319edo: Difference between revisions

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== Theory ==

319edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway its steps. 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]].

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.

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.

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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<br>8ve ~~Stretch~~ (¢)

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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

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 == Theory ==
-edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway its steps. 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]].
+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.
+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.
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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<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
 ! [[TE simple badness|Relative]] (%)