277edo: Difference between revisions

Line 1:

~~==Theory==~~

~~277et tempers out 32805/32768 (schisma) and |-11 -37 30> in the 5-limit.~~

~~Using the patent val~~, it ~~tempers out 4375/4374, 65625/65536, and 829440/823543~~ in the 7-limit~~; 540~~/~~539, 6250/6237, 15488/15435,~~ and ~~35937/35840 in the~~ 11-~~limit; 625/624, 729/728, 1573/1568, 2080/2079, and 2200/2197~~ in the 13-limit.

== Theory ==

277edo is a good 5-limit tuning; however, it is in[[consistent]] in the [[7-odd-limit]]. The equal temperament [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and {{monzo| -11 -37 30 }} in the 5-limit.

~~Using the 277d~~ val~~, it~~ tempers out ~~1029~~/~~1024~~, ~~10976~~/~~10935~~, and ~~48828125~~/~~48771072~~ in the 7-limit; ~~385~~/~~384~~, ~~441~~/~~440~~, ~~19712~~/~~19683~~, and ~~234375~~/~~234256~~ in the 11-limit; 625/624, ~~847~~/~~845~~, ~~1001~~/~~1000~~, and ~~1575~~/~~1573~~ in the 13-limit.

The [[patent val]] {{val| 277 439 643 778 }} tempers out [[4375/4374]], [[65625/65536]], and 829440/823543 in the 7-limit; [[540/539]], [[6250/6237]], 15488/15435, and 35937/35840 in the 11-limit; [[625/624]], [[729/728]], [[1573/1568]], [[2080/2079]], and [[2200/2197]] in the 13-limit. It [[support]]s [[pontiac]].

The ~~patent~~ val [[~~support~~]]s the [[~~pontiac~~]], and the ~~277d val~~ supports ~~the~~ [[guiron]] and ~~the~~ [[widefourth]].

The 277d val {{val| 277 439 643 '''777''' }} tempers out [[1029/1024]], [[10976/10935]], and 48828125/48771072 in the 7-limit; [[385/384]], [[441/440]], [[19712/19683]], and 234375/234256 in the 11-limit; 625/624, [[847/845]], [[1001/1000]], and [[1575/1573]] in the 13-limit. It supports [[guiron]] and [[widefourth]].

===Prime harmonics===

=== Prime harmonics ===

===Subsets and supersets===

277edo is the 59th [[prime ~~EDO~~]].

=== Subsets and supersets ===

==Regular temperament properties==

277edo is the 59th [[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|-439 277}}

| {{monzo| -439 277 }}

|{{~~val~~|277 439}}

| {{mapping| 277 439 }}

| 0.0473

| 1.09

|-

|2.3.5

| 2.3.5

|32805/32768, {{monzo|-11 -37 30}}

| 32805/32768, {{monzo| -11 -37 30 }}

|{{~~val~~|277 439 643}}

| {{mapping| 277 439 643 }}

| 0.1398

| 0.1364

| 3.15

|}

=== 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

|115\277

| 115\277

|498.19

| 498.19

|4/3

| 4/3

|[[Helmholtz]]

| [[Helmholtz]]

|}

[[~~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|277}}
-==Theory==
-et tempers out 32805/32768 (schisma) and |-11 -37 30&gt; in the 5-limit.
-Using the patent val, it tempers out 4375/4374, 65625/65536, and 829440/823543 in the 7-limit; 540/539, 6250/6237, 15488/15435, and 35937/35840 in the 11-limit; 625/624, 729/728, 1573/1568, 2080/2079, and 2200/2197 in the 13-limit.
+== Theory ==
+edo is a good 5-limit tuning; however, it is in[[consistent]] in the [[7-odd-limit]]. The equal temperament [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and {{monzo| -11 -37 30 }} in the 5-limit.
-Using the 277d val, it tempers out 1029/1024, 10976/10935, and 48828125/48771072 in the 7-limit; 385/384, 441/440, 19712/19683, and 234375/234256 in the 11-limit; 625/624, 847/845, 1001/1000, and 1575/1573 in the 13-limit.
+The [[patent val]] {{val| 277 439 643 778 }} tempers out [[4375/4374]], [[65625/65536]], and 829440/823543 in the 7-limit; [[540/539]], [[6250/6237]], 15488/15435, and 35937/35840 in the 11-limit; [[625/624]], [[729/728]], [[1573/1568]], [[2080/2079]], and [[2200/2197]] in the 13-limit. It [[support]]s [[pontiac]].
-The patent val [[support]]s the [[pontiac]], and the 277d val supports the [[guiron]] and the [[widefourth]].
+The 277d val {{val| 277 439 643 '''777''' }} tempers out [[1029/1024]], [[10976/10935]], and 48828125/48771072 in the 7-limit; [[385/384]], [[441/440]], [[19712/19683]], and 234375/234256 in the 11-limit; 625/624, [[847/845]], [[1001/1000]], and [[1575/1573]] in the 13-limit. It supports [[guiron]] and [[widefourth]].
-===Prime harmonics===
+=== Prime harmonics ===
 {{Harmonics in equal|277}}
-===Subsets and supersets===
-edo is the 59th [[prime EDO]].
+=== Subsets and supersets ===
-==Regular temperament properties==
+edo is the 59th [[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|-439 277}}
+| {{monzo| -439 277 }}
-|{{val|277 439}}
+| {{mapping| 277 439 }}
 | 0.0473
 | 0.0473
 | 1.09
 |-
-|2.3.5
+| 2.3.5
-|32805/32768, {{monzo|-11 -37 30}}
+| 32805/32768, {{monzo| -11 -37 30 }}
-|{{val|277 439 643}}
+| {{mapping| 277 439 643 }}
 | 0.1398
 | 0.1364
 | 3.15
 |}
 === 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
-|115\277
+| 115\277
-|498.19
+| 498.19
-|4/3
+| 4/3
-|[[Helmholtz]]
+| [[Helmholtz]]
 |}
-[[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]]