Jubilismic–augmented equivalence continuum - Revision history

FloraC: Cleanup and note the restrictions in the tables

2025-09-11T05:50:48Z

Cleanup and note the restrictions in the tables

← Older revision		Revision as of 05:50, 11 September 2025
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	The '''jubilismic–augmented equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[2.5.7 subgroup]] temperaments which equate a number of [[50/49\|jubilismas (50/49)]] with the [[128/125\|lesser diesis (128/125)]].		The '''jubilismic–augmented equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[2.5.7 subgroup]] temperaments which equate a number of [[50/49\|jubilismas (50/49)]] with the [[128/125\|lesser diesis (128/125)]].

	All temperaments in the continuum satisfy {{nowrap\|(50/49)<sup>''n''</sup> ~ 128/125}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[~~Jubilismic clan#Jubilic\|~~jubilic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.5.7 subgroup temperaments supported by [[6edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); due to 6edo representing this subgroup modestly well for its size, this continuum is structurally important to 2.5.7. The just value of ''n'' is 1.1739…, and temperaments near this tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(50/49)<sup>''n''</sup> ~ 128/125}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[jubilic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.5.7 subgroup temperaments supported by [[6edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); due to 6edo representing this subgroup modestly well for its size, this continuum is structurally important to 2.5.7. The just value of ''n'' is 1.1739…, and temperaments near this tend to be the most accurate ones.

	128/125 is the characteristic 2.5 comma tempered out in [[6edo]]. In each case, we notice that ''n'' equals half the order of harmonic 7 in the corresponding comma (noting that 6edo's ring of [[5/4]]s inherits from [[3edo]], 6edo therefore has two rings and any comma involving 7 therefore has 7 to an even power), and equals the number of generators to obtain a harmonic 5 in the ~~MOS~~ scale.		128/125 is the characteristic 2.5 comma tempered out in [[6edo]]. In each case, we notice that ''n'' equals half the order of harmonic 7 in the corresponding comma (noting that 6edo's ring of [[5/4]]'s inherits from [[3edo]], 6edo therefore has two rings and any comma involving 7 therefore has 7 to an even power), and equals the number of generators to obtain a harmonic 5 in the mos scale.

	Note temperaments linked to in the below are generally 2.5.7 subgroup restrictions of full 7-limit temperaments.		Note temperaments linked to in the below are generally 2.5.7 subgroup restrictions of full 7-limit temperaments.

	{\| class="wikitable center-1 ~~center-2~~"		{\| class="wikitable center-1"
	\|+ style="font-size: 105%;" \| Temperaments in the continuum		\|+ style="font-size: 105%;" \| Temperaments in the continuum
	\|-		\|-
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	\|-		\|-
	\| -2		\| -2
	\| [[~~Ripple family#Septimal ripple\|Ripple~~]]		\| [[Rip]] restriction
	\| 2560/2401		\| 2560/2401
	\| {{~~monzo~~\| 9 1 -4 }}		\| {{Monzo\| 9 1 -4 }}
	\|-		\|-
	\| -1		\| -1
	\| [[~~Bapbo clan#Bapbo\|~~Bapbo]]		\| [[Bapbo]]
	\| [[256/245]]		\| [[256/245]]
	\| {{~~monzo~~\| 8 -1 -2 }}		\| {{Monzo\| 8 -1 -2 }}
	\|-		\|-
	\| 0		\| 0
	\| [[~~No-threes subgroup temperaments#Augment\|~~Augment]]		\| [[Augment]]
	\| [[128/125]]		\| [[128/125]]
	\| {{~~monzo~~\| 7 -3 }}		\| {{Monzo\| 7 -3 }}
	\|-		\|-
	\| 1/2		\| 1/2
	\| [[~~Diaschismic family#Septimal diaschismic\|~~Diaschismic]]		\| [[Diaschismic]] restriction
	\| 401408/390625		\| 401408/390625
	\| {{~~monzo~~\| 13 -8 2 }}		\| {{Monzo\| 13 -8 2 }}
	\|-		\|-
	\| 1		\| 1
	\| [[~~Hemimean clan#Didacus\|~~Didacus]]		\| [[Didacus]]
	\| [[3136/3125]]		\| [[3136/3125]]
	\| {{~~monzo~~\| 6 -5 2 }}		\| {{Monzo\| 6 -5 2 }}
	\|-		\|-
	\| 3/2		\| 3/2
	\| [[Compton ~~family#Septimal compton\|Waage~~]]		\| [[Compton]] restriction
	\| 244140625/240945152		\| 244140625/240945152
	\| {{~~monzo~~\| -11 12 -6 }}		\| {{Monzo\| -11 12 -6 }}
	\|-		\|-
	\| 2		\| 2
	\| [[~~No-threes subgroup temperaments#Frostburn\|~~Frostburn]]		\| [[Frostburn]]
	\| 78125/76832		\| 78125/76832
	\| {{~~monzo~~\| -5 7 -4 }}		\| {{Monzo\| -5 7 -4 }}
	\|-		\|-
	\| 3		\| 3
	\| [[~~Misty family#Fog\|~~Fog]]		\| [[Fog]] restriction
	\| 1953125/1882384		\| 1953125/1882384
	\| {{~~monzo~~\| -4 9 -6 }}		\| {{Monzo\| -4 9 -6 }}
	\|-		\|-
	\| …		\| …
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	\|-		\|-
	\| ∞		\| ∞
	\| [[~~Jubilismic clan#Jubilic\|~~Jubilic]]		\| [[Jubilic]]
	\| [[50/49]]		\| [[50/49]]
	\| {{~~monzo~~\| 1 2 -2 }}		\| {{Monzo\| 1 2 -2 }}
	\|}		\|}

