Sensamagic–gariboh equivalence continuum - Revision history

ArrowHead294 at 13:04, 10 March 2025

2025-03-10T13:04:05Z

← Older revision		Revision as of 13:04, 10 March 2025
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	</div>}}		</div>}}

	[[Category:~~Bohlen-Pierce~~]]		[[Category:Bohlen–Pierce]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

ArrowHead294 at 14:38, 20 December 2024

2024-12-20T14:38:08Z

← Older revision		Revision as of 14:38, 20 December 2024
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	\|-		\|-
	\| 1		\| 1
	\| [[~~BPS~~]]		\| [[Bohlen–Pierce–Stearns]]
	\| [[245/243]]		\| [[245/243]]
	\| {{monzo\| -5 1 2 }}		\| {{monzo\| -5 1 2 }}

ArrowHead294 at 19:31, 19 December 2024

2024-12-19T19:31:51Z

← Older revision		Revision as of 19:31, 19 December 2024
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	[[Category:Bohlen-Pierce]]		[[Category:Bohlen-Pierce]]
	~~[[Category:13edt]]~~
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

Lériendil: added link to procyon

2024-12-19T18:35:33Z

added link to procyon

← Older revision		Revision as of 18:35, 19 December 2024
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	\| 2		\| 2
	\| 3		\| 3
	\| Procyon ~~([https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=823543%2F820125&submit_comma=submit b13 & b144~~])		\| [[Procyon]]
	\| 823543/820125		\| 823543/820125
	\| {{monzo\| -8 -3 7 }}		\| {{monzo\| -8 -3 7 }}

ArrowHead294: Broken - fixed

2024-12-19T14:30:06Z

Broken - fixed

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	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"
	\|+ style="font-size: 105%;" Temperaments in the continuum		\|+ style="font-size: 105%;" \| Temperaments in the continuum
	\|-		\|-
	! rowspan="2" \| ''m''		! rowspan="2" \| ''m''

ArrowHead294: /* Projective tuning space */

2024-12-19T14:29:47Z

Projective tuning space

← Older revision		Revision as of 14:29, 19 December 2024
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	== Projective tuning space ==		== Projective tuning space ==

	Below is a depiction of the temperaments of this continuum in 3.5.7 [[projective tuning space]].		Below is a depiction of the temperaments of this continuum in 3.5.7 [[projective tuning space]].

ArrowHead294 at 16:09, 18 December 2024

2024-12-18T16:09:21Z

← Older revision		Revision as of 16:09, 18 December 2024
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	All temperaments in the continuum satisfy {{nowrap\|(245/243)<sup>''n''</sup> ~ 3125/3087}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Bohlen–Pierce–Stearns]] as ''n'' approaches infinity and [[Sirius]] as ''n'' approaches 0. If we allow non-integer and infinite ''n'', the continuum describes the set of all 3.5.7 subgroup temperaments supported by [[13edt]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); additionally, if prime 2 is added, this describes the set of all rank-3 7-limit temperaments supported by [[bohpier]]. Due to 13edt, the equal-tempered [[Bohlen–Pierce scale]], being a very good representation of the subgroup, and many named temperaments within the subgroup being supported by 13edt, this continuum is structurally important. The just value of ''n'' is 1.4926…, and temperaments near this tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(245/243)<sup>''n''</sup> ~ 3125/3087}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Bohlen–Pierce–Stearns]] as ''n'' approaches infinity and [[Sirius]] as ''n'' approaches 0. If we allow non-integer and infinite ''n'', the continuum describes the set of all 3.5.7 subgroup temperaments supported by [[13edt]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); additionally, if prime 2 is added, this describes the set of all rank-3 7-limit temperaments supported by [[bohpier]]. Due to 13edt, the equal-tempered [[Bohlen–Pierce scale]], being a very good representation of the subgroup, and many named temperaments within the subgroup being supported by 13edt, this continuum is structurally important. The just value of ''n'' is 1.4926…, and temperaments near this tend to be the most accurate ones.

	An alternative definition of this equivalence continuum satisfies (245/243)<sup>''k''</sup> ~ 7<sup>13</sup>/3<sup>23</sup> = 96889010407/94143178827, which sets ''k'' = 5 - ''n''. This definition might be advantageous as ''k'' gives the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 7 in the MOS scale.		An alternative definition of this equivalence continuum satisfies {{nowrap\|(245/243)<sup>''k''</sup> ~ 7<sup>13</sup>/3<sup>23</sup>}} = 96889010407/94143178827, which sets {{nowrap\|''k'' {{=}} 5 − ''n''}}. This definition might be advantageous as ''k'' gives the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 7 in the MOS scale.

