Minimal consistent EDOs: Difference between revisions

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An [[EDO|edo]] N is [[~~consistent|~~consistent]] with respect to a set of rational numbers s if the [[Patent_val|patent val]] mapping of every element of s is the nearest N-edo approximation. It is ''uniquely consistent'' if every element of s is mapped to a unique value. If the set s is the q [[Odd_limit|odd limit]], we say N is q-limit consistent and q-limit uniquely consistent, respectively. Below is a table of the least consistent, and least uniquely consistent, edo for every odd number up to 135.

An [[EDO|edo]] N is [[consistent]] with respect to a set of rational numbers s if the [[Patent_val|patent val]] mapping of every element of s is the nearest N-edo approximation. It is ''uniquely consistent'' if every element of s is mapped to a unique value. If the set s is the q [[Odd_limit|odd limit]], we say N is q-limit consistent and q-limit uniquely consistent, respectively. Below is a table of the least consistent, and least uniquely consistent, edo for every odd number up to 135.

{| class="wikitable"

|-

| | Odd limit

! | Odd limit

| | Smallest consistent

! | Smallest consistent

| | Smallest uniquely consistent

! | Smallest uniquely consistent

|-

| | 1

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|-

| | 87

| | 2901533

|-

| | 89

| | 2901533

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=OEIS integer sequences links=

* {{OEIS|A116474|Equal divisions of the octave with progressively increasing consistency levels.}}

* {{OEIS|A116474|Equal divisions of the octave with progressively increasing consistency levels}}

* {{OEIS|A116475|Equal divisions of the octave with progressively increasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit.}}

* {{OEIS|A116475|Equal divisions of the octave with progressively increasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit}}

* {{OEIS|A117577|Equal divisions of the octave with nondecreasing consistency levels.}}

* {{OEIS|A117578|Equal divisions of the octave with nondecreasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit.}}

* {{OEIS|A117578|Equal divisions of the octave with nondecreasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit}}

@@ Line 1: / Line 1: @@
-An [[EDO|edo]] N is [[consistent|consistent]] with respect to a set of rational numbers s if the [[Patent_val|patent val]] mapping of every element of s is the nearest N-edo approximation. It is ''uniquely consistent'' if every element of s is mapped to a unique value. If the set s is the q [[Odd_limit|odd limit]], we say N is q-limit consistent and q-limit uniquely consistent, respectively. Below is a table of the least consistent, and least uniquely consistent, edo for every odd number up to 135.
+An [[EDO|edo]] N is [[consistent]] with respect to a set of rational numbers s if the [[Patent_val|patent val]] mapping of every element of s is the nearest N-edo approximation. It is ''uniquely consistent'' if every element of s is mapped to a unique value. If the set s is the q [[Odd_limit|odd limit]], we say N is q-limit consistent and q-limit uniquely consistent, respectively. Below is a table of the least consistent, and least uniquely consistent, edo for every odd number up to 135.
 {| class="wikitable"
 |-
-| | Odd limit
+! | Odd limit
-| | Smallest consistent
+! | Smallest consistent
-| | Smallest uniquely consistent
+! | Smallest uniquely consistent
 |-
 | | 1
@@ Line 180: / Line 180: @@
 |-
 | | 87
+| | 2901533
+| | 2901533
+|-
+| | 89
 | | 2901533
 | | 2901533
@@ Line 317: / Line 321: @@
 =OEIS integer sequences links=
-* {{OEIS|A116474|Equal divisions of the octave with progressively increasing consistency levels.}}
+* {{OEIS|A116474|Equal divisions of the octave with progressively increasing consistency levels}}
-* {{OEIS|A116475|Equal divisions of the octave with progressively increasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit.}}
+* {{OEIS|A116475|Equal divisions of the octave with progressively increasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit}}
 * {{OEIS|A117577|Equal divisions of the octave with nondecreasing consistency levels.}}
-* {{OEIS|A117578|Equal divisions of the octave with nondecreasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit.}}
+* {{OEIS|A117578|Equal divisions of the octave with nondecreasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit}}