589edo: Difference between revisions

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== Theory ==

~~589et~~ tempers out 420175/419904 in the 7-limit; ~~55296000~~/~~55240493~~, 117649/117612, ~~107495424~~/~~107421875~~, ~~[[1953125~~/~~1948617]]~~, ~~4302592~~/~~4296875~~, ~~85937500~~/~~85766121, 422576/421875~~, [[~~3025~~/~~3024~~]], ~~766656~~/~~765625~~, ~~456533~~/~~455625~~, ~~3294225~~/~~3294172~~, ~~644204~~/~~643125~~ and 781258401/781250000 in the 11-limit.

589edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. As every other step of [[1178edo]], the approximations to lower harmonics are not impressive, making it only suitable for a 2.9.15.21.19 [[subgroup]] interpretation, in which case it is identical to 1178edo. The full 17-limit [[patent val]], however, is plausible since all the harmonics from 3 to 17 are tuned sharp. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] 420175/419904 in the 7-limit; [[3025/3024]], 117649/117612, 422576/421875, 456533/455625, 644204/643125, 766656/765625, [[1953125/1948617]], 3294225/3294172, 4302592/4296875, 55296000/55240493, 85937500/85766121, 107495424/107421875 and 781258401/781250000 in the 11-limit.

=== Odd harmonics ===

=== Subsets and supersets ===

589 factors into ~~19 × 31 with subset edos~~ {{~~EDOs~~| 19, and ~~31}}~~. [[1178edo]], which doubles it, gives a good ~~correction~~ to ~~the 3rd harmonic~~.

Since 589 factors into {{factorization|589}}, 589edo contains [[19edo]] and [[31edo]] as subsets. [[1178edo]], which doubles it, gives good corrections to harmonics 3, 5, 7, 11, 13, and 17.

== Regular temperament properties ==

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

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

|-

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

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

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

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

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

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

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

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

! colspan="2" | Tuning error

|-

![[TE error|Absolute]] (¢)

! [[TE error|Absolute]] (¢)

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

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

|-

|2.9

| 2.9

|{{monzo|-1867 589}}

| {{monzo| -1867 589 }}

|{{~~val~~|589 1867}}

| {{mapping| 589 1867 }}

| +0.0276

| 0.0276

| 1.35

|-

|2.9.5

| 2.9.5

|{{monzo|-37 19 -10}}, {{monzo|72 0 -31}}

| {{monzo| -37 19 -10 }}, {{monzo| 72 0 -31 }}

|{{~~val~~|589 1867 1368}}

| {{mapping| 589 1867 1368 }}

| -0.0940

| −0.0940

| 0.1734

| 8.51

|}

== Scales ==

* [[Kryptonite10]]

* [[Kryptonite19]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|589}}
+{{ED intro}}
 == Theory ==
-et tempers out 420175/419904 in the 7-limit; 55296000/55240493, 117649/117612, 107495424/107421875, [[1953125/1948617]], 4302592/4296875, 85937500/85766121, 422576/421875, [[3025/3024]], 766656/765625, 456533/455625, 3294225/3294172, 644204/643125 and 781258401/781250000 in the 11-limit.
+edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. As every other step of [[1178edo]], the approximations to lower harmonics are not impressive, making it only suitable for a 2.9.15.21.19 [[subgroup]] interpretation, in which case it is identical to 1178edo. The full 17-limit [[patent val]], however, is plausible since all the harmonics from 3 to 17 are tuned sharp. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] 420175/419904 in the 7-limit; [[3025/3024]], 117649/117612, 422576/421875, 456533/455625, 644204/643125, 766656/765625, [[1953125/1948617]], 3294225/3294172, 4302592/4296875, 55296000/55240493, 85937500/85766121, 107495424/107421875 and 781258401/781250000 in the 11-limit.
 === Odd harmonics ===
 {{Harmonics in equal|589}}
 === Subsets and supersets ===
-factors into 19 × 31 with subset edos {{EDOs| 19, and 31}}. [[1178edo]], which doubles it, gives a good correction to the 3rd harmonic.
+Since 589 factors into {{factorization|589}}, 589edo contains [[19edo]] and [[31edo]] as subsets. [[1178edo]], which doubles it, gives good corrections to harmonics 3, 5, 7, 11, 13, and 17.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
+|-
-! rowspan="2" |[[Comma list|Comma List]]
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" |[[Mapping]]
+! rowspan="2" | [[Comma list]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
+! rowspan="2" | [[Mapping]]
-! colspan="2" |Tuning Error
+! rowspan="2" | Optimal<br>8ve stretch (¢)
+! colspan="2" | Tuning error
 |-
-![[TE error|Absolute]] (¢)
+! [[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
+! [[TE simple badness|Relative]] (%)
 |-
-|2.9
+| 2.9
-|{{monzo|-1867 589}}
+| {{monzo| -1867 589 }}
-|{{val|589 1867}}
+| {{mapping| 589 1867 }}
 | +0.0276
 | 0.0276
 | 1.35
 |-
-|2.9.5
+| 2.9.5
-|{{monzo|-37 19 -10}}, {{monzo|72 0 -31}}
+| {{monzo| -37 19 -10 }}, {{monzo| 72 0 -31 }}
-|{{val|589 1867 1368}}
+| {{mapping| 589 1867 1368 }}
-| -0.0940
+| −0.0940
 | 0.1734
 | 8.51
 |}
 == Scales ==
 * [[Kryptonite10]]
 * [[Kryptonite19]]