Tarot equivalence continuum: Difference between revisions
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{{Mathematical interest}} | |||
The Tarot equivalence continuum is a continuum of 2.3.5 subgroup temperaments which all equate a number of [[22nd-octave temperaments#Major Arcana|Major Arcana]] commas with the [[barium]] comma, which due to it being period-56 can also be called Minor Arcana comma. Since there's 22 cards in the Major Arcana, and 56 in the Minor Arcana, the natural name that arises is Tarot equivalence continuum. | The Tarot equivalence continuum is a continuum of 2.3.5 subgroup temperaments which all equate a number of [[22nd-octave temperaments#Major Arcana|Major Arcana]] commas with the [[barium]] comma, which due to it being period-56 can also be called Minor Arcana comma. Since there's 22 cards in the Major Arcana, and 56 in the Minor Arcana, the natural name that arises is Tarot equivalence continuum. | ||
All temperaments in this continuum satisfy (2<sup> | All temperaments in this continuum satisfy {{nowrap|(2<sup>−193</sup> 3<sup>154</sup> 5<sup>−22</sup>)<sup>''n''</sup> ~ (2<sup>−225</sup> 3<sup>224</sup> 5<sup>−56</sup>)}}. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by [[1848edo]], 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 2.0070839..., and temperaments having ''n'' near this value tend to be the most accurate ones. | ||
The continuum contains two notable 5-limit temperaments supported by 1848edo, minortone and atomic. | The continuum contains two notable 5-limit temperaments supported by 1848edo, minortone and atomic. | ||
{| class="wikitable center-1 center-2 center-4 center-5" | {| class="wikitable center-1 center-2 center-4 center-5" | ||
|- | |- | ||
!''n'' | ! ''n'' | ||
!Temperament | ! Temperament | ||
!Comma (monzo) | ! Comma (monzo) | ||
|- | |- | ||
|0 | | 0 | ||
|[[22nd-octave temperaments#Major Arcana|Major Arcana]] | | [[22nd-octave temperaments#Major Arcana|Major Arcana]] | ||
|{{Monzo|-193 154 -22}} | | {{Monzo|-193 154 -22}} | ||
|- | |- | ||
|1 | | 1 | ||
|[[Minortone]] | | [[Minortone]] | ||
|{{Monzo|16 35 -17}} | | {{Monzo|-16 35 -17}} | ||
|- | |- | ||
|2 | | 2 | ||
|[[Atomic]] | | [[Atomic]] | ||
|[[Kirnberger's atom|[161 -84 -12⟩]] | | [[Kirnberger's atom|[161 -84 -12⟩]] | ||
|- | |- | ||
|3 | | 3 | ||
|1848 & 53 | | [[#Empress|Empress]] (1848 & 53) | ||
|{{Monzo|-177 119 -5}} | | {{Monzo|-177 119 -5}} | ||
|- | |- | ||
|4 | | 4 | ||
|1848 & 3296 | | 1848 & 3296 | ||
|{{Monzo|-547 392 -32}} | | {{Monzo|-547 392 -32}} | ||
|- | |- | ||
|5 | | 5 | ||
|1848 & 3243 | | 1848 & 3243 | ||
|{{Monzo|-370 273 -27}} | | {{Monzo|-370 273 -27}} | ||
|- | |- | ||
|… | | … | ||
|… | | … | ||
|… | | … | ||
|- | |- | ||
|∞ | | ∞ | ||
|[[Barium]] | | [[Barium]] | ||
| | | | ||
|} | |} | ||
== Empress == | == Empress == | ||
Empress tempers out | Empress tempers out {{Monzo|-177 119 -5}}, and since it's number 3 in the equivalence continuum, it is named after the 3rd Major Arcana card. | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
| Line 54: | Line 56: | ||
Optimal tuning (CTE): ~343597383680/282429536481 = 339.611 | Optimal tuning (CTE): ~343597383680/282429536481 = 339.611 | ||
{{Optimal ET sequence|legend=1|53, 1530, 1583, 1636, 1689, 1742, 1795, 1848, 1901, 1954}}, ... | |||
[[Category:1848edo]] | [[Category:1848edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
{{Todo| cleanup | discuss title }} | |||