319edo: Difference between revisions

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==Theory==

== Theory ==

~~319et~~ is consistent to the [[7-odd-limit]] ~~and the~~ [[harmonic]] 3 is about halfway its steps. ~~Using~~ the patent val~~, it tempers out 156250000/155649627, 1220703125/1219784832, [[6144/6125]], [[10976/10935]] and 420175/419904 in the 7~~-limit~~; 95703125/95664294, 161280/161051, 35156250/35153041,~~ [[~~4000/3993~~]], ~~117649/117612, 1296000/1294139,~~ [[~~6250/6237~~]], 107495424/107421875, 4302592/4296875, 825000/823543, 422576/421875, 15488/15435, [[3025/3024]], 59290/59049, 766656/765625, 456533/455625, 202397184/201768035, 3294225/3294172, 585640/583443 and 644204/643125 in the 11-limit. It [[support]]s [[mystery]] in the 5-limit ~~and [[protolangwidge]]~~.

319edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway its steps. Nonetheless, it provides the optimal patent val for 5-limit [[mystery]] temperament, which tempers out the [[29-comma]], despite poor 5-limit harmonic approximation.

===Odd harmonics===

The full 11-limit [[patent val]] is nonetheless a reasonable interpretation since the lower harmonics all tend sharp. Using this val, it [[tempering out|tempers out]] [[6144/6125]], [[10976/10935]], and {{monzo| 9 -10 9 -5 }} in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], 15488/15435, 59290/59049, and [[65536/65219]] in the 11-limit. It [[support]]s [[mystery]] in the 5-limit and [[protolangwidge]].

If we instead adopt the 2.9.… [[subgroup]] interpretation, then 2.9.15.21 is a good subgroup to start with.

=== Odd harmonics ===

===Subsets and supersets===

=== Subsets and supersets ===

319 factors into 11 × 29, ~~with~~ [[11edo]] and [[29edo]] as its ~~subset edos~~. [[638edo]], which doubles it, gives a good correction to the ~~harmonic~~ 3.

Since 319 factors into 11 × 29, 319edo has [[11edo]] and [[29edo]] as its subsets. [[638edo]], which doubles it, gives a good correction to the harmonics 3, 5, and 7.

==Regular temperament properties==

== Regular temperament properties ==

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

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

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

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

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

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

|-

~~![[TE error|Absolute]] (¢)~~

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

|-

|2.9

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

|{{monzo|-1011 319}}

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

|{{mapping|319 1011}}

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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

|-

| 2.9

| {{monzo| -1011 319 }}

| {{mapping| 319 1011 }}

| +0.1223

| 0.1223

| 3.25

|-

|2.9.5

| 2.9.15

|~~32805/32768~~, {{monzo|~~54 35~~ -71}}

| {{monzo| -51 5 9 }}, {{monzo| -16 26 -17 }}

|{{mapping|319 1011 ~~741~~}}

| {{mapping| 319 1011 1246 }}

| -0.~~0832~~

| +0.1869

| 0.~~3072~~

| 0.1353

| 8.17

| 3.60

|}

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|319}}
+{{ED intro}}
-==Theory==
+== Theory ==
-et is consistent to the [[7-odd-limit]] and the [[harmonic]] 3 is about halfway its steps. Using the patent val, it tempers out 156250000/155649627, 1220703125/1219784832, [[6144/6125]], [[10976/10935]] and 420175/419904 in the 7-limit; 95703125/95664294, 161280/161051, 35156250/35153041, [[4000/3993]], 117649/117612, 1296000/1294139, [[6250/6237]], 107495424/107421875, 4302592/4296875, 825000/823543, 422576/421875, 15488/15435, [[3025/3024]], 59290/59049, 766656/765625, 456533/455625, 202397184/201768035, 3294225/3294172, 585640/583443 and 644204/643125 in the 11-limit. It [[support]]s [[mystery]] in the 5-limit and [[protolangwidge]].
+edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway its steps. Nonetheless, it provides the optimal patent val for 5-limit [[mystery]] temperament, which tempers out the [[29-comma]], despite poor 5-limit harmonic approximation.
-===Odd harmonics===
+The full 11-limit [[patent val]] is nonetheless a reasonable interpretation since the lower harmonics all tend sharp. Using this val, it [[tempering out|tempers out]] [[6144/6125]], [[10976/10935]], and {{monzo| 9 -10 9 -5 }} in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], 15488/15435, 59290/59049, and [[65536/65219]] in the 11-limit. It [[support]]s [[mystery]] in the 5-limit and [[protolangwidge]].
+If we instead adopt the 2.9.… [[subgroup]] interpretation, then 2.9.15.21 is a good subgroup to start with.
+=== Odd harmonics ===
 {{Harmonics in equal|319}}
-===Subsets and supersets===
+=== Subsets and supersets ===
-factors into 11 × 29, with [[11edo]] and [[29edo]] as its subset edos. [[638edo]], which doubles it, gives a good correction to the harmonic 3.
+Since 319 factors into 11 × 29, 319edo has [[11edo]] and [[29edo]] as its subsets. [[638edo]], which doubles it, gives a good correction to the harmonics 3, 5, and 7.
-==Regular temperament properties==
+== Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
-! rowspan="2" |[[Comma list|Comma List]]
-! rowspan="2" |[[Mapping]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
-! colspan="2" |Tuning Error
-|-
-![[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
 |-
-|2.9
+! rowspan="2" | [[Subgroup]]
-|{{monzo|-1011 319}}
+! rowspan="2" | [[Comma list]]
-|{{mapping|319 1011}}
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.9
+| {{monzo| -1011 319 }}
+| {{mapping| 319 1011 }}
 | +0.1223
 | 0.1223
 | 3.25
 |-
-|2.9.5
+| 2.9.15
-|32805/32768, {{monzo|54 35 -71}}
+| {{monzo| -51 5 9 }}, {{monzo| -16 26 -17 }}
-|{{mapping|319 1011 741}}
+| {{mapping| 319 1011 1246 }}
-| -0.0832
+| +0.1869
-| 0.3072
+| 0.1353
-| 8.17
+| 3.60
 |}