5-odd-limit: Difference between revisions

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The smallest [[equal division of the octave]] which is [[consistent]] in the 5-odd-limit is [[3edo]]; that which is distinctly consistent in the same is [[9edo]].

The smallest [[equal division of the octave]] which is [[consistent]] in the 5-odd-limit is [[3edo]]; that which is distinctly consistent in the same is [[9edo]]. The {{w|natural density|density}} of edos consistent in the 5-odd-limit is expected to be 3/4.

{{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''''A''(''x'') {{=}} 1 }} if {{nowrap| −1/2 ≤ ''x'' ≤ +1/2 }} and 0 otherwise, {{nowrap| ''f''''B''(''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''''C''(''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''''C''(''z'')''dz'')/({{subsup|∫|−∞|+∞}} ''f''''C''(''z'')''dz''), which evaluates to 3/4.

}}

== See also ==

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-The smallest [[equal division of the octave]] which is [[consistent]] in the 5-odd-limit is [[3edo]]; that which is distinctly consistent in the same is [[9edo]].
+The smallest [[equal division of the octave]] which is [[consistent]] in the 5-odd-limit is [[3edo]]; that which is distinctly consistent in the same is [[9edo]]. The {{w|natural density|density}} of edos consistent in the 5-odd-limit is expected to be 3/4.
+{{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 ==