Pathology of enfactoring - Revision history

Dave Keenan: /* Enfactored comma bases vs. periodicity blocks with torsion */ Simplified the wording of the most recent change.

2025-03-25T08:41:13Z

Enfactored comma bases vs. periodicity blocks with torsion: Simplified the wording of the most recent change.

← Older revision		Revision as of 08:41, 25 March 2025
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	Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector\|-6 4}} was because the tempering: if {{vector\|-6 4}} is made to vanish, then {{vector\|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.		Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector\|-6 4}} was because the tempering: if {{vector\|-6 4}} is made to vanish, then {{vector\|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.

	Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: [[2048/2025]] ({{vector\|11 -4 -2}}) and 625/324 ({{vector\|-2 -4 4}}). The value at the fourth corner, 12800/6561, has vector {{vector\|9 -8 2}}. ~~You may recognize this as~~ the meantone comma ~~(its inverse, anyway)~~ squared, (80~~/81~~)^2 = {{vector\|4 -4 1}}×2~~, but with an extra factor of 2 introduced in order to octave-reduce (well, in this case, "octave-increase") the ratio to between 1 and 2. The most important part is that the vector is 2-enfactored~~. You can see that the node at the very center of this block is 160/81 ~~= {{vector\|5 -4 1}}~~, ~~which is unsurprisingly half of~~ the ~~ratio at the fourth corner, i.e. the ''non-enfactored'' version~~ of the meantone comma ~~(though, again, inverted and octave-adjusted to between 1 and 2)~~.		Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: [[2048/2025]] ({{vector\|11 -4 -2}}) and 625/324 ({{vector\|-2 -4 4}}). The value at the fourth corner, 12800/6561, is the product of those two and is the octave complement of 6561/6400 which has vector {{vector\|-8 8 -2}}. This is the meantone comma squared, (81/80)^2, and its vector is 2-enfactored, {{vector\|-4 4 -1}}×2. You can see that the node at the very center of this block is 160/81, the octave complement of the meantone comma.

	The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset.		The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset.

Cmloegcmluin: /* Enfactored comma bases vs. periodicity blocks with torsion */ clarifications per Kite's feedback

2025-03-20T23:39:50Z

Enfactored comma bases vs. periodicity blocks with torsion: clarifications per Kite's feedback

← Older revision		Revision as of 23:39, 20 March 2025
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	Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector\|-6 4}} was because the tempering: if {{vector\|-6 4}} is made to vanish, then {{vector\|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.		Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector\|-6 4}} was because the tempering: if {{vector\|-6 4}} is made to vanish, then {{vector\|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.

	Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: [[2048/2025]] ({{vector\|11 -4 -2}}) and 625/324 ({{vector\|-2 -4 4}}). The value at the fourth corner, 12800/6561, has vector 2×{{vector\|-8 ~~8 -~~2}}. ~~The first~~ 2 ~~is just~~ to octave-reduce ~~it to being positive~~, ~~but you may recognize~~ the ~~actual vector part as~~ 2 ~~times the meantone comma~~. The most important part is that the vector is 2-enfactored. You can see that the node at the very center of this block is 160/81~~, which again is 2×~~{{vector\|-4 ~~4 -~~1}}, or the ~~octave-reduced~~ non-enfactored version of ~~that same~~ comma.		Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: [[2048/2025]] ({{vector\|11 -4 -2}}) and 625/324 ({{vector\|-2 -4 4}}). The value at the fourth corner, 12800/6561, has vector {{vector\|9 -8 2}}. You may recognize this as the meantone comma (its inverse, anyway) squared, (80/81)^2 = {{vector\|4 -4 1}}×2, but with an extra factor of 2 introduced in order to octave-reduce (well, in this case, "octave-increase") the ratio to between 1 and 2. The most important part is that the vector is 2-enfactored. You can see that the node at the very center of this block is 160/81 = {{vector\|5 -4 1}}, which is unsurprisingly half of the ratio at the fourth corner, i.e. the ''non-enfactored'' version of the meantone comma (though, again, inverted and octave-adjusted to between 1 and 2).

	The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset.		The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset.

TallKite: /* Enfactored comma bases vs. periodicity blocks with torsion */ typo

2025-03-20T10:45:10Z

Enfactored comma bases vs. periodicity blocks with torsion: typo

← Older revision		Revision as of 10:45, 20 March 2025
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	Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector\|-6 4}} was because the tempering: if {{vector\|-6 4}} is made to vanish, then {{vector\|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.		Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector\|-6 4}} was because the tempering: if {{vector\|-6 4}} is made to vanish, then {{vector\|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.

	Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: [[2048/2025]] ({{vector\|11 -4 -4}}) and 625/324 ({{vector\|-2 -4 4}}). The value at the fourth corner, 12800/6561, has vector 2×{{vector\|-8 8 -2}}. The first 2 is just to octave-reduce it to being positive, but you may recognize the actual vector part as 2 times the meantone comma. The most important part is that the vector is 2-enfactored. You can see that the node at the very center of this block is 160/81, which again is 2×{{vector\|-4 4 -1}}, or the octave-reduced non-enfactored version of that same comma.		Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: [[2048/2025]] ({{vector\|11 -4 -2}}) and 625/324 ({{vector\|-2 -4 4}}). The value at the fourth corner, 12800/6561, has vector 2×{{vector\|-8 8 -2}}. The first 2 is just to octave-reduce it to being positive, but you may recognize the actual vector part as 2 times the meantone comma. The most important part is that the vector is 2-enfactored. You can see that the node at the very center of this block is 160/81, which again is 2×{{vector\|-4 4 -1}}, or the octave-reduced non-enfactored version of that same comma.

	The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset.		The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset.

TallKite: added some musical terminology

2025-03-20T09:54:03Z

added some musical terminology

← Older revision		Revision as of 09:54, 20 March 2025
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	First, let's look at a [[defactoring\|defactored]] mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.		First, let's look at a [[defactoring\|defactored]] mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.

	This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket\|{{map\|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector\|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector\|-3 2}}].		This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket\|{{map\|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector\|-3 2}}, AKA 9/8, the 3-limit major 2nd. And so the comma basis for this temperament is [{{vector\|-3 2}}]. The generator is both 4/3 and 3/2. In musical terms, both the fourth and the fifth are so heavily tempered that they each become a half-octave.

	We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is made the comma {{vector\|-3 2}} vanish so that any nodes in this lattice which are 2 over and 3 up from each other are equivalent. Therefore we only need to consider a thin swath of the lattice anymore, specifically, a swath which connects the origin {{vector\|0 0}}, AKA 1/1, to {{vector\|-3 2}}, and then runs perpendicularly to infinity in either direction.		We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is made the comma {{vector\|-3 2}} vanish so that any nodes in this lattice which are 2 over and 3 up from each other are equivalent. Therefore we only need to consider a thin swath of the lattice anymore, specifically, a swath which connects the origin {{vector\|0 0}}, AKA 1/1, to {{vector\|-3 2}}, and then runs perpendicularly to infinity in either direction.
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	[[File:2-enfactored comma-basis.png\|365px\|thumb\|left\|enfactored comma bases are garbage]]		[[File:2-enfactored comma-basis.png\|365px\|thumb\|left\|enfactored comma bases are garbage]]

	Here's where things get kind of nuts. Most recently we experimented with enfactoring our healthy temperament's mapping. Now let's experiment with enfactoring its comma basis. In the defactored situation, if our comma basis was [{{vector\|-3 2}}], then 2-enfactoring it produces 2×[{{vector\|-3 2}}] = [{{vector\|-6 4}}].		Here's where things get kind of nuts. Most recently we experimented with enfactoring our healthy temperament's mapping. Now let's experiment with enfactoring its comma basis. In the defactored situation, if our comma basis was [{{vector\|-3 2}}] = 9/8, then 2-enfactoring it produces 2×[{{vector\|-3 2}}] = [{{vector\|-6 4}}] = 81/64, the 3-limit major 3rd.

	We know that in the original diagram, the large-labelled {{vector\|-3 2}} represented our comma, and this was the point that our dotted line ran through, the one that represented our boundary of warp/wrap. So our first thought should be: we must alter our diagram so that now {{vector\|-6 4}} is that point instead. Fine.		We know that in the original diagram, the large-labelled {{vector\|-3 2}} represented our comma, and this was the point that our dotted line ran through, the one that represented our boundary of warp/wrap. So our first thought should be: we must alter our diagram so that now {{vector\|-6 4}} is that point instead. Fine.

