559edo: Difference between revisions

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== Theory ==

559edo is a strong 5-limit system. ~~The equal temperament~~ [[tempering out|tempers out]] the luna comma, {{monzo| 38 -2 -15 }} and the minortone comma, {{monzo| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]].

559edo is a very strong 5-limit system. It [[tempering out|tempers out]] the luna comma, {{monzo| 38 -2 -15 }} and the minortone comma, {{monzo| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]].

=== Prime harmonics ===

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== Regular temperament properties ==

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

|-

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

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

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

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

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

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

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| {{monzo| 886 -559 }}

| {{mapping| 559 886 }}

| -0.0040

| −0.0040

| 0.0040

| 0.19

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| {{monzo| 38 -2 -15 }}, {{monzo| -16 35 -17 }}

| {{mapping| 559 886 1298 }}

| -0.0157

| −0.0157

| 0.0168

| 0.78

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| 5.48

|}

* 559et has a lower relative error than any previous equal temperaments in the 5-limit, past [[441edo|441]] and before [[612edo|612]].

=== Rank-2 temperaments ===

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

|+Table of rank-2 temperaments by generator

|+ 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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| [[Acrokleismic]]

|-

|13

| 13

|232\559 (17\559)

| 232\559 (17\559)

|498.03 (36.494)

| 498.03 (36.494)

|4/3 (?)

| 4/3 (?)

|[[Aluminium]]

| [[Aluminium]]

|-

| 43

| 232\559 (2\559)

| 232\559 (2\559)

| 498.03 (4.29)

| 498.03 (4.29)

| 4/3 (385/384)

| 4/3 (385/384)

| [[Meridic]]

|}

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

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|559}}
+{{ED intro}}
 == Theory ==
-edo is a strong 5-limit system. The equal temperament [[tempering out|tempers out]] the luna comma, {{monzo| 38 -2 -15 }} and the minortone comma, {{monzo| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]].
+edo is a very strong 5-limit system. It [[tempering out|tempers out]] the luna comma, {{monzo| 38 -2 -15 }} and the minortone comma, {{monzo| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]].
 === Prime harmonics ===
@@ Line 13: / Line 13: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 25: / Line 26: @@
 | {{monzo| 886 -559 }}
 | {{mapping| 559 886 }}
-| -0.0040
+| −0.0040
 | 0.0040
 | 0.19
@@ Line 32: / Line 33: @@
 | {{monzo| 38 -2 -15 }}, {{monzo| -16 35 -17 }}
 | {{mapping| 559 886 1298 }}
-| -0.0157
+| −0.0157
 | 0.0168
 | 0.78
@@ Line 50: / Line 51: @@
 | 5.48
 |}
+* 559et has a lower relative error than any previous equal temperaments in the 5-limit, past [[441edo|441]] and before [[612edo|612]].
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ 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 84: / Line 87: @@
 | [[Acrokleismic]]
 |-
-|13
+| 13
-|232\559<br>(17\559)
+| 232\559<br />(17\559)
-|498.03<br>(36.494)
+| 498.03<br />(36.494)
-|4/3<br>(?)
+| 4/3<br />(?)
-|[[Aluminium]]
+| [[Aluminium]]
 |-
 | 43
-| 232\559<br>(2\559)
+| 232\559<br />(2\559)
-| 498.03<br>(4.29)
+| 498.03<br />(4.29)
-| 4/3<br>(385/384)
+| 4/3<br />(385/384)
 | [[Meridic]]
 |}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct