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] (~193 ¢), semisixth [math]\displaystyle{ \sqrt{\frac{5}{3}} = \frac{\sqrt{5}}{\sqrt{3}} }[/math] (~442 ¢), contrasemisixth (semi-minor-third) [math]\displaystyle{ \sqrt{\frac{6}{5}} = \frac{\sqrt{6}}{\sqrt{5}} }[/math] (~158 ¢), and contrasemithird (semi-minor-sixth) [math]\displaystyle{ \sqrt{\frac{8}{5}} = \frac{2\sqrt{2}}{\sqrt{5}} }[/math] (~407 ¢).
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
Edos which support hemipent by patent val include 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.