263edo: Difference between revisions

Line 1:

~~==Theory==~~

263et tempers out 393216/390625 (Würschmidt comma) and |50 -33 1> in the 5-limit. Using the patent val, it tempers out 4375/4374, 50421/50000, and 458752/455625 in the 7-limit; 441/440, 3388/3375, 16384/16335, and 26411/26244 in the 11-limit; 364/363, 2080/2079, 2197/2187, and 3584/3575 in the 13-limit; 595/594, 833/832, 936/935, and 1156/1155 in the 17-limit.

Using the ~~263d~~ val, it tempers out ~~5120~~/~~5103~~, ~~16875~~/~~16807~~, and ~~1959552~~/~~1953125~~ in the 7-limit; ~~540~~/~~539~~, ~~1375~~/~~1372~~, 16384/16335, and ~~43923~~/~~43750~~ in the 11-limit; ~~351/350, 1001~~/~~1000~~, ~~1573~~/~~1568~~, 2197/2187, and ~~4225~~/~~4224~~ in the 13-limit.

== Theory ==

263et [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and {{monzo| 50 -33 1 }} in the 5-limit. Using the [[patent val]], it tempers out [[4375/4374]], [[50421/50000]], and 458752/455625 in the 7-limit; [[441/440]], [[3388/3375]], [[16384/16335]], and 26411/26244 in the 11-limit; [[364/363]], [[2080/2079]], [[2197/2187]], and 3584/3575 in the 13-limit; [[595/594]], [[833/832]], [[936/935]], and [[1156/1155]] in the 17-limit.

Using the 263df val, it tempers out 352/351, 640/637, 729/728, and 3584/3575 in the 13-limit.

Using the 263d val, it tempers out [[5120/5103]], [[16875/16807]], and 1959552/1953125 in the 7-limit; [[540/539]], 1375/1372, 16384/16335, and 43923/43750 in the 11-limit; [[351/350]], [[1001/1000]], [[1573/1568]], 2197/2187, and [[4225/4224]] in the 13-limit.

===Prime harmonics===

Using the 263df val, it tempers out [[352/351]], [[640/637]], [[729/728]], and 3584/3575 in the 13-limit.

Finally, it is accurate for the 17th harmonic, as the denominator of a convergent to log217, after [[80edo|80]] and before [[343edo|343]].

=== Prime harmonics ===

===Subsets and supersets===

263edo is the 56th [[prime ~~EDO]]. It is accurate for the 17th harmonic, as the denominator of a convergent to log217, after [[80edo|80]] and before [[343edo|343~~]].

=== Subsets and supersets ===

==Regular temperament properties==

263edo is the 56th [[prime edo]].

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" |[[Subgroup]]

! rowspan="2" | [[Subgroup]]

! rowspan="2" |[[Comma list|Comma List]]

! rowspan="2" | [[Comma list|Comma List]]

! rowspan="2" |[[Mapping]]

! rowspan="2" | [[Mapping]]

! rowspan="2" |Optimal 8ve Stretch (¢)

! rowspan="2" | Optimal 8ve Stretch (¢)

! colspan="2" |Tuning Error

! colspan="2" | Tuning Error

|-

![[TE error|Absolute]] (¢)

! [[TE error|Absolute]] (¢)

![[TE simple badness|Relative]] (%)

! [[TE simple badness|Relative]] (%)

|-

|2.3

| 2.3

|{{monzo|417 -263}}

| {{monzo| 417 -263 }}

|{{val|263 417}}

| {{val| 263 417 }}

| -0.2229

| 0.2229

| 4.89

|-

|2.3.5

| 2.3.5

|393216/390625, {{monzo|50 -33 1}}

| 393216/390625, {{monzo| 50 -33 1 }}

|{{val|263 417 611}}

| {{val| 263 417 611 }}

| -0.3666

| 0.2728

| 5.98

|}

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperaments

|-

|1

| 1

|40\263

| 40\263

|182.51

| 182.51

|10/9

| 10/9

|[[Minortone]]

| [[Minortone]]

|-

|1

| 1

|85\263

| 85\263

|387.83

| 387.83

|5/4

| 5/4

|[[Würschmidt]]

| [[Würschmidt]]

|}

[[~~Category:Equal divisions of the~~ octave~~|###~~]] <!-~~- 3-digit number -->~~

