]].


- '''mattyhawthorn''' April 04, 2015, 05:08:22 PM UTC-0700
– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 00:53, 6 May 2026 (UTC)
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Latest revision as of 00:53, 6 May 2026

This page also contains archived Wikispaces discussion.

Cleanup and notes

I cleaned up a lot of the formatting on this page. It is in a much better state now.

There's some genuinely cool stuff buried in here, and I haven't really taken the time to understand all the details. I believe this material can be presented in much better ways. Not sure exactly how to go about this. A good start would maybe be to split off approximants into a more encyclopedic article rather than this essay format. The section on argent tuning is also of high interest, since it relates to what we now call the argent comma.

– Sintel🎏 (talk) 00:53, 6 May 2026 (UTC)

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