Pathology of enfactoring: Difference between revisions

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Starting from the origin, we can see that it takes us 2 moves of the generator to reach the approximation of {{vector|-1 1}}, AKA 3/2, where before we made that step in one go. Then another 2 moves to reach the approximation of {{vector|-2 2}}, AKA 9/4, for a total of 4 moves, where before it only took us 2 steps. As you keep going, you'll see that each node it has taken us 2x as many steps as before to reach it.

And for what? What happens in the steps that are halfway between nodes that were on the JI lattice? These are shown with hollow blue circles instead of filled blue circles, to indicate that there's no JI lattice node underneath them. In other words, while these are legitimate musical intervals, there is no JI interval which would be said to temper to them. In other words, since this is 4-ET, that first generator step is to a node {{vector|1}} that's about 300¢. But {{vector|0 0}} tempers to {{vector|0}} and {{vector|-1 1}} tempers to {{vector|2}}; nothing tempers to {{vector|1}}. It's an interval that can certainly at least be heard and understood musically, but has no meaning with respect to tempering JI, or ~~in other words~~, no RTT purpose.

And for what? What happens in the steps that are halfway between nodes that were on the JI lattice? These are shown with hollow blue circles instead of filled blue circles, to indicate that there's no JI lattice node underneath them. In other words, while these are legitimate musical intervals, there is no JI interval which would be said to temper to them. In still other words, since this is 4-ET, that first generator step is to a node {{vector|1}} that's about 300¢. But {{vector|0 0}} tempers to {{vector|0}} and {{vector|-1 1}} tempers to {{vector|2}}; nothing tempers to {{vector|1}}. It's an interval that can certainly at least be heard and understood musically, but it has no meaning with respect to tempering JI, or said another way, it has no RTT purpose.

And so this 4-ET doesn't bring anything to the table that isn't already brought by 2-ET. And so it is fitting to consider it only a temperoid, rather than a true temperament. Were this as bad as things got, it might not be worth pushing for distinguishing temperoids from temperaments. But once we look at enfactored comma bases, we'll see why things get pretty pathological.

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 Starting from the origin, we can see that it takes us 2 moves of the generator to reach the approximation of {{vector|-1 1}}, AKA 3/2, where before we made that step in one go. Then another 2 moves to reach the approximation of {{vector|-2 2}}, AKA 9/4, for a total of 4 moves, where before it only took us 2 steps. As you keep going, you'll see that each node it has taken us 2x as many steps as before to reach it.
-And for what? What happens in the steps that are halfway between nodes that were on the JI lattice? These are shown with hollow blue circles instead of filled blue circles, to indicate that there's no JI lattice node underneath them. In other words, while these are legitimate musical intervals, there is no JI interval which would be said to temper to them. In other words, since this is 4-ET, that first generator step is to a node {{vector|1}} that's about 300¢. But {{vector|0 0}} tempers to {{vector|0}} and {{vector|-1 1}} tempers to {{vector|2}}; nothing tempers to {{vector|1}}. It's an interval that can certainly at least be heard and understood musically, but has no meaning with respect to tempering JI, or in other words, no RTT purpose.
+And for what? What happens in the steps that are halfway between nodes that were on the JI lattice? These are shown with hollow blue circles instead of filled blue circles, to indicate that there's no JI lattice node underneath them. In other words, while these are legitimate musical intervals, there is no JI interval which would be said to temper to them. In still other words, since this is 4-ET, that first generator step is to a node {{vector|1}} that's about 300¢. But {{vector|0 0}} tempers to {{vector|0}} and {{vector|-1 1}} tempers to {{vector|2}}; nothing tempers to {{vector|1}}. It's an interval that can certainly at least be heard and understood musically, but it has no meaning with respect to tempering JI, or said another way, it has no RTT purpose.
 And so this 4-ET doesn't bring anything to the table that isn't already brought by 2-ET. And so it is fitting to consider it only a temperoid, rather than a true temperament. Were this as bad as things got, it might not be worth pushing for distinguishing temperoids from temperaments. But once we look at enfactored comma bases, we'll see why things get pretty pathological.