Tarot equivalence continuum: Difference between revisions
will fill in the table on the visual editor |
m Mark as mathematical interest. Todo |
||
| (9 intermediate revisions by 4 users not shown) | |||
| Line 1: | Line 1: | ||
{{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. | |||
{| class="wikitable center-1 center-2 center-4 center-5" | {| class="wikitable center-1 center-2 center-4 center-5" | ||
|- | |- | ||
! | ! ''n'' | ||
! | ! Temperament | ||
! | ! Comma (monzo) | ||
|- | |- | ||
| 0 | |||
| [[22nd-octave temperaments#Major Arcana|Major Arcana]] | |||
| {{Monzo|-193 154 -22}} | |||
|- | |- | ||
| | | 1 | ||
| | | [[Minortone]] | ||
| | | {{Monzo|-16 35 -17}} | ||
| | |||
|- | |- | ||
| | | 2 | ||
| | | [[Atomic]] | ||
| | | [[Kirnberger's atom|[161 -84 -12⟩]] | ||
| | |||
|- | |- | ||
| | | 3 | ||
| | | [[#Empress|Empress]] (1848 & 53) | ||
| {{Monzo|-177 119 -5}} | |||
| | |||
| | |||
| | |||
|- | |- | ||
| | | 4 | ||
| | | 1848 & 3296 | ||
| | | {{Monzo|-547 392 -32}} | ||
| | |||
|- | |- | ||
| | | 5 | ||
| | | 1848 & 3243 | ||
| | | {{Monzo|-370 273 -27}} | ||
| | |||
|- | |- | ||
| | | … | ||
| | | … | ||
| | | … | ||
| | |- | ||
|∞ | | ∞ | ||
| | | [[Barium]] | ||
| | |||
| | |||
|} | |} | ||
== Empress == | |||
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 | |||
Comma list: {{Monzo|-177 119 -5}} | |||
Mapping: [{{val|1 3 36}}, {{val|0 -5 -119}}] | |||
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:Equivalence continua]] | |||
{{Todo| cleanup | discuss title }} | |||