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The '''Jubilismic-augmented equivalence continuum''' is a 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). 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, 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"
|+Temperaments in the continuum
|+ style="font-size: 105%;" | Temperaments in the continuum
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 16: Line 16:
! Ratio
! Ratio
! Monzo (2.5.7 subgroup)
! Monzo (2.5.7 subgroup)
|-
| -2
| [[Rip]] restriction
| 2560/2401
| {{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
| [[Augmented]]
| [[Augment]]
| [[128/125]]
| [[128/125]]
| {{monzo| 7 -3 }}
| {{Monzo| 7 -3 }}
|-
|-
| 1/2
| 1/2
| [[Diaschismic family#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
| [[Sensamagic clan#Superthird|Superthird]]
| [[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]]
| {{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.
 
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments in the continuum
|-
! rowspan="2" | ''m''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo (2.5.7 subgroup)
|-
| -1
| [[Diaschismic]] restriction
| 401408/390625
| {{monzo| 13 -8 2 }}
|-
| 0
| [[Augment]]
| [[128/125]]
| {{monzo| 7 -3 }}
|-
| 1
| [[Jubilic]]
| [[50/49]]
| [[50/49]]
| {{monzo| 1 2 -2 }}
| {{monzo| 1 2 -2 }}
|-
| 2
| [[Superthird]] restriction
| 78125/76832
| {{monzo| -5 7 -4 }}
|-
| 3
| [[Compton]] restriction
| 244140625/240945152
| {{monzo| -11 12 -6 }}
|-
| 4
| [[Quintupole]] restriction
| 762939453125/755603996672
| {{monzo| -17 17 -8 }}
|-
| 5
| [[Undim]] restriction
| (32 digits)
| {{monzo| -23 22 -10 }}
|-
| 6
| [[Term (temperament)|Term]] restriction
| (38 digits)
| {{monzo| -29 27 -12 }}
|-
| 7
| [https://sintel.pythonanywhere.com/result?subgroup=2.9.5.7&reduce=on&weights=weil&target=&edos=6+%26+190&commas=321489%2F320000%2C+703125%2F702464&submit_comma=submit 6 & 190]
| (46 digits)
| {{monzo| 35 -32 14 }}
|-
| …
| …
| …
|
|-
| ∞
| [[Didacus]]
| [[3136/3125]]
| {{monzo| 6 -5 2 }}
|}
|}
== Graphs ==
[[File:Jubilaug5.png|alt=Jubilaug5.png|745x470px]]
Squared [[DKW theory|DKW]] error of temperaments in the equivalence continuum as a function of tuning of [[5/4]].
[[File:Jubilaug7.png|alt=Jubilaug7.png|745x470px]]
Squared [[DKW theory|DKW]] error of temperaments in the equivalence continuum as a function of tuning of [[8/7]].


[[Category:6edo]]
[[Category:6edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
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