	But here's the problem. It simply doesn't make sense to double the width of our swath/tube! If {{vector\|-6 4}} is made to vanish, then so is {{vector\|-3 2}}. That is, while nothing would stop you from drawing a diagram with a double-width swath/tube, the musical reality is that it is impossible to make {{vector\|-6 4}} vanish without also making {{vector\|-3 2}} vanish. And so there is no meaning or purpose to the comma basis {{vector\|-6 4}}, whether RTT-wise or musically in general. It is garbage.		But here's the problem. It simply doesn't make sense to double the width of our swath/tube! If {{vector\|-6 4}} is made to vanish, then so is {{vector\|-3 2}}. That is, while nothing would stop you from drawing a diagram with a double-width swath/tube, the musical reality is that it is impossible to make {{vector\|-6 4}} vanish without also making {{vector\|-3 2}} vanish. In musical terms, the 3-limit major 3rd is the sum of two 3-limit major 2nds. If the major 3rd vanishes, the major 2nd must also. And so there is no meaning or purpose to the comma basis {{vector\|-6 4}}, whether RTT-wise or musically in general. It is garbage.

	And so our lattice for an enfactored comma basis looks almost identical to the original defactored lattice. The only difference here is that we've drawn a "supposed (but false)" tube circumference out to {{vector\|-6 4}}, while the half of this length which is real is now labelled the "true" circumference.		And so our lattice for an enfactored comma basis looks almost identical to the original defactored lattice. The only difference here is that we've drawn a "supposed (but false)" tube circumference out to {{vector\|-6 4}}, while the half of this length which is real is now labelled the "true" circumference.

Cmloegcmluin: improvements suggested (verbatim) by Dave

2025-03-18T13:10:01Z

improvements suggested (verbatim) by Dave

← Older revision		Revision as of 13:10, 18 March 2025
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	In this article~~, we will use lattices~~ to ~~visualize~~ [[~~enfactoring\|enfactored~~]] ~~temperaments~~, to demonstrate the musical implications of mappings ~~with~~ common factors, and the lack of musical implications of comma bases ~~with~~ common factors.		In this article that relates to [[regular temperaments]], we will use lattices to demonstrate the musical implications of [[mappings]] that contain common factors, and the lack of musical implications of [[comma basis\|comma bases]] that contain common factors.

	== Defactored case ==		== Defactored case ==
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	[[File:Unenfactored mapping.png\|365px\|thumb\|right\|A 3-limit tempered lattice, superimposed on the JI lattice]]		[[File:Unenfactored mapping.png\|365px\|thumb\|right\|A 3-limit tempered lattice, superimposed on the JI lattice]]

	First, let's look at a defactored mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.		First, let's look at a [[defactoring\|defactored]] mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.

	This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket\|{{map\|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector\|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector\|-3 2}}].		This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket\|{{map\|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector\|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector\|-3 2}}].
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	== Enfactored mappings vs. enfactored comma bases ==		== Enfactored mappings vs. enfactored comma bases ==

	One may pose the question: ~~what~~ is ~~the~~ relationship between an enfactored mapping and an enfactored comma basis? ~~Can you have~~ one ~~but not~~ the other~~? Must you? Or must you not~~? Or ~~does the question even make sense? Certainly at least some have suggested these cases~~ are ~~meaningfully~~ independent~~<ref>such as the page [[Color_notation/Temperament_Names\|color notation]], which reads "it's possible that there is both torsion and contorsion"</ref>.~~		One may pose the question: is there any relationship between an enfactored mapping and an enfactored comma basis? Does one imply the other. Does one imply the absence of the other? Or are they completely independent?

	~~The conclusion we arrive at here~~ is ~~that because~~ enfactored ~~comma bases don't make any sense~~, ~~or at least don't represent any legitimately new musical information of any kind that their defactored version doesn't already represent~~, ~~it is not generally useful~~ to ~~think of enfactored mappings~~ and ~~enfactored comma bases~~ as independent ~~phenomena~~. ~~It only makes sense to speak~~ of ~~enfactored temperaments~~. ~~Of course~~, ~~one will often use~~ the ~~term "~~enfactored ~~mapping" because~~ enfactored mappings ~~are the kind~~ which ~~do have some musical purpose~~, ~~and often~~ the ~~enfactored mapping will be being used to represent~~ the ~~enfactored temperament — or~~ temperoid, that is.		We note that there is no such thing as an enfactored temperament, only enfactored matrices. And these essentially result from an arithmetic oversight, namely failing to defactor, and as such they are completely independent<ref>as observed on the page [[Color_notation/Temperament_Names\|color notation]], which reads "it's possible that there is both torsion and contorsion"</ref>. When we use matrix math for RTT we defactor as a matter of course. For example, defactoring is built in to the NullSpace matrix operation in the Wolfram Language.