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

[[~~Category:Prime EDO~~]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
 {{EDO intro|263}}
-==Theory==
-et tempers out 393216/390625 (Würschmidt comma) and |50 -33 1&gt; in the 5-limit. Using the patent val, it tempers out 4375/4374, 50421/50000, and 458752/455625 in the 7-limit; 441/440, 3388/3375, 16384/16335, and 26411/26244 in the 11-limit; 364/363, 2080/2079, 2197/2187, and 3584/3575 in the 13-limit; 595/594, 833/832, 936/935, and 1156/1155 in the 17-limit.
-Using the 263d val, it tempers out 5120/5103, 16875/16807, and 1959552/1953125 in the 7-limit; 540/539, 1375/1372, 16384/16335, and 43923/43750 in the 11-limit; 351/350, 1001/1000, 1573/1568, 2197/2187, and 4225/4224 in the 13-limit.
+== Theory ==
+et [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and {{monzo| 50 -33 1 }} in the 5-limit. Using the [[patent val]], it tempers out [[4375/4374]], [[50421/50000]], and 458752/455625 in the 7-limit; [[441/440]], [[3388/3375]], [[16384/16335]], and 26411/26244 in the 11-limit; [[364/363]], [[2080/2079]], [[2197/2187]], and 3584/3575 in the 13-limit; [[595/594]], [[833/832]], [[936/935]], and [[1156/1155]] in the 17-limit.
-Using the 263df val, it tempers out 352/351, 640/637, 729/728, and 3584/3575 in the 13-limit.
+Using the 263d val, it tempers out [[5120/5103]], [[16875/16807]], and 1959552/1953125 in the 7-limit; [[540/539]], 1375/1372, 16384/16335, and 43923/43750 in the 11-limit; [[351/350]], [[1001/1000]], [[1573/1568]], 2197/2187, and [[4225/4224]] in the 13-limit.
-===Prime harmonics===
+Using the 263df val, it tempers out [[352/351]], [[640/637]], [[729/728]], and 3584/3575 in the 13-limit.
+Finally, it is accurate for the 17th harmonic, as the denominator of a convergent to log<sub>2</sub>17, after [[80edo|80]] and before [[343edo|343]].
+=== Prime harmonics ===
 {{Harmonics in equal|263}}
-===Subsets and supersets===
-edo is the 56th [[prime EDO]]. It is accurate for the 17th harmonic, as the denominator of a convergent to log<sub>2</sub>17, after [[80edo|80]] and before [[343edo|343]].
+=== Subsets and supersets ===
-==Regular temperament properties==
+edo is the 56th [[prime edo]].
+== Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" |[[Comma list|Comma List]]
+! rowspan="2" | [[Comma list|Comma List]]
-! rowspan="2" |[[Mapping]]
+! rowspan="2" | [[Mapping]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" |Tuning Error
+! colspan="2" | Tuning Error
 |-
-![[TE error|Absolute]] (¢)
+! [[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
+! [[TE simple badness|Relative]] (%)
 |-
-|2.3
+| 2.3
-|{{monzo|417 -263}}
+| {{monzo| 417 -263 }}
-|{{val|263 417}}
+| {{val| 263 417 }}
 | -0.2229
 | 0.2229
 | 4.89
 |-
-|2.3.5
+| 2.3.5
-|393216/390625, {{monzo|50 -33 1}}
+| 393216/390625, {{monzo| 50 -33 1 }}
-|{{val|263 417 611}}
+| {{val| 263 417 611 }}
 | -0.3666
 | 0.2728
 | 5.98
 |}
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
-|1
+| 1
-|40\263
+| 40\263
-|182.51
+| 182.51
-|10/9
+| 10/9
-|[[Minortone]]
+| [[Minortone]]
 |-
-|1
+| 1
-|85\263
+| 85\263
-|387.83
+| 387.83
-|5/4
+| 5/4
-|[[Würschmidt]]
+| [[Würschmidt]]
 |}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
-[[Category:Prime EDO]]