Hemipent - Revision history

Overthink: + cent values

2026-03-12T03:40:11Z

+ cent values

← Older revision		Revision as of 03:40, 12 March 2026
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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>.		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>, 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>.		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\|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.		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.

Overthink: /* Temperament interpretations */ lowercase

2026-03-11T22:10:43Z

Temperament interpretations: lowercase

← Older revision		Revision as of 22:10, 11 March 2026
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	== Temperament interpretations ==		== Temperament interpretations ==
	~~EDOs~~ which support hemipent by patent val include {{Edos\| 6, 20, 24, 30, 38, 44, 62, and 68}}.		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.		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.

Overthink: note edos

2026-03-11T22:10:30Z

note edos

← Older revision		Revision as of 22:10, 11 March 2026
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	== Temperament interpretations ==		== 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.		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.

Xenllium at 23:24, 5 February 2026

2026-02-05T23:24:44Z

← Older revision		Revision as of 23:24, 5 February 2026
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	== Temperament interpretations ==		== 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.		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 ==		== See also ==

Xenllium at 22:57, 5 February 2026

2026-02-05T22:57:11Z

← Older revision		Revision as of 22:57, 5 February 2026
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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>.		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>, 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\|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 ==
			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 ==

Overthink: use template (also there's no "the" irrational interval here)

2026-02-05T16:23:33Z

use template (also there's no "the" irrational interval here)

← Older revision		Revision as of 16:23, 5 February 2026
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	~~: ''"~~Hemipental~~" redirects here; this page is about the irrational interval. For~~ the regular temperament~~, see [[~~Quintile family #Hemiquintile~~]].''~~		{{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>.

Xenllium: 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 $\sqrt{2}\,.\sqrt{3}\,.\sqrt{5}$ 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..."

2026-02-05T11:36:22Z

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..."

New page

: ''"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 [[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>.

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.

== See also ==
* [[Hemipyth]]

[[Category:Rank 3]]
[[Category:Subgroup]]