Jubilismic–augmented equivalence continuum: Difference between revisions
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The ''' | 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. | ||
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{| 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'' | ||
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|} | |} | ||
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'' | ||
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== Graphs == | == Graphs == | ||
[[File:Jubilaug5.png|alt=Jubilaug5.png|745x470px]] | [[File:Jubilaug5.png|alt=Jubilaug5.png|745x470px]] | ||