789edo: Difference between revisions

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789edo is notable for an extremely good approximation of the [[2.5.7]] ~~subgroup~~, unbeaten until [[5902edo]].

789edo is notable for an extremely good approximation of the [[2.5.7 subgroup]], unbeaten until [[5902edo]]; notably, it is the denominator of logarithmic [[semiconvergent]]s to both [[5/4]] and [[7/5]], and remarkably tempers out the [[Don Page comma]] between the intervals of [[128/125]] and [[50/49]], mapping each to 27 and 23 steps respectively, and additionally tempers out 281484423828125/281474976710656 ([-48 0 11 8⟩), supporting exodia, the subgroup restriction of mohajira. It also has a very accurate representation of the 17th harmonic (due to tempering out S49/S50 = [[2000033/2000000]], equating [[17/16]] to three [[50/49]] intervals) and has a good 9th and 23rd harmonic as well; there is a common flat tendency allowing consistency to high distance in the 2.9.5.7.33.17.23 subgroup.

It is also the most accurate edo below 10000 to approximate [[quarter-comma meantone]], with an approximation of {{monzo| 0 0 1/4 }} by 458\789 with a relative error of only 0.03%.

[[1578edo]], which doubles it, provides good corrections for the 3rd and 11th harmonics, making for a very strong [[11-limit]] and higher-limit system.

=== Odd harmonics ===

~~{{stub}}~~

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 {{Infobox ET}}
-{{EDO intro|789}}
+{{ED intro}}
-edo is notable for an extremely good approximation of the [[2.5.7]] subgroup, unbeaten until [[5902edo]].
+edo is notable for an extremely good approximation of the [[2.5.7 subgroup]], unbeaten until [[5902edo]]; notably, it is the denominator of logarithmic [[semiconvergent]]s to both [[5/4]] and [[7/5]], and remarkably tempers out the [[Don Page comma]] between the intervals of [[128/125]] and [[50/49]], mapping each to 27 and 23 steps respectively, and additionally tempers out 281484423828125/281474976710656 ([-48 0 11 8⟩), supporting exodia, the subgroup restriction of mohajira. It also has a very accurate representation of the 17th harmonic (due to tempering out S49/S50 = [[2000033/2000000]], equating [[17/16]] to three [[50/49]] intervals) and has a good 9th and 23rd harmonic as well; there is a common flat tendency allowing consistency to high distance in the 2.9.5.7.33.17.23 subgroup.
+It is also the most accurate edo below 10000 to approximate [[quarter-comma meantone]], with an approximation of {{monzo| 0 0 1/4 }} by 458\789 with a relative error of only 0.03%.
+[[1578edo]], which doubles it, provides good corrections for the 3rd and 11th harmonics, making for a very strong [[11-limit]] and higher-limit system.
 === Odd harmonics ===
 {{Harmonics in equal|789}}
-{{stub}}