	We may invert the continuum by setting ''m'' such that {{nowrap\|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''~~didacus-augmented~~ equivalence continuum'', as temperaments satisfy {{nowrap\|(3136/3125)<sup>''m''</sup> ~ 128/125}}. The just value of ''m'' is 6.7495…, and temperaments close to this value are the most accurate.		We may invert the continuum by setting ''m'' such that {{nowrap\| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''didacus–augmented equivalence continuum'', as temperaments satisfy {{nowrap\| (3136/3125)<sup>''m''</sup> ~ 128/125 }}. The just value of ''m'' is 6.7495…, and temperaments close to this value are the most accurate.

	{\| class="wikitable center-1 ~~center-2~~"		{\| class="wikitable center-1"
	\|+ style="font-size: 105%;" \| Temperaments in the continuum		\|+ style="font-size: 105%;" \| Temperaments in the continuum
	\|-		\|-
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	\|-		\|-
	\| -1		\| -1
	\| [[~~Diaschismic family#Septimal diaschismic\|~~Diaschismic]]		\| [[Diaschismic]] restriction
	\| 401408/390625		\| 401408/390625
	\| {{monzo\| 13 -8 2 }}		\| {{monzo\| 13 -8 2 }}
	\|-		\|-
	\| 0		\| 0
	\| [[~~Augmented~~]]		\| [[Augment]]
	\| [[128/125]]		\| [[128/125]]
	\| {{monzo\| 7 -3 }}		\| {{monzo\| 7 -3 }}
	\|-		\|-
	\| 1		\| 1
	\| [[~~Jubilismic clan#Jubilic\|~~Jubilic]]		\| [[Jubilic]]
	\| [[50/49]]		\| [[50/49]]
	\| {{monzo\| 1 2 -2 }}		\| {{monzo\| 1 2 -2 }}
	\|-		\|-
	\| 2		\| 2
	\| [[~~Sensamagic clan#Superthird\|~~Superthird]]		\| [[Superthird]] restriction
	\| 78125/76832		\| 78125/76832
	\| {{monzo\| -5 7 -4 }}		\| {{monzo\| -5 7 -4 }}
	\|-		\|-
	\| 3		\| 3
	\| [[Compton ~~family#Septimal compton\|Waage~~]]		\| [[Compton]] restriction
	\| 244140625/240945152		\| 244140625/240945152
	\| {{monzo\| -11 12 -6 }}		\| {{monzo\| -11 12 -6 }}
	\|-		\|-
	\| 4		\| 4
	\| [[~~Quintaleap family#Quintupole\|~~Quintupole]]		\| [[Quintupole]] restriction
	\| 762939453125/755603996672		\| 762939453125/755603996672
	\| {{monzo\| -17 17 -8 }}		\| {{monzo\| -17 17 -8 }}
	\|-		\|-
	\| 5		\| 5
	\| [[~~Undim family#Septimal undim\|~~Undim]]		\| [[Undim]] restriction
	\| (32 digits)		\| (32 digits)
	\| {{monzo\| -23 22 -10 }}		\| {{monzo\| -23 22 -10 }}
	\|-		\|-
	\| 6		\| 6
	\| [[~~Schismatic family#~~Term\|Term]]		\| [[Term (temperament)\|Term]] restriction
	\| (38 digits)		\| (38 digits)
	\| {{monzo\| -29 27 -12 }}		\| {{monzo\| -29 27 -12 }}
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	\|-		\|-
	\| ∞		\| ∞
	\| [[~~Hemimean clan#Didacus\|~~Didacus]]		\| [[Didacus]]
	\| [[3136/3125]]		\| [[3136/3125]]
	\| {{monzo\| 6 -5 2 }}		\| {{monzo\| 6 -5 2 }}

