Minimal consistent EDOs: Difference between revisions

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~~<noinclude>~~{{Idiosyncratic terms}}

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 ''accurately consistent''{{idiosyncratic}} if the edo is consistent to [[Consistency #Generalization|distance 2]], or alternatively put, every ''q''-odd-limit interval in the edo has 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'''.~~</noinclude>~~

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%. 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'''.

{| class="wikitable center-all"

<onlyinclude>{| class="wikitable center-all"

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

|-

! Odd limit !! Smallest consistent edo~~{{asterisk}}~~ !! Smallest distinctly consistent edo !! Smallest purely consistent edo !! Smallest ~~accurately~~ consistent ~~edo~~ !! Smallest distinctly ~~accurate edo~~

! Odd limit !! Smallest consistent edo* !! Smallest distinctly consistent edo !! Smallest purely consistent edo* !! Smallest edo consistent to [[Consistency #Generalization|distance 2]]* !! Smallest edo distinctly consistent to distance 2

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

| 1 || 1 || 1 || 1 || 1 || 1

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| 21 || 94 || 282 || 311 || 8539 || 8539

|-

| 23 || 94 || 282 || 311 || 16808?

| 23 || 94 || 282 || 311 || 16808 || 16808

| 16808?

|-

| 25 || 282 || 388 || 311 || 16808?

| 25 || 282 || 388 || 311 || 16808 || 16808

| 16808?

|-

| 27 || 282 || 388 || 311 || 16808?

| 27 || 282 || 388 || 311 || 16808 || 16808

| 16808?

|-

| 29 || 282 || 1323 || 311 || 16808?

| 29 || 282 || 1323 || 311 || 16808 || 16808

| 16808?

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

| 31 || 311 || 1600 || 311 || 16808?

| 31 || 311 || 1600 || 311 || 16808 || 16808

| 16808?

|-

| 33 || 311 || 1600 || 311 || 16808?

| 33 || 311 || 1600 || 311 || 16808 || 16808

| 16808?

|-

| 35 || 311 || 1600 || 311 || 16808?

| 35 || 311 || 1600 || 311 || 16808 || 16808

| 16808?

|-

| 37 || 311 || 1600 || 311 || ||

| 37 || 311 || 1600 || 311 || 324296 || 324296

|-

| 39 || 311 || 2554 || 311 || ||

| 39 || 311 || 2554 || 311 || 2398629 || 2398629

|-

| 41 || 311 || 2554 || 311 || ||

| 41 || 311 || 2554 || 311 || 19164767 || 19164767

|-

| 43 || 17461 || 17461 || 20567 || ||

| 43 || 17461 || 17461 || 20567 || 19735901 || 19735901

|-

| 45 || 17461 || 17461 || 20567 || ||

| 45 || 17461 || 17461 || 20567 || 19735901 || 19735901

|-

| 47 || 20567 || 20567 || 20567 || ||

| 47 || 20567 || 20567 || 20567 || 152797015 || 152797015

|-

| 49 || 20567 || 20567 || 459944 || ||

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<nowiki />* Apart from 0edo

<nowiki />** Purely consistent to the 137-odd-limit<~~noinclude~~>

<nowiki />** Purely consistent to the 137-odd-limit</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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== See also ==

* [[Consistency limits of small EDOs]]

* {{u|ArrowHead294|Purely consistent EDOs by odd limit}}

[[Category:Mapping]]

[[Category:Consistency]]

[[Category:Odd limit]]

~~</noinclude>~~

@@ Line 1: / Line 1: @@
-<noinclude>{{Idiosyncratic terms}}
+{{Idiosyncratic terms}}
-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 ''accurately consistent''{{idiosyncratic}} if the edo is consistent to [[Consistency #Generalization|distance 2]], or alternatively put, every ''q''-odd-limit interval in the edo has 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'''.</noinclude>
+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%. 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'''.
-{| class="wikitable center-all"
+<onlyinclude>{| class="wikitable center-all"
 |+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit
 |-
-! Odd<br />limit !! Smallest<br />consistent edo{{asterisk}} !! Smallest distinctly<br />consistent edo !! Smallest purely<br />consistent edo !! Smallest accurately<br />consistent edo !! Smallest distinctly<br />accurate edo
+! Odd<br>limit !! Smallest<br>consistent edo* !! Smallest distinctly<br>consistent edo !! Smallest purely<br>consistent edo* !! Smallest edo<br>consistent to<br>[[Consistency #Generalization|distance 2]]* !! Smallest edo<br>distinctly consistent<br>to distance 2
 |- style="font-weight: bold; background-color: #dddddd;"
 | 1 || 1 || 1 || 1 || 1 || 1
@@ Line 29: / Line 29: @@
 | 21 || 94 || 282 || 311 || 8539 || 8539
 |-
-| 23 || 94 || 282 || 311 || 16808?
+| 23 || 94 || 282 || 311 || 16808 || 16808
-| 16808?
 |-
-| 25 || 282 || 388 || 311 || 16808?
+| 25 || 282 || 388 || 311 || 16808 || 16808
-| 16808?
 |-
-| 27 || 282 || 388 || 311 || 16808?
+| 27 || 282 || 388 || 311 || 16808 || 16808
-| 16808?
 |-
-| 29 || 282 || 1323 || 311 || 16808?
+| 29 || 282 || 1323 || 311 || 16808 || 16808
-| 16808?
 |- style="font-weight: bold; background-color: #dddddd;"
-| 31 || 311 || 1600 || 311 || 16808?
+| 31 || 311 || 1600 || 311 || 16808 || 16808
-| 16808?
 |-
-| 33 || 311 || 1600 || 311 || 16808?
+| 33 || 311 || 1600 || 311 || 16808 || 16808
-| 16808?
 |-
-| 35 || 311 || 1600 || 311 || 16808?
+| 35 || 311 || 1600 || 311 || 16808 || 16808
-| 16808?
 |-
-| 37 || 311 || 1600 || 311 ||  ||
+| 37 || 311 || 1600 || 311 || 324296 || 324296
 |-
-| 39 || 311 || 2554 || 311 ||  ||
+| 39 || 311 || 2554 || 311 || 2398629 || 2398629
 |-
-| 41 || 311 || 2554 || 311 ||  ||
+| 41 || 311 || 2554 || 311 || 19164767 || 19164767
 |-
-| 43 || 17461 || 17461 || 20567 ||  ||
+| 43 || 17461 || 17461 || 20567 || 19735901 || 19735901
 |-
-| 45 || 17461 || 17461 || 20567 ||  ||
+| 45 || 17461 || 17461 || 20567 || 19735901 || 19735901
 |-
-| 47 || 20567 || 20567 || 20567 ||  ||
+| 47 || 20567 || 20567 || 20567 || 152797015 || 152797015
 |-
 | 49 || 20567 || 20567 || 459944 ||  ||
@@ Line 152: / Line 145: @@
 <nowiki />* Apart from 0edo
-<nowiki />** Purely consistent to the 137-odd-limit<noinclude>
+<nowiki />** Purely consistent to the 137-odd-limit</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.
@@ Line 165: / Line 157: @@
 == See also ==
 * [[Consistency limits of small EDOs]]
+* {{u|ArrowHead294|Purely consistent EDOs by odd limit}}
 [[Category:Mapping]]
 [[Category:Consistency]]
 [[Category:Odd limit]]
-</noinclude>