875/864

Interval information

Ratio	875/864
Factorization	2-5 × 3-3 × 53 × 7
Monzo	[-5 -3 3 1⟩
Size in cents	21.90205¢
Name	keema
Color name	zy31, zotriyo 1sn, Zotriyo comma
FJS name	[math]\displaystyle{ \text{A1}^{5,5,5,7} }[/math]
Special properties	reduced
Tenney height (log2 nd)	19.528
Weil height (log2 max(n, d))	19.5463
Wilson height (sopfr(nd))	41
Harmonic entropy (Shannon, [math]\displaystyle{ \sqrt{nd} }[/math])	~3.98367 bits
Comma size	small
S-expression	S5 / S6
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875/864, the keema, is a small 7-limit comma measuring about 21.9 cents. It marks the difference between the septimal minor seventh (7/4) and a stack of three just minor thirds (6/5), or between a septimal major seventh (27/14) and a stack of three 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 EDOs 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 into this equidistance). EDOs with this structure include 15, 19, 22, and 26 (though 15 tunes 9/7 very poorly).

Tempering it out leads to the supermagic temperament. See Keemic family for the rank-3 temperament family where it is tempered out. See Keemic temperaments for a collection of rank-2 temperaments where it is tempered out.

Etymology

This comma was first named as supermagic by Gene Ward Smith in 2005 as a contraction of superkleismic and magic, hence the name of the corresponding rank-3 temperament. It is not clear how it later became keema, but the root of keema is obvious, being a contraction of keemun and magic.