Lériendil at 14:32, 29 May 2025

2025-05-29T14:32:21Z

← Older revision		Revision as of 14:32, 29 May 2025
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	\|-		\|-
	\| 2		\| 2
	\| [[~~Magic family~~#~~Quadrimage~~\|~~Quadrimage~~]]		\| [[No-threes subgroup temperaments#Frostburn\|Frostburn]]
	\| 78125/76832		\| 78125/76832
	\| {{monzo\| -5 7 -4 }}		\| {{monzo\| -5 7 -4 }}

Lériendil: Text replacement - "#Diaschismic" to "#Septimal diaschismic"

2025-04-27T22:02:07Z

Text replacement - "#Diaschismic" to "#Septimal diaschismic"

← Older revision		Revision as of 22:02, 27 April 2025
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	\|-		\|-
	\| 1/2		\| 1/2
	\| [[Diaschismic family#~~Diaschismic~~\|Diaschismic]]		\| [[Diaschismic family#Septimal diaschismic\|Diaschismic]]
	\| 401408/390625		\| 401408/390625
	\| {{monzo\| 13 -8 2 }}		\| {{monzo\| 13 -8 2 }}
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	\|-		\|-
	\| -1		\| -1
	\| [[Diaschismic family#~~Diaschismic~~\|Diaschismic]]		\| [[Diaschismic family#Septimal diaschismic\|Diaschismic]]
	\| 401408/390625		\| 401408/390625
	\| {{monzo\| 13 -8 2 }}		\| {{monzo\| 13 -8 2 }}

Lériendil: replaced superthird with the non-contorted quadrimage

2025-03-16T17:24:48Z

replaced superthird with the non-contorted quadrimage

← Older revision		Revision as of 17:24, 16 March 2025
Line 48:		Line 48:
	\|-		\|-
	\| 2		\| 2
	\| [[~~Sensamagic clan~~#~~Superthird~~\|~~Superthird~~]]		\| [[Magic family#Quadrimage\|Quadrimage]]
	\| 78125/76832		\| 78125/76832
	\| {{monzo\| -5 7 -4 }}		\| {{monzo\| -5 7 -4 }}

ArrowHead294 at 14:50, 18 December 2024

2024-12-18T14:50:44Z

← Older revision		Revision as of 14:50, 18 December 2024
Line 1:		Line 1:
	The '''~~jubilismic-augmented~~ equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[2.5.7 subgroup]] temperaments which equate a number of [[50/49\|jubilismas (50/49)]] with the [[128/125\|lesser diesis (128/125)]].		The '''jubilismic–augmented equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[2.5.7 subgroup]] temperaments which equate a number of [[50/49\|jubilismas (50/49)]] with the [[128/125\|lesser diesis (128/125)]].

	All temperaments in the continuum satisfy (50/49)<sup>''n''</sup> ~ 128/125. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Jubilismic clan#Jubilic\|jubilic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.5.7 subgroup temperaments supported by [[6edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); due to 6edo representing this subgroup modestly well for its size, this continuum is structurally important to 2.5.7. The just value of ''n'' is 1.1739…, and temperaments near this tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(50/49)<sup>''n''</sup> ~ 128/125}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Jubilismic clan#Jubilic\|jubilic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.5.7 subgroup temperaments supported by [[6edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); due to 6edo representing this subgroup modestly well for its size, this continuum is structurally important to 2.5.7. The just value of ''n'' is 1.1739…, and temperaments near this tend to be the most accurate ones.

