Talk:31st-octave temperaments
sad enfactoring?
Trying to understand this statement in the Birds section: "It also tempers out the 31-7 comma, but sadly, combining the two commas leads to torsion."
When the commas listed — 3136/3125 and 823543/819200 — are expressed as the comma basis for this temperament, we get ⟨[6 0 -5 2⟩ [-15 0 -2 7⟩]. This matches with the mapping provided [⟨31 49 72 87] ⟨0 1 0 0]⟩, i.e. it is its null-space. In canonical form this mapping and comma basis are [⟨31 0 72 87] ⟨0 1 0 0]⟩ and ⟨[-72 0 31 0⟩ [-33 0 13 1⟩], respectively.
Elsewhere description claims that this temperament could be defined by tempering out the 31-5 and the 31-7 commas, were it not for torsion. I don't know what is "sad" about the torsion. Simply remove it by defactoring, right? When these two commas are expressed as a comma basis for a temperament it looks like ⟨[-87 0 0 31⟩ [72 0 -31 0⟩], and then if we put it in canonical form (which defactors it), we get the same thing ⟨[-72 0 31 0⟩ [-33 0 13 1⟩] as what's there.
So can't we just remove the part where it says "but sadly, combining the two commas leads to torsion"? Otherwise, can we clarify what is sad?
If this clause is retained, then I have a revision request. As you can read about on the page re: defactoring, I am recommending we not use the word "torsion" for temperaments, but only for periodicity blocks. A temperament may "be enfactored", but it shouldn't be said to "have torsion". --Cmloegcmluin (talk) 16:24, 30 September 2021 (UTC)