Hemipent: Difference between revisions
Created page with ": ''"Hemipental" redirects here; this page is about the irrational interval. For the regular temperament, see Quintile family #Hemiquintile.'' 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 hemipythagorean, by adding a generator representin..." Tags: Mobile edit Mobile web edit |
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{{Redirect|Hemipental|the regular temperament|Quintile family #Hemiquintile}} | |||
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>. | 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>. | ||
Revision as of 16:23, 5 February 2026
- "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], semi-minor-third [math]\displaystyle{ \sqrt{\frac{6}{5}} = \frac{\sqrt{6}}{\sqrt{5}} }[/math], semisixth [math]\displaystyle{ \sqrt{\frac{5}{3}} = \frac{\sqrt{5}}{\sqrt{3}} }[/math], and 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, ~6/5, ~5/3, or ~8/5 exactly in half and can thus be interpreted as semithirds, semi-minor-thirds, semisixths, or semi-minor-sixths within the temperament.