5-odd-limit: Difference between revisions

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-{{odd-limit navigation}}
+{{Odd-limit navigation|5}}
-This is a list of '''5-[[odd-limit]]''' intervals. To [[3-odd-limit]], it adds 2 interval pairs involving 5.
+[[File:5-odd-limit.png|480px|thumb|right|5-odd-limit intervals within an octave]]
+{{Odd-limit intro|5}}
 * [[1/1]]
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 * [[4/3]], [[3/2]]
-[[Category:Just interval]]
+{| class="wikitable center-all right-2 left-5"
-[[Category:Odd limit]]
+! Ratio
+! Size ([[cents|¢]])
+! colspan="2" | [[Color name]]
+! Name
+|-
+| [[6/5]]
+| 315.641
+| g3
+| gu 3rd
+| minor third
+|-
+| [[5/4]]
+| 386.314
+| y3
+| yo 3rd
+| major third
+|-
+| [[8/5]]
+| 813.686
+| g6
+| gu 6th
+| minor sixth
+|-
+| [[5/3]]
+| 884.359
+| y6
+| yo 6th
+| major sixth
+|}
+The smallest [[equal division of the octave]] which is [[consistent]] in the 5-odd-limit is [[3edo]].
+The one which is distinctly consistent in the same is [[9edo]].
+The {{W|Natural density|density}} of edos consistent in the 5-odd-limit is 3/4. See below for proof.
+{{Proof
+| title = Proof for the density of edos consistent in the 5-odd-limit
+| contents = Let the error of harmonic 3 be ''A''. Let the error of harmonic 5 be ''B''. By the {{w|equidistribution theorem}}, the relative error of any individual interval is {{w|equidistribution|equidistributed}} from −50% to +50%, so we have the probability density functions {{nowrap| ''f''<sub>''A''</sub>(''x'') {{=}} 1 }} if {{nowrap| −1/2 ≤ ''x'' ≤ +1/2 }} and 0 otherwise, {{nowrap| ''f''<sub>''B''</sub>(''y'') {{=}} 1 }} if {{nowrap| −1/2 ≤ ''y'' ≤ +1/2 }} and 0 otherwise.
+An edo is consistent in the 5-odd-limit if the error {{nowrap| ''C'' {{=}} ''B'' − ''A'' }} of the only compound interval – 5/3 – falls into the range from −50% to +50%.
+It is easy to show the probability density function ''f''<sub>''C''</sub>(''z'') is the {{w|triangular distribution}} {{nowrap| 1 − {{!}}''z''{{!}} }} if {{nowrap| −1 ≤ ''z'' ≤ +1 }} and 0 otherwise.
+The density of edos consistent in the 5-odd-limit therefore equals ({{subsup|∫|−1/2|+1/2}} ''f''<sub>''C''</sub>(''z'')''dz'')/({{subsup|∫|−∞|+∞}} ''f''<sub>''C''</sub>(''z'')''dz''), which evaluates to 3/4.
+}}
+== See also ==
+* [[5-limit]] ([[prime limit]])
+* [[Diamond5]] – as a scale
+[[Category:5-odd-limit| ]] <!-- main article -->