Minimal consistent EDOs: Difference between revisions

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An [[edo]] ''N'' is ''[[consistent]]'' with respect to the [[Odd limit|''q''-odd-limit]] if the closest approximations of the odd harmonics of the q-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics. It is ''[[distinctly consistent]]'' if every one of those closest approximations is a distinct value, and ''purely consistent'' if its [[relative interval error|relative errors]] on odd harmonics up to and including ''q'' never exceed 25%. It is accurate if every interval in that odd-limit are consistent to at least [[Harmonic distance|distance 2]], or at most 25% relative error. Below is a table of the smallest consistent, and the smallest distinctly consistent, edo for every odd number up to 135. Odd limits of {{nowrap|2''n'' − 1}} are '''highlighted'''.

An [[edo]] ''N'' is ''[[consistent]]'' with respect to the [[Odd limit|''q''-odd-limit]] if the closest approximations of the odd harmonics of the q-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics. It is ''[[distinctly consistent]]'' if every one of those closest approximations is a distinct value, and ''purely consistent''{{idiosyncratic}} if its [[relative interval error|relative errors]] on odd harmonics up to and including ''q'' never exceed 25%. It is accurate{{idiosyncratic}} if every interval in that odd-limit are consistent to at least [[Harmonic distance|distance 2]], or at most 25% relative error. Below is a table of the smallest consistent, and the smallest distinctly consistent, edo for every odd number up to 135. Odd limits of {{nowrap|2''n'' − 1}} are '''highlighted'''.

~~<onlyinclude><includeonly>~~

~~|+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit~~

~~</includeonly>~~

{| class="wikitable center-all"

|+ Smallest consistent EDOs per odd limit

|-

! Odd limit !! Smallest consistent edo* !! Smallest distinctly consistent edo !! Smallest ''purely consistent''** edo

!Smallest ''accurate'' edo

!Smallest

!Smallest distinctly ''accurate'' edo

accurate~~****~~ edo

!Smallest distinctly

accurate~~****~~ edo

|- style="font-weight: bold; background-color: #dddddd;"

| 1 || 1 || 1 || 1

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|

|-

| 131 || 2901533 || 2901533 || 93678217813***

| 131 || 2901533 || 2901533 || 93678217813**

|

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|

|{{Table notes|cols=6

{{Table notes|cols=6

| Apart from 0edo

~~| ''Purely consistent'' is an {{idiosyncratic}}~~

| Purely consistent to the 137-odd-limit

~~| Accurate is an {{idiosyncratic}}~~

}}

|}

~~</onlyinclude>~~

The last entry, 70910024edo, is consistent up to the 135-odd-limit. The next edo is [[5407372813edo|5407372813]], reported to be consistent to the 155-odd-limit.

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-An [[edo]] ''N'' is ''[[consistent]]'' with respect to the [[Odd limit|''q''-odd-limit]] if the closest approximations of the odd harmonics of the q-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics. It is ''[[distinctly consistent]]'' if every one of those closest approximations is a distinct value, and ''purely consistent'' if its [[relative interval error|relative errors]] on odd harmonics up to and including ''q'' never exceed 25%. It is accurate if every interval in that odd-limit are consistent to at least [[Harmonic distance|distance 2]], or at most 25% relative error. Below is a table of the smallest consistent, and the smallest distinctly consistent, edo for every odd number up to 135. Odd limits of {{nowrap|2<sup>''n''</sup> &minus; 1}} are '''highlighted'''.
+An [[edo]] ''N'' is ''[[consistent]]'' with respect to the [[Odd limit|''q''-odd-limit]] if the closest approximations of the odd harmonics of the q-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics. It is ''[[distinctly consistent]]'' if every one of those closest approximations is a distinct value, and ''purely consistent''{{idiosyncratic}} if its [[relative interval error|relative errors]] on odd harmonics up to and including ''q'' never exceed 25%. It is accurate{{idiosyncratic}} if every interval in that odd-limit are consistent to at least [[Harmonic distance|distance 2]], or at most 25% relative error. Below is a table of the smallest consistent, and the smallest distinctly consistent, edo for every odd number up to 135. Odd limits of {{nowrap|2<sup>''n''</sup> &minus; 1}} are '''highlighted'''.
-<onlyinclude><includeonly>
-|+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit
-</includeonly>
 {| class="wikitable center-all"
+|+ Smallest consistent EDOs per odd limit
 |-
 ! Odd<br />limit !! Smallest<br />consistent edo&#42; !! Smallest distinctly<br />consistent edo !! Smallest ''purely<br />consistent''&#42;&#42; edo
+!Smallest<br />''accurate'' edo
-!Smallest
+!Smallest distinctly<br />''accurate'' edo
-accurate**** edo
-!Smallest distinctly
-accurate**** edo
 |- style="font-weight: bold; background-color: #dddddd;"
 | 1 || 1 || 1 || 1
@@ Line 339: / Line 333: @@
 |
 |-
-| 131 || 2901533 || 2901533 || 93678217813***
+| 131 || 2901533 || 2901533 || 93678217813**
 |
@@ Line 352: / Line 346: @@
 |
-|{{Table notes|cols=6
+{{Table notes|cols=6
 | Apart from 0edo
-| ''Purely consistent'' is an {{idiosyncratic}}
 | Purely consistent to the 137-odd-limit
-| Accurate is an {{idiosyncratic}}
 }}
 |}
-</onlyinclude>
 The last entry, 70910024edo, is consistent up to the 135-odd-limit. The next edo is [[5407372813edo|5407372813]], reported to be consistent to the 155-odd-limit.