309edo: Difference between revisions

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== Theory ==

309edo is ~~only~~ consistent to the 3-limit, with ~~five~~ possible mappings in the 7-limit:

309edo is in[[consistent]] to the [[5-odd-limit]], with four possible mappings in the [[7-limit]]:

* {{val|309 490 717 867}} (patent val)

* {{val| 309 490 717 867 }} ([[patent val]])

* {{val|309 '''489''' 717 867}} (309b)

* {{val| 309 490 '''718''' 867 }} (309c)

* {{val|309 490 '''718''' 867}} (309c)

* {{val| 309 490 717 '''868''' }} (309d)

* {{val|309 490 717 '''868'''}} (309d)

* {{val| 309 490 '''718''' '''868''' }} (309cd)

* {{val|309 490 '''718''' '''868'''}} (309cd)

~~Using the patent val, it tempers out [[5120/5103]], 117649/116640, 390625/387072, 537824/531441, [[823543/819200]] and 1953125/1928934 in the 7-limit, supporting the [[hemifamity]] temperament.~~

~~Using the 309b val, it tempers out [[78732/78125]] in the 5-limit; [[225/224]], [[1029/1024]], [[2401/2400]], [[16875/16807]] and 33075/32768 in the 7-limit.~~

~~Using the 309c val, it tempers out [[4375/4374]], 153664/151875, 395136/390625, 537824/531441, [[2100875/2097152]] and 5250987/5242880 in the 7-limit, supporting [[mitonic]] and [[ragismic]].~~

Using the 309d val, it tempers out [[2109375/2097152]] in the 5-limit; [[4000/3969]], 420175/419904, 829440/823543, 1071875/1062882 and 9765625/9633792 in the 7-limit. It supports the [[~~octagari~~]] ~~temperament~~.

Using the patent val, it tempers out [[5120/5103]], 117649/116640, 390625/387072, 537824/531441, [[823543/819200]] and 1953125/1928934 in the 7-limit. Using the 309c val, it tempers out [[4375/4374]], 153664/151875, 395136/390625, 537824/531441, [[2100875/2097152]] and 5250987/5242880 in the 7-limit. Using the 309d val, it tempers out [[2109375/2097152]] in the 5-limit; [[4000/3969]], [[420175/419904]], 829440/823543, 1071875/1062882 and 9765625/9633792 in the 7-limit. Using the 309cd val, it tempers out [[67108864/66430125]] in the 5-limit; [[3136/3125]], [[5120/5103]], [[250047/250000]], 458752/455625, 49009212/48828125 and 725594112/720600125 in the 7-limit. It supports [[misty]]. Note that the 309b val {{val| 309 '''489''' 717 867 }} (309b) is [[enfactoring|enfactored]] [[103edo]].

~~Using the 309cd val, it tempers out [[67108864/66430125]]~~ in the 5~~-limit; [[3136~~/~~3125]], [[5120~~/~~5103]], [[250047~~/~~250000]]~~, ~~458752/455625, 49009212/48828125 and 725594112/720600125 in~~ the ~~7-limit. It supports [[misty]], [[hemifamity]]~~ and [[~~landscape~~]].

This system is quite accurate in the 2.81.45.63.11.117.17 (or alternatively 2.81.5/9.7/9.11.13/9.17) subgroup, with the same tuning and commas as [[1236edo]].

=== Prime harmonics ===

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=== Subsets and supersets ===

309 factors into 3 × 103, ~~with~~ [[3edo]] and [[103edo]] as its subset edos. [[~~927edo~~]], which ~~triples~~ it, gives a good correction to the ~~harmonic~~ 5.

309 factors into prime as {{nowrap| 3 × 103 }}, so 309edo has [[3edo]] and [[103edo]] as its subset edos. [[1236edo]], which quadruples it, gives a good correction to the harmonics 3, 5, 7, and 13.

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|309}}
+{{ED intro}}
 == Theory ==
-edo is only consistent to the 3-limit, with five possible mappings in the 7-limit:
+edo is in[[consistent]] to the [[5-odd-limit]], with four possible mappings in the [[7-limit]]:
-* {{val|309 490 717 867}} (patent val)
+* {{val| 309 490 717 867 }} ([[patent val]])
-* {{val|309 '''489''' 717 867}} (309b)
+* {{val| 309 490 '''718''' 867 }} (309c)
-* {{val|309 490 '''718''' 867}} (309c)
+* {{val| 309 490 717 '''868''' }} (309d)
-* {{val|309 490 717 '''868'''}} (309d)
+* {{val| 309 490 '''718''' '''868''' }} (309cd)
-* {{val|309 490 '''718''' '''868'''}} (309cd)
-Using the patent val, it tempers out [[5120/5103]], 117649/116640, 390625/387072, 537824/531441, [[823543/819200]] and 1953125/1928934 in the 7-limit, supporting the [[hemifamity]] temperament.
-Using the 309b val, it tempers out [[78732/78125]] in the 5-limit; [[225/224]], [[1029/1024]], [[2401/2400]], [[16875/16807]] and 33075/32768 in the 7-limit.
-Using the 309c val, it tempers out [[4375/4374]], 153664/151875, 395136/390625, 537824/531441, [[2100875/2097152]] and 5250987/5242880 in the 7-limit, supporting [[mitonic]] and [[ragismic]].
-Using the 309d val, it tempers out [[2109375/2097152]] in the 5-limit; [[4000/3969]], 420175/419904, 829440/823543, 1071875/1062882 and 9765625/9633792 in the 7-limit. It supports the [[octagari]] temperament.
+Using the patent val, it tempers out [[5120/5103]], 117649/116640, 390625/387072, 537824/531441, [[823543/819200]] and 1953125/1928934 in the 7-limit. Using the 309c val, it tempers out [[4375/4374]], 153664/151875, 395136/390625, 537824/531441, [[2100875/2097152]] and 5250987/5242880 in the 7-limit. Using the 309d val, it tempers out [[2109375/2097152]] in the 5-limit; [[4000/3969]], [[420175/419904]], 829440/823543, 1071875/1062882 and 9765625/9633792 in the 7-limit. Using the 309cd val, it tempers out [[67108864/66430125]] in the 5-limit; [[3136/3125]], [[5120/5103]], [[250047/250000]], 458752/455625, 49009212/48828125 and 725594112/720600125 in the 7-limit. It supports [[misty]]. Note that the 309b val {{val| 309 '''489''' 717 867 }} (309b) is [[enfactoring|enfactored]] [[103edo]].
-Using the 309cd val, it tempers out [[67108864/66430125]] in the 5-limit; [[3136/3125]], [[5120/5103]], [[250047/250000]], 458752/455625, 49009212/48828125 and 725594112/720600125 in the 7-limit. It supports [[misty]], [[hemifamity]] and [[landscape]].
+This system is quite accurate in the 2.81.45.63.11.117.17 (or alternatively 2.81.5/9.7/9.11.13/9.17) subgroup, with the same tuning and commas as [[1236edo]].
 === Prime harmonics ===
@@ Line 23: / Line 17: @@
 === Subsets and supersets ===
-factors into 3 × 103, with [[3edo]] and [[103edo]] as its subset edos. [[927edo]], which triples it, gives a good correction to the harmonic 5.
+factors into prime as {{nowrap| 3 × 103 }}, so 309edo has [[3edo]] and [[103edo]] as its subset edos. [[1236edo]], which quadruples it, gives a good correction to the harmonics 3, 5, 7, and 13.