243edo: Difference between revisions

Line 28:

|-

| 2.3

| {{~~monzo~~| -385 243 }}

| {{Monzo| -385 243 }}

| {{~~mapping~~| 243 385 }}

| {{Mapping| 243 385 }}

| +0.227

| 0.227

Line 36:

| 2.3.5

| 2109375/2097152, {{monzo| 1 -27 18 }}

| {{~~mapping~~| 243 385 564 }}

| {{Mapping| 243 385 564 }}

| +0.314

| 0.222

Line 43:

| 2.3.5.7

| 2401/2400, 4375/4374, 2109375/2097152

| {{~~mapping~~| ~~241~~ 385 564 682 }}

| {{Mapping| 243 385 564 682 }}

| +0.318

| 0.192

| 3.90

|}

* 243et (243e val) has a lower absolute ~~error~~ than any previous equal temperaments in the 19-limit, despite inconsistency in the corresponding odd limit~~. The same subgroup~~ is only ~~better tuned~~ by [[270edo|270et]]. It is much stronger in the no-11 19-limit~~, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is~~ [[354edo|354et]] in terms of absolute error and [[935edo|935et]] in terms of relative error.

* 243et (243e val) has lower absolute errors than any previous equal temperaments in the 19-, 23-limit, and somewhat beyond, despite inconsistency in the corresponding odd limits. In both the 19- and 23-limit, it beats [[217edo|217]] and is only bettered by [[270edo|270et]].

* It is much stronger in the no-11 subgroups of the limits above, holding the record of lowest relative errors until being bettered in the no-11 19-limit by [[354edo|354et]] in terms of absolute error and [[935edo|935et]] in terms of relative error, and in the no-11 23-limit by [[422edo|422]] in terms of absolute error and [[2460edo|2460]] in terms of relative error.

=== Rank-2 temperaments ===

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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Associated ratio*

! Temperaments

|-

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Line 92:

|-

| 9

| 64\243 (10\243)

| 64\243 (10\243)

| 316.05 (49.38)

| 316.05 (49.38)

| 6/5 (36/35)

| 6/5 (36/35)

| [[Ennealimmal]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal~~ lists|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct

[[Category:Ennealimmal]]

[[Category:Jove]]

@@ Line 28: / Line 28: @@
 |-
 | 2.3
-| {{monzo| -385 243 }}
+| {{Monzo| -385 243 }}
-| {{mapping| 243 385 }}
+| {{Mapping| 243 385 }}
 | +0.227
 | 0.227
@@ Line 36: / Line 36: @@
 | 2.3.5
 | 2109375/2097152, {{monzo| 1 -27 18 }}
-| {{mapping| 243 385 564 }}
+| {{Mapping| 243 385 564 }}
 | +0.314
 | 0.222
@@ Line 43: / Line 43: @@
 | 2.3.5.7
 | 2401/2400, 4375/4374, 2109375/2097152
-| {{mapping| 241 385 564 682 }}
+| {{Mapping| 243 385 564 682 }}
 | +0.318
 | 0.192
 | 3.90
 |}
-* 243et (243e val) has a lower absolute error than any previous equal temperaments in the 19-limit, despite inconsistency in the corresponding odd limit. The same subgroup is only better tuned by [[270edo|270et]]. It is much stronger in the no-11 19-limit, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is [[354edo|354et]] in terms of absolute error and [[935edo|935et]] in terms of relative error.
+* 243et (243e val) has lower absolute errors than any previous equal temperaments in the 19-, 23-limit, and somewhat beyond, despite inconsistency in the corresponding odd limits. In both the 19- and 23-limit, it beats [[217edo|217]] and is only bettered by [[270edo|270et]].
+* It is much stronger in the no-11 subgroups of the limits above, holding the record of lowest relative errors until being bettered in the no-11 19-limit by [[354edo|354et]] in terms of absolute error and [[935edo|935et]] in terms of relative error, and in the no-11 23-limit by [[422edo|422]] in terms of absolute error and [[2460edo|2460]] in terms of relative error.
 === Rank-2 temperaments ===
@@ Line 54: / Line 55: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br />per 8ve
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br />ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 91: / Line 92: @@
 |-
 | 9
-| 64\243<br />(10\243)
+| 64\243<br>(10\243)
-| 316.05<br />(49.38)
+| 316.05<br>(49.38)
-| 6/5<br />(36/35)
+| 6/5<br>(36/35)
 | [[Ennealimmal]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
 [[Category:Ennealimmal]]
 [[Category:Jove]]