	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"

ArrowHead294 at 16:08, 18 December 2024

2024-12-18T16:08:42Z

← Older revision		Revision as of 16:08, 18 December 2024
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	The '''sensamagic–gariboh equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[3.5.7 subgroup]] temperaments which equate a number of [[245/243\|sensamagic commas (245/243)]] with the [[3125/3087\|greater BP diesis (3125/3087)]].		The '''sensamagic–gariboh equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[3.5.7 subgroup]] temperaments which equate a number of [[245/243\|sensamagic commas (245/243)]] with the [[3125/3087\|greater BP diesis (3125/3087)]].

	All temperaments in the continuum satisfy {{nowrap\|(245/243)<sup>''n''</sup> ~ 3125/3087}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Bohlen–Pierce–Stearns]] as ''n'' approaches infinity and [[Sirius]] as ''n'' approaches 0. If we allow non-integer and infinite ''n'', the continuum describes the set of all 3.5.7 subgroup temperaments supported by [[13edt]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); additionally, if prime 2 is added, this describes the set of all rank-3 7-limit temperaments supported by [[bohpier]]. Due to 13edt, the equal-tempered [[~~Bohlen-Pierce~~ scale]], being a very good representation of the subgroup, and many named temperaments within the subgroup being supported by 13edt, this continuum is structurally important. The just value of ''n'' is 1.4926…, and temperaments near this tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(245/243)<sup>''n''</sup> ~ 3125/3087}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Bohlen–Pierce–Stearns]] as ''n'' approaches infinity and [[Sirius]] as ''n'' approaches 0. If we allow non-integer and infinite ''n'', the continuum describes the set of all 3.5.7 subgroup temperaments supported by [[13edt]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); additionally, if prime 2 is added, this describes the set of all rank-3 7-limit temperaments supported by [[bohpier]]. Due to 13edt, the equal-tempered [[Bohlen–Pierce scale]], being a very good representation of the subgroup, and many named temperaments within the subgroup being supported by 13edt, this continuum is structurally important. The just value of ''n'' is 1.4926…, and temperaments near this tend to be the most accurate ones.

	An alternative definition of this equivalence continuum satisfies (245/243)<sup>''k''</sup> ~ 7<sup>13</sup>/3<sup>23</sup> = 96889010407/94143178827, which sets ''k'' = 5 - ''n''. This definition might be advantageous as ''k'' gives the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 7 in the MOS scale.		An alternative definition of this equivalence continuum satisfies (245/243)<sup>''k''</sup> ~ 7<sup>13</sup>/3<sup>23</sup> = 96889010407/94143178827, which sets ''k'' = 5 - ''n''. This definition might be advantageous as ''k'' gives the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 7 in the MOS scale.

ArrowHead294: /* Projective tuning space */

2024-12-18T15:16:34Z

Projective tuning space

← Older revision		Revision as of 15:16, 18 December 2024
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	{{!}} Labelled by name		{{!}} Labelled by name
	{{!)}}		{{!)}}
	</div><div style{{=}}"display: inline-grid~~; margin-right: 25px~~;">		</div><div style{{=}}"display: inline-grid;">
	{{(!}} class{{=}}"wikitable"		{{(!}} class{{=}}"wikitable"
	{{!-}}		{{!-}}

ArrowHead294: /* Projective tuning space */

2024-12-18T15:16:16Z

Projective tuning space

← Older revision		Revision as of 15:16, 18 December 2024
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	Below is a depiction of the temperaments of this continuum in 3.5.7 [[projective tuning space]].		Below is a depiction of the temperaments of this continuum in 3.5.7 [[projective tuning space]].

	{{center\|<div style="display: inline-grid; margin-right: 25px;">		{{center\|<div style{{=}}"display: inline-grid; margin-right: 25px;">
	{\| class="wikitable"		{{(!}} class{{=}}"wikitable"
	\|-		{{!-}}
	\| [[File:357ptslines1n.png\|320px]]		{{!}} [[File:357ptslines1n.png\|320px]]
	\|-		{{!-}}
	\| Labelled by name		{{!}} Labelled by name
	\|}		{{!)}}
	</div><div style="display: inline-grid; margin-right: 25px;">		</div><div style{{=}}"display: inline-grid; margin-right: 25px;">
	{\| class="wikitable"		{{(!}} class{{=}}"wikitable"
	\|-		{{!-}}
	\| [[File:357ptslines1c.png\|320px]]		{{!}} [[File:357ptslines1c.png\|320px]]
	\|-		{{!-}}
	\| Labelled by comma		{{!}} Labelled by comma
	\|}		{{!)}}
	</div>}}		</div>}}