875/864: Difference between revisions

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'''875/864''', the '''keema''', is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 21.9 [[cent]]s. It marks the difference between the [[7/4|septimal minor seventh (7/4)]] and a stack of three [[6/5|just minor thirds (6/5)]], or between a [[27/14|septimal major seventh (27/14)]] and a stack of three [[5/4|just major thirds (5/4)]]. It is also the sum of [[245/243]] and [[225/224]], the sum of [[100/99]] and [[385/384]], the sum of [[105/104]] and [[325/324]], and the difference between [[49/48]] and [[126/125]].

== Temperaments ==

Tempering out the keema is an important way that septimal harmony is organized in [[~~EDO~~]]s of medium size; keemic sets [[25/24]] and [[36/35]] equal, resulting in the distance between the classical major and minor thirds being narrowed, so that [[7/6]] ~~- 6~~/~~5 - 5~~/~~4 -~~ [[9/7]] are made equidistant (an "opposite" option to [[myna]], which widens that distance instead to insert a neutral third, [[49/40]], into this equidistance). ~~EDOs~~ with this structure include {{EDOs|15, 19, 22, 26, and 41}} (though 15edo tunes 9/7 very poorly).

[[Tempering out]] the keema is an important way that septimal harmony is organized in [[edo]]s of medium size; keemic sets [[25/24]] and [[36/35]] equal, resulting in the distance between the classical major and minor thirds being narrowed, so that [[7/6]]–6/5–5/4–[[9/7]] are made equidistant (an "opposite" option to [[myna]], which widens that distance instead to insert a neutral third, [[49/40]], into this equidistance). Edos with this structure include {{EDOs| 15, 19, 22, 26, and 41 }} (though 15edo tunes 9/7 very poorly).

Tempering it out alone in the 7-limit leads to the [[~~supermagic~~]] rank-3 temperament, whose extensions form the rank-3 [[keemic family]], enabling [[keemic chords]]. See [[Keemic temperaments]] for a collection of rank-2 temperaments where it is tempered out.

Tempering it out alone in the 7-limit leads to the [[keemic]] rank-3 temperament, whose extensions form the rank-3 [[keemic family]], enabling [[keemic chords]]. See [[Keemic temperaments]] for a collection of rank-2 temperaments where it is tempered out.

== Etymology ==

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 '''875/864''', the '''keema''', is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 21.9 [[cent]]s. It marks the difference between the [[7/4|septimal minor seventh (7/4)]] and a stack of three [[6/5|just minor thirds (6/5)]], or between a [[27/14|septimal major seventh (27/14)]] and a stack of three [[5/4|just major thirds (5/4)]]. It is also the sum of [[245/243]] and [[225/224]], the sum of [[100/99]] and [[385/384]], the sum of [[105/104]] and [[325/324]], and the difference between [[49/48]] and [[126/125]].
 == Temperaments ==
-Tempering out the keema is an important way that septimal harmony is organized in [[EDO]]s of medium size; keemic sets [[25/24]] and [[36/35]] equal, resulting in the distance between the classical major and minor thirds being narrowed, so that [[7/6]] - 6/5 - 5/4 - [[9/7]] are made equidistant (an "opposite" option to [[myna]], which widens that distance instead to insert a neutral third, [[49/40]], into this equidistance). EDOs with this structure include {{EDOs|15, 19, 22, 26, and 41}} (though 15edo tunes 9/7 very poorly).
+[[Tempering out]] the keema is an important way that septimal harmony is organized in [[edo]]s of medium size; keemic sets [[25/24]] and [[36/35]] equal, resulting in the distance between the classical major and minor thirds being narrowed, so that [[7/6]]–6/5–5/4–[[9/7]] are made equidistant (an "opposite" option to [[myna]], which widens that distance instead to insert a neutral third, [[49/40]], into this equidistance). Edos with this structure include {{EDOs| 15, 19, 22, 26, and 41 }} (though 15edo tunes 9/7 very poorly).
-Tempering it out alone in the 7-limit leads to the [[supermagic]] rank-3 temperament, whose extensions form the rank-3 [[keemic family]], enabling [[keemic chords]]. See [[Keemic temperaments]] for a collection of rank-2 temperaments where it is tempered out.
+Tempering it out alone in the 7-limit leads to the [[keemic]] rank-3 temperament, whose extensions form the rank-3 [[keemic family]], enabling [[keemic chords]]. See [[Keemic temperaments]] for a collection of rank-2 temperaments where it is tempered out.
 == Etymology ==