Hemipent: Difference between revisions
use template (also there's no "the" irrational interval here) |
No edit summary Tags: Mobile edit Mobile web edit |
||
| Line 3: | Line 3: | ||
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>. | ||
Notable hemipent intervals include the semithird <math>\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}</math>, | Notable hemipent intervals include the semithird <math>\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}</math>, semisixth <math>\sqrt{\frac{5}{3}} = \frac{\sqrt{5}}{\sqrt{3}}</math>, contrasemisixth (semi-minor-third) <math>\sqrt{\frac{6}{5}} = \frac{\sqrt{6}}{\sqrt{5}}</math>, and contrasemithird (semi-minor-sixth) <math>\sqrt{\frac{8}{5}} = \frac{2\sqrt{2}}{\sqrt{5}}</math>. | ||
Many temperaments naturally produce intervals that split ~{{sfrac|5|4}}, ~{{sfrac| | 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 == | |||
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. | |||
== See also == | == See also == | ||
Revision as of 22:57, 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], 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.