View source for 5-odd-limit

{{Odd-limit navigation|5}}
{{Odd-limit intro|5}}

* [[1/1]]
* '''[[6/5]], [[5/3]]'''
* '''[[5/4]], [[8/5]]'''
* [[4/3]], [[3/2]]

{| class="wikitable center-all right-2 left-5"
! 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 -->