Hemipent
- "Hemipental" redirects here. For the regular temperament, see Quintile family #Hemiquintile.
A hemipent (or "hemipental") interval is an interval in the [math]\displaystyle{ \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 hemipythagorean, by adding a generator representing [math]\displaystyle{ \sqrt{5} }[/math].
Notable hemipent intervals include the semithird [math]\displaystyle{ \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2} }[/math], semisixth [math]\displaystyle{ \sqrt{\frac{5}{3}} = \frac{\sqrt{5}}{\sqrt{3}} }[/math], contrasemisixth (semi-minor-third) [math]\displaystyle{ \sqrt{\frac{6}{5}} = \frac{\sqrt{6}}{\sqrt{5}} }[/math], and contrasemithird (semi-minor-sixth) [math]\displaystyle{ \sqrt{\frac{8}{5}} = \frac{2\sqrt{2}}{\sqrt{5}} }[/math].
Many temperaments naturally produce intervals that split ~5/4, ~5/3, ~6/5, or ~8/5 exactly in half and can thus be interpreted as semithirds, semisixths, contrasemisixths, or contrasemithirds within the temperament.
Temperament interpretations
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.