Saturation, torsion, and contorsion: Difference between revisions

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'''Torsion''' is used to refer to the opposite of saturation, but only regarding comma bases; in other words, a comma basis is either saturated, or it ''has torsion''.

A comma basis with torsion is rarely useful at all. It states that a power of a ratio is ~~tempered out~~ but does not explicitly state that the ratio itself is ~~tempered out~~ (for instance, (81/80)^2 is ~~tempered out~~ but 81/80 is not). From a mathematical standpoint, there are multiple ways to interpret this situation. Historically, a group theory formalism was used, leading to the interpretation that the ratio itself was indeed ''not'' ~~tempered out~~, despite this being musically absurd. Using a linear algebra formalism as is preferred now, however, no such absurdity is suggested. This historical usage of the group theory formalism is another reason why it is bad practice to use comma bases with torsion.

A comma basis with torsion is rarely useful at all. It states that a power of a ratio is made to [[vanish]] but does not explicitly state that the ratio itself is made to vanish (for instance, (81/80)^2 is made to vanish but 81/80 is not). From a mathematical standpoint, there are multiple ways to interpret this situation. Historically, a group theory formalism was used, leading to the interpretation that the ratio itself was indeed ''not'' made to vanish, despite this being musically absurd. Using a linear algebra formalism as is preferred now, however, no such absurdity is suggested. This historical usage of the group theory formalism is another reason why it is bad practice to use comma bases with torsion.

Torsion also refers to a similar situation that occurs for the list of commas defining a [[periodicity block]]; in fact, this was its original use case.

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 '''Torsion''' is used to refer to the opposite of saturation, but only regarding comma bases; in other words, a comma basis is either saturated, or it ''has torsion''.
-A comma basis with torsion is rarely useful at all. It states that a power of a ratio is tempered out but does not explicitly state that the ratio itself is tempered out (for instance, (81/80)^2 is tempered out but 81/80 is not). From a mathematical standpoint, there are multiple ways to interpret this situation. Historically, a group theory formalism was used, leading to the interpretation that the ratio itself was indeed ''not'' tempered out, despite this being musically absurd. Using a linear algebra formalism as is preferred now, however, no such absurdity is suggested. This historical usage of the group theory formalism is another reason why it is bad practice to use comma bases with torsion.
+A comma basis with torsion is rarely useful at all. It states that a power of a ratio is made to [[vanish]] but does not explicitly state that the ratio itself is made to vanish (for instance, (81/80)^2 is made to vanish but 81/80 is not). From a mathematical standpoint, there are multiple ways to interpret this situation. Historically, a group theory formalism was used, leading to the interpretation that the ratio itself was indeed ''not'' made to vanish, despite this being musically absurd. Using a linear algebra formalism as is preferred now, however, no such absurdity is suggested. This historical usage of the group theory formalism is another reason why it is bad practice to use comma bases with torsion.
 Torsion also refers to a similar situation that occurs for the list of commas defining a [[periodicity block]]; in fact, this was its original use case.