Ennealimmal–enneadecal equivalence continuum - Revision history

ArrowHead294 at 19:48, 17 December 2024

2024-12-17T19:48:09Z

← Older revision		Revision as of 19:48, 17 December 2024
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	The '''ennealimmal–enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[ennealimma\|ennealimmas ({{monzo\|1 -27 18}})]] with [[Enneadeca\|enneadeca comma ({{monzo\|-14 -19 19}})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments.		The '''ennealimmal–enneadecal equivalence continuum''' is a [[equivalence continuum\|continuum]] of 5-limit temperaments which equate a number of [[ennealimma\|ennealimmas ({{monzo\|1 -27 18}})]] with [[Enneadeca\|enneadeca comma ({{monzo\|-14 -19 19}})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments.

	All temperaments in the continuum satisfy {{nowrap\|(2･3<sup>−27</sup>･5<sup>18</sup>)<sup>''n''</sup> ~ (2<sup>−14</sup>･3<sup>−19</sup>･5<sup>19</sup>)}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[ennealimmal]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[171edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 3.2669545024..., and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(2･3<sup>−27</sup>･5<sup>18</sup>)<sup>''n''</sup> ~ (2<sup>−14</sup>･3<sup>−19</sup>･5<sup>19</sup>)}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[ennealimmal]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[171edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 3.2669545024..., and temperaments having ''n'' near this value tend to be the most accurate ones.

ArrowHead294 at 19:47, 17 December 2024

2024-12-17T19:47:53Z

← Older revision		Revision as of 19:47, 17 December 2024
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	The '''~~ennealimmal-enneadecal~~ equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[ennealimma\|ennealimmas ({{monzo\|1 -27 18}})]] with [[Enneadeca\|enneadeca comma ({{monzo\|-14 -19 19}})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments.		The '''ennealimmal–enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[ennealimma\|ennealimmas ({{monzo\|1 -27 18}})]] with [[Enneadeca\|enneadeca comma ({{monzo\|-14 -19 19}})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments.

	All temperaments in the continuum satisfy (2･3<sup>-27</sup>･5<sup>18</sup>)<sup>''n''</sup> ~ (2<sup>-14</sup>･3<sup>-19</sup>･5<sup>19</sup>). Varying ''n'' results in different temperaments listed in the table below. It converges to [[ennealimmal]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[171edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 3.2669545024..., and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(2･3<sup>−27</sup>･5<sup>18</sup>)<sup>''n''</sup> ~ (2<sup>−14</sup>･3<sup>−19</sup>･5<sup>19</sup>)}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[ennealimmal]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[171edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 3.2669545024..., and temperaments having ''n'' near this value tend to be the most accurate ones.

