12edf: Difference between revisions

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'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]]. This is similar to every second step of [[41edo]], and thus to the fretting of one string on the [[Kite Guitar]]. It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2. It contains the [[macrodiatonic and microdiatonic scales|microdiatonic]] scale that corresponds to 12edo's [[5L 2s|diatonic scale]] with [[2/1]] compressed to [[3/2]].

==Harmonics==

==Intervals==

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| | 1

| | 58.49625

| | 91/88, 88/85

| | [[28/27]], 91/88, 88/85

| |

|-

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| | 3

| | 175.48875

| | [[21/19]]

| | [[10/9]], [[21/19]]

| |

|-

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| | 7

| | 409.47375

| | [[19/15]]

| | [[19/15]], [[63/50]]

| |

|-

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| | 11

| | 643.4588

| |13/9

| | [[13/9]]

| |

|-

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|

|}

~~[[Category:Edf]]~~

~~[[Category:Edonoi]]~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.
+'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]]. This is similar to every second step of [[41edo]], and thus to the fretting of one string on the [[Kite Guitar]]. It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.
+It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2. It contains the [[macrodiatonic and microdiatonic scales|microdiatonic]] scale that corresponds to 12edo's [[5L 2s|diatonic scale]] with [[2/1]] compressed to [[3/2]].
+==Harmonics==
+{{Harmonics in equal|12|3|2}}
+{{Harmonics in equal|12|3|2|start=12|collapsed=1}}
 ==Intervals==
@@ Line 16: / Line 22: @@
 | | 1
 | | 58.49625
-| | 91/88, 88/85
+| | [[28/27]], 91/88, 88/85
 | |
 |-
@@ Line 26: / Line 32: @@
 | | 3
 | | 175.48875
-| | [[21/19]]
+| | [[10/9]], [[21/19]]
 | |
 |-
@@ Line 46: / Line 52: @@
 | | 7
 | | 409.47375
-| | [[19/15]]
+| | [[19/15]], [[63/50]]
 | |
 |-
@@ Line 66: / Line 72: @@
 | | 11
 | | 643.4588
-| |13/9
+| | [[13/9]]
 | |
 |-
@@ Line 134: / Line 140: @@
 |
 |}
-[[Category:Edf]]
-[[Category:Edonoi]]
+{{todo|expand}}