User talk:FloraC: Difference between revisions

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: —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 14:12, 31 December 2025 (UTC)

:: Degeneracy is normally understood to be limiting behavior; the case of degeneracy being a removal of a dimension is specific to linear algebra, according to the broader mathematical definition, these temperaments can still be considered degenerate cases, hear me out:

:: Assuming 1edo is a trivial temperament of the 2-limit:

:: Let's think about the simplest case scenario: Going from the 3-limit, tempering out 4/3 to get a rank-1 temperament. This "rank-1 Bixby" can only be represented by patent val by 1edo, where each step is an (tempered) octave, and fifths are rounded to octaves, same with 5/4s and all other primes. Here, the structure has collapsed, since only the 2-limit is left and only the trivial temperament (1edo) can represent it.

:: Even in the 5-limit extension (Bixby proper), no other type of temperament collapses its structure so much that it is ''only'' representable by patent val by 1edo, not even Antitonic. Since the 3-structure has collapsed to the 2-limit and only the 2.5 subgroup makes sense, I believe that the only reasonable way to look at this, is to see it as a degenerate case.

:: In Antitonic, you can temper the period to get a bit closer to "fifths" and "octaves", but the structure has not collapsed. In fact, Antitonic is the first proper 3-limit temperament, it even demonstrates 3-2 telicity!

:: I believe this is a useful case of degeneracy that should be used to categorize these limiting temperaments. I wouldn't deem them exotemperaments, no that's far too generous... just degenerate temperaments.

:: --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 16:55, 31 December 2025 (UTC)

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 : —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 14:12, 31 December 2025 (UTC)
+:: Degeneracy is normally understood to be limiting behavior; the case of degeneracy being a removal of a dimension is specific to linear algebra, according to the broader mathematical definition, these temperaments can still be considered degenerate cases, hear me out:
+:: Assuming 1edo is a trivial temperament of the 2-limit:
+:: Let's think about the simplest case scenario: Going from the 3-limit, tempering out 4/3 to get a rank-1 temperament. This "rank-1 Bixby" can only be represented by patent val by 1edo, where each step is an (tempered) octave, and fifths are rounded to octaves, same with 5/4s and all other primes. Here, the structure has collapsed, since only the 2-limit is left and only the trivial temperament (1edo) can represent it.
+:: Even in the 5-limit extension (Bixby proper), no other type of temperament collapses its structure so much that it is ''only'' representable by patent val by 1edo, not even Antitonic. Since the 3-structure has collapsed to the 2-limit and only the 2.5 subgroup makes sense, I believe that the only reasonable way to look at this, is to see it as a degenerate case.
+:: In Antitonic, you can temper the period to get a bit closer to "fifths" and "octaves", but the structure has not collapsed. In fact, Antitonic is the first proper 3-limit temperament, it even demonstrates 3-2 telicity!
+:: I believe this is a useful case of degeneracy that should be used to categorize these limiting temperaments. I wouldn't deem them exotemperaments, no that's far too generous... just degenerate temperaments.
+:: --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 16:55, 31 December 2025 (UTC)