Hemipent: Difference between revisions

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A '''hemipent''' (or '''"hemipental"''') interval is an [[interval]] in the <math>\sqrt{2}\,.\sqrt{3}\,.\sqrt{5}</math> [[subgroup]] i.e. intervals that can be constructed by multiplying half-integer powers of 2, 3, and 5. Hemipental system is an [[expansion]] of [[Hemipyth|hemipythagorean]], by adding a generator representing <math>\sqrt{5}</math>.

Notable hemipent intervals include the semithird <math>\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}</math>, ~~semi-minor-third~~ <math>\sqrt{\frac{6}{5}} = \frac{\sqrt{6}}{\sqrt{5}}</math>, ~~semisixth~~ <math>\sqrt{\frac{5}{3}} = \frac{\sqrt{5}}{\sqrt{3}}</math>, and semi-minor-sixth <math>\sqrt{\frac{8}{5}} = \frac{2\sqrt{2}}{\sqrt{5}}</math>.

Notable hemipent intervals include the semithird <math>\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}</math> (~193{{C}}), semisixth <math>\sqrt{\frac{5}{3}} = \frac{\sqrt{5}}{\sqrt{3}}</math> (~442{{C}}), contrasemisixth (semi-minor-third) <math>\sqrt{\frac{6}{5}} = \frac{\sqrt{6}}{\sqrt{5}}</math> (~158{{C}}), and contrasemithird (semi-minor-sixth) <math>\sqrt{\frac{8}{5}} = \frac{2\sqrt{2}}{\sqrt{5}}</math> (~407{{C}}).

Many temperaments naturally produce intervals that split ~{{sfrac|5|4}}, ~{{sfrac|6|5}}, ~{{sfrac|5|3}}, or ~{{sfrac|8|5}} exactly in half and can thus be interpreted as semithirds, ~~semi~~-~~minor~~-~~thirds~~, ~~semisixths~~, ~~or semi~~-~~minor~~-~~sixths within~~ the temperament.

Many temperaments naturally produce intervals that split ~{{sfrac|5|4}}, ~{{sfrac|5|3}}, ~{{sfrac|6|5}}, or ~{{sfrac|8|5}} exactly in half and can thus be interpreted as semithirds, semisixths, contrasemisixths, or contrasemithirds within the temperament.

== Temperament interpretations ==

Edos which support hemipent by patent val include {{Edos| 6, 20, 24, 30, 38, 44, 62, and 68}}.

A rank-3 temperament which supports full hemipent must temper out at least three commas: one to equate an interval to its octave-complement, one to equate another interval to its fifth-complement, and one to equate another interval to its 5/4-complement. As a result, hemipent-based temperaments must be at least in the 13-limit or another 6-prime subgroup. As such, there are few specifically defined interpretations of hemipent structure as a temperament.

== See also ==

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 A '''hemipent''' (or '''"hemipental"''') interval is an [[interval]] in the <math>\sqrt{2}\,.\sqrt{3}\,.\sqrt{5}</math> [[subgroup]] i.e. intervals that can be constructed by multiplying half-integer powers of 2, 3, and 5. Hemipental system is an [[expansion]] of [[Hemipyth|hemipythagorean]], by adding a generator representing <math>\sqrt{5}</math>.
-Notable hemipent intervals include the semithird <math>\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}</math>, semi-minor-third <math>\sqrt{\frac{6}{5}} = \frac{\sqrt{6}}{\sqrt{5}}</math>, semisixth <math>\sqrt{\frac{5}{3}} = \frac{\sqrt{5}}{\sqrt{3}}</math>, and semi-minor-sixth <math>\sqrt{\frac{8}{5}} = \frac{2\sqrt{2}}{\sqrt{5}}</math>.
+Notable hemipent intervals include the semithird <math>\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}</math> (~193{{C}}), semisixth <math>\sqrt{\frac{5}{3}} = \frac{\sqrt{5}}{\sqrt{3}}</math> (~442{{C}}), contrasemisixth (semi-minor-third) <math>\sqrt{\frac{6}{5}} = \frac{\sqrt{6}}{\sqrt{5}}</math> (~158{{C}}), and contrasemithird (semi-minor-sixth) <math>\sqrt{\frac{8}{5}} = \frac{2\sqrt{2}}{\sqrt{5}}</math> (~407{{C}}).
-Many temperaments naturally produce intervals that split ~{{sfrac|5|4}}, ~{{sfrac|6|5}}, ~{{sfrac|5|3}}, or ~{{sfrac|8|5}} exactly in half and can thus be interpreted as semithirds, semi-minor-thirds, semisixths, or semi-minor-sixths within the temperament.
+Many temperaments naturally produce intervals that split ~{{sfrac|5|4}}, ~{{sfrac|5|3}}, ~{{sfrac|6|5}}, or ~{{sfrac|8|5}} exactly in half and can thus be interpreted as semithirds, semisixths, contrasemisixths, or contrasemithirds within the temperament.
+== Temperament interpretations ==
+Edos which support hemipent by patent val include {{Edos| 6, 20, 24, 30, 38, 44, 62, and 68}}.
+A rank-3 temperament which supports full hemipent must temper out at least three commas: one to equate an interval to its octave-complement, one to equate another interval to its fifth-complement, and one to equate another interval to its 5/4-complement. As a result, hemipent-based temperaments must be at least in the 13-limit or another 6-prime subgroup. As such, there are few specifically defined interpretations of hemipent structure as a temperament.
 == See also ==