Timeline for What does the space induced by this unusual metric(?) on R/Z look like?

5 events

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May 11, 2011 at 17:52	comment	added	Roland Bacher		Right. Add $\epsilon$ everywhere. (I guess one can find a nice way related to proportions but this requires probably some trial and error. My first attempt was too naive.)
May 11, 2011 at 15:47	comment	added	Alexander Woo		These choices don't satisfy the triangle inequality: $d(0,1/12)>d(0,5/12)+d(1/12,5/12)$.
May 11, 2011 at 12:18	comment	added	Roland Bacher		A perhaps pleasant choice is given by $\alpha_5=2\epsilon$, $\alpha_4=3\epsilon$, $\alpha_2=4\epsilon$, $\alpha_3=5\epsilon$, $\alpha_1=6\epsilon$, $\alpha_6=7\epsilon$ where $\epsilon$ is small and positive. Justification: One wants a large (relative) distance if one is out of pitch (this is achieved by $\epsilon$ small). A more or less "rational" choice of distance is then to consider a distance proportional to $a+b$ if an interval consists of two notes with frequencies (roughly) proportional to $a/b2^{\mathbb Z}$.
May 11, 2011 at 9:32	history	edited	Roland Bacher	CC BY-SA 3.0	added 198 characters in body
May 11, 2011 at 9:13	history	answered	Roland Bacher	CC BY-SA 3.0