	128/125 is the characteristic 2.5 comma tempered out in [[6edo]]. In each case, we notice that ''n'' equals half the order of harmonic 7 in the corresponding comma (noting that 6edo's ring of [[5/4]]s inherits from [[3edo]], 6edo therefore has two rings and any comma involving 7 therefore has 7 to an even power), and equals the number of generators to obtain a harmonic 5 in the MOS scale.		128/125 is the characteristic 2.5 comma tempered out in [[6edo]]. In each case, we notice that ''n'' equals half the order of harmonic 7 in the corresponding comma (noting that 6edo's ring of [[5/4]]s inherits from [[3edo]], 6edo therefore has two rings and any comma involving 7 therefore has 7 to an even power), and equals the number of generators to obtain a harmonic 5 in the MOS scale.
Line 8:		Line 8:

	{\| 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 68:		Line 68:
	\|}		\|}

	We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''didacus-augmented equivalence continuum'', as temperaments satisfy (3136/3125)<sup>''m''</sup> ~ 128/125. The just value of ''m'' is 6.7495…, and temperaments close to this value are the most accurate.		We may invert the continuum by setting ''m'' such that {{nowrap\|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''didacus-augmented equivalence continuum'', as temperaments satisfy {{nowrap\|(3136/3125)<sup>''m''</sup> ~ 128/125}}. The just value of ''m'' is 6.7495…, and temperaments close to this value are the most accurate.

	{\| 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" \| ''m''		! rowspan="2" \| ''m''
Line 137:		Line 137:

	== Graphs ==		== Graphs ==

	[[File:Jubilaug5.png\|alt=Jubilaug5.png\|745x470px]]		[[File:Jubilaug5.png\|alt=Jubilaug5.png\|745x470px]]

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

2024-12-18T14:49:43Z

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

← Older revision	Revision as of 14:49, 18 December 2024
(No difference)

Lériendil: replaced Augmented label with Augment

2024-12-12T11:41:39Z

replaced Augmented label with Augment

← Older revision		Revision as of 11:41, 12 December 2024
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	\|-		\|-
	\| 0		\| 0
	\| [[~~Augmented~~]]		\| [[No-threes subgroup temperaments#Augment\|Augment]]
	\| [[128/125]]		\| [[128/125]]
	\| {{monzo\| 7 -3 }}		\| {{monzo\| 7 -3 }}

Lériendil at 19:54, 10 September 2024

2024-09-10T19:54:47Z

← Older revision		Revision as of 19:54, 10 September 2024
Line 16:		Line 16:
	! Ratio		! Ratio
	! Monzo (2.5.7 subgroup)		! Monzo (2.5.7 subgroup)
			\|-
			\| -2
			\| [[Ripple family#Septimal ripple\|Ripple]]
			\| 2560/2401
			\| {{monzo\| 9 1 -4 }}
	\|-		\|-
	\| -1		\| -1

Lériendil at 19:46, 10 September 2024

2024-09-10T19:46:45Z

← Older revision		Revision as of 19:46, 10 September 2024
Line 68:		Line 68:
	\|+Temperaments in the continuum		\|+Temperaments in the continuum
	\|-		\|-
	! rowspan="2" \| ''n''		! rowspan="2" \| ''m''
	! rowspan="2" \| Temperament		! rowspan="2" \| Temperament
	! colspan="2" \| Comma		! colspan="2" \| Comma

Lériendil at 10:57, 10 September 2024

2024-09-10T10:57:52Z

← Older revision		Revision as of 10:57, 10 September 2024
Line 130:		Line 130:
	\| {{monzo\| 6 -5 2 }}		\| {{monzo\| 6 -5 2 }}
	\|}		\|}

			== Graphs ==

	[[File:Jubilaug5.png\|alt=Jubilaug5.png\|745x470px]]		[[File:Jubilaug5.png\|alt=Jubilaug5.png\|745x470px]]