			However, if you have an artistic or other reason to generate scales using generators derived directly from enfactored mappings, and therefore to treat enfactored mappings as representing different musical objects, which we call temperoids, not temperaments, then you might choose to encode the same common factor into the comma basis representation of the temperoid with the understanding that this is a mere bookkeeping exercise and has no mathematical basis.

	=Footnotes=		=Footnotes=

Cmloegcmluin: /* Enfactored comma bases vs. periodicity blocks with torsion */

2025-03-18T12:56:00Z

Enfactored comma bases vs. periodicity blocks with torsion

← Older revision		Revision as of 12:56, 18 March 2025
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	Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector\|-6 4}} was because the tempering: if {{vector\|-6 4}} is made to vanish, then {{vector\|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.		Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector\|-6 4}} was because the tempering: if {{vector\|-6 4}} is made to vanish, then {{vector\|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.

	Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: 2048/2025 and 625/324. The value at the fourth corner, 12800/6561, has vector 2×{{vector\|-8 8 -2}}. The first 2 is just to octave-reduce it to being positive, but you may recognize the actual vector part as 2 times the meantone comma. The most important part is that the vector is 2-enfactored. You can see that the node at the very center of this block is 160/81, which again is 2×{{vector\|-4 4 -1}}, or the octave-reduced non-enfactored version of that same comma.		Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: [[2048/2025]] ({{vector\|11 -4 -4}}) and 625/324 ({{vector\|-2 -4 4}}). The value at the fourth corner, 12800/6561, has vector 2×{{vector\|-8 8 -2}}. The first 2 is just to octave-reduce it to being positive, but you may recognize the actual vector part as 2 times the meantone comma. The most important part is that the vector is 2-enfactored. You can see that the node at the very center of this block is 160/81, which again is 2×{{vector\|-4 4 -1}}, or the octave-reduced non-enfactored version of that same comma.

	The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset.		The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset.

Cmloegcmluin: /* Enfactored mapping */

2024-06-16T14:58:55Z

Enfactored mapping

← Older revision		Revision as of 14:58, 16 June 2024
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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.		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.		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.

Cmloegcmluin: /* Enfactored mapping */

2024-06-16T14:58:01Z

Enfactored mapping

← Older revision		Revision as of 14:58, 16 June 2024
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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.		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 ~~not~~ 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 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 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.		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.

Cmloegcmluin: /* Defactored case */

2024-06-16T14:55:32Z

Defactored case

← Older revision		Revision as of 14:55, 16 June 2024
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	There's a couple good ways to interpret this situation:		There's a couple good ways to interpret this situation:
	# We've turned on a teleportation field for every point outside this swath, so that it moves by these (-3,2) intervals until it finds its way inside the swath.		# We've turned on a teleportation field for every point outside this swath, so that it moves by these (-3,2) intervals until it finds its way inside the swath.
	# We've rolled up space, so that the line from 1/1 to 9/~~8 ₋ the~~ width of our ~~swath ₋ is~~ like the circumference of a tube. In this case, since we're only working with 3-limit JI, "space" was only ever 2D, so we can just think of it as paper that we've rolled up, so the pair of dotted lines visualized here are touching along their entire lengths (and if you wanted to imagine further copies of these dotted lines at every (-3,2) interval farther out in either direction, and the paper is infinitely thin, just stack all the dotted lines on top of each other forever).		# We've rolled up space, so that the line from 1/1 to 9/8 — the width of our swath — is like the circumference of a tube. In this case, since we're only working with 3-limit JI, "space" was only ever 2D, so we can just think of it as paper that we've rolled up, so the pair of dotted lines visualized here are touching along their entire lengths (and if you wanted to imagine further copies of these dotted lines at every (-3,2) interval farther out in either direction, and the paper is infinitely thin, just stack all the dotted lines on top of each other forever).

	Either way, then we just superimpose the new tempered lattice on top. It's drawn in blue.		Either way, then we just superimpose the new tempered lattice on top. It's drawn in blue.

Dave Keenan: /* Defactored case */ Changed "an defactored mapping" to "a defactored mapping".

2023-12-20T02:07:48Z

Defactored case: Changed "an defactored mapping" to "a defactored mapping".

← Older revision		Revision as of 02:07, 20 December 2023
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	[[File:Unenfactored mapping.png\|365px\|thumb\|right\|A 3-limit tempered lattice, superimposed on the JI lattice]]		[[File:Unenfactored mapping.png\|365px\|thumb\|right\|A 3-limit tempered lattice, superimposed on the JI lattice]]

	First, let's look at an defactored mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.		First, let's look at a defactored mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.