	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"
	\|+ Temperaments in the continuum		\|+ style="font-size: 105%" \| Temperaments in the continuum
	\|-		\|-
	! rowspan="2" \| ''n''		! rowspan="2" \| ''n''
Line 13:		Line 13:
	! Monzo		! Monzo
	\|-		\|-
	\| -4		\| −4
	\| 171 & 2429		\| 171 & 2429
	\|		\|
	\| {{monzo\|-10 -127 91}}		\| {{monzo\|-10 -127 91}}
	\|-		\|-
	\| -3		\| −3
	\| 171 & 1817		\| 171 & 1817
	\|		\|
	\| {{monzo\|-11 -100 73}}		\| {{monzo\|-11 -100 73}}
	\|-		\|-
	\| -2		\| −2
	\| [[Ragismic microtemperaments #Semidimi\|Semidimi]]		\| [[Ragismic microtemperaments #Semidimi\|Semidimi]]
	\|		\|
	\| {{monzo\|-12 -73 55}}		\| {{monzo\|-12 -73 55}}
	\|-		\|-
	\| -1		\| −1
	\| [[Ragismic microtemperaments #Supermajor\|Supermajor]]		\| [[Ragismic microtemperaments #Supermajor\|Supermajor]]
	\|		\|
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	Examples of temperaments with fractional values of ''n'':		Examples of temperaments with fractional values of ''n'':
	* 171 & 3193 (''n'' = -5/2 = -2.5)		* 171 & 3193 ({{nowrap\|''n'' {{=}} −5/2 {{=}} −2.5}})
	* 171 & 2140 (''n'' = -3/2 = -1.5)		* 171 & 2140 ({{nowrap\|''n'' {{=}} −3/2 {{=}} −1.5}})
	* 171 & 1087 (''n'' = -1/2 = -0.5)		* 171 & 1087 ({{nowrap\|''n'' {{=}} −1/2 {{=}} −0.5}})
	* [[Landscape microtemperaments #Pnict\|Pnict]] (''n'' = 1/3 = 0.{{overline\|3}})		* [[Landscape microtemperaments #Pnict\|Pnict]] ({{nowrap\|''n'' {{=}} 1/3 {{=}} 0.{{overline\|3}}}})
	* [[Gammic family\|Gammic]] (''n'' = 1/2 = 0.5)		* [[Gammic family\|Gammic]] ({{nowrap\|''n'' {{=}} 1/2 {{=}} 0.5}})
	* [[Horwell temperaments #Mutt\|Mutt]] (''n'' = 2/3 = 0.{{overline\|6}})		* [[Horwell temperaments #Mutt\|Mutt]] ({{nowrap\|''n'' {{=}} 2/3 {{=}} 0.{{overline\|6}}}})
	* [[Landscape microtemperaments #Septichrome\|Septichrome]] (''n'' = 4/3 = 1.{{overline\|3}})		* [[Landscape microtemperaments #Septichrome\|Septichrome]] ({{nowrap\|''n'' {{=}} 4/3 {{=}} 1.{{overline\|3}}}})
	* [[Metric microtemperaments #Geb\|Geb]] (''n'' = 3/2 = 1.5)		* [[Metric microtemperaments #Geb\|Geb]] ({{nowrap\|''n'' {{=}} 3/2 {{=}} 1.5}})
	* 171 & 1901 (''n'' = 5/2 = 2.5)		* 171 & 1901 ({{nowrap\|''n'' {{=}} 5/2 {{=}} 2.5}})
	* 171 & 4125 (''n'' = 10/3 = 3.{{overline\|3}})		* 171 & 4125 ({{nowrap\|''n'' {{=}} 10/3 {{=}} 3.{{overline\|3}}}})
	* 171 & 3125 (''n'' = 7/2 = 3.5)		* 171 & 3125 ({{nowrap\|''n'' {{=}} 7/2 {{=}} 3.5}})

	[[Category:171edo]]		[[Category:171edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

ArrowHead294: ArrowHead294 moved page Ennealimmal-enneadecal equivalence continuum to Ennealimmal–enneadecal equivalence continuum: The dash in titles like these should be an en dash, not a hyphen-minus, since "ennealimmal" does not modify "enneadecal"

2024-12-17T19:45:11Z

ArrowHead294 moved page Ennealimmal-enneadecal equivalence continuum to Ennealimmal–enneadecal equivalence continuum: The dash in titles like these should be an en dash, not a hyphen-minus, since "ennealimmal" does not modify "enneadecal"

← Older revision	Revision as of 19:45, 17 December 2024
(No difference)

Fredg999 category edits: Removing from Category:Temperament using Cat-a-lot

2022-02-28T20:06:45Z

Removing from Category:Temperament using Cat-a-lot

← Older revision		Revision as of 20:06, 28 February 2022
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	[[Category:171edo]]		[[Category:171edo]]
	~~[[Category:Temperament]]~~
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2022-02-28T20:05:57Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 20:05, 28 February 2022
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	[[Category:171edo]]		[[Category:171edo]]
	~~[[Category:Theory]]~~
	[[Category:Temperament]]		[[Category:Temperament]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

Xenllium at 11:09, 25 May 2021

2021-05-25T11:09:04Z

← Older revision		Revision as of 11:09, 25 May 2021
Line 75:		Line 75:

	Examples of temperaments with fractional values of ''n'':		Examples of temperaments with fractional values of ''n'':
			* 171 & 3193 (''n'' = -5/2 = -2.5)
			* 171 & 2140 (''n'' = -3/2 = -1.5)
			* 171 & 1087 (''n'' = -1/2 = -0.5)
			* [[Landscape microtemperaments #Pnict\|Pnict]] (''n'' = 1/3 = 0.{{overline\|3}})
			* [[Gammic family\|Gammic]] (''n'' = 1/2 = 0.5)
			* [[Horwell temperaments #Mutt\|Mutt]] (''n'' = 2/3 = 0.{{overline\|6}})
			* [[Landscape microtemperaments #Septichrome\|Septichrome]] (''n'' = 4/3 = 1.{{overline\|3}})
			* [[Metric microtemperaments #Geb\|Geb]] (''n'' = 3/2 = 1.5)
			* 171 & 1901 (''n'' = 5/2 = 2.5)
	* 171 & 4125 (''n'' = 10/3 = 3.{{overline\|3}})		* 171 & 4125 (''n'' = 10/3 = 3.{{overline\|3}})
			* 171 & 3125 (''n'' = 7/2 = 3.5)

	[[Category:171edo]]		[[Category:171edo]]

Xenllium: Created page with "The '''ennealimmal-enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of 1 -27 18}}) with E..."

2021-05-23T22:47:40Z

Created page with "The '''ennealimmal-enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of ennealimmas ({{monzo|1 -27 18}}) with E..."

New page

The '''ennealimmal-enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[ennealimma|ennealimmas ({{monzo|1 -27 18}})]] with [[Enneadeca|enneadeca comma ({{monzo|-14 -19 19}})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments.

All temperaments in the continuum satisfy (2･3-27･518)''n'' ~ (2-14･3-19･519). Varying ''n'' results in different temperaments listed in the table below. It converges to [[ennealimmal]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[171edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 3.2669545024..., and temperaments having ''n'' near this value tend to be the most accurate ones.

{| class="wikitable center-1 center-2"
|+ Temperaments in the continuum
|-
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| -4
| 171 & 2429
|
| {{monzo|-10 -127 91}}
|-
| -3
| 171 & 1817
|
| {{monzo|-11 -100 73}}
|-
| -2
| [[Ragismic microtemperaments #Semidimi|Semidimi]]
|
| {{monzo|-12 -73 55}}
|-
| -1
| [[Ragismic microtemperaments #Supermajor|Supermajor]]
|
| {{monzo|-13 -46 37}}
|-
| 0
| [[Enneadecal]]
| 19073486328125/19042491875328
| {{monzo|-14 -19 19}}
|-
| 1
| [[Schismic]]
| 32805/32768
| {{monzo|-15 8 1}}
|-
| 2
| [[Minortonic family|Minortone]]
| 50031545098999707/50000000000000000
| {{monzo|-16 35 -17}}
|-
| 3
| [[Very high accuracy temperaments #Senior|Senior]]
|
| {{monzo|-17 62 -35}}
|-
| 4
| 171 & 1783
|
| {{monzo|18 -89 53}}
|-
| 5
| 171 & 2395
|
| {{monzo|19 -116 71}}
|-
| …
| …
| …
| …
|-
| ∞
| [[Ennealimmal]]
| 7629394531250/7625597484987
| {{monzo|1 -27 18}}
|}

Examples of temperaments with fractional values of ''n'':
* 171 & 4125 (''n'' = 10/3 = 3.{{overline|3}})

[[Category:171edo]]
[[Category:Theory]]
[[Category:Temperament]]
[[Category:Equivalence continua]]