	This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket\|{{map\|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector\|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector\|-3 2}}].		This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket\|{{map\|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector\|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector\|-3 2}}].

← Older revision		Revision as of 09:54, 20 March 2025
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	First, let's look at a [[defactoring\|defactored]] mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.		First, let's look at a [[defactoring\|defactored]] mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.

	This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket\|{{map\|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector\|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector\|-3 2}}].		This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket\|{{map\|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector\|-3 2}}, AKA 9/8, the 3-limit major 2nd. And so the comma basis for this temperament is [{{vector\|-3 2}}]. The generator is both 4/3 and 3/2. In musical terms, both the fourth and the fifth are so heavily tempered that they each become a half-octave.

	We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is made the comma {{vector\|-3 2}} vanish so that any nodes in this lattice which are 2 over and 3 up from each other are equivalent. Therefore we only need to consider a thin swath of the lattice anymore, specifically, a swath which connects the origin {{vector\|0 0}}, AKA 1/1, to {{vector\|-3 2}}, and then runs perpendicularly to infinity in either direction.		We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is made the comma {{vector\|-3 2}} vanish so that any nodes in this lattice which are 2 over and 3 up from each other are equivalent. Therefore we only need to consider a thin swath of the lattice anymore, specifically, a swath which connects the origin {{vector\|0 0}}, AKA 1/1, to {{vector\|-3 2}}, and then runs perpendicularly to infinity in either direction.
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	[[File:2-enfactored comma-basis.png\|365px\|thumb\|left\|enfactored comma bases are garbage]]		[[File:2-enfactored comma-basis.png\|365px\|thumb\|left\|enfactored comma bases are garbage]]

	Here's where things get kind of nuts. Most recently we experimented with enfactoring our healthy temperament's mapping. Now let's experiment with enfactoring its comma basis. In the defactored situation, if our comma basis was [{{vector\|-3 2}}], then 2-enfactoring it produces 2×[{{vector\|-3 2}}] = [{{vector\|-6 4}}].		Here's where things get kind of nuts. Most recently we experimented with enfactoring our healthy temperament's mapping. Now let's experiment with enfactoring its comma basis. In the defactored situation, if our comma basis was [{{vector\|-3 2}}] = 9/8, then 2-enfactoring it produces 2×[{{vector\|-3 2}}] = [{{vector\|-6 4}}] = 81/64, the 3-limit major 3rd.

	We know that in the original diagram, the large-labelled {{vector\|-3 2}} represented our comma, and this was the point that our dotted line ran through, the one that represented our boundary of warp/wrap. So our first thought should be: we must alter our diagram so that now {{vector\|-6 4}} is that point instead. Fine.		We know that in the original diagram, the large-labelled {{vector\|-3 2}} represented our comma, and this was the point that our dotted line ran through, the one that represented our boundary of warp/wrap. So our first thought should be: we must alter our diagram so that now {{vector\|-6 4}} is that point instead. Fine.

	But here's the problem. It simply doesn't make sense to double the width of our swath/tube! If {{vector\|-6 4}} is made to vanish, then so is {{vector\|-3 2}}. That is, while nothing would stop you from drawing a diagram with a double-width swath/tube, the musical reality is that it is impossible to make {{vector\|-6 4}} vanish without also making {{vector\|-3 2}} vanish. And so there is no meaning or purpose to the comma basis {{vector\|-6 4}}, whether RTT-wise or musically in general. It is garbage.		But here's the problem. It simply doesn't make sense to double the width of our swath/tube! If {{vector\|-6 4}} is made to vanish, then so is {{vector\|-3 2}}. That is, while nothing would stop you from drawing a diagram with a double-width swath/tube, the musical reality is that it is impossible to make {{vector\|-6 4}} vanish without also making {{vector\|-3 2}} vanish. In musical terms, the 3-limit major 3rd is the sum of two 3-limit major 2nds. If the major 3rd vanishes, the major 2nd must also. And so there is no meaning or purpose to the comma basis {{vector\|-6 4}}, whether RTT-wise or musically in general. It is garbage.

	And so our lattice for an enfactored comma basis looks almost identical to the original defactored lattice. The only difference here is that we've drawn a "supposed (but false)" tube circumference out to {{vector\|-6 4}}, while the half of this length which is real is now labelled the "true" circumference.		And so our lattice for an enfactored comma basis looks almost identical to the original defactored lattice. The only difference here is that we've drawn a "supposed (but false)" tube circumference out to {{vector\|-6 4}}, while the half of this length which is real is now labelled the